Patent Publication Number: US-2021165121-A1

Title: Systems and Methods for Determining Properties of Porous, Fluid-Filled Geological Formations Based on Multi-Frequency Measurements

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This is a continuation in part of U.S. patent application Ser. No. 16/940,492, which is based on and claims priority to U.S. Provisional Application Ser. No. 62/879,882, filed Jul. 29, 2019, which is incorporated herein by reference in its entirety. 
    
    
     BACKGROUND 
     This disclosure relates to determining properties of porous, fluid-fluid geological formations based on multi-frequency electromagnetic measurements. 
     This section is intended to introduce the reader to various aspects of art that may be related to various aspects of the present techniques, which are described and/or claimed below. This discussion is believed to be helpful in providing the reader with background information to facilitate a better understanding of the various aspects of the present disclosure. Accordingly, it should be understood that these statements are to be read in this light, and not as admissions of prior art. 
     Producing hydrocarbons from a wellbore drilled into a geological formation is a remarkably complex endeavor. In many cases, decisions involved in hydrocarbon exploration and production may be informed by measurements from downhole well-logging tools that are conveyed deep into the wellbore. The measurements may be used to infer properties and characteristics of the geological formation surrounding the wellbore. The discovery and observation of resources using downhole techniques generally takes place down in the wellbore with certain sensors. Electromagnetic well-logging sensors or induction well-logging sensors use electromagnetic waves to acquire measurements, which may inform the decisions involved in hydrocarbon exploration and production. The composition of the geological formation may increase the complexity of the measurements by adding artifacts. 
     SUMMARY 
     A summary of certain embodiments disclosed herein is set forth below. It should be understood that these aspects are presented merely to provide the reader with a brief summary of these certain embodiments and that these aspects are not intended to limit the scope of this disclosure. Indeed, this disclosure may encompass a variety of aspects that may not be set forth below. 
     One embodiment of the present disclosure relates to a method for determining a wettability of one or more types of solid particles within a geological formation. The method includes identifying at least one type of solid particle within the geological formation. The method also includes identifying a frequency range for an electromagnetic measurement based on the identified at least one type of solid particle within the geological formation. Further, the method includes receiving a plurality of electromagnetic (EM) measurements associated with the geological formation, wherein the plurality of EM measurements are within the identified frequency range. Further still, the method includes determining a contact angle associated with solid particles within the geological formation based on the received plurality of EM measurements. 
     Another embodiment of the present disclosure relates to a non-transitory, computer-readable medium comprising instructions that, when executed by at least one processor, cause the at least one processor to receive an input indicative of a conductivity of at least one solid particle present within a geological formation. The instructions may also cause the processor to retrieve a mechanistic model based on a relative conductivity of the at least one solid particle. Further, the instructions may cause the processor to identify a frequency range for an electromagnetic measurement based on the mechanistic model. Even further, the instructions may cause the processor to receive a plurality of electromagnetic (EM) measurements associated with the geological formation, wherein the plurality of EM measurements are within the identified frequency range. Further still, the instructions may cause the processor to determine a contact angle associated with solid particles within the geological formation based on the received plurality of EM measurements. 
     Another embodiment of the present disclosure relates to a system. The system includes a non-transitory machine-readable medium storing a first mechanistic model and a second mechanistic model. The system also includes a processor configured to execute instructions stored in the non-transitory, machine readable medium to perform operations. The operations include identifying a type of solid particle present within a geological formation. The operations also include identifying at least one model to use based on a relative conductivity of the type of the solid particle, wherein the model comprises the first mechanistic model, the second mechanistic model, or both. Further, the operations include receiving, as an input to the identified at least one model, one or more inputs indicative of estimated properties of the porous, fluid-filled geological formation, wherein the mechanistic model correlates one or more fluid phases, compositions, or both, to a contact angle of at least one type of solid particle and correlates an interfacial polarization of the at least one type of solid particle to the contact angle of the at least one type of solid particle. Further still, the operations include generating, as an output by the identified at least one model, a set of frequencies to measure by a downhole tool, wherein the set of frequencies corresponds to where frequency dispersions in conductivity, permittivity, or both are measureable. 
     Various refinements of the features noted above may be undertaken in relation to various aspects of the present disclosure. Further features may also be incorporated in these various aspects as well. These refinements and additional features may exist individually or in any combination. For instance, various features discussed below in relation to one or more of the illustrated embodiments may be incorporated into any of the above-described aspects of the present disclosure alone or in any combination. The brief summary presented above is intended to familiarize the reader with certain aspects and contexts of embodiments of the present disclosure without limitation to the claimed subject matter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Various aspects of this disclosure may be better understood upon reading the following detailed description and upon reference to the drawings in which: 
         FIG. 1  is an example of a neutron-induced gamma-ray spectroscopy system, in accordance with an embodiment; 
         FIG. 2  is an example of a neutron-induced gamma-ray spectroscopy downhole tool, in accordance with an embodiment; 
         FIG. 3  is an example of a process for determining properties of a fluid-filled formation, in accordance with an embodiment; 
         FIG. 4  is an example illustration of a cross section of a volume that includes a solid suspended in an oil-water media, in accordance with an embodiment; 
         FIG. 5A  shows a graph illustrating an example of determined effective conductivity for fluid-filled porous material for different contact angles with a 10% oil saturation, in accordance with an embodiment; 
         FIG. 5B  shows a graph illustrating an example of determined effective conductivity for fluid-filled porous material for different contact angles with a 90% oil saturation, in accordance with an embodiment; 
         FIG. 5C  shows a graph illustrating an example of determined effective permittivity for fluid-filled porous material for different contact angles with a 10% oil saturation, in accordance with an embodiment; 
         FIG. 5D  shows a graph illustrating an example of determined effective permittivity for fluid-filled porous material for different contact angles with a 90% oil saturation, in accordance with an embodiment; 
         FIG. 6A  shows a graph illustrating an example of determined effective conductivity for fluid-filled porous material for different oil saturations with a contact angle of 30°, in accordance with an embodiment; 
         FIG. 6B  shows a graph illustrating an example of determined effective conductivity for fluid-filled porous material for different oil saturations with a contact angle of 90°, in accordance with an embodiment; 
         FIG. 6C  shows a graph illustrating an example of determined effective conductivity for fluid-filled porous material for different oil saturations with a contact angle of 150°, in accordance with an embodiment; 
         FIG. 6D  shows a graph illustrating an example of determined effective permittivity for fluid-filled porous material for different oil saturations with a contact angle of 30°, in accordance with an embodiment; 
         FIG. 6E  shows a graph illustrating an example of determined effective permittivity for fluid-filled porous material for different oil saturations with a contact angle of 90°, in accordance with an embodiment; 
         FIG. 6F  shows a graph illustrating an example of determined effective permittivity for fluid-filled porous material for different oil saturations with a contact angle of 150°, in accordance with an embodiment; 
         FIG. 7A  shows graphs depicting a history of Markov Chain Monte Carlo (MCMC) inversion-derived estimates of clay surface conductance for oil-water-filled porous material containing water-wet sands, clays, and graphite, in accordance with an embodiment; 
         FIG. 7B  shows graphs depicting a history of Markov Chain Monte Carlo (MCMC) inversion-derived estimates of graphite contact angle for oil-water-filled porous material containing water-wet sands, clays, and graphite, in accordance with an embodiment; 
         FIG. 7C  shows graphs depicting a history of Markov Chain Monte Carlo (MCMC) inversion-derived estimates of water conductivity for oil-water-filled porous material containing water-wet sands, clays, and graphite, in accordance with an embodiment; 
         FIG. 8A  shows graphs depicting a histogram of MCMC inversion-derived estimates of clay surface conductance of  FIG. 7A , in accordance with an embodiment; 
         FIG. 8B  shows graphs depicting a histogram of MCMC inversion-derived estimates of graphite contact angle of  FIG. 7B , in accordance with an embodiment; 
         FIG. 8C  shows graphs depicting a histogram of MCMC inversion-derived estimates of water conductivity of  FIG. 7C , in accordance with an embodiment; 
         FIG. 9A  shows multi-frequency electromagnetic (EM) measurements and model predictions based on inversion-derived estimates of effective conductivity associated with  FIGS. 7A, 7B, and 7C  and  FIGS. 8A, 8B, and 8C , in accordance with an embodiment; 
         FIG. 9B  shows multi-frequency electromagnetic (EM) measurements and model predictions based on inversion-derived estimates of effective permittivity associated with  FIGS. 7A, 7B, and 7C  and  FIGS. 8A, 8B, and 8C , in accordance with an embodiment; 
         FIG. 10A  shows graphs depicting a history of Markov Chain Monte Carlo (MCMC) inversion-derived estimates for clay surface conductance for oil-water-filled porous material containing water-wet sands, clays, and slightly oil-wet graphite, in accordance with an embodiment; 
         FIG. 10B  shows graphs depicting a history of Markov Chain Monte Carlo (MCMC) inversion-derived estimates for graphite contact angle for oil-water-filled porous material containing water-wet sands, clays, and slightly oil-wet graphite, in accordance with an embodiment; 
         FIG. 10C  shows graphs depicting a history of Markov Chain Monte Carlo (MCMC) inversion-derived estimates for water conductivity for oil-water-filled porous material containing water-wet sands, clays, and slightly oil-wet graphite, in accordance with an embodiment; 
