Abstract:
Methods and related systems are described relating to an inversion approach for interpreting the geophysical electromagnetic data. The inversion can be constrained by using a multiphase fluid flow simulator (incorporating pressure data if available) which simulates the fluid flow process and calculates the spatial distribution of the water saturation and the salt concentration, which are in turn transformed into the formation conductivity using a resistivity-saturation formula. In this way, the inverted invasion profile is consistent with the fluid flow physics and moreover accounts for gravity segregation effects. Jointly with the pressure data, the inversion estimates a parametric one-dimensional distribution of permeability and porosity. The fluid flow volume is directly inverted from the fluid-flow-constrained inversion of the electromagnetic data. The approach is not limited by the traditional interpretation of the formation test, which is based on a single-phase model without taking into account invasion or assuming that the fluid, for example mud-filtrate, has been cleaned up from the formation testing zone. The joint inversion of the electromagnetic and pressure data provides for a more reliable interpretation of formation permeability. One advantage of the approaches described herein, is its possible generalization to three-dimensional geometries, for example dipping beds and highly deviated wells.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
       [0001]    This patent application is a divisional application of U.S. patent application Ser. No. 12/607,708, filed Oct. 28, 2009, which claims the benefit of U.S. Provisional Patent Application Ser. No. 61/145,658, filed Jan. 19, 2009, both of which are hereby incorporated by reference in their entireties. 
     
    
     FIELD OF THE INVENTION 
       [0002]    This invention relates broadly to the investigation of geological formations traversed by a borehole. More particularly, this invention relates to a method for integrating electromagnetics and multi-phase fluid flow in order to provide a robust, physically consistent interpretation for reservoir characterization. 
       BACKGROUND OF THE INVENTION 
       [0003]    Induction logging measurements are sensitive to water saturation and brine concentration in the rock pores. In an active reservoir, the formation&#39;s petrophysical parameters can have a strong imprint on the temporal and spatial distribution of water saturation and salt concentration, which in turn can be transformed into the distribution of formation conductivity using an appropriate saturation-resistivity equation. This relationship between induction measurements and the petrophysical parameters offers an opportunity to integrate electromagnetics and multi-phase fluid flow to provide a robust, physically consistent interpretation for reservoir characterization. 
         [0004]    Induction logging tools can be used to determine the formation resistivity and invasion profile via a model-based inversion approach. See, Habashy, T. M. and A. Abubakar, 2004, A general framework for constraint minimization for the inversion of electromagnetic measurements: Progress In Electromagnetics Research, 46, 265-312 (hereinafter “Habashy and Abubakar”); Barber, T., B. Anderson, A. Abubakar, T. Broussard, K. Chen, S. Davydycheva, V. Druskin, T. Habashy, D. Homan, G. Minerbo, R. Rosthal, R. Schlein, and H. Wang, 2004, Determining formation resisitivty anisotropy in the presence of invasion, in SPE Annual Technical Conference and Exhibition, paper 90526; Abubakar, A., T. M. Habashy, V. Druskin, L. Knizhnerman, and S. Davydycheva, 2006, A 3d parametric inversion algorithm for triaxial induction data: Geophysics, 71, G1-G9; and Abubakar, A., T. M. Habashy, V. Druskin, and L. Knizhnerman, 2006, An enhanced gauss-newton inversion algorithm using a dual-optimal grid approach: IEEE Transactions on Geoscience and Remote Sensing, 44, 1419-1427. However, this application is constrained by some simplifying assumptions, for example, assuming a well penetrating a layer-cake formation with a step-profile three-parameter (R xo , R t , D i ) invasion model. The quality and accuracy of the inversion results are affected by the complexity of the reservoir configuration and the actual invasion profile. In horizontal or highly deviated wells or in anisotropic formations, invasion profiles become too complex to be described by a simple invasion model. An alternative inversion approach is to employ the so-called pixel-based inversion method. See: Abubakar, A. and P. M. van den Berg, 2000, Nonlinear inversion in electrode logging in a highly deviated formation with invasion using an oblique coordinate system: IEEE Transactions on Geoscience and Remote Sensing, 38, 25-38; Alumbaugh, D. and M. Wilt, 2001, A numerical sensitivity study of three-dimensional imaging from a single borehole: Petrophysics, 42, 19-31; Abubakar, A. and P. van den Berg, 2002, Application of a non-orthogonal coordinate system to inverse single-well electromagnetic logging problem: Three-Dimensional Electromagnetics, eds. M. S. Zhdanov and P. E. Wannamaker, 215-231; Abubakar, A. and T. Habashy, 2006, Three-dimensional single-well imaging of the multi-array triaxial induction logging data, in SEG Technical Program Expanded Abstracts, volume 25, 411-415. A disadvantage of this pixel-based inversion method is that the number of unknowns used to describe the configuration is large and hence, a large number of unknown model parameters often need to be inverted. This approach will only provide a qualitative inverted resistivity map (an image) around the well-bore. 
         [0005]    On the other hand, the mud-filtrate invasion affects the interpretation of well testing or wireline formation test. Although some researchers studied and discussed the influence of invasion to the formation test interpretation, practically, the interpretation of formation test usually still employs a single-phase fluid flow model regardless of the mud-filtrate invasion. The error associates with this can be neglected only when the mobility of the mud-filtrate is very close to that of the formation fluid and the capillary pressure is negligible or the invaded mud-filtrate has already been cleaned up from the test target zone by a drawdown, which may be time-consuming. A more realistic invasion profile will benefit both induction data interpretation and formation test interpretation. 
         [0006]    Some previous works have already exploited the benefit of integrating the electromagnetic model with the multiphase fluid flow model. In Semmelbeck, M. E. and S. A. Holditch, 1988, The effects of mud-filtrate invasion on the interpretation of induction logs: SPE (hereinafter “Semmelbeck and Holditch (1988)”), an attempt is made to develop a model to simulate the mud-filtrate invasion process, which deals with the mud-cake and the formation simultaneously, and incorporated the capability to simulate the salt transport to calculate the formation resistivity. This model was then used to analyze the effect of the mud-cake permeability, the formation permeability, the porosity and the overbalance pressure on the induction logging data. In Tobola, D. P. and S. A. Holditch, 1991, Determination of reservoir permeability from repeated induction logging: SPE Formation Evaluation, 20-26 (hereinafter “Tobola and Holditch (1991)”), the approach described in Semmelbeck and Holditch (1988) was applied to a low permeability reservoir and determined the reservoir permeability by history matching the change in the induction logging data over time. The effect of the initial water saturation on the induction logging data was also analyzed. In this approach, the mud-cake permeability profile over time must be accurately simulated to properly interpret the induction logging data. In Yao, C. Y. and S. A. Holditch, 1996, Reservoir permeability estimation from time-lapse log data: SPE Formation Evaluation, 69-74 (hereinafter “Yao and Holditch (1996)”), the approach described in Semmelbeck and Holditch (1988) was extended to history match both the time-lapse resistivity and neutron logging data. This technique was used to estimate the reservoir permeability, which was verified against production data and core analysis. 
         [0007]    The approaches described in Semmelbeck and Holditch (1988), Tobola and Holditch (1991) and Yao and Holditch (1996) simulate the mud-filtrate invasion assuming a mud-cake with a constant thickness and a variable permeability, which need to be accurately determined from other independent measurement data. See: Ferguson, C. K. and J. A. Klotz, 1954, Filtration from mud during drilling: Trans., AIME 201, 29-42; and Williams, M. and G. E. Cannon, 1938, Evaluation of filtration properties of drilling muds: The Oil Weekly, 25-32. These approaches ignore gravity and diffusion effects, and aim to estimate the permeability in tight formations. However, this estimation heavily depends on the precision of determining mud-cake properties, which is another difficult problem to be resolved. 
         [0008]    Semmelbeck, M. E., J. T. Dewan, and S. A. Holditch, 1995, Invasion-based method for estimating permeability from logs: SPE 30581 presented at the 1995 SPE Annual Technical Conference and Exhibition, Dallas, 22-25 describes an extension of their previous work to integrate the fluid flow model with the Dewan&#39;s mudcake growth model, which is an experimentally-verified model for predicting the mud-cake thickness and the permeability during static and dynamic filtration conditions. See: Dewan, J. T. and M. E. Chenevert, 1993, Mudcake buildup and invasion in low permeability formations: application to permeability determination by measurement while drilling: SPWLA 34th Annual Logging Symposium, 13-16. This approach was applied to the new generation of array induction logging data. From a linear covariance analysis, they concluded that their method could provide reasonable estimates of the permeability in low to moderate permeable formations, but the method could not provide accurate estimates of the saturation-dependent properties. 
         [0009]    Ramakrishnan, T. S. and D. J. Wilkinson, 1997, Formation producibility and fractional flow curves from radial resistivity variation caused by drilling fluid invasion: Physics of Fluids, 9, 833-844, and Ramakrishnan, T. S. and D. J. Wilkinson, Water-cut and fractional flow logs from array-induction measurements: SPE Reservoir Evaluation &amp; Engineering, 2, 85-94 discuss establishing a mathematical model to estimate the fractional flow and the relative permeability curves from the array induction logging measurements. Based on the work described in the two Ramakrishnan papers, Zeybek, M., T. Ramakrishnan, S. AI-Otaibi, S. Salamy, and F. Kuchuk, 2001, Estimating multiphase flow properties using pressure and flowline water-cut data from dual packer formation tester interval tests and openhole array resistivity measurements: SPE 71568 discusses a methodology for estimating the horizontal and vertical layer permeabilities and the relative permeabilities using array induction logging tool measurements, pressure transient measurements and water-cut measurements from a packer-probe wireline formation tester. This joint inversion was done in a sequential mode. In this method, fractional flow parameters (connate water saturation, residual water saturation, maximum residual oil saturation, filtrate loss and pore size distribution) are estimated by matching resistivity measurements. These estimated fractional flow parameters are then input into a numerical model for the sampling process to ultimately match the observed and simulated water cut and pressure data. The final outputs are the relative, the horizontal and vertical permeabilities. 
         [0010]    Alpak, F. O., T. M. Habashy, C. Torres-Verdin, and V. Dussan, 2004, Joint inversion of pressure and time-lapse electromagnetic logging measurements: Petrophysics, 45, 251-267 (hereinafter “Alpak, et al. (2004)”); Alpak, F. O., C. Torres-Verdin, and T. M. Habashy, 2004, Joint inversion of pressure and dc resistivity measurements acquired with in-situ permanent sensors: a numerical study: Geophysics, 69, 1173-1191; Alpak, F. O., C. Torres-Verdin, and T. M. Habashy 2006, Petrophysical inversion of borehole array-induction logs: Part I—numerical examples: Geophysics, 71, F101-F119; and Alpak, F. O., C. Torres-Verdin, T. M. Habashy, and K. Sephernoori, 2004, Simultaneous estimation of in-situ multiphase petrophysical properties of rock formations from wireline formation tester and induction logging measurements: SPE 90960 all discuss the simultaneous estimation of in-situ multiphase petrophysical properties of rock formations from wireline formation tester and induction logging measurements. The papers also discuss extensive work to assess the sensitivity of array induction measurements to the presence of the water-based mud-filtrate invasion for various combinations of petrophysical parameters and fluid properties. Torres-Verdin, C., F. O. Alpak, and T. M. Habashy, 2006, Petrophysical inversion of borehole array-induction log: Part II—field data examples: Geophysics, 71, G261-G268; and Salazar, J. M., C. Torres-Verdin, F. O. Alpak, T. M. Habashy, and J. D. Klein, 2006, Estimation of permeability from borehole array induction measurements: Application to the petrophysical appraisal of tight gas sands: Petrophysics, 47, 527-544 discuss applying method described by Alpak to some low or moderate permeability gas reservoirs for estimating the permeability and the porosity. This method employs a modified UTCHEM code (INVADE) (see Wu, J., C. Torres-Verdin, K. Sepehrnoori, and M. Proett, 2005, The influence of water-base mud properties and petrophysical parameters on mudcake growth, filtrate invasion, and formation pressure: Petrophysics, 46, 14-32) to calculate the time-dependent mud-filtrate loss rate, and takes the average of this rate as the injection rate to run a two-phase three-component fluid flow model for simulating the mud-filtrate invasion process, which consequently affects the resistivity measurements. Therefore, the accuracy of the inverted petrophysical parameters can depend on the mud-filtrate loss calculation, which is still a difficult problem at present. Angeles, R., J. Skolnakorn, F. Antonsen, A. Chandler, and C. Torres-Verdin, 2008, Advantages in joint-inversions of resistivity and formation-tester measurements: Examples from a norwegian field: SPWLA 49th Annual Logging Symposium, Edinburgh, Scotland discusses advancing previous work by using a reservoir simulator dynamically-coupled to a mud-filtrate invasion model. Hence they integrated the calculation of mud-filtrate invasion rate into the inversion process. 
         [0011]    Pereira, N., R. Altman, J. Rasmus, and J. Oliveira, 2008, Estimation of permeability and permeability anisotropy in horizontal wells through numerical simulation of mud filtrate invasion: Rio Oil &amp; Gas Expo and Conference, Rio de Janeiro. discusses an approach to estimate formation permeability and permeability anisotropy using dual inversion of time-lapse azimuthal laterolog-while-drilling and near well-bore numerical simulation of the water-based mud-filtrate invasion. This method requires a priori bottom hole pressure while drilling and assumes a known constant mud-cake permeability. Kuchuk, F., L. Zhan, S. M. Ma, A. M. Al-Shahri, T. S. Ramakrishnan, B. Altundas, M. Zeybek, R. Loubens, and N. Chugunov, 2008, Determination of in-situ two-phase flow properties through downhole fluid movement monitoring: SPE 116068 presented at the 2008 SPE Annual Technical Conference and Exhibition held in Denver, Colo., USA, 21-24 discussses method for in-situ estimation of two-phase transport properties of porous media using the time-lapse DC resistivity data, pressure and flow rate data. The time-lapse DC resistivity are recorded using a permanent downhole electrode resistivity array. The pressure and flow rate data are obtained from the injection test regardless of the mud-filtrate invasion. The integrated interpretation is based on an inversion workflow, which employs a multi-layered integrated flow/electrical numerical simulator to solve the fluid flow, the salt transport and the DC electrical array response. A field experiment conducted in a carbonate reservoir has been used to verify the proposed method. 
         [0012]    Most of the above works involve the simulation of the mud-filtrate invasion process to obtain the invasion rates as accurately as possible. However, the mud-filtrate invasion is a dynamic and complex process. It starts with a high invasion rate, which is mainly controlled by the overbalance pressure and the formation permeability when a well is drilled. After the mud-cake is formed, the invasion rate will tend to be controlled by the mud-cake since the permeability of mud-cake is normally much lower than that of the formation. There have been extensive works on the simulation of the mud-filtrate invasion process, which although possible, is still too complex for any practical application, especially when considering the complicated drilling and reaming conditions, such as the mud circulation rate, workover, mud-cake scraped by the tools, and so on. 
       SUMMARY OF THE INVENTION 
       [0013]    This method according to some embodiments of the invention paper includes the general framework developed in Alpak et al. (2004). However, the method of the invention propose to directly invert the average mud-filtrate invasion rates for each flow unit or each depth interval, in order to avoid using complicated, time-consuming mud-cake growth and invasion models. Moreover, based on the derived invasion profile, the joint inversion technique according to some embodiments of the invention also provide a more reliable interpretation of formation test compared to the conventional methods. The method according to the invention can be applied to support horizontal and deviated wells with dipping formation layers. 
         [0014]    According to some embodiments, a method of estimating fluid invasion rate for portions of a subterranean formation of interest surrounding a borehole is provided. The method includes receiving electromagnetic survey data of the subterranean formation of interest and receiving fluid flow characteristics relating to the formation of interest. A direct inversion is performed of the electromagnetic data and constrained by the fluid flow simulator for a fluid invasion rate, and the fluid invasion rate is thereby estimated based. The fluid can be borehole originating fluids such as mud filtrate or completion fluids. The fluid flow simulator can also be used to generate the pressure transient data made in the borehole. If the pressure transient measurements are available to be compared with the one generated from the fluid flow simulator then permeability can be estimated. According to some embodiments porosity is also estimated based on the inversion. A quantitative predictive formation model can be generated based on the inversion. The electromagnetic data can be obtained using various tools operated in a single well. The same or similar workflow or process can be applied to different configurations such as cross-well, surface to borehole, borehole to surface and surface to surface. 
         [0015]    According to some embodiments, a method of estimating porosity for a subterranean formation of interest including one or more geologic beds is provided. The method includes receiving electromagnetic survey data and fluid flow characteristics of the formation of interest, and estimating porosity within at least a portion of the formation of interest based at least in part on the electromagnetic survey data and the fluid flow characteristics. The estimation has a spatial resolution such that at least two porosity values are estimated for each of the one or more geologic beds. 
         [0016]    According to some embodiments, systems are provided for estimating fluid invasion rate and for estimating porosity for portions of a subterranean formation of interest surrounding a borehole. 
         [0017]    Additional objects and advantages of the invention will become apparent to those skilled in the art upon reference to the detailed description taken in conjunction with the provided figures. 
     
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         [0018]      FIG. 1  is a flow chart showing workflow steps of a joint inversion, according to some embodiments; 
           [0019]      FIGS. 2   a  and  2   b  are diagrams showing certain configurations, according to some embodiments; 
           [0020]      FIGS. 3   a ,  3   b  and  3   c  are plots illustrating the saturation dependent functions used in certain described examples; 
           [0021]      FIGS. 4   a - 4   h , are vertical cross section diagrams show the results for oil-water example shown in  FIG. 2   a;    
           [0022]      FIGS. 5   a  and  5   b  are log-log plots of the pressure and the pressure derivative for the build-up for the example shown in  FIG. 2   a;    
           [0023]      FIG. 6  is a two-dimensional vertical cross-section of a reservoir having an injector and producer well, according to some embodiments; 
           [0024]      FIG. 7  shows the source-receiver setup and associated surface equipment according to the example shown in  FIG. 6 ; and 
           [0025]      FIGS. 8   a - h  are vertical cross-sections of spatial distributions of the porosity, the conductivity, the water saturation and the salt concentration, according to some embodiments. 
       
    
    
       [0026]    It will be recognized by the person of ordinary skill in the art, given the benefit of this disclosure, that certain dimensions, features, components, and the like in the figures may have been enlarged, distorted or otherwise shown in a non-proportional or non-conventional manner to facilitate a better understanding of the technology disclosed herein. 
       DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
       [0027]    The following description provides exemplary embodiments only, and is not intended to limit the scope, applicability, or configuration of the disclosure. Rather, the following description of the exemplary embodiments will provide those skilled in the art with an enabling description for implementing one or more exemplary embodiments. It being understood that various changes may be made in the function and arrangement of elements without departing from the spirit and scope of the invention as set forth in the appended claims. 
         [0028]    Specific details are given in the following description to provide a thorough understanding of the embodiments. However, it will be understood by one of ordinary skill in the art that the embodiments may be practiced without these specific details. For example, systems, processes, and other elements in the invention may be shown as components in block diagram form in order not to obscure the embodiments in unnecessary detail. In other instances, well-known processes, structures, and techniques may be shown without unnecessary detail in order to avoid obscuring the embodiments. Further, like reference numbers and designations in the various drawings indicated like elements. 
         [0029]    Also, it is noted that individual embodiments may be described as a process which is depicted as a flowchart, a flow diagram, a data flow diagram, a structure diagram, or a block diagram. Although a flowchart may describe the operations as a sequential process, many of the operations can be performed in parallel or concurrently. In addition, the order of the operations may be re-arranged. A process may be terminated when its operations are completed, but could have additional steps not discussed or included in a figure. Furthermore, not all operations in any particularly described process may occur in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, etc. When a process corresponds to a function, its termination corresponds to a return of the function to the calling function or the main function. 
         [0030]    Furthermore, embodiments of the invention may be implemented, at least in part, either manually or automatically. Manual or automatic implementations may be executed, or at least assisted, through the use of machines, hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware or microcode, the program code or code segments to perform the necessary tasks may be stored in a machine readable medium. A processor(s) may perform the necessary tasks. 
         [0031]    Fluid Flow Simulation. 
         [0032]    Further detail relating to fluid flow simulation, according to some embodiments, will now be provided. After a well is drilled the mud-filtrate invades into the surrounding rock formation. A three-phase, four-component (oil, water, gas, salt) model is employed to describe the mud-filtrate invasion process. The mud-filtrate invasion is simulated as an injection process by solving the pressure diffusive equation derived from Darcy&#39;s Law and the mass balance equation. Ignoring the chemical reactions and diffusive/dispersive transport, the mass balance equation for the fluid phase can be stated as follows (see Aziz and Settari (1979)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ∂ 
                           
                             ( 
                             
                               
                                 ρ 
                                 l 
                               
                                
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                     1 
                   
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                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the subscript l is the index of phase, which can be oil, water or gas. The symbols ρ, v, φ, q and S denote the fluid density, fluid velocity, porosity, source term, and phase saturation, respectively. The symbol ∇ denotes the spatial differentiation with respect to spatial position (x,y,z). Darcy&#39;s law is used as the constitutive equation for the velocity vector, i.e., 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       v 
                       l 
                     
                     = 
                     
                       
                         
                           - 
                           k 
                         
                         · 
                         
                           
                             k 
                             
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                               ; 
                               i 
                             
                           
                           