         FIG. 11A  shows graphs depicting a histogram of MCMC inversion-derived estimates of clay surface conductance of  FIG. 10A , in accordance with an embodiment; 
         FIG. 11B  shows graphs depicting a histogram of MCMC inversion-derived estimates of graphite contact angle of  FIG. 10B , in accordance with an embodiment; 
         FIG. 11C  shows graphs depicting a histogram of MCMC inversion-derived estimates of water conductivity of  FIG. 10C , in accordance with an embodiment; 
         FIG. 12A  shows multi-frequency EM measurements and model predictions for effective conductivity based on inversion-derived estimates associated with  FIGS. 10A, 10B, and 10C  and  FIGS. 11A, 11B, and 11C , in accordance with an embodiment; 
         FIG. 12B  shows multi-frequency EM measurements and model predictions for effective permittivity based on inversion-derived estimates associated with  FIGS. 10A, 10 , and IOC and  FIGS. 11A, 11B, and 11C , in accordance with an embodiment; 
         FIG. 13A  shows a graph depicting a history of Markov Chain Monte Carlo (MCMC) inversion-derived estimates for contact angle for oil/water-filled porous material containing water-wet sand and clays and oil-wet graphite, in accordance with an embodiment; 
         FIG. 13B  shows a graph depicting a history of Markov Chain Monte Carlo (MCMC) inversion-derived estimates for contact angle for oil saturation oil/water-filled porous material containing water-wet sand and clays and oil-wet graphite, in accordance with an embodiment; 
         FIG. 14A  shows a graph depicting a histogram of MCMC inversion-derived estimates of clay surface conductance of  FIG. 13A , in accordance with an embodiment; 
         FIG. 14B  shows a graph depicting a histogram of MCMC inversion-derived estimates of oil saturation of  FIG. 13B , in accordance with an embodiment; 
         FIG. 15A  shows multi-frequency EM measurements and model predictions for effective conductance based on inversion-derived estimates associated with  FIGS. 13A and 13B , and  FIGS. 14A and 14B , in accordance with an embodiment; 
         FIG. 15B  shows multi-frequency EM measurements and model predictions for effective permittivity based on inversion-derived estimates associated with  FIGS. 13A and 13B , and  FIGS. 14A and 14B , in accordance with an embodiment; 
         FIG. 16  is an example illustration of a cross section of a volume that includes a solid suspended in an oil-water media, in accordance with an embodiment; 
         FIG. 17A  shows a graph illustrating an example of determined effective conductivity for fluid-filled porous material for different contact angles with a 10% oil saturation, in accordance with an embodiment; 
         FIG. 17B  shows a graph illustrating an example of determined effective conductivity for fluid-filled porous material for different contact angles with a 90% oil saturation, in accordance with an embodiment; 
         FIG. 17C  shows a graph illustrating an example of determined effective permittivity for fluid-filled porous material for different contact angles with a 10% oil saturation, in accordance with an embodiment; 
         FIG. 17D  shows a graph illustrating an example of determined effective permittivity for fluid-filled porous material for different contact angles with a 90% oil saturation, in accordance with an embodiment; 
         FIG. 18A  shows a graph illustrating an example of determined effective conductivity for fluid-filled porous material for different amounts of oil saturation for a contact angle of 30 degrees, in accordance with an embodiment; 
         FIG. 18B  shows a graph illustrating an example of determined effective conductivity for fluid-filled porous material for different amounts of oil saturation for a contact angle of 150 degrees, in accordance with an embodiment; 
         FIG. 18C  shows a graph illustrating an example of determined effective permittivity for fluid-filled porous material for different amounts of oil saturation for a contact angle of 30 degrees, in accordance with an embodiment; 
         FIG. 18D  shows a graph illustrating an example of determined effective permittivity for fluid-filled porous material for different amounts of oil saturation for a contact angle of 150 degrees, in accordance with an embodiment; 
         FIG. 19A  shows a graph illustrating an example of determined effective conductivity for fluid-filled porous material for different surface conductance with a contact angle of 30 degrees, in accordance with an embodiment; 
         FIG. 19B  shows a graph illustrating an example of determined effective conductivity for fluid-filled porous material for different surface conductance with a contact angle of 150 degrees, in accordance with an embodiment; 
         FIG. 19C  shows a graph illustrating an example of determined effective permittivity for fluid-filled porous material for different surface conductance with a contact angle of 30 degrees, in accordance with an embodiment; 
         FIG. 19D  shows a graph illustrating an example of determined effective permittivity for fluid-filled porous material for different surface conductance with a contact angle of 150 degrees, in accordance with an embodiment; and 
         FIG. 20  is a second example of a process for determining properties of a fluid-filled formation, in accordance with an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     One or more specific embodiments of the present disclosure will be described below. These described embodiments are examples of the presently disclosed techniques. Additionally, in an effort to provide a concise description of these embodiments, all features of an actual implementation may not be described in the specification. It should be appreciated that in the development of any such actual implementation, as in any engineering or design project, numerous implementation-specific decisions must be made to achieve the developers&#39; specific goals, such as compliance with system-related and business-related constraints, which may vary from one implementation to another. Moreover, it should be appreciated that such a development effort might be complex and time consuming, but would nevertheless be a routine undertaking of design, fabrication, and manufacture for those of ordinary skill having the benefit of this disclosure. 
     When introducing elements of various embodiments of the present disclosure, the articles “a,” “an,” and “the” are intended to mean that there are one or more of the elements. The terms “comprising,” “including,” and “having” are intended to be inclusive and mean that there may be additional elements other than the listed elements. Additionally, it should be understood that references to “one embodiment” or “an embodiment” of the present disclosure are not intended to be interpreted as excluding the existence of additional embodiments that also incorporate the recited features. 
     As used herein, “wettability” refers to a tendency of one fluid to spread on and/or adhere to a solid surface in the presence of other immiscible fluids. “Wettability” may be quantified by a contact angle where a liquid interface meets a solid surface. 
     As discussed above, electromagnetic well-logging or induction well-logging may inform certain decisions related to hydrocarbon exploration and production. Certain existing techniques in electromagnetic well logging may use models that assume that conductive particles, like graphite and pyrite, and surface-charge-bearing nonconductive particles, like quartz, calcite and clays, are completely water wet (e.g., the contact angle between a liquid and a solid surface is zero). It is presently noted that wettability of conductive particles and surface-charge-bearing nonconductive particles governs the preferential spreading of fluids on the surface of the particles that influences the interfacial polarization phenomena and charge transport/accumulation around the particles. Consequently, wettability of conductive particles influences the electromagnetic properties of fluid-filled porous materials. Further, the wettability and the electrical properties are closely related such that wettability can be estimated using the electromagnetic properties. For example, the dielectric permittivity of oil-wet sand is smaller than that of the water-wet sand at low water saturation, while the dielectric permittivity of oil-wet sand becomes much larger than that of the water-wet sand at higher water saturation. Additionally, it is noted that both resistivity and magnitude of the phase increase with the increase of oil saturation for sand saturated with non-wetting oil, while they both decrease with the increase of oil saturation for sand partially saturated with wetting oil. 
     Accordingly, one aspect of the present disclosure relates to systems and methods for using a material and subsurface characterization model to quantify the effects of wettability of conductive particles. Moreover, the model may be implemented to determine the wettability effects of solid particles that produce interfacial polarization phenomena on multi-frequency electromagnetic measurements. Further, the material and subsurface characterization model, in accordance with the present disclosure, provides a novel technique for identifying a range of operating frequencies for electromagnetic measurements to characterize the contact angle of solid particles that are present within a subsurface formation. 
     With this in mind,  FIG. 1  illustrates an electromagnetic well-logging system  10  that may employ the systems and methods of this disclosure. The electromagnetic well-logging system  10  may be used to convey an electromagnetic well-logging tool  12  through a geological formation  14  via a wellbore  16 . The electromagnetic well-logging tool  12  may be conveyed on a cable  18  via a logging winch system  20 . Although the logging winch system  20  is schematically shown in  FIG. 1  as a mobile logging winch system carried by a truck, the logging winch system  20  may be substantially fixed (e.g., a long-term installation that is substantially permanent or modular). Any suitable cable  18  for well logging may be used. The cable  18  may be spooled and unspooled on a drum  22  and an auxiliary power source  24  may provide energy to the logging winch system  20  and/or the electromagnetic well-logging tool  12 . 
     Moreover, although the electromagnetic well-logging tool  12  is described as a wireline downhole tool, it should be appreciated that any suitable conveyance may be used. For example, the electromagnetic well-logging tool  12  may instead be conveyed as a logging-while-drilling (LWD) tool as part of a bottom hole assembly (BHA) of a drill string, conveyed on a slickline or via coiled tubing, and so forth. For the purposes of this disclosure, the electromagnetic well-logging tool  12  may be any suitable measurement tool that obtains electromagnetic logging measurements through depths of the wellbore  16 . 
     Many types of electromagnetic well-logging tools  12  may obtain electromagnetic logging measurements in the wellbore  16 . These include, for example, the Rt Scanner, AIT, and Thrubit Electromagnetic tools by Schlumberger Technology Corporation, but electromagnetic logging measurements from other downhole tools by other manufacturers may also be used. The electromagnetic well-logging tool  12  may provide electromagnetic logging measurements  26  to a data processing system  28  via any suitable telemetry (e.g., via electrical signals pulsed through the geological formation  14  or via mud pulse telemetry). The data processing system  28  may process the electromagnetic logging measurements  26  to identify a contact angel and/or wettability at various depths of the geological formation  14  in the wellbore  16 . 