                             μ 
                             l 
                           
                         
                       
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         [0000]    where k is the tensor absolute permeability, k r  is the phase relative permeability, μ is the phase viscosity, p is the phase pressure, γ is the phase specific gravity and D z  is the vertical depth below a reference level. We assume that the fluid (oil and water) and rock compressibilities are defined by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       c 
                       fluid 
                     
                     = 
                     
                       
                         - 
                         
                           1 
                           V 
                         
                       
                        
                       
                         
                           ∂ 
                           V 
                         
                         
                           ∂ 
                           p 
                         
                       
                        
                       
                         
                           
                              
                             T 
                           
                            
                           
                             = 
                             
                               
                                 1 
                                 ρ 
                               
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                                   ∂ 
                                   ρ 
                                 
                                 
                                   ρ 
                                    
                                   
                                     ∂ 
                                     p 
                                   
                                 
                               
                             
                           
                            
                         
                         T 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   3 
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         [0000]    They are constant over the pressure range of interest in the simulation. The gas compressibility is treated as a function of the pressure-volume-temperature (PVT) properties of gas for each time-step. Additional relations about capillary pressures and fluid saturations are given by: 
         [0000]        p   c   =p   nw   −p   w ,  (5)
 
         [0000]      and 
         [0000]      Σ i   S   1 =1  (6)
 
         [0000]    where the subscripts nw and w stand for the nonwetting and the wetting phase. 
         [0033]    Salt is the forth component which is simulated, according to some embodiments. Assuming there is a high salt concentration contrast between the injected fluid and the in-situ formation brine and ignoring the diffusive/dispersive transport, the isothermal salt mixing in the aqueous-phase is solved by using the convective transport equation as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ∂ 
                           
                             ( 
                             
                               
                                 ρ 
                                 w 
                               
                                
                               φ 
                                
                               
                                   
                               
                                
                               
                                 S 
                                 w 
                               
                                
                               
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                             ) 
                           
                         
                         
                           ∂ 
                           t 
                         
                       
                       + 
                       
                         ∇ 
                         
                           · 
                           
                             ( 
                             
                               
                                 ρ 
                                 w 
                               
                                
                               
                                 v 
                                 w 
                               
                                
                               
                                 C 
                                 w 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           C 
                           wi 
                         
                       
                        
                       
                         q 
                         i 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where C w , C wi  and q i  is the salt concentration of formation brine, the salt concentration of invading fluid and the fluid injection rate, respectively. In our study the above equations are solved numerically using a finite-difference based reservoir simulator in fully implicit, black-oil mode with brine tracking option (e.g. ECLIPSE™). The pressure transient in the well/formation test can be simulated as well as the temporal and spatial distribution of the phase saturation and the salt concentration. In this description, focus is made on the scenario where a single well penetrates layered hydrocarbon-bearing formations. In the case of a vertical well, a cylindrical axisymmetric model is used in the numerical simulation and a logarithmic radial grid with very fine cells around the borehole is used to honor the mud-filtrate invasion and accurately catch the pressure transient phenomena in the well/formation test. 
         [0034]    Electromagnetic Simulation. 
         [0035]    Further detail relating to electromagnetic simulation, according to some embodiments, will now be provided. Electromagnetic responses measured by the array induction tools can be numerically simulated by solving a frequency-domain electromagnetic induction problem, which is described by the following diffusive Maxwell&#39;s equations: 
         [0000]      ∇× E+iωμH= 0,  (8)
 
         [0000]      ∇× H−σE=−J,   (9)
 
         [0000]    where E is the electric field vector, H is the magnetic field vector and J is the external electric current source vector. The symbols σ=σ(x,y,z) and μ denote the electrical conductivity and the magnetic permeability (which is constant). The symbol ω denotes the radial frequency and i=√{square root over (−1)}. By substituting equation (8) into equation (9) we obtain: 
         [0000]      σ −1   ∇×∇×E+iωE=−iωσ   −1   J   (10)
 
         [0036]    A frequency-domain finite-difference algorithm such as described in Druskin, V., L. Knizhnerman, and P. Lee, 1999, New spectral lanczos decomposition method for induction modeling in arbitrary 3d geometry: Geophysics, 64, 701-706 (herein after “Druskin et al. (1999)”), incorporated herein by reference, is employed to numerically solve equation (10) on a staggered Yee grid (See, Yee, K. S., 1966, Numerical solution of initial boundary value problems involving maxwell&#39;s equations in isotropic media: IEEE Transactions on Antennas and Propagation, AP-14, 302-307). This approach uses a spectral Lanczos decomposition method (SLDM) with Krylov subspaces generated from the inverse powers of the Maxwell operator. Compared to the standard SLDM method, this so-called SLDMINV method is significantly more efficient. As shown in Druskin et al. (1999), the convergence rate of the SLDMINV method increases as the frequency decreases, this makes SLDMINV particularly attractive for low-frequency electromagnetic simulations. 
         [0037]    Petrophysical Transform. 
         [0038]    Further detail relating to petrophysical transforms, according to some embodiments, will now be provided. According to some embodimetns, the spatial distribution of water saturation and salt concentration at logging times are computed by a multiphase fluid flow simulator such as ECLIPSE™ from Schlumberger. These are then transformed to a conductivity map using Archie&#39;s law (see, Archie, G. E., 1942, The electrical resistivity log as an aid in determining some reservoir characteristics: Petroleum Transactions of the AIME, 146, 54-62): 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     = 
                     
                       
                         
                           σ 
                           w 
                         
                          
                         
                           φ 
                           m 
                         
                          
                         
                           S 
                           w 
                           n 
                         
                       
                       a 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where σ w  is the brine conductivity and S w  is the water saturation. Archie&#39;s parameters m, n and a are obtained from core measurements. The brine conductivity is a function of the salt concentration and temperature and is given by the following equation (see, Zhang, J.-H., Q. Hu, and Z.-H. Liu, 1999, Estimation of true formation resistivity and water saturation with a time-lapse induction logging method: The Log Analyst, 40, 138-148): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       σ 
                       w 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             ( 
                             
                               0.0123 
                               + 
                               
                                 3647.5 
                                 
                                   C 
                                   w 
                                   0.0955 
                                 
                               
                             
                             ) 
                           
                            
                           
                             82 
                             
                               
                                 1.8 
                                  
                                 T 
                               
                               + 
                               39 
                             
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where C w  is the salt concentration (ppm) and T is the formation temperature (° C.). In this example, The multiphase fluid flow model is solved on a cylindrical grid while solving the electromagnetic model on a Cartesian grid. A robust homogenization approach described in Moskow, S., V. Druskin, T. M. Habashy, P. Lee, and S. Davydycheva, 1999, A finite difference scheme for elliptic equations with rough coefficients using a cartesian grid nonconforming to interfaces: SIAM Journal on Numerical Analysis, 36, 442-464, incorporated herein by reference, is employed to map the conductivities from the three-dimensional (3D) cylindrical grid to the 3D Cartesian grid. This feature gives us more flexibility on constructing the model. 
         [0039]    Inversion Algorithm. 
         [0040]    Further detail relating to inversion algorithms, according to some embodiments, will now be provided. In the inversion process the electromagnetic data and pressure data (whenever available) are simultaneously inverted. After spatial discretization, the nonlinear inverse problem can be described by the following operator notations: 
         [0000]          M     e   =  S     e (    m   )+  ε   e ,  (13)
 
         [0000]          M     p   =  S     p (    m   )+  ε   p ,  (14)
 
         [0000]    where  M   e  is the measured electromagnetic data while  M   p  is the measured pressure data. The components of these vectors are given by 
         [0000]          M     e   =[M (   r     i   s   ,  r     j   r   ,t   l ), i= 1, . . . ,  N   s   ;j= 1, . . . ,  N   r   ;l= 1, . . . ,  N   l   ]T,   (15)
 
         [0000]          M     p   =[M (   r     i   g   ,t   j ), i= 1, . . . ,  N   g   ;j= 1, . . . ,  N   t ] T ,  (16)
 
         [0000]    where N s , N r  and N l  are the number of sources, receivers and logging snapshots in the electromagnetic measurements while N g  and N t  are the number of pressure gauges and time steps in the pressure measurements. In equations (13) and (14),  S   e  and  S   p  are the vector of simulated electromagnetic and pressure data as functions of the vector model parameter  m . The parametric model according to this example is layer by layer horizontal permeabilities, vertical permeabilities, porosities and average mud-filtrate invasion rates denoted by k h , k v , φ and q. The vector of model parameters is defined as follows: 
         [0000]          m =[k   h;1   ,k   h;2   , . . . ,k   h;L   ,k   v;1   ,k   v;2   , . . . ,k   v;L ,φ 1 ,φ 2 , . . . ,φ L   ,q   1   ,q   2   , . . . ,q   L ] T   (17)
 
         [0000]    where L is the number of layers in our one-dimensional (1D) parametric model. In equations (13) and (14),  ε   e  and  ε   p  denote the measurement errors in the electromagnetic and pressure data. 
         [0041]    The inverse problems are posed as a minimization problem of the following cost function: 
         [0000]        C   n (    m   )=∥   W     d   e   ·[  M     e   −  S     e (    m   )] 2   ∥+∥  W     d   p   ·[  M     p   −  S     p (    m   )]∥ 2 +λ n ∥ln(    m   )−ln(   m     n )∥ 2 ,  (18)
 
         [0000]    where λ n  is the regularization parameter at iteration n. In the regularization term, the logarithmic functions are employed since the unknown model parameters have different dimensions and can have different orders of magnitude. The diagonal matrices  W   d   e  and  W   d   p  are the data weighting matrices. There are several options for choosing these data weighting matrices. One particular choice is to account for all data points on equal footing. Hence, the data weighting matrices can be defined as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     W 
                     
                       
                         d 
                         ; 
                         i 
                       
                       , 
                       i 
                     
                     e 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               1 
                               
                                 
                                   
                                     N 
                                     e 
                                   
                                    
                                   
                                      
                                     
                                       M 
                                       i 
                                       e 
                                     
                                      
                                   
                                 
                               
                             
                             , 
                           
                            
                           
                               
                           
                         
                       
                       
                         
                           
                             i 
                             = 
                             1 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             N 
                             e 
                           
                           , 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     W 
                     
                       
                         d 
                         ; 
                         i 
                       
                       , 
                       i 
                     
                     p 
                   
                   = 
                   
                     
                       
                         
                           1 
                           
                             
                               
                                 N 
                                 p 
                               
                                
                               
                                  
                                 
                                   M 
                                   i 
                                   p 
                                 
                                  
                               
                             
                           
                         
                       
                       
                         
                           
                             i 
                             = 
                             1 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             N 
                             p 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where N e =N s ×N r ×N l  and N p =N g ×N t . Substituting equations (19) and (20) into equation (18) we arrive at 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         C 
                         n 
                       
                        
                       
                         ( 
                         
                           m 
                           _ 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           1 
                           
                             N 
                             e 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             
                               N 
                               e 
                             
                           
                            
                           
                             
                                
                               
                                 
                                   
                                     M 
                                     i 
                                     e 
                                   
                                   - 
                                   
                                     
                                       S 
                                       i 
                                       e 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         _ 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   M 
                                   i 
                                   e 
                                 
                               
                                
                             
                             2 
                           
                         
                       
                       + 
                       
                         
                           1 
                           
                             N 
                             p 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             
                               N 
                               p 
                             
                           
                            
                           
                             
                                
                               
                                 
                                   
                                     M 
                                     i 
                                     p 
                                   
                                   - 
                                   
                                     
                                       S 
                                       i 
                                       p 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         _ 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   M 
                                   i 
                                   p 
                                 
                               
                                
                             
                             2 
                           
                         
                       
                       + 
                       
                         
                           λ 
                           n 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               1 
                             
                             
                               4 
                                
                               
                                   
                               
                                
                               L 
                             
                           
                            
                           
                             
                                
                               
                                 
                                   ln 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       l 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   ln 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       
                                         n 
                                         , 
                                         l 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                             
                             2 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Another choice is to apply uniform weighting on all data points as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       W 
                       
                         
                           d 
                           ; 
                           i 
                         
                         , 
                         i 
                       
                       e 
                     
                     = 
                     
                       1 
                       
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             
                               N 
                               e 
                             
                           
                            
                           
                             
                                
                               
                                 M 
                                 j 
                                 e 
                               
                                
                             
                             2 
                           
                         
                       
                     
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     N 
                     e 
                   
                   , 
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       W 
                       
                         
                           d 
                           ; 
                           i 
                         
                         , 
                         i 
                       
                       p 
                     
                     = 
                     
                       1 
                       
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             
                               N 
                               p 
                             
                           
                            
                           
                             
                                
                               
                                 M 
                                 j 
                                 p 
                               
                                
                             
                             2 
                           
                         
                       
                     
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     N 
                     p 
                   
                   , 
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Substituting equations (22) and (23) into equation (18) we arrive at 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       C 
                       n 
                     
                      
                     
                       ( 
                       
                         m 
                         _ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           
                             N 
                             e 
                           
                         
                          
                         
                           
                              
                             
                               
                                 M 
                                 i 
                                 e 
                               
                               - 
                               
                                 
                                   S 
                                   i 
                                   e 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     _ 
                                   
                                   ) 
                                 
                               
                             
                              
                           
                           2 
                         
                       
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           
                             N 
                             e 
                           
                         
                          
                         
                           
                              
                             
                               M 
                               i 
                               e 
                             
                              
                           
                           2 
                         
                       
                     
                     + 
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           
                             N 
                             p 
                           
                         
                          
                         
                           
                              
                             
                               
                                 M 
                                 i 
                                 p 
                               
                               - 
                               
                                 
                                   S 
                                   i 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     _ 
                                   
                                   ) 
                                 
                               
                             
                              
                           
                           2 
                         
                       
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           
                             N 
                             p 
                           
                         
                          
                         
                           
                              
                             
                               M 
                               i 
                               p 
                             
                              
                           
                           2 
                         
                       
                     
                     + 
                     
                       
                         λ 
                         n 
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             1 
                           
                           
                             4 
                              
                             
                                 
                             
                              
                             L 
                           
                         
                          
                         
                           
                             
                                
                               
                                 
                                   ln 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       l 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   ln 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       
                                         n 
                                         , 
                                         l 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                             
                             2 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0042]    The minimization process is done by using the Gauss-Newton approach described in Habashy and Abubakar, incorporated herein by reference. The gradient vector of the cost function in equation (18) is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             g 
                             i 
                           
                            
                           
                             ( 
                             
                               m 
                               _ 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             ∇ 
                             
                               m 
                               i 
                             
                           
                            
                           
                             
                               C 
                               n 
                             
                              
                             
                               ( 
                               
                                 m 
                                 _ 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               
                                 N 
                                 e 
                               
                             
                              
                             
                               
                                 
                                   J 
                                   
                                     j 
                                     , 
                                     i 
                                   
                                   e 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     _ 
                                   
                                   ) 
                                 
                               
                                
                               
                                 W 
                                 
                                   
                                     d 
                                     ; 
                                     j 
                                   
                                   , 
                                   j 
                                 
                                 e 
                               
                                
                               
                                 
                                   W 
                                   
                                     
                                       d 
                                       ; 
                                       j 
                                     
                                     , 
                                     j 
                                   
                                   e 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       M 
                                       j 
                                       e 
                                     
                                     - 
                                     
                                       
                                         S 
                                         j 
                                         e 
                                       
                                        
                                       
                                         ( 
                                         
                                           m 
                                           _ 
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               
                                 N 
                                 p 
                               
                             
                              
                             
                               
                                 
                                   J 
                                   
                                     j 
                                     , 
                                     i 
                                   
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     _ 
                                   
                                   ) 
                                 
                               
                                
                               
                                 W 
                                 
                                   
                                     d 
                                     ; 
                                     j 
                                   
                                   , 
                                   j 
                                 
                                 p 
                               
                                
                               
                                 
                                   W 
                                   
                                     
                                       d 
                                       ; 
                                       j 
                                     
                                     , 
                                     j 
                                   
                                   p 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       M 
                                       j 
                                       p 
                                     
                                     - 
                                     
                                       
                                         S 
                                         j 
                                         p 
                                       
                                        
                                       
                                         ( 
                                         
                                           m 
                                           _ 
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               λ 
                               n 
                             
                              
                             
                               
                                 
                                   1 
                                   
                                     m 
                                     i 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       ln 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           i 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       ln 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           
                                             n 
                                             , 
                                             i 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               2 
                             
                           
                           , 
                           
                             i 
                             = 
                             1 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             4 
                              
                             
                                 
                             
                              
                             L 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Hence, the gradient vector at the n-th iteration is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     g 
                     
                       n 
                       , 
                       i 
                     
                   
                   = 
                   
                     
                       
                         g 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             m 
                             _ 
                           
                           n 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           
                             N 
                             e 
                           
                         
                          
                         
                           
                             
                               J 
                               
                                 j 
                                 , 
                                 i 
                               
                               e 
                             
                              
                             
                               ( 
                               
                                 
                                   m 
                                   _ 
                                 
                                 n 
                               
                               ) 
                             
                           
                            
                           
                             W 
                             
                               
                                 d 
                                 ; 
                                 j 
                               
                               , 
                               j 
                             
                             e 
                           
                            
                           
                             
                               W 
                               
                                 
                                   d 
                                   ; 
                                   j 
                                 
                                 , 
                                 j 
                               
                               e 
                             
                              
                             
                               [ 
                               
                                 
                                   M 
                                   j 
                                   e 
                                 
                                 - 
                                 
                                   
                                     S 
                                     j 
                                     e 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         m 
                                         _ 
                                       
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           
                             N 
                             p 
                           
                         
                          
                         
                           
                             
                               J 
                               
                                 j 
                                 , 
                                 i 
                               
                               P 
                             
                              
                             
                               ( 
                               
                                 
                                   m 
                                   _ 
                                 
                                 n 
                               
                               ) 
                             
                           
                            
                           
                             W 
                             
                               
                                 d 
                                 ; 
                                 j 
                               
                               , 
                               j 
                             
                             p 
                           
                            
                           
                             
                               
                                 W 
                                 
                                   
                                     d 
                                     ; 
                                     j 
                                   
                                   , 
                                   j 
                                 
                                 p 
                               
                                
                               
                                 [ 
                                 
                                   
                                     M 
                                     j 
                                     p 
                                   
                                   - 
                                   
                                     
                                       S 
                                       j 
                                       p 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           m 
                                           _ 
                                         
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The jacobian matrices are calculated using finite-difference approximations as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       J 
                                       
                                         j 
                                         , 
                                         i 
                                       
                                       e 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           m 
                                           _ 
                                         
                                         n 
                                       
                                       ) 
                                     
                                   
                                   = 
                                     
                                    
                                   
                                     
                                       ∂ 
                                       
                                         
                                           S 
                                           j 
                                           e 
                                         
                                          
                                         
                                           ( 
                                           
                                             m 
                                             _ 
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       ∂ 
                                       
                                         m 
                                         i 
                                       
                                     
                                   
                                 
                                  
                               
                               
                                 
                                   m 
                                   _ 
                                 
                                 = 
                                 
                                   
                                     m 
                                     _ 
                                   
                                   n 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ≈ 
                                   
                                  
                                 
                                   - 
                                   
                                     
                                       
                                         
                                           S 
                                           j 
                                           e 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               
                                                 m 
                                                 _ 
                                               
                                               n 
                                             
                                             + 
                                             
                                               
                                                 δ 
                                                 
                                                   i 
                                                   , 
                                                   k 
                                                 
                                               
                                                
                                               Δ 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 
                                                   m 
                                                   _ 
                                                 
                                                 n 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         
                                           S 
                                           j 
                                           e 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               m 
                                               _ 
                                             
                                             n 
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         m 
                                         
                                           n 
                                           , 
                                           i 
                                         
                                       
                                     
                                   
                                 
                               
                               , 
                             
                           
                         
                       
                     
                     
                       
                         ( 
                         27 
                         ) 
                       
                     
                   
                   
                     
                       
                         
                           j 
                           = 
                           1 
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           N 
                           e 
                         
                         , 
                         i 
                         , 
                         
                           k 
                           = 
                           1 
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           4 
                            
                           
                               
                           
                            
                           L 
                         
                       
                     
                     
                       
                           
                       
                     
                   
                   
                     
                       and 
                     
                     
                       
                           
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 J 
                                 
                                   j 
                                   , 
                                   i 
                                 
                                 p 
                               
                                
                               
                                 ( 
                                 
                                   
                                     m 
                                     _ 
                                   
                                   n 
                                 
                                 ) 
                               
                             
                             = 
                               
                              
                             
                               - 
                               
                                 
                                   ∂ 
                                   
                                     
                                       S 
                                       j 
                                       p 
                                     
                                      
                                     
                                       ( 
                                       
                                         m 
                                         _ 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   ∂ 
                                   
                                     m 
                                     i 
                                   
                                 
                               
                             
                           
                            
                         
                         
                           
                             m 
                             _ 
                           
                           = 
                           
                             
                               m 
                               _ 
                             
                             n 
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                          
                         
                           - 
                           
                             
                               
                                 
                                   S 
                                   j 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         m 
                                         _ 
                                       
                                       n 
                                     
                                     + 
                                     
                                       
                                         δ 
                                         
                                           i 
                                           , 
                                           k 
                                         
                                       
                                        
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           m 
                                           _ 
                                         
                                         n 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   S 
                                   j 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       m 
                                       _ 
                                     