     To this end, the data processing system  28  thus may be any electronic data processing system that can be used to carry out the systems and methods of this disclosure. For example, the data processing system  28  may include a processor  30 , which may execute instructions stored in memory  32  and/or storage  34 . As such, the memory  32  and/or the storage  34  of the data processing system  28  may be any suitable article of manufacture that can store the instructions. The memory  32  and/or the storage  34  may be ROM memory, random-access memory (RAM), flash memory, an optical storage medium, or a hard disk drive, to name a few examples. A display  36 , which may be any suitable electronic display, may provide a visualization, a well log, or other indication of properties in the geological formation  14  or the wellbore  16  using the electromagnetic logging measurements  26 . 
       FIG. 2  shows an example of an electromagnetic well-logging tool  12  that may acquire electromagnetic measurements. The illustrated embodiment of the electromagnetic well-logging tool  12  includes a transmitter  40  and a receiver  42 . While only one transmitter  40  and one receiver  42  are shown, it should be noted that the number of transmitters and receivers is not a limit on the scope of the present disclosure. Generally speaking, the transmitter  40  induces electric eddy currents to produce electromagnetic waves  44  having a set of frequencies in a direction of the magnetic dipole moment of the transmitter  40 . The electromagnetic waves  44  that interact with the geological formation  14  are subsequently received by the receiver  42  to generate electromagnetic measurements. 
     As shown in  FIG. 2 , the illustrated embodiment of the electromagnetic well-logging tool  12  is communicatively coupled to the data processing system  28 , which includes a material and subsurface characterization model  46  stored in the memory  32 . As discussed in further detail below, the material and subsurface characterization model  46  may be utilized by the processor  30  of the data processing system  28  to determine a set of frequencies that the electromagnetic well-logging tool  12  may operate to acquire electromagnetic measurements. Further, the electromagnetic measurements may be processed according to the present disclosure to quantify the wettability effects of graphite, clays and other conductive or surface-charge-bearing nonconductive particles for improving subsurface electromagnetic log measurement interpretation in various subsurface geological formations to better quantify the water content/saturation in the subsurface. 
       FIG. 3  illustrates a process  50  for determining one or more physical properties of a fluid-filled geological formation. Although described in a particular order, which represents a particular embodiment, it should be noted that the process  50  may be performed in any suitable order. Additionally, embodiments of the process  50  may omit process blocks and/or include additional process blocks. Moreover, in some embodiments, the process  50  may be implemented at least in part by executing instructions stored in a tangible, non-transitory, computer-readable medium, such as memory  32  implemented in a data processing system  28 , using processing circuitry, such as a processor  30  implemented in the data processing system  28 . 
     In general, the illustrated process  50  includes receiving (process block  52 ) electromagnetic measurements from a set of frequencies (e.g., emitted by the electromagnetic well-logging tool  12 ), and determining (process block  54 ) one or more physical properties of the geological formation. 
     As described herein, in some embodiments, the set of frequencies emitted by the electromagnetic well-logging tool  12  may be determined based on the material and subsurface characterization model  46 . For example, an operator may determine a number (e.g., 1, 2, 3, 4, 5 etc.) properties of a fluid-filled porous material to be estimated (e.g., determined) and provide these at inputs to a suitable computing system (e.g., the data processing system  28 ). In some embodiments, the properties to be estimated may include contact angle of conductive particles, contact angle of surface-charge-bearing particles, fluid saturations, fluid conductivity/salinity, surface conductance of solid particles, diffusion coefficients of charge carriers in various components of the material, and volume fractions of fluid and solid components in the materials. Further, the operator may provide an initial assumption of the composition of the geological formation, and the properties of the fluid and solid components in the geological formation. Based on the initial assumption, the operator may apply the material and subsurface characterization model  46  to identify the set of frequencies where frequency dispersions in conductivity and/or permittivity will be dominant and measureable (e.g., absent certain effects related to the complex conductivity and/or complex permittivity as described herein). In some embodiments, the set of frequencies may be a range of frequencies or one or more discrete frequencies. 
     The identified frequency range may be provided as an output to the electromagnetic well-logging tool  12 . For example, the data processing system  28  may provide an output that instructs the electromagnetic well-logging tool  12  to tune the electromagnetic (EM) measurement to measure multi-frequency complex conductivity/permittivity of the fluid-filled porous material within the identified frequency ranges, or at specific frequencies in the frequency range. In some embodiments, the number of discrete frequencies included in the identified frequency range or the number of specific frequencies may be at least 3 times the number of physical properties to be estimated as described above. 
     As such, the electromagnetic well-logging tool  12  may perform the electromagnetic (EM) measurements of multi-frequency complex conductivity/permittivity on the fluid-filled porous material using the measurement settings tuned and finalized in the steps as described above. In some embodiments, a Markov-Chain Monte-Carlo may be applied to the EM measurements received in process block  52  to determine properties such as the contact angles and other physical properties as described herein. 
     The material and subsurface characterization model  46  may include multiple relationships, or be generated based on multiple models. For example, the material and subsurface characterization model  46  may include a first mechanistic model for a solid particle being preferentially surrounded by one of the fluid phases or fluid components surrounding the solid particle as a function of the contact angle of the solid particle. Further, the material and subsurface characterization model  46  may include a second mechanistic model that quantifies the interfacial polarization due to a solid particle (conductive or surface-charge-bearing nonconductive particle) preferentially surrounded by one of the fluid phases/components surrounding the solid particle as function of the contact angle of the solid particle and the operating frequency of the externally applied electromagnetic field. 
     The material and subsurface characterization model  46  may be developed by solving the Young-Laplace equation for a spherical grain in a mixture of wetting and non-wetting fluids with a known proportion of the two fluids. For example, Young-Laplace equation may be used to compute the shape of the wetting and non-wetting fluid interface (meniscus) at equilibrium by applying appropriate boundary conditions. In this way, the following expressions may be obtained: the wetting angle of the conductive or surface-charge-bearing nonconductive particle as a function of contact angle of the solid particle and the properties of fluid phases/components surrounding the solid particle. It should be noted that the interfacial polarization due to conductive and surface-charge-bearing nonconductive solid particles depends on the nature of preferential wetting of the solid particle. As such, the subsurface characterization model  46  may be used to quantify the effects of contact angle (wettability) of solid grains/particles (conductive or surface-charge-bearing nonconductive particle) on the net charge transport and net charge accumulation as a function of the frequency of the external electromagnetic field at various fluid saturations and solid wettability. The net charge transport determines the conductivity and net charge accumulation determines the permittivity that govern the electromagnetic measurements and log responses of the fluid-filled porous material. 
     At the representative volume level, developing the material and subsurface characterization model  46 , in accordance with the techniques of the present disclosure, may include assuming the non-wetting layer (e.g., oil) stays at the top, wetting layer (e.g., water) goes to the bottom, the two layers have (e.g., the non-wetting layer and the wetting layer) one common interface, and the two layers are spread across a length scale that is orders of magnitude larger than the size of the spherical solid particle. The height of these two layers are in proportion to the corresponding fluid saturations. The solid particle suspends at the interface of wetting and non-wetting fluids, as shown in the  FIG. 4 , is discussed below. The wetting phase may surround the solid particle to satisfy the contact angle. The climb or height of the interface between the wetting layer and the non-wetting layer generates a wetting angle, which represents the degree of exposure of the particle to the wetting phase. The interfacial polarization phenomena due to such solid particles are entirely governed by the extent to which the solid particle is surround by the wetting phase versus non-wetting phase, which is governed by the wettability and contact angle of the solid particle. For example, when water wets a conductive mineral, its interfacial polarization effects on the complex conductivity/permittivity measurements will be enhanced. In another example, when the conductive mineral is preferentially oil wet, its interfacial polarization effects on the complex conductivity/permittivity measurements will diminish. 
       FIG. 4  is an example illustration of a cross-section of a volume  56  (e.g., within a geological formation) that includes a solid particle  57  suspended in an oil-water media, in accordance with an embodiment. In general, the volume  56  may be assumed for developing the model, as discussed herein. As shown, the solid particle  57  is a circle (e.g., a cross-section of a sphere); however, it should be noted that, in some embodiments, the solid particle may be ellipsoidal (e.g., a diameter  58  of the solid particle  57  may be greater than or less than a diameter  59  of the solid particle  57 ) or have a radial normal distribution of radii. 
     In the illustrated cross-section of the volume  56  shown in  FIG. 4 , C denotes the point where the oil-water interface (e.g., interface between the non-wetting layer and the wetting layer) contacts the particle surface; θ is the contact angle of conductive particle; φ is the wetting angle; ψ is the angle between oil-water interface and the horizon (x-axis) at point C; R is the radius of conductive particle; h i  is the uniform height of oil-water interface in the absence of wetting of the conductive particle (far-field height); h c  is the height where the oil-water interface contacts the particle surface, such that h c =R(1−cos φ); r is the horizontal distance perpendicular to the vertical axis z; and h(r) is the height of oil-water interface at any distance r away from the vertical axis z. 
     Young-Laplace Equation 
     As discussed herein, developing the material and subsurface characterization model  46  may include solving the Young-Laplace equation to quantify the shape of the oil-water interface. For example, the shape of the oil-water interface, where oil is non-wetting phase and water is the wetting phase, at equilibrium may be described by the Young-Laplace equation: 
       (ρ w −ρ o ) g [ h ( r )− h   i ]=2 Hσ 
 