                                     n 
                                   
                                   ) 
                                 
                               
                             
                             
                               Δ 
                                
                               
                                   
                               
                                
                               
                                 m 
                                 
                                   n 
                                   , 
                                   i 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   
                     j 
                     = 
                     1 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     N 
                     p 
                   
                   , 
                   i 
                   , 
                   
                     k 
                     = 
                     1 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     4 
                      
                     
                         
                     
                      
                     L 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    The Hessian matrix is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             H 
                             
                               i 
                               , 
                               k 
                             
                           
                            
                           
                             ( 
                             
                               m 
                               _ 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             ∇ 
                             
                               m 
                               q 
                             
                           
                            
                           
                             
                               ∇ 
                               
                                 m 
                                 i 
                               
                             
                              
                             
                               
                                 C 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   _ 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               
                                 N 
                                 e 
                               
                             
                              
                             
                               
                                 
                                   J 
                                   
                                     j 
                                     , 
                                     i 
                                   
                                   e 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     _ 
                                   
                                   ) 
                                 
                               
                                
                               
                                 W 
                                 
                                   
                                     d 
                                     ; 
                                     j 
                                   
                                   , 
                                   j 
                                 
                                 e 
                               
                                
                               
                                 W 
                                 
                                   
                                     d 
                                     ; 
                                     j 
                                   
                                   , 
                                   j 
                                 
                                 e 
                               
                                
                               
                                 
                                   J 
                                   
                                     j 
                                     , 
                                     k 
                                   
                                   e 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     _ 
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               
                                 N 
                                 p 
                               
                             
                              
                             
                               
                                 
                                   J 
                                   
                                     j 
                                     , 
                                     i 
                                   
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     _ 
                                   
                                   ) 
                                 
                               
                                
                               
                                 W 
                                 
                                   
                                     d 
                                     ; 
                                     j 
                                   
                                   , 
                                   j 
                                 
                                 p 
                               
                                
                               
                                 W 
                                 
                                   
                                     d 
                                     ; 
                                     j 
                                   
                                   , 
                                   j 
                                 
                                 p 
                               
                                
                               
                                 
                                   J 
                                   
                                     j 
                                     , 
                                     k 
                                   
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     m 
                                     _ 
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               λ 
                               n 
                             
                              
                             
                               1 
                               
                                 m 
                                 i 
                               
                             
                              
                             
                               1 
                               
                                 m 
                                 k 
                               
                             
                           
                           , 
                           
                               
                           
                            
                           i 
                           , 
                           
                             k 
                             = 
                             1 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             4 
                              
                             
                                 
                             
                              
                             L 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Hence, the Hessian matrix at the n-th iteration is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       H 
                       
                         i 
                         , 
                         k 
                       
                     
                      
                     
                       ( 
                       
                         
                           m 
                           _ 
                         
                         n 
                       
                       ) 
                     
                   
                   ≈ 
                   
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         
                           N 
                           e 
                         
                       
                        
                       
                         
                           
                             J 
                             
                               j 
                               , 
                               i 
                             
                             e 
                           
                            
                           
                             ( 
                             
                               
                                 m 
                                 _ 
                               
                               n 
                             
                             ) 
                           
                         
                          
                         
                           W 
                           
                             
                               d 
                               ; 
                               j 
                             
                             , 
                             j 
                           
                           e 
                         
                          
                         
                           W 
                           
                             
                               d 
                               ; 
                               j 
                             
                             , 
                             j 
                           
                           e 
                         
                          
                         
                           
                             J 
                             
                               j 
                               , 
                               k 
                             
                             e 
                           
                            
                           
                             ( 
                             
                               
                                 m 
                                 _ 
                               
                               n 
                             
                             ) 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         
                           N 
                           p 
                         
                       
                        
                       
                         
                           
                             J 
                             
                               j 
                               , 
                               i 
                             
                             p 
                           
                            
                           
                             ( 
                             
                               
                                 m 
                                 _ 
                               
                               n 
                             
                             ) 
                           
                         
                          
                         
                           W 
                           
                             
                               d 
                               ; 
                               j 
                             
                             , 
                             j 
                           
                           p 
                         
                          
                         
                           W 
                           
                             
                               d 
                               ; 
                               j 
                             
                             , 
                             j 
                           
                           p 
                         
                          
                         
                           
                             J 
                             
                               j 
                               , 
                               k 
                             
                             p 
                           
                            
                           
                             ( 
                             
                               
                                 m 
                                 _ 
                               
                               n 
                             
                             ) 
                           
                         
                       
                     
                     + 
                     
                       
                         λ 
                         n 
                       
                        
                       
                         1 
                         
                           m 
                           
                             n 
                             , 
                             i 
                           
                         
                       
                        
                       
                         
                           1 
                           
                             m 
                             
                               n 
                               , 
                               k 
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In the Gauss-Newton minimization process, the search vector p n  can be constructed from the following linear equation: 
         [0000]          H     n   ·  p     n   =−  g     n .  (31)
 
         [0043]    Equation (31) is solved via a Gauss elimination procedure. The regularization parameter λ n  is chosen as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       λ 
                       n 
                     
                     = 
                     
                       β 
                        
                       
                           
                       
                        
                       max 
                        
                       
                         { 
                         
                           τ 
                           i 
                         
                         } 
                       
                     
                   
                   , 
                   
                     
                       if 
                        
                       
                           
                       
                        
                       
                         
                           min 
                            
                           
                             { 
                             
                               τ 
                               i 
                             
                             } 
                           
                         
                         
                           max 
                            
                           
                             { 
                             
                               τ 
                               i 
                             
                             } 
                           
                         
                       
                     
                     &lt; 
                     β 
                   
                   , 
                   
                       
                   
                    
                   
                     i 
                     = 
                     1 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     4 
                      
                     
                         
                     
                      
                     L 
                   
                   , 
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where β is a constant value determined from numerical experiments and τ are the eigenvalues of the Hessian matrix  H . 
         [0044]    We employ a line-search procedure as in Habashy and Abubakar to enforce the reduction of cost function in each iteration as follows: 
         [0000]          m     n+1   =  m     n   +v  p     n ,  (33)
 
         [0000]    where 0&lt;v n ≦1 is the step-length. The step-length is calculated by solving the minimization problem: 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                     n 
                   
                   = 
                   
                     arg 
                      
                     
                         
                     
                      
                     
                       
                         min 
                         v 
                       
                        
                       
                         
                           { 
                           
                             C 
                              
                             
                               ( 
                               
                                 
                                   
                                     m 
                                     _ 
                                   
                                   n 
                                 
                                 + 
                                 
                                   v 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       p 
                                       _ 
                                     
                                     n 
                                   
                                 
                               
                               ) 
                             
                           
                           } 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
         [0045]    In order to avoid expensive evaluation of the cost function C(  m ), we adopt an algorithm (see Dennis Jr., J. E. and R. B. Schnabel, 1983, Numerical methods for unconstrained optimization and nonlinear equations: Prentice Hall, Inc., New Jersey) whereby v n  is selected such that the average rate of decrease from C(  m   n ) to C(  m   n +v n   p   n ) is at least some prescribed fraction, α, of the initial rate of decrease at  m   n  along the direction  p   n , i.e., 
         [0000]        C (   m     n   +v   n     p     n )≦ C (   m     n )+α v   n   δC   n+1 ,  (35)
 
         [0000]    where 0&lt;α&lt;1 is a fractional number which, in practice, is set quite small (we use 10 −4  herein) so that hardly more than a decrease in function value is required. δC n+1  is the rate of decrease of C(  m ) at  m   n  along the direction  p   n  and is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           δ 
                            
                           
                               
                           
                            
                           
                             C 
                             
                               n 
                               + 
                               1 
                             
                           
                         
                         = 
                         
                           
                             ∂ 
                             
                               ∂ 
                               v 
                             
                           
                            
                           
                             C 
                              
                             
                               ( 
                               
                                 
                                   
                                     m 
                                     _ 
                                   
                                   n 
                                 
                                 + 
                                 
                                   v 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       p 
                                       _ 
                                     
                                     n 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                        
                     
                     
                       v 
                       = 
                       0 
                     
                   
                   = 
                   
                     
                       
                         
                           g 
                           _ 
                         
                         T 
                       
                        
                       
                         ( 
                         
                           
                             m 
                             _ 
                           
                           n 
                         
                         ) 
                       
                     
                     · 
                     
                       
                         
                           p 
                           _ 
                         
                         n 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
         [0000]    We first employ the full Gauss-Newton search step in the minimization process. If v n =1 fails to satisfy the criterion given by equation (35), then we reduce v n  to backtrack along the direction  p   n  until an acceptable next  m   n+1  is found. Assume, at the (n+1)-th Gauss-Newton iteration, v n   k  is the current step-length that does not satisfy equation (35), we compute the next backtracking step-length, v n   k+1 , by searching for the minimum of the cost function, f(v)≡C(  m   n +v  p   n ), which can be approximated by a quadratic expression. Thus, v n   k+1 , which is the minimum of f(v), for k=0, 1, 2, . . . is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                     n 
                     
                       k 
                       + 
                       1 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                             ( 
                             
                               v 
                               n 
                               k 
                             
                             ) 
                           
                           2 
                         
                         2 
                       
                     
                      
                     
                       
                         
                           δ 
                            
                           
                               
                           
                            
                           
                             C 
                             
                               n 
                               + 
                               1 
                             
                           
                         
                         
                           
                             C 
                              
                             
                               ( 
                               
                                 
                                   
                                     m 
                                     _ 
                                   
                                   n 
                                 
                                 + 
                                 
                                   
                                     v 
                                     n 
                                     k 
                                   
                                    
                                   
                                     
                                       p 
                                       _ 
                                     
                                     n 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             C 
                              
                             
                               ( 
                               
                                 
                                   m 
                                   _ 
                                 
                                 n 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               v 
                               n 
                               k 
                             
                              
                             δ 
                              
                             
                                 
                             
                              
                             
                               C 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0046]    To impose a priori information of maximum and minimum bounds on the unknown parameters, we constrained them using a non-linear transformation of the form: 
         [0000]    
       
         
           
             
               
                 
                   
                     m 
                     = 
                     
                       
                         f 
                          
                         
                           ( 
                           
                             c 
                             , 
                             
                               m 
                               min 
                             
                             , 
                             
                               m 
                               max 
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           m 
                           min 
                         
                         + 
                         
                           
                             
                               
                                 m 
                                 max 
                               
                               - 
                               
                                 m 
                                 min 
                               
                             
                             
                               
                                 c 
                                 2 
                               
                               + 
                               1 
                             
                           
                            
                           
                             c 
                             2 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       - 
                       ∞ 
                     
                     &lt; 
                     c 
                     &lt; 
                     
                       + 
                       
                         ∞ 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
         [0047]    The Gauss-Newton search step in m, p n =m n+1 −m n , is related to the step in c, q n =c n+1 −c n , by 
         [0000]    
       
         
           
             
               
                 
                   p 
                   = 
                   
                     
                       
                         
                            
                           m 
                         
                         
                            
                           c 
                         
                       
                        
                       q 
                     
                     = 
                     
                       2 
                        
                       
                         
                           
                             m 
                             max 
                           
                           - 
                           m 
                         
                         
                           
                             m 
                             max 
                           
                           - 
                           
                             m 
                             min 
                           
                         
                       
                        
                       
                         
                           
                             ( 
                             
                               
                                 m 
                                 max 
                               
                               - 
                               m 
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               m 
                               - 
                               
                                 m 
                                 min 
                               
                             
                             ) 
                           
                         
                       
                        
                       
                         q 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The inverted parameter m is then updated using the following equation, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       m 
                       
                         n 
                         + 
                         1 
                       
                     
                     = 
                     
                       
                         m 
                         min 
                       
                       + 
                       
                         
                           
                             
                               m 
                               max 
                             
                             - 
                             
                               m 
                               min 
                             
                           
                           
                             
                               α 
                               n 
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     m 
                                     n 
                                   
                                   - 
                                   
                                     m 
                                     min 
                                   
                                 
                                 ) 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       m 
                                       max 
                                     
                                     - 
                                     
                                       m 
                                       n 
                                     
                                   
                                   ) 
                                 
                                 3 
                               
                             
                           
                         
                          
                         
                           α 
                           n 
                           2 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
             
               
                 
                   
                     α 
                     n 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             m 
                             n 
                           
                           - 
                           
                             m 
                             min 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             m 
                             max 
                           
                           - 
                           
                             m 
                             n 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             m 
                             max 
                           
                           - 
                           
                             m 
                             min 
                           
                         
                         ) 
                       
                        
                       
                         v 
                         n 
                       
                        
                       
                         
                           p 
                           n 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
         [0048]      FIG. 1  is a flow chart showing workflow steps of a joint inversion, according to some embodiments. The initial guess of the unknown parameters (permeability, porosity and mud-filtrate invasion rate)  112  and additional information such as PVT, Kr, Pc, formation geometry, mud-filtrate and formation brine concentration et al.  110  are used for building a reservoir model. A multiphase fluid flow simulator  114  is run on the reservoir model generated from the data  110  and the initial parameters  112 , thereby generating spatial distributions  116  of water saturation and salt concentration. A conductivity-saturation model  118  is used to transform the spatial distributions  116  into a conductivity distribution  120 . Electromagnetic simulator  122  generates simulated electromagnetic data  126  based on the input conductivity distribution  120 . Simulated pressure and flow rate data  124  is generated by multiphase fluid flow simulator  114 . The pressure and flow rate data  124 , electromagnetic data  126 , measured pressure and flow rate data  128 , and measured electromagnetic data  130  are input into cost function  132 . Cost function  132  is combination of differences of  124  to  128 , and  126  to  130 . In step  136 , values are updated for permeability, porosity and mud-filtrate invasion rate  112  until the cost function  132  becomes less than a predefined tolerance, thereby yielding the inverted parameters and derived distributions of water saturation and conductivity  134 . 
         [0049]    Test Cases. 
         [0050]    Further detail relating to two test cases, according to some embodiments, will now be provided. To demonstrate the workflow, according to some embodiments, two synthetic examples were employed consisting of a vertical well penetrating layered hydrocarbon-bearing horizontal formations with sealing upper and lower shoulder beds. For reasons of facilitating comparisons with known results, the same examples are employed which can be referred to in Alpak, et al. (2004). 
         [0051]    In this example, it is assumed that there is no-flux surrounding the boundaries. The water-based mud-filtration invasion is taking place since the well is drilled. The induction logging data are assumed to be recorded using an array induction tool, which has one transmitter and an eight-receiver array, providing fourteen pairs of apparent conductivity logs for different depths of investigation. Immediately after the first induction logging, a formation test is carried out using a packer-probe formation tester such as with Schlumberger&#39;s Modular Formation Dynamics Tester (MDT) tools. Three time-series of pressure transient measurements are recorded at two probes and the packer interval. This is followed by a second induction logging to sense the perturbed resistivity around the packer interval caused by the formation test. This process is referred to as a log-test-log mode. If the perturbation caused by the formation test is not significant, the second induction logging can be discarded, so the process is simplified to the log-test mode. In these two modes there are two measurements, namely the induction logging and pressure transient data, which are then used for the joint inversion. According to some embodiments, another mode is provided that only uses two induction logging data collected at different times. This mode is referred to herein as a time-lapse log mode. In a time-lapse log mode, the multiphase fluid flow simulator is only used to constrain the inversion of the induction logging data. 
         [0052]      FIGS. 2   a  and  2   b  are diagrams showing certain configurations (refer to Alpak, et al. (2004)), according to some embodiments. For  FIGS. 2   a  and  2   b  the drainage radii for wells  230  and  228  respectively, are both about 300 m.  FIGS. 2   a  and  2   b  are two-dimensional vertical cross-sections of the reservoir model configuration. Impermeable shoulder beds  208  and  210  with known homogeneous formation conductivities are superimposed on the top and bottom of the reservoirs  212  and  214  respectively. The permeable zones of interest are subject to the water-based mud-filtrate invasion. In  FIG. 2   a , the reservoir  212  comprises a single-layer TI-anisotropic formation  216  with oil-water system. Mud filtrate invasion zones  232  for well  230 , and  234  for well  228  are shown in  FIGS. 2   a  and  2   b  respectively. 
         [0053]    The case shown in  FIG. 2   a  is a single-layer model with a TI-anisotropic formation and oil-water system, while the case shown in  FIG. 2   b  is a three-layer model with isotropic formation and gas-water system. In these examples, the fluid properties, saturation dependent functions and the parameters in Archie&#39;s equation are available from laboratory and core measurements. Both the oleic and aqueous phases are slightly compressible while the gaseous phase is compressible with its compressibility dependent on pressure. 
         [0054]      FIGS. 3   a ,  3   b  and  3   c  are plots illustrating the saturation dependent functions used in certain described examples (refer to Alpak, et al. (2004)).  FIG. 3   a  shows the relative permeability for the oil-water case shown in  FIG. 2   a , where the capillary pressure is negligible. Curve  310  is the relative permeability of water and curve  312  is the relative permeability of oil in the formation layer  214 .  FIG. 3   b  shows the layer-by-layer relative permeability for the gas-water case shown in  FIG. 2   b . Curve  314  shows the relative permeability for gas in the upper and lower layers  218  and  222 ; curve  316  shows the relative permeability for water in the upper and lower layers  218  and  222 ; curve  318  shows the relative permeability of gas for the middle layer  220 ; and curve  320  shows the relative permeability of water for the middle layer  220 .  FIG. 3   c  shows the layer-by-layer capillary pressure for gas-water case shown in  FIG. 2   c . Curve  324  shows the capillary pressure for the upper layer  118 ; curve  326  shows the capillary pressure for the middle layer  220 ; and curve  328  shows the capillary pressure for the lower layer  222 . 
         [0055]    For the oil-water case shown in  FIG. 2   a , the following operation schedule is assumed: The well is subjected to the mud-filtrate invasion after drilling. The first induction log is recorded on the third day and then immediately followed by the wireline formation test, which is composed of a draw-down at a constant liquid rate of 4.76962 m 3 /d for 100 minutes followed by a build-up for 200 minutes. In this example, the pressure transient data is only used in the build-up period for the inversion. Another induction log is then recorded to capture the perturbation caused by the formation test. For the time-lapse log mode, the second induction log is recorded at the tenth day. 
         [0056]    In the fluid flow simulator we used a 31×6×30 grid with the finest radial cell size of about 0.0305 m, adjacent to the borehole. The model is initialized with the irreducible water saturation. Since the fluid and salt movement are convective and the diffusion and salt dispersion can be neglected, the mud-filtrate invasion process is calculated using an average mud-filtrate invasion rate. It is assumed that the average invasion rate is 0.0067 m 3 /d/m for the first three days. For the time-lapse log mode, an invasion rate of 0.00577 m 3 /d/m between the third and tenth day is also assumed. Other specific model parameters are listed in Table 1. 
         [0000]    
       
         
               
             
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Petrophysical and fluid parameters and other relevant 
               
               
                 information employed in oil-water example shows in 
               
               
                 FIG. 2a. (also, Refer to Alpak, et al. (2004)) 
               
             
          
           
               
                 Variable 
                 Value 
               
               
                   
               
             
          
           
               
                 Formation rock compressibility 
                 7.25′10 −8   
                 [bar −1 ] 
               
               
                 Aqueous phase viscosity 
                 1.274 
                 [cp] 
               
               
                 Aqueous phase density 
                 1001.074 
                 [kg/m 3 ] 
               
               
                 Aqueous phase formation volume factor 
                 0.996 
                 [rm 3 /sm 3 ] 
               
               
                 Aqueous phase compressibility 
                 3.7′10 −5   
                 [bar −1 ] 
               
               
                 Oleic phase viscosity 
                 0.355 
                 [cp] 
               
               
                 Oleic phase API density 
                 42 
                 [° API] 
               
               
                 Oleic phase density 
                 815.564 
                 [kg/m 3 ] 
               
               
                 Oleic phase formation volume factor 
                 1.471 
                 [rm 3 /sm 3 ] 
               
               
                 Oleic phase compressibility 
                 2.76′10 −4   
                 [bar −1 ] 
               
               
                 Formation pressure (refer to formation top) 
                 206.84 
                 [bar] 
               
               
                 Wellbore radius 
                 0.108 
                 [m] 
               
               
                 Formation horizontal permeability 
                 100 
                 [mD] 
               
               
                 Formation vertical permeability 
                 20 
                 [mD] 
               
               
                 Formation porosity 
                 0.25 
                 [fraction] 
               
               
                 Formation temperature 
                 220 
                 [° F.] 
               