     where ρ w  and ρ 0  are the density of water and oil, respectively; g denotes gravitational acceleration; H is mean curvature of the meniscus surface; and σ is interfacial tension between oil and water. 
     Under a small slope assumption, where the Bond number, 
     
       
         
           
             
               
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                       r 
                       ) 
                     
                   
                   - 
                   
                     h 
                     i 
                   
                 
                 
                   L 
                   c 
                 
               
             
             , 
             
               
                 where 
                  
                 
                     
                 
                  
                 
                   L 
                   c 
                 
               
               = 
               
                 
                   σ 
                   
                     
                       ( 
                       
                         
                           ρ 
                           w 
                         
                         - 
                         
                           ρ 
                           o 
                         
                       
                       ) 
                     
                      
                     g 
                   
                 
               
             
           
         
       
     
     is capillary length, the Young-Laplace equation becomes a modified Bessel differential equation: 
     
       
         
           
             
               
                 G 
                 ″ 
               
               + 
               
                 
                   G 
                   ′ 
                 
                 
                   r 
                   ^ 
                 
               
               - 
               G 
             
             = 
             0 
           
         
       
     
     where G′ and G″ represents and 
     
       
         
           
             
               
                 dG 
                 dr 
               
                
               
                   
               
                
               and 
                
               
                 
                     
                 
                  
                 
                     
                 
               
                
               
                 
                   
                     d 
                     2 
                   
                    
                   G 
                 
                 
                   dr 
                   2 
                 
               
             
             , 
           
         
       
     
     respectively. 
     Boundary Conditions (BC) 
     As discussed herein, developing the material and subsurface characterization model  46  may include solving the Young-Laplace equation with certain boundary conditions. For example, a first boundary condition may be the height of oil-water interface at infinite distance, h(r)| r→∞ , is equal to h i . 
                 lim       r   ^     →   ∞          G     =   0         (lim) T ( r {circumflex over ( )}→∞) G= 0
 
     A secondary boundary condition may be the height of oil-water interface at distance r=R sin φ is h c . 
         G ( {circumflex over (r)}=B   o  sin φ)= ĥ   c   −ĥ   i  
 
     Shape of the Oil-Water Interface 
     The Young-Laplace equation is solved using the boundary conditions to obtain the expression for the shape of the oil-water interface: 
     
       
         
           
             
               h 
               ^ 
             
             = 
             
               
                 
                   h 
                   ^ 
                 
                 i 
               
               + 
               
                 
                   
                     
                       
                         h 
                         ^ 
                       
                       c 
                     
                     - 
                     
                       
                         h 
                         ^ 
                       
                       i 
                     
                   
                   
                     
                       K 
                       0 
                     
                      
                     
                       ( 
                       
                         
                           
                             B 
                             o 
                           
                         
                          
                         
                           sin 
                            
                           ϕ 
                         
                       
                       ) 
                     
                   
                 
                  
                 
                   
                     K 
                     0 
                   
                    
                   
                     ( 
                     
                       r 
                       ^ 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     where K_0 is modified Bessel function of the second kind of order 0. 
     An Expression of Wetting Angle 
     Wetting angle may be expressed as: 
     
       
         
           
             ϕ 
             = 
             
               180 
               - 
               θ 
               - 
               
                 
                   
                     
                       
                         h 
                         ^ 
                       
                       c 
                     
                     - 
                     
                       
                         h 
                         ^ 
                       
                       i 
                     
                   
                   
                     
                       K 
                       0 
                     
                      
                     
                       ( 
                       
                         
                           
                             B 
                             o 
                           
                         
                          
                         
                           sin 
                            
                           ϕ 
                         
                       
                       ) 
                     
                   
                 
                  
                 
                   
                     K 
                     1 
                   
                    
                   
                     ( 
                     
                       
                         
                           B 
                           o 
                         
                       
                        
                       
                         sin 
                          
                         ϕ 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     where K_1 is modified Bessel function of the second kind of order 1. 
     Effective Medium Model 
     In some embodiments, developing the material and subsurface characterization model  46  may include using an effective medium model. For example, to simulate the wettability effects of solid particles constituting a fluid-filled porous material on the electromagnetic properties of the material (e.g., multi-frequency complex conductivity and complex permittivity), the newly developed model of wetting angle of a solid particle may include a petrophysical model to express the complex conductivity/permittivity due to the interfacial polarization of the solid particles at various saturations, wettability, and operating frequencies. 
     From an effective medium standpoint, the effective complex conductivity of a porous fluid-filled geomaterial containing conductive particles of any wettability (e.g., graphite particle) and fully wetted surface-charge-bearing nonconductive particles (e.g., water-wet sand and clay particles) at any saturation of the wetting phase (e.g., water) may be expressed as: 
     
       
         
           
             
               
                 
                   K 
                   eff 
                 
                 - 
                 
                   K 
                   w 
                 
               
               
                 
                   K 
                   eff 
                 
                 + 
                 
                   2 
                    
                   
                     K 
                     w 
                   
                 
               
             
             = 
             
               
                 
                   φ 
                   c 
                 
                  
                 
                   p 
                   w 
                 
                  
                 
                   
                     f 
                     
                       c 
                       , 
                       w 
                     
                   
                    
                   
                     ( 
                     ω 
                     ) 
                   
                 
               
               + 
               
                 
                   
                     φ 
                     c 
                   
                    
                   
                     ( 
                     
                       1 
                       - 
                       
                         p 
                         w 
                       
                     
                     ) 
                   
                 
                  
                 
                   
                     f 
                     
                       c 
                       , 
                       nw 
                     
                   
                    
                   
                     ( 
                     ω 
                     ) 
                   
                 
               
               + 
               
                 
                   φ 
                   
                     n 
                      
                     1 
                   
                 
                  
                 
                   
                     f 
                     
                       n 
                        
                       1 
                     
                   
                    
                   
                     ( 
                     ω 
                     ) 
                   
                 
               
               + 
               
                 
                   φ 
                   
                     n 
                      
                     2 
                   
                 
                  
                 
                   
                     f 
                     
                       n 
                        
                       2 
                     
                   
                    
                   
                     ( 
                     ω 
                     ) 
                   
                 
               
               + 
               
                 
                   φ 
                   nw 
                 
                  
                 
                   
                     f 
                     nw 
                   
                    
                   
                     ( 
                     ω 
                     ) 
                   
                 
               
             
           
         
       
     
     Where K eff  is the effective complex conductivity of the porous fluid-filled geomaterial; K w  is the complex conductivity of pore-filling wetting phase, which may be brine or saline water in some cases, with an assumption that the complex conductivity of pore-filling non-wetting phase, which is oil in in the illustration of the cross-section of the volume  56 , is negligible; f is the dipolarizability due to interfacial polarization of solid particle; ω is the angular frequency of the external EM field; ϕ is the volume fraction of solid particles or the fluid phases; p w  is the proportion of a single solid particle surface that is covered by wetting phase (water) determined using the newly developed model of wetting angle of a solid particle; and subscripts c, n1, n2, nw, and w represent the conductive particle of any wettability (e.g., graphite), water-wet surface-charge-bearing nonconductive particle #1 (e.g., sand), water-wet surface-charge-bearing nonconductive particle #2 (e.g., clay), non-wetting phase (e.g., oil), and wetting phase (e.g., water), respectively. 
     When a solid particle is not fully wet, the interfacial polarization effect of such a solid particle is determined as a volumetric mixing of interfacial polarization when the solid particle is completely surrounded by non-wetting fluid phase, f c,n,w , and that when completely surround by wetting fluid phase, f c,w , expressed as pp w f c,w (ω)+ϕ c (1−p w )f c,nw (ω), where p w  is the proportion of the solid particle surface that is covered by wetting phase (water) determined using the newly developed model of wetting angle of a solid particle. 
     The proportion of a single graphite surface that covered by water or oil may be expressed as: 
     
       
         
           
             
               p 
               w 
             
             = 
             
               
                 1 
                 - 
                 cosϕ 
               
               2 
             
           
         
       
     
     where φ is the wetting angle. 
     Dipolarizability of conductive particle (e.g., graphite) completely immersed in wetting phase may be expressed as: 
     
       
         
           
             
               
                 f 
                 c 
               
                
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               
                 - 
                 
                   1 
                   2 
                 
               
               + 
               
                 
                   3 
                   2 
                 
                  
                 
                   
                     i 
                      
                     ω 
                   
                   
                     [ 
                     
                       
                         
                           2 
                           a 
                         
                          
                         
                           
                             σ 
                             w 
                           
                           
                             ɛ 
                             w 
                           
                         
                          
                         
                           
                             E 
                             w 
                           
                           
                             G 
                             w 
                           
                         
                       
                       - 
                       
                         
                           2 
                           a 
                         
                          
                         
                           
                             K 
                             w 
                           
                           
                             K 
                             c 
                           
                         
                          
                         
                           
                             σ 
                             c 
                           
                           
                             ɛ 
                             c 
                           
                         
                          
                         
                           
                             F 
                             c 
                           
                           
                             H 
                             c 
                           
                         
                       
                       + 
                       
                         
                           i 
                            
                           ω 
                         
                          
                         
                           ( 
                           
                             
                               
                                 2 
                                  
                                 
                                   K 
                                   w 
                                 
                               
                               
                                 K 
                                 c 
                               
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
       
         
           
             
               where 
               : 
               
                 
 
               
                
               
                 E 
                 w 
               
             
             = 
             
               
                 q 
                 
                   
                     γ 
                     w 
                     2 
                   
                    
                   
                     ɛ 
                     w 
                   
                 
               
                
               
                 
                   e 
                   
                     - 
                     
                       
                         a 
                          
                         γ 
                       
                       w 
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       1 
                       
                         
                           a 
                            
                           γ 
                         
                         w 
                       
                     
                     + 
                     
                       1 
                       
                         
                           ( 
                           
                             
                               a 
                                
                               γ 
                             
                             w 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               G 
               w 
             
             = 
             
               
                 q 
                 
                   
                     γ 
                     w 
                   
                    
                   
                     ɛ 
                     w 
                   
                 
               
                
               
                 
                   e 
                   
                     - 
                     
                       
                         a 
                          
                         γ 
                       
                       w 
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       1 
                       
                         
                           a 
                            
                           γ 
                         
                         w 
                       
                     
                     + 
                     
                       2 
                       
                         
                           ( 
                           
                             
                               a 
                                
                               γ 
                             
                             w 
                           
                           ) 
                         
                         2 
                       
                     
                     + 
                     
                       2 
                       
                         
                           ( 
                           
                             
                               a 
                                
                               γ 
                             
                             w 
                           
                           ) 
                         
                         3 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               F 
               c 
             
             = 
             
               
                 q 
                 
                   
                     γ 
                     c 
                   
                    
                   
                     ɛ 
                     c 
                   
                 
               
                
               
                 [ 
                 
                   
                     
                       cosh 
                        
                       
                         ( 
                         
                           
                             a 
                              
                             γ 
                           
                           c 
                         
                         ) 
                       
                     
                     
                       
                         a 
                          
                         γ 
                       
                       c 
                     
                   
                   - 
                   
                     
                       sinh 
                        
                       
                         ( 
                         
                           
                             a 
                              
                             γ 
                           
                           c 
                         
                         ) 
                       
                     
                     
                       
                         ( 
                         
                           
                             a 
                              
                             γ 
                           
                           c 
                         
                         ) 
                       
                       2 
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               F 
               c 
             
             = 
             
               
                 q 
                 
                   
                     γ 
                     c 
                   
                    
                   
                     ɛ 
                     c 
                   
                 
               
                
               
                 [ 
                 
                   
                     
                       2 
                        
                       
                         cosh 
                          
                         
                           ( 
                           
                             
                               a 
                                
                               γ 
                             
                             c 
                           
                           ) 
                         
                       
                     
                     
                       
                         ( 
                         
                           
                             a 
                              
                             γ 
                           
                           c 
                         
                         ) 
                       
                       2 
                     
                   
                   - 
                   
                     
                       sinh 
                        
                       
                         ( 
                         
                           
                             a 
                              
                             γ 
                           
                           c 
                         
                         ) 
                       
                     
                     
                       
                         a 
                          
                         γ 
                       
                       c 
                     
                   
                   - 
                   
                     
                       2 
                        
                       
                         sinh 
                          
                         
                           ( 
                           
                             
                               a 
                                
                               γ 
                             
                             c 
                           
                           ) 
                         
                       
                     
                     
                       
                         ( 
                         
                           
                             a 
                              
                             γ 
                           
                           c 
                         
                         ) 
                       
                       3 
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               
                 γ 
                 j 
               
               = 
               
                 
                   
                     
                       i 
                        
                       ω 
                     
                     
                       D 
                       j 
                     
                   
                   + 
                   
                     
                       σ 
                       j 
                     
                     
                       
                         ɛ 
                         j 
                       
                        
                       
                         D 
                         j 
                       
                     
                   
                 
               
             
             , 
             
               
                 for 
                  
                 
                     
                 
                  
                 j 
               
               = 
               
                 w 
                  
                 
                     
                 
                  
                 or 
                  
                 
                     
                 
                  
                 c 
               
             
           
         
       
     
     where ω is the angular frequency of the electric field; i is square root of −1; a is characteristic length of inclusion phase; λ is surface conductance of nonconductive particle; σ is electrical conductivity; ε is dielectric permittivity; and D is diffusion coefficient of charge carriers. 
     Dipolarizability of nonconductive particle (e.g., clay, sand, oil) completely immersed in wetting phase may be expressed as: 
     
       
         
           
             
               
                 f 
                 ncond 
               
                
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               
                 
                   Q 
                    
                   
                     ( 
                     
                       R 
                       + 
                       A 
                     
                     ) 
                   
                 
                 - 
                 P 
               
               
                 
                   Q 
                    
                   
                     ( 
                     
                       R 
                       - 
                       
                         2 
                          
                         A 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   2 
                    
                   P 
                 
               
             
           
         
       
       
         
           
             
               where 
               : 
               
                 
 
               
                
               A 
             
             = 
             
               1 
               
                 a 
                 2 
               
             
           
         
       
       
         
           
             P 
             = 
             
               
                 γ 
                 w 
                 2 
               
               + 
               
                 
                   ξ 
                   w 
                   2 
                 
                  
                 
                   
                     G 
                     * 
                   
                   
                     H 
                     * 
                   
                 
               
               + 
               
                 
                   2 
                    
                   
                     G 
                     * 
                   
                 
                 
                   
                     a 
                     2 
                   
                    
                   L 
                 
               
             
           
         
       
       
         
           
             Q 
             = 
             
               
                 1 
                 
                   iF 
                   + 
                   1 
                 
               
                
               
                 [ 
                 
                   2 
                   - 
                   
                     
                       
                         
                           a 
                           2 
                         
                          
                         
                           ξ 
                           h 
                           2 
                         
                       
                       
                         H 
                         * 
                       
                     
                      
                     
                       ( 
                       
                         
                           L 
                           iF 
                         
                         + 
                         E 
                       
                       ) 
                     
                   
                   - 
                   
                     
                       2 
                        
                       E 
                     
                     L 
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             R 
             = 
             
               
                 P 
                 Q 
               
                
               
                 ( 
                 
                   
                     iFE 
                     + 
                     L 
                   
                   
                     iF 
                     + 
                     1 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 H 
                 * 
               
               = 
               
                 
                   aL 
                   w 
                 
                 
                   F 
                   w 
                 
               
             
             , 
             
               
                 G 
                 * 
               
               = 
               
                 
                   aG 
                   w 
                 
                 
                   E 
                   w 
                 
               
             