               
                 Formation brine salinity 
                 120000 
                 [ppm] 
               
               
                 Mud-filtrate salinity 
                 5000 
                 [ppm] 
               
               
                 a-constant in the Archie&#39;s equation 
                 1.00 
                 [dimensionless] 
               
               
                 m-cementation exponent in the Archie&#39;s 
                 2.00 
                 [dimensionless] 
               
               
                 equation 
               
               
                 n-water saturation exponent in the 
                 2.00 
                 [dimensionless] 
               
               
                 Archie&#39;s equation 
               
               
                 Mud conductivity 
                 2631.58 
                 [mS/m] 
               
               
                 Upper and lower shoulder bed conductivities 
                 1000.00 
                 [mS/m] 
               
               
                 Logging interval 
                 0.61 
                 [m] 
               
               
                   
               
             
          
         
       
     
         [0057]    In the gas-water three-layer example shown in  FIG. 2   b , a similar operation schedule is used. However, the first induction log is recorded at the first day, the draw-down for the formation test lasts 100 minutes at a constant rate of 0.2385 m 3 /d, then the tool is shut down for another 200 minutes for pressure build-up. The second induction logging was not carried out since the perturbation in formation resistivity caused by the formation test is not significant. The fluid flow simulator uses a 91×6×30 grid with the finest radial cell size of about 0.0061 m. The model is initialized with the irreducible water saturation. Different average invasion rates, permeabilities and porosities are used for the three layers. Referring to  FIG. 2   b , for the first-layer  218 : k=6.83 mD, φ=0.139, q=0.42523 m 3 /d/m; for the second-layer  220 : k=40.98 mD, φ=0.137, q=0.42485 m 3 /d/m; and for the third-layer  222 : k=14.33 mD, φ=0.125, q=0.42548 m 3 /d/m. Other specific model parameters can be found in Table 2. Some parameters are the same as in oil-water example of  FIG. 2   a . 
         [0000]    
       
         
               
             
               
               
             
               
               
               
             
               
               
             
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Petrophysical and fluid parameters and other relevant 
               
               
                 information employed in example shown in FIG. 2b. 
               
               
                 (Also refer to Alpak, et al. (2004)) 
               
             
          
           
               
                 Variable 
                 Value 
               
               
                   
               
             
          
           
               
                 Aqueous phase viscosity 
                 1.000 
                 [cp] 
               
               
                 Aqueous phase density 
                 1186.007 
                 [kg/m 3 ] 
               
               
                 Aqueous phase formation volume factor 
                 1.000 
                 [rm 3 /sm 3 ] 
               
               
                 Aqueous phase compressibility 
                 1.45′10 −5   
                 [bar −1 ] 
               
             
          
           
               
                 Gaseous phase viscosity [f(p) for T = const.] 
                 0.01087 [cp] @1 bar 
               
               
                 Gaseous phase density [f(p) for T = const.] 
                 0.977 [kg/m 3 ] @1 bar 
               
               
                 Gaseous phase formation volume factor [f(p) 
                 0.782 [rm 3 /sm 3 ] @1 bar 
               
               
                 for T = const.] 
               
             
          
           
               
                 Formation pressure (refer to formation top) 
                 6.62 
                 [bar] 
               
               
                 Wellbore radius 
                 0.107 
                 [m] 
               
               
                 Formation temperature 
                 96 
                 [° F.] 
               
               
                 Formation brine salinity 
                 200000 
                 [ppm] 
               
               
                 Mud-filtrate salinity 
                 2000 
                 [ppm] 
               
               
                 Mud conductivity 
                 387.60 
                 [mS/m] 
               
               
                 Upper and lower shoulder bed conductivities 
                 400.00 
                 [mS/m] 
               
               
                   
               
             
          
         
       
     
         [0058]    Inversion Results. 
         [0059]    In this section, the inversion results are described for the synthetic induction logging data and pressure transient data, which are generated by forward modelling based on pre-defined true petrophysical parameters and invasion rates, with added zero-mean Gauss random noise.  FIGS. 4-6 , show the results for oil-water example shown in  FIG. 2   a . In  FIG. 4   a , diagram  410  shows the vertical cross-section of spatial distributions of the reconstructed water saturation for the first logging run for the oil-water example shown in  FIG. 2   a . In  FIG. 4   b , diagram  412  shows the vertical cross-section of spatial distributions of the reconstructed water saturation for the second logging run for the oil-water example shown in  FIG. 2   a . In  FIG. 4   c , diagram  414  shows the vertical cross-section of spatial distributions of the reconstructed salt concentration for the first logging run for the oil-water example shown in  FIG. 2   a . In  FIG. 4   d , diagram  416  shows the vertical cross-section of spatial distributions of the reconstructed salt concentration for the second logging run for the oil-water example shown in  FIG. 2   a . In  FIG. 4   e , diagram  418  shows the vertical cross-section of spatial distributions of the reconstructed formation conductivity for the first logging run for the oil-water example shown in  FIG. 2   a . In  FIG. 4   f , diagram  420  shows the vertical cross-section of spatial distributions of the reconstructed formation conductivity for the second logging run for the oil-water example shown in  FIG. 2   a . In all cases there was Gaussian random noise of up to 7%. A high conductivity annulus is observed, which is caused by the invasion of the low salinity water-based mud-filtrate into the hydrocarbon-bearing formation with high salinity connate water.  FIGS. 4   b,    4   d  and  4   f  show oil breaking through the high conductivity annulus caused by draw-down in the formation test. In  FIG. 4   g , diagram  422  plots the difference between the true conductivity and the reconstructed conductivity distributions for the first logging run for the oil-water example shown in  FIG. 2   a . In  FIG. 4   h , diagram  424  plots the difference between the true conductivity and the reconstructed conductivity distributions for the second logging run for the oil-water example shown in  FIG. 2   a . It can be seen from  FIGS. 4   h  and  4   g  that there is good agreement between the true conductivity and reconstructed conductivity distributions. 
         [0060]      FIGS. 5   a  and  5   b  are log-log plots of the pressure and the pressure derivative for the build-up for the example shown in  FIG. 2   a . In  FIG. 5   a  plots results with 0% noise, and in  FIG. 5   b  plots results with 7% noise. In  FIG. 5   a , curve  510  plots the measured pressure, curve  512  plots the measured derivative pressure, curve  514  plots the post-inversion pressure, and curve  516  plots the post-inversion derivative pressure. In  FIG. 5   b , curve  520  plots the measured pressure, curve  522  plots the measured derivative pressure, curve  524  plots the post-inversion pressure, and curve  526  plots the post-inversion derivative pressure. 
         [0061]    The inverted permeability, porosity and average mud-filtrate invasion rate are listed together with the number of Gauss-Newton iterations in Tables 3 and 4. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Log-test-log inversion results for example shown in FIG. 2a 
               
               
                 Log-test-log 
               
             
          
           
               
                   
                   
                   
                   
                 φ 
                   
                 Num. of 
               
               
                   
                 Add. Gauss 
                 k h   
                 k v   
                 (frac- 
                 q 
                 itera- 
               
               
                   
                 noise level 
                 (md) 
                 (md) 
                 tion) 
                 (m 3 /d/m) 
                 tions 
               
               
                   
                   
               
             
          
           
               
                 Init. 
                   
                 40 
                 40 
                 0.12 
                 0.00052 
                   
               
               
                 guess 
               
               
                 True 
                   
                 100 
                 20 
                 0.25 
                 0.00670 
               
               
                 Inverted 
                 0% 
                 100.00 
                 20.00 
                 0.250 
                 0.00670 
                 8 
               
               
                 Inverted 
                 1% 
                 100.78 
                 19.08 
                 0.250 
                 0.00670 
                 6 
               
               
                 Inverted 
                 3% 
                 102.43 
                 17.02 
                 0.250 
                 0.00670 
                 5 
               
               
                 Inverted 
                 5% 
                 104.24 
                 14.86 
                 0.250 
                 0.00672 
                 7 
               
               
                 Inverted 
                 7% 
                 105.99 
                 13.54 
                 0.250 
                 0.00672 
                 6 
               
               
                 Analytic 
                 0% 
                 80.26 
                 18.32 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 Time-lapse log inversion results for example shown in FIG. 2a 
               
               
                 Time-lapse log 
               
             
          
           
               
                   
                   
                   
                 φ 
                   
                   
                 Num. of 
               
               
                   
                 Add. Gauss 
                 k 
                 (frac- 
                 q 1   
                 q 2   
                 itera- 
               
               
                   
                 noise level 
                 (md) 
                 tion) 
                 (m 3 /d/m) 
                 (m 3 /d/m) 
                 tions 
               
               
                   
                   
               
             
          
           
               
                 Init. 
                   
                 N/A 
                 0.12 
                 0.001 
                 0.00052 
                   
               
               
                 guess 
               
               
                 True 
                   
                 N/A 
                 0.25 
                 0.00670 
                 0.00577 
               
               
                 Inverted 
                 0% 
                 N/A 
                 0.250 
                 0.00670 
                 0.00577 
                 4 
               
               
                 Inverted 
                 1% 
                 N/A 
                 0.250 
                 0.00677 
                 0.00562 
                 3 
               
               
                 Inverted 
                 3% 
                 N/A 
                 0.249 
                 0.00721 
                 0.00526 
                 2 
               
               
                 Inverted 
                 5% 
                 N/A 
                 0.249 
                 0.00732 
                 0.00495 
                 2 
               
               
                   
               
             
          
         
       
     
         [0062]    The inversion results demonstrate that we can obtain reliable horizontal permeability, porosity and average invasion rate for up to 7% noise level. The inversion of the porosity is very robust with little sensitivity to the noise level. This is because the induction logging data is governed by the formation conductivity, which is strongly affected by the porosity. With increasing noise level, the quality of the inverted vertical permeability quickly deteriorates. This is expected because the sensitivity of pressure transient measurements to the vertical permeability are relatively low and would be overwhelmed by noise. The inversion of the horizontal permeability shows good accuracy. In the time-lapse log mode, the formation permeability cannot be inverted from induction measurements only and without the pressure data since the invasion process is mainly governed by the mud-cake and moreover we are using average invasion rates as the boundary condition in the fluid flow simulation. 
         [0063]    The horizontal and vertical permeabilities can also be independently obtained from the formation test by using the analytic straight-line computation, which is based on the single-phase model without taking into account the invasion profile. This approach can be applied on the pressure transient data, which are generated using the fluid flow simulator for the oil-water example shown in  FIG. 2   a . Only the build-up section of the data are interpreted. But the build-up time is prolonged to reach a stable radial flow status ( FIGS. 5   a  and  5   b  show that the radial flow has not formed yet). No noise was introduced in this case to the synthetic pressure transient data. The analytical interpretation results of the build-up section are also listed in Table 3 for comparison. It can be observed that the joint inversion produces more accurate estimation of formation permeabilities. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 5 
               
             
             
               
                   
               
               
                 Log-test inversion results for example shown in FIG. 2b. 
               
             
          
           
               
                   
                   
                   
                 φ 
                   
                 Num. of 
               
               
                   
                 Add. Gauss 
                 k 
                 (frac- 
                 q 
                 itera- 
               
               
                   
                 noise level 
                 (md) 
                 tion) 
                 (m 3 /d/m) 
                 tions 
               
               
                   
                   
               
             
          
           
               
                   
                 Init. 
                   