             , 
             
               L 
               = 
               
                 
                   2 
                    
                   λ 
                 
                 
                   
                     a 
                      
                     σ 
                   
                   w 
                 
               
             
             , 
             
               E 
               = 
               
                 
                   ɛ 
                   n 
                 
                 
                   ɛ 
                   w 
                 
               
             
             , 
             
               F 
               = 
               
                 
                   ωɛ 
                   w 
                 
                 
                   σ 
                   w 
                 
               
             
           
         
       
       
         
           
             
               F 
               w 
             
             = 
             
               
                 q 
                 
                   
                     ξ 
                     w 
                     2 
                   
                    
                   
                     ɛ 
                     w 
                   
                 
               
                
               
                 
                   e 
                   
                     - 
                     
                       
                         a 
                          
                         ξ 
                       
                       w 
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       1 
                       
                         
                           a 
                            
                           ξ 
                         
                         w 
                       
                     
                     + 
                     
                       1 
                       
                         
                           ( 
                           
                             
                               a 
                                
                               ξ 
                             
                             w 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               L 
               w 
             
             = 
             
               
                 q 
                 
                   
                     ξ 
                     w 
                   
                    
                   
                     ɛ 
                     w 
                   
                 
               
                
               
                 
                   e 
                   
                     - 
                     
                       
                         a 
                          
                         ξ 
                       
                       w 
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       1 
                       
                         
                           a 
                            
                           ξ 
                         
                         w 
                       
                     
                     + 
                     
                       2 
                       
                         
                           ( 
                           
                             
                               a 
                                
                               ξ 
                             
                             w 
                           
                           ) 
                         
                         2 
                       
                     
                     + 
                     
                       2 
                       
                         
                           ( 
                           
                             
                               a 
                                
                               ξ 
                             
                             w 
                           
                           ) 
                         
                         3 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               
                 ξ 
                 j 
               
               = 
               
                 
                   
                     i 
                      
                     ω 
                   
                   
                     D 
                     j 
                   
                 
               
             
             , 
             
               
                 for 
                  
                 
                     
                 
                  
                 j 
               
               = 
               
                 n 
                  
                 
                   
                       
                   
                    
                   
                       
                   
                 
                  
                 or 
                  
                 
                     
                 
                  
                 w 
               
             
           
         
       
     
     Based on certain assumptions of the properties of solid particles and fluid phases in the fluid-filled porous material and the list of unknown properties to be estimated, the new mechanistic model is used to identify the frequency range where frequency dispersions in conductivity and/or permittivity will be dominant and measurable for purposes of desired estimations. Electromagnetic (EM) measurements in all the following cases (presented in  FIGS. 6 to 8 ) were tuned to be within the frequency range identified using the mechanistic model, such that the number of discrete frequencies as which the measurements were acquired is at least 3 times the number of physical properties to be estimated. The following case demonstrates the use of mechanistic model to plan the electromagnetic (EM) data acquisition procedure. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Example properties of wetting phase 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 D w  (m 2 /s) 
                 ε r, w   
                 σ w  (S/m) 
                 ρ w  (kg/m 3 ) 
                 σ (N/m) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 Water 
                 10 −9   
                 70 
                 0.1 
                 1000 
                 0.05 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Example properties of solid particles, i is c for conductive 
               
               
                 particle (e.g., graphite), i is n for surface-charge- 
               
               
                 bearing nonconductive particle (e.g., clay and sand), 
               
               
                 and i is nw for non-wetting phase (e.g., oil). 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                   
                   
                   
                   
                   
                 σ i   
                   
                   
               
               
                   
                 ϕ i  (%) 
                 α i  (μm) 
                 D i  (m 2 /s) 
                 ε r, i   
                 (S/m) 
                 λ (S) 
                 θ (°) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Graphite 
                 10 
                 200 
                 5 × 10 −5   
                 12 
                 500 
                 — 
                 0~180 
               
               
                 Sand 
                 70 
                 1000 
                 — 
                 4 
                 — 
                 10 −9   
                 — 
               
               
                 Clay 
                 10 
                 100 
                 — 
                 8 
                 — 
                 10 −8   
                 — 
               
               
                 Oil 
                 1~9 
                 100 
                 — 
                 2 
                 — 
                     10 −30   
                 — 
               
               
                   
               
            
           
         
       
     