                 12.39 
                 0.18 
                 0.26081 
                   
               
               
                   
                 guess 
               
               
                 Layer 1 
                 True 
                   
                 6.83 
                 0.139 
                 0.42523 
               
               
                   
                 Inverted 
                 0% 
                 6.83 
                 0.139 
                 0.42523 
                 8 
               
               
                   
                 Inverted 
                 5% 
                 6.61 
                 0.139 
                 0.41693 
                 10 
               
               
                 Layer 2 
                 True 
                   
                 40.98 
                 0.137 
                 0.42485 
               
               
                   
                 Inverted 
                 0% 
                 40.98 
                 0.137 
                 0.42486 
                 8 
               
               
                   
                 Inverted 
                 5% 
                 47.15 
                 0.139 
                 0.47190 
                 10 
               
               
                 Layer 3 
                 True 
                   
                 14.33 
                 0.125 
                 0.42548 
               
               
                   
                 Inverted 
                 0% 
                 14.33 
                 0.125 
                 0.42547 
                 8 
               
               
                   
                 Inverted 
                 5% 
                 14.33 
                 0.123 
                 0.40225 
                 10 
               
               
                   
               
             
          
         
       
     
         [0064]    Thus, according to some embodiments, a parametric inversion algorithm is described to jointly invert the induction logging and pressure transient measurements, which can be used for the determination of the horizontal and vertical permeability, the porosity and the average mud-filtrate invasion rate for each flow unit or unit height. When using pressure measurements, this algorithm can reliably invert the permeability without incorporating a mud-cake growth and invasion model. Additionally, the porosity can be inverted robustly either using the induction log together with pressure measurement, or only by using induction logs but constrained by the fluid flow simulator. 
         [0065]    Traditionally, the mud-filtrate invasion profile can be derived from the inversion of induction measurements, which are based on a simplified step-profile invasion model. Inversion of the induction logs constrained by the fluid flow simulator improves the estimation of the mud-filtrate invasion profile, which becomes consistent with the fluid flow physics. On the other hand, the conventional interpretation of formation test is based on the single-phase flow model without taking into account fluid invasion or the need to pump a long time to clean up the contamination of the mud-filtrate before the test procedure. The joint inversion of induction logging and pressure measurements enable a more reliable result. This technique can accommodate the existence of the mud-filtrate in the region of formation test, and hence can save costly rig time for clean up. 
         [0066]    As a by-product, through the fluid-flow constrained time-lapse induction log inversion, the temporal and spatial distributions of the saturation and conductivity can be obtained, which can be used to monitor the fluid front movement. According to some embodiments, the inversion techniques described herein are used to generate a quantitative predictive formation model that is dynamic, rather than static, to perform fluid flow simulations. 
         [0067]    According to some embodiments, the inversion procedures described herein with respect to mud-filtrate invasion rates are applied to estimate completion fluid invasion rates. For example the invasion zones  232  and  234  in  FIGS. 2   a  and  2   b  respectively, according to these embodiments, represent completion fluid invasion zones. According to some embodiments, a subsequent electromagnetic survey can be conducted for the purpose of estimating the completion fluid invasion rates. However, according to other embodiments, a quantitative predictive formation model based on earlier electromagnetic surveys for mud filtrate invasion can also be used to directly predict the completion fluid invasion. 
         [0068]    According to some embodiments, the result of the inversion procedures is used to aid well clean up job design. A “clean up” job can be performed following a drilling procedure and/or completion procedure to reduce the effects that the drilling fluid and/or completion fluid has on the producing formation. Based on the invasion profile and the quantitative predictive formation model derived from the inversion techniques described herein, a fluid flow simulation is performed to find out how the clean up procedure should be operated. 
         [0069]    According to some embodiments, the described workflow can be applied to deviated and horizontal wells with dipping layers and cross-well applications. In these cases the cylindrical grid in the fluid flow simulator should be replaced by the Cartesian grid. 
         [0070]    In addition to the 1D parametric inversions of porosity for each layer, according to some embodiments, the approach can also be applied to invert a spatial distribution of porosity.  FIGS. 6 and 7  illustrate an example used to demonstrate this capability.  FIG. 6  is a two-dimensional vertical cross-section of a reservoir having an injector and producer well, according to some embodiments. The reservoir  614  is composed of three layers,  618 ,  620  and  622 , with a total thickness of 25 meters. The sealing upper and lower shoulder beds  610  and  612  allow for no-flux surrounding the boundaries. The permeability and porosity of each layer are different. The upper layer  618  has k h =30 mD, the middle layer  620  has k h =100 mD, and the lower layer  622  has k h =70 mD. It is assumed that the formations are TI-anisotropic with constant anisotropy ratio, k v /k h =0.1. An injection well  630  and a production well  634  penetrate all the layers  610 ,  618 ,  620 ,  622  and  612 . Both wells  630  and  634  are vertical and the separation is 80 meters. An oil-water system is used and it is assumed that the fluid properties, saturation dependent functions and coefficients used in the Archie&#39;s equation are available from laboratory and core measurements. For simplicity, it is assumed that the injection well  630  is operated under bottomhole pressure control, which is a constant 241.32 bar. The production well  634  produces a constant oil rate of 158.99 m 3 /d. After 5 days injection and production, a crosswell electromagnetic logging is conducted. Water flooding zone  632  is also shown. 
         [0071]      FIG. 7  shows the source-receiver setup and associated surface equipment according to the example shown in  FIG. 6 . In the crosswell electromagnetic logging process, sources  702  are located in the injection well  630 , while receivers  708  are located in the production well  634 . Although only 14 source stations and 14 receiver stations are shown on the toolstrings in  FIG. 7  for simplicity, in this example there are 29 source stations  702  and 29 receiver stations  708 . Thus there are 29×29 source-receiver combinations. For example, each of the receivers  708  receiver signals form source  704  and from source  706  as shown by the dashed and solid lines in  FIG. 7 . The data are the vertical component of the magnetic fields and the sources operate at 500 Hz. For the fluid flow modelling, a 198×11×31 grid is used according to some embodiments. The grid is uniform and fine at the center region then logarithmically coarsened towards the boundaries in the x and y directions, however the grid is completely uniform in the z direction. The model is initialized with the irreducible water saturation. For the electromagnetic modelling, an 84×84 grid is used where the inner inversion domain is uniform and the outer domain is optimized automatically. It is assumed that the conductivities of shoulder beds are 1.183 S/m. According to some embodiments, the electromagnetic measurements are acquired using a tool or tools such as Schlumberger&#39;s Array Induction Imager Tools (AIT) in either a single borehole (such as  FIG. 2   a  and/or  2   b , or in two boreholes as shown in  FIGS. 6 and 7 . 
         [0072]    Also shown in  FIG. 7  is related surface equipment, according to some embodiments. Sources  702  are deployed in well  630  on a tool string suspended from a wireline cable  764  from logging truck  762 . Logging truck  762  also is used to control the sources  702  and in communication with a processing center  750 . Similarly receivers  708  are deployed in well  634  on a tool string suspended from a wireline cable  774  from logging truck  772 . Measurements from receivers  708  are recorded and can be processed in one of the logging trucks  762  or  772  or can be transmitted directly to a processing center  750 . Processing center  750  includes one or more central processing units  744  for carrying out the inversion procedures as described herein, as well as other processing. Processing center  750  also includes a storage system  742 , communications and input/output modules  740 , a user display  746  and a user input system  748 . According to some embodiments, processing center  750  can be included in one or both of the logging trucks  762  and  772 , or may be located in a location remote from the wellsites. Also shown in  FIG. 7  is a surface electromagnetic array  780  for obtaining electromagnetic data. Thus the electromagnetic data used in the inversion procedures described herein can be obtained using various arrangements of equipment including single well (as in  FIGS. 2   a  and  2   b ), cross well, surface to borehole, borehole to surface and surface to surface. Although wireline logging trucks are shown deploying the sources and receivers in  FIG. 7 , when deployed in a marine environment the sources and receivers may be deployed from a platform or vessel as well known in the art. Furthermore, according to other embodiments the sources and/or receivers can be deployed on a drill collar during part of a drilling operation. In the case of surface deployed electromagnetic arrays in the marine environment, according to some embodiments, surface array  780  can be implemented using deep-towed electric dipole antenna such as used with Controlled Source Electromagnetics (CSEM) surveys from WesterGeco, a business unit of Schlumbeger. 
         [0073]    Because the crosswell electromagnetic inversion is not very sensitive to permeability, we assume the formation permeabilities are acquired from other independent measurements.  FIGS. 8   a - h  are vertical cross-sections of spatial distributions of the porosity, the conductivity, the water saturation and the salt concentration, according to some embodiments. In  FIG. 8   a  diagram  810  shows the true distribution of porosity, while in  FIG. 8   b  diagram  812  shows the inverted distribution of porosity. In  FIG. 8   c , diagram  814  shows the true distribution for conductivity, while in  FIG. 8   d  diagram  816  shows the reconstructed distribution for conductivity. In  FIG. 8   e , diagram  818  shows the true distribution for water saturation while in  FIG. 8   f , diagram  820  shows the reconstructed distribution for water saturation. In  FIG. 8   g , diagram  822  shows the true distribution for salt concentration, while in  FIG. 8   h , diagram  824  shows the reconstructed distribution for salt concentration. In all cases the data is corrupted with 2% Gauss random noise in the inversion. Compared to the true distribution, also as shown in  FIG. 8 , the inverted porosity recovers the true distribution to some extent. However, the reconstructed conductivity and water saturation are very close to the true distributions, and from which we can clearly see the fluid front, which is the main interest for the fluid movement monitoring. 
         [0074]    According to some embodiments, a second example was conducted composed of 20 dipping layers, which were categorized into 4 rock types. Each rock type had different properties such as permeability, porosity and relative permeability. Similar to the example shown in  FIGS. 6 and 7 , water was simultaneously injected and oil produced in certain zones. It was assumed that the injection well was operated under bottomhole pressure control and the production well produces a constant oil rate. The same inversion procedure was implemented with additional 2% Gauss random noise added to the data. The spatial porosity distribution was recovered to some extent, and the water front was clearly seen from the reconstructed conductivity distribution. 
         [0075]    Whereas many alterations and modifications of the present invention will no doubt become apparent to a person of ordinary skill in the art after having read the foregoing description, it is to be understood that the particular embodiments shown and described by way of illustration are in no way intended to be considered limiting. Further, the invention has been described with reference to particular preferred embodiments, but variations within the spirit and scope of the invention will occur to those skilled in the art. It is noted that the foregoing examples have been provided merely for the purpose of explanation and are in no way to be construed as limiting of the present invention. While the present invention has been described with reference to exemplary embodiments, it is understood that the words, which have been used herein, are words of description and illustration, rather than words of limitation. Changes may be made, within the purview of the appended claims, as presently stated and as amended, without departing from the scope and spirit of the present invention in its aspects. Although the present invention has been described herein with reference to particular means, materials and embodiments, the present invention is not intended to be limited to the particulars disclosed herein; rather, the present invention extends to all functionally equivalent structures, methods and uses, such as are within the scope of the appended claims.