       FIGS. 5A and 5B  show graphs of effective conductivity and  FIGS. 5C and 5D  show graphs of effective permittivity for mixtures including a conductive solid particle. More specifically, the curves in the graph  62  and graph  64  show effective conductivity, and the curves in the graph  66  and graph  68  show effective permittivity. When comparing the different curves in graph  62  and graph  66 , or graph  64  and graph  68 , the frequency dispersion reduces as contact angle increases, which means the conductive particle becomes oil wet. This is because, as contact angle increases, the graphite surface is covered more by oil, which has much fewer charge carriers than water and impedes the interfacial polarization in the fluid phase which lowers charge accumulation. As oil saturation increases, both σ_eff and ε_(r,eff) will reduce due to the increase in the volume fraction of oil as nonconductive inclusion. Both σ_eff and ε_(r,eff) will converge to a single value at high frequency because the charge carriers rapidly respond to the alternating external EM field and there is no net accumulation around particles, resulting in an apparent increase in conductivity. Consequently, conductivity reaches to a high value and permittivity reaches to a low value (representing only dipole moment of water) at high frequency close to 1 GHz. In the contrast, at low frequency, the charge carriers quickly reach the equilibrium distribution around the conductive particles&#39; interface, so that the polarized particles act as insulators, which lead to lower σ_eff and higher ε_(r,eff). σ_eff at low frequency can be modeled using effective medium model assuming the conductive particles to be insulators. 
       FIGS. 6A, 6B, and 6C  (i.e.,  FIGS. 6A-C ) show graphs of effective conductivity and  FIGS. 6D, 6E, and 6F  (i.e.,  FIGS. 6D-F ) effective permittivity for mixtures including a conductive solid particle. More specifically, the curves in the graph  70 , graph  72 , and graph  74  show effective conductivity, and the curves in the graph  76 , graph  78 , and graph  80  show effective permittivity. When comparing the different curves in graph  70  and graph  76 , graph  72  and graph  78 , and graph  74  and graph  80 , the frequency dispersion reduces as oil saturation increases because graphite surface is covered more by oil, similar to the effect of contact angle. It should be noted that as oil saturation increases, both σ_eff and ε_(r,eff) will reduce due to the increase in the volume fraction of oil because the oil behaves as nonconductive inclusion. Also, by comparing the rate of change among curves in graph  62  and graph  66  (e.g., as shown in  FIGS. 5A and 5C ) and graph  70  and graph  76  (e.g., as shown in  FIGS. 6A and 6D ), it is evident that the effect of change in oil saturation from 10% to 70% on the frequency dispersion of conductivity and permittivity is much lower than the effect of change in contact angle from 0° to 180°. This indicates that the contact angle plays a primary effect and oil saturation plays a secondary effect in controlling the multi-frequency behavior. 
     The following three cases demonstrate the efficacy of the electromagnetic (EM) data acquisition procedure followed by data processing workflow. Based on some assumption of the properties of solid particles and fluid phases in the fluid-filled porous material and the list of unknown properties to be estimated, the new mechanistic model is used to identify the frequency range where frequency dispersions in conductivity and/or permittivity will be dominant and measurable for purposes of desired estimations. Electromagnetic (EM) measurements in all the following cases (presented in  FIGS. 5A, 5B, 5C, and 5D ,  FIGS. 6A-C ,  FIGS. 6D-F , and  FIGS. 7A, 7B, and 7C ) were tuned to be within the frequency range identified using the mechanistic model, such that the number of discrete frequencies at which the measurements were acquired is at least 3 times the number of physical properties to be estimated. Using the MCMC inversion coupled with the mechanistic model, several properties of the fluid-filled porous materials were estimated, the primary being the simultaneous estimations of oil saturation (or water saturation) and contact angle (or wettability). 
       FIGS. 7A, 7B, and 7C  (i.e.,  FIGS. 7A-C ) show a graph  90 , a graph  92 , and a graph  94 , which each depict a history of Markov Chain Monte Carlo (MCMC) inversion-derived estimates. More specifically, the graph  90 , the graph  92 , and the graph  94  illustrate the MCMC inversion of multi-frequency EM measurements of conductivity and permittivity (shown in  FIG. 5.3 ) to estimate clay surface conductance λ c  (e.g., shown in graph  90 ), graphite contact angle θ (e.g., shown in graph  92 ), and water conductivity σ w  (e.g., shown in graph  90 ) for oil/water-filled porous material containing water-wet sand, clays, and graphite. 
       FIGS. 8A, 8B, and 8C  (i.e.,  FIGS. 8A-C ) show a graph  96 , a graph  98 , and a graph  100 , which each depict a histogram of MCMC inversion-derived estimates of clay surface conductance, graphite contact angle, and water conductivity. The graph  96 , the graph  98 , and the graph  100  represent histograms of MCMC inversion-derived estimates of clay surface conductance λ c  (e.g., shown in graph  96 ), graphite contact angle θ (e.g., shown in graph  98 ), and water conductivity σ w  (e.g., shown in graph  100 ). A line  102  represents the original values of the properties and the region between lines  104  and  106  represent 90% highest posterior density (HPD) interval of the inversion-derived estimates. 
       FIGS. 9A and 9B  show a graph  108  of multi-frequency EM measurements and a graph  110  of model predictions based on inversion-derived estimates. More specifically, the graph  108  and the graph  110  illustrates a comparison of the multi-frequency EM measurements against the mechanistic model predictions for effective permittivity based on the inversion-derived estimates for effective conductance and effective permittivity. 
       FIGS. 10A, 10B, and 10C  show a graph  112 , a graph  114 , and a graph  116 , which each depict a history of Markov Chain Monte Carlo (MCMC) inversion-derived estimates. More specifically, the graph  112 , the graph  114 , and the graph  116  illustrate the MCMC inversion of multi-frequency EM measurements of conductivity and permittivity (shown in  FIG. 6.3 ) to estimate clay surface conductance λ c  (e.g., shown in graph  112 ), graphite contact angle θ (e.g., shown in graph  114 ), and water conductivity σ w  (e.g., shown in graph  116 ) for oil/water-filled porous material containing water-wet sand and clays and slightly oil-wet graphite. 
       FIGS. 11A, 11B, and 11C  show a graph  118 , a graph  120 , and a graph  122 , which each depict a histogram of MCMC inversion-derived estimates of clay surface conductance, graphite contact angle, and water conductivity. More specifically, the graph  118 , the graph  120 , and the graph  120  illustrate a histogram of MCMC inversion-derived estimates of clay surface conductance λ c  (e.g., shown in graph  118 ), graphite contact angle θ (e.g., shown in graph  120 ), and water conductivity σ w  (e.g., shown in graph  122 ). The line  102  represents the original values of the properties and the region between lines  104  and  106  represent 90% HPD interval of the inversion-derived estimates. 
       FIGS. 12A and 12B  show a graph  124  of multi-frequency MS measurements and a graph  126  of model predictions based on inversion-derived estimates. More specifically, the graph  124  and graph  126  illustrate a comparison of the multi-frequency EM measurements against the mechanistic model predictions based on the inversion-derived estimates of effective conductance and effective permittivity. 
       FIGS. 13A and 13B  show a graph  128  and a graph  130  that each history of Markov Chain Monte Carlo (MCMC) inversion-derived estimates. More specifically, the graph  128  and the graph  130  illustrate the MCMC inversion of multi-frequency EM measurements of conductivity and permittivity (shown in  FIG. 7.3 ) to estimate graphite contact angle θ and oil saturation S o  for oil/water-filled porous material containing water-wet sand and clays and oil-wet graphite. 
       FIGS. 14A and 14B  show a graph  132  and a graph  134  that each depict a third histogram of MCMC inversion-derived estimates of clay surface conductance, graphite contact angle, and water conductivity. More specifically, the graph  132  and the graph  134  illustrate the histogram of MCMC inversion-derived estimates of graphite contact angle θ and oil saturation S o . The line  102  represents the original values of the properties and the region between lines  104  and  106  represent 90% HPD interval of the inversion-derived estimates. 
       FIGS. 15A and 15B  show a graph  136  of multi-frequency MS measurements and a graph  138  model predictions based on inversion-derived estimates. More specifically, the graph  136  and the graph  138  illustrate a comparison of the multi-frequency EM measurements against the mechanistic model predictions based on the inversion-derived estimates of effective conductance and effective permittivity. 
     Accordingly, the present disclosure is directed to techniques for quantitatively determining effects of wettability (e.g., contact angle) of conductive particles on the multi-frequency complex conductivity of fluid-filled porous materials, such as a geological formation. In some embodiments, the techniques include developing a material and subsurface characterization model  46 . The material and subsurface characterization model  46  may be developed by solving the Young-Laplace equation as discussed herein. Additionally, the material and subsurface characterization model  46  may be developed by applying, invoking, or utilizing the Poisson-Nernst-Planck (PNP) equation to quantify dipolarizability of a partially wetted graphite particle. Further, developing the model may include using an effective medium model to combine the interfacial polarization effects of nonconductive particles (e.g., sand and clay) and conductive particles (e.g., graphite and pyrite) to compute the complex conductivity of fluid-filled porous material containing strongly water-wet nonconductive particles and conductive particles of any wettability. 
     Nomenclature 
     PPIP model=perfectly polarized interfacial polarization model 
     SCAIP model=surface-conductance-assisted interfacial polarization model 
     a=characteristic length of inclusion phase (m) 
     A o =surface area of graphite particle covered by oil (m 2 ) 
     A s =surface area of graphite particle (m 2 ) 
     A w =surface area of graphite particle covered by water (m 2 ) 
     B o =Bond number 
     D j =diffusion coefficient of charge carriers of medium j (m 2 /s) 
     e=Euler&#39;s number 
     E 0 =amplitude of the electric field (V) 
     E 0 =vacuum permittivity (8.854×10 −12  F/m) 
     ε eff =effective dielectric permittivity of the mixture (F/m) 
     ε j =dielectric permittivity of medium j (F/m) 
     ε r,j =relative permittivity of medium j 
     f=frequency (Hz) 
     f j  (ω)=dipolarizability (dipolar field coefficient) of medium j 
     f(φ)=a function of wetting angle φ 
     g=gravitational acceleration (N/kg) 
     G=dimensionless form of h−h i    
     h(r)=height of oil-water interface at any distance r away from the vertical axis z (m) 
     ĥ=dimensionless form of h 
     h c =height where the oil-water interface contacts the particle surface (m) 
     h i =height of oil-water interface in the absence of wetting of graphite (far-field height) (m) 
     H=mean curvature of the meniscus surface (m −1 ) 
     i=square root of −1 
     I 0 =modified Bessel function of the first kind of order 0 
     K 0 =modified Bessel function of the second kind of order 0 
     K 1 =modified Bessel function of the second kind of order 1 
     L c =capillary length (m) 
     λ=surface conductance (S) 
     ω=angular frequency of the electric field (rad/s) 
     Δp=Laplace pressure (Pa) 
     p o =proportion of graphite surface that covered by oil (%) 
     p w =proportion of graphite surface that covered by water (%) 
     φ=wetting angle (°) 
     ϕ=porosity of the porous media (%) 
     ϕ j =volume fraction of medium j in the mixture (%) 
     ϕ o =volume fraction of oil in the mixture (%) 
     ψ=angle between oil-water interface and the horizon (x-axis) at contact point (°) 
     q=elementary charge (1.6×10 −19  C) 
     r=distance from vertical axis z (m) 
     {circumflex over (r)}=dimensionless form of r 
     R=radius of graphite particle (m) 
     ρ o =density of oil (kg/m 3 ) 
     ρ w =density of water (kg/m 3 ) 
     S o =oil saturation (%) 
     σ=interfacial tension between oil and water (N/m) 
     σ eff =effective electrical conductivity of the mixture (S/m) 
     σ eff *=effective complex electrical conductivity of the mixture (S/m) 
     σ j =electrical conductivity of medium j (S/m) 
     σ j *=complex electrical conductivity of medium j (S/m) 
     θ=contact angle (°) 
     Another aspect of the present disclosure relates to systems and methods for using a material and subsurface characterization model to quantify the effects of wettability of nonconductive particles. Moreover, the model may be implemented to determine the wettability effects of the solid particles that produce interfacial polarization phenomena on multi-frequency electromagnetic measurements. Further, the material and subsurface characterization model, in accordance with the present disclosure, provides a novel technique for identifying a range of operating frequencies for electromagnetic measurements to characterize the contact angle of solid particles that are present within a subsurface formation. 
     With the foregoing in mind,  FIG. 16  is an example illustration of a cross-section of a volume  140  (e.g., within a geological formation) that includes a solid particle  142  suspended in an oil-water media, in accordance with an embodiment. In general, the volume  140  may be assumed for developing the model, as discussed herein. As shown, the solid particle  142  is a circle (e.g., a cross-section of a sphere); however, it should be noted that, in some embodiments, the solid particle may be ellipsoidal (e.g., a first diameter  144  of the solid particle  142  may be greater than or less than a diameter  146  of the solid particle  142 ) or have a radial normal distribution of radii. 
     In the illustrated cross-section of the volume  140  shown in  FIG. 16 , C denotes the point where the oil-water interface (e.g., interface between the non-wetting layer and the wetting layer) contacts the particle surface; θ is the contact angle of conductive particle; φ is the wetting angle; ψ is the angle between oil-water interface and the horizon (x-axis) at point C; R is the radius of conductive particle; h i  is the uniform height of oil-water interface in the absence of wetting of the conductive particle (far-field height); h c  is the height where the oil-water interface contacts the particle surface, such that h c =R(1−cos φ); r is the horizontal distance perpendicular to the vertical axis z; and h(r) is the height of oil-water interface at any distance r away from the vertical axis z. 
     The preferential spread/wetting of the wetting/non-wetting interface generates a wetting angle, which represents the surface area of the solid particle in contact with each of the two fluid phases. The interfacial polarization the phenomena due to such solid particle in contact with two distinct fluid types are entirely governed by the extent to which solid particle is surround by the wetting phase versus non-wetting phase, which is governed by the wettability and contact angle of the solid particle. For example, when water wets a clay particle, the interfacial polarization effects on the complex conductivity/permittivity measurements will be enhanced. In another example, when the clay particle is preferentially oil wet, its interracial polarization effects on the complex conductivity/permittivity measurements will diminish. 
     From an effective medium standpoint, the effective complex conductivity of a porous fluid-filled geomaterial containing surface-charge-bearing nonconductive particles (e.g., water-wet sand and mixed-wet clay particles) at any saturation of the wetting phase (e.g., water) may be expressed as: 
     
       
         
           
             
               
                 
                   K 
                   eff 
                 
                 - 
                 
                   K 
                   w 
                 
               
               
                 
                   K 
                   eff 
                 
                 + 
                 
                   2 
                    
                   
                     K 
                     w 
                   
                 
               
             
             = 
             
               
                 ∑ 
                 
                   
                     φ 
                     
                       n 
                        
                       1 
                     
                   
                    
                   
                     
                       f 
                       
                         n 
                          
                         1 
                       
                     
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                 
               
               + 
               
                 ∑ 
                 
                   
                     φ 
                     
                       n 
                        
                       2 
                     
                   
                    
                   
                     
                       f 
                       
                         
                           n 
                            
                           2 
                         
                         , 
                         w 
                       
                     
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                    
                   
                     p 
                     w 
                   
                 
               
               + 
               
                 ∑ 
                 
                   
                     φ 
                     
                       n 
                        
                       2 
                     
                   
                    
                   
                     
                       f 
                       
                         
                           n 
                            
                           2 
                         
                         , 
                         nw 
                       
                     
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                    
                   
                     ( 
                     
                       1 
                       - 
                       
                         p 
                         w 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   φ 
                   nw 
                 
                  
                 
                   
                     f 
                     nw 
                   
                    
                   
                     ( 
                     ω 
                     ) 
                   
                 
               
             
           
         
       
     
     K eff  is the effective complex conductivity of the porous fluid-filled geomaterial; K w  is the complex conductivity of pore-filling wetting phase, which is brine or saline water in our case, with an assumption that the complex conductivity of pore-filling non-wetting phase, which is oil in our case, is negligible; f is the dipolarizability due to interfacial polarization of solid particle; ωω is the angular frequency of the external EM field; ϕ is the volume fraction of solid particles or the fluid phases; p w  is the proportion of a single solid particle surface that is covered by wetting phase (water) determined using the newly developed model of wetting angle of a solid particle; and Subscripts n1, n2, nw, and w represent water-wet surface-charge-bearing nonconductive particle #1 (e.g. sand), surface-charge-bearing nonconductive particle #2 of any wettability (e.g. clay), non-wetting phase (e.g. oil), and wetting phase (e.g. water), respectively. 
     When a surface-charge-bearing nonconductive solid particle is not fully wet, the interfacial polarization effect of the surface-charge-bearing nonconductive solid particle may be determined as a volumetric mixing of interfacial polarization when the solid particle is completely surrounded by non-wetting fluid phase, f c,nw,  and that when completely surround by wetting fluid phase, f c,w , expressed as ϕ n2 f n2,w (ω)p w +ϕ n2 f n2,nw (ω)(1−p w ), where p w  is the proportion of the solid particle surface that is covered by wetting phase (water) determined using the newly developed model of wetting angle of a solid particle. For example, the portion of a single clay surface that is covered by a wetting phase may be expressed as: 
     
       
         
           
             
               p 
               w 
             
             = 
             
               
                 1 
                 - 
                 cosϕ 
               
               2 
             
           
         
       
     
     where φ is the wetting angle. 
     Dipolarizability of Nonconductive Particle (eg., Clay, Sand, Oil) Completely Immersed in Wetting Phase 
     
       
         
           
             
               
                 f 
                 
                   non 
                   - 
                   conductive 
                 
               
                
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               
                 
                   Q 
                    
                   
                     ( 
                     
                       R 
                       + 
                       A 
                     
                     ) 
                   
                 
                 - 
                 P 
               
               
                 
                   Q 
                    
                   
                     ( 
                     
                       R 
                       - 
                       
                         2 
                          
                         A 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   2 
                    
                   P 
                 
               
             
           
         
       
       
         
           
             
               where 
               : 
               
                 
 
               
                
               A 
             
             = 
             
               1 
               
                 a 
                 2 
               
             
           
         
       
       
         
           
             P 
             = 
             
               
                 γ 
                 w 
                 2 
               
               + 
               
                 
                   ξ 
                   w 
                   2 
                 
                  
                 
                   
                     G 
                     * 
                   
                   
                     H 
                     * 
                   
                 
               
               + 
               
                 
                   2 
                    
                   
                     G 
                     * 
                   
                 
                 
                   
                     a 
                     2 
                   
                    
                   L 
                 
               
             
           
         
       
       
         
           
             Q 
             = 
             
               
                 1 
                 
                   iF 
                   + 
                   1 
                 
               
                
               
                 [ 
                 
                   2 
                   - 
                   
                     
                       
                         
                           a 
                           2 
                         
                          
                         
                           ξ 
                           h 
                           2 
                         
                       
                       
                         H 
                         * 
                       
                     
                      
                     
                       ( 
                       
                         
                           L 
                           iF 
                         
                         + 
                         E 
                       
                       ) 
                     
                   
                   - 
                   
                     
                       2 
                        
                       E 
                     
                     L 
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             R 
             = 
             
               
                 P 
                 Q 
               
                
               
                 ( 
                 
                   
                     iFE 
                     + 
                     L 
                   
                   
                     iF 
                     + 
                     1 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 H 
                 * 
               
               = 
               
                 
                   aL 
                   w 
                 
                 
                   F 
                   w 
                 
               
             
             , 
             
               
                 G 
                 * 
               
               = 
               
                 
                   aG 
                   w 
                 
                 
                   E 
                   w 
                 
               
             
             , 
             
               L 
               = 
               
                 
                   2 
                    
                   λ 
                 
                 
                   
                     a 
                      
                     σ 
                   
                   w 
                 
               
             
             , 
             
               E 
               = 
               
                 
                   ϵ 
                   n 
                 
                 
                   ɛ 
                   w 
                 
               
             
             , 
             
               F 
               = 
               
                 
                   σϵ 
                   w 
                 
                 
                   σ 
                   w 
                 
               
             
           
         
       
       
         
           
             
               F 
               w 
             
             = 
             
               
                 q 
                 
                   
                     γ 
                     w 
                     2 
                   
                    
                   
                     ϵ 
                     w 
                   
                 
               
                
               
                 
                   e 
                   
                     - 
                     
                       
                         a 
                          
                         γ 
                       
                       w 
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       1 
                       
                         
                           a 
                            
                           γ 
                         
                         w 
                       
                     
                     + 
                     
                       1 
                       
                         
                           ( 
                           
                             
                               a 
                                
                               γ 
                             
                             w 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               g 
               w 
             
             = 
             
               
                 q 
                 
                   
                     γ 
                     w 
                   
                    
                   
                     ϵ 
                     w 
                   
                 
               
                
               
                 
                   e 
                   
                     - 
                     
                       
                         a 
                          
                         ξ 
                       
                       w 
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       1 
                       
                         
                           a 
                            
                           γ 
                         
                         w 
                       
                     
                     + 
                     
                       2 
                       
                         
                           ( 
                           
                             
                               a 
                                
                               γ 
                             
                             w 
                           
                           ) 
                         
                         2 
                       
                     
                     + 
                     
                       2 
                       
                         
                           ( 
                           
                             
                               a 
                                
                               γ 
                             
                             w 
                           
                           ) 
                         
                         3 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               L 
               w 
             
             = 
             
               
                 q 
                 
                   
                     ξ 
                     w 
                   
                    
                   
                     ϵ 
                     w 
                   
                 
               
                
               
                 
                   e 
                   
                     - 
                     
                       
                         a 
                          
                         ξ 
                       
                       w 
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       1 
                       
                         
                           a 
                            
                           ξ 
                         
                         w 
                       
                     
                     + 
                     
                       2 
                       
                         
                           ( 
                           
                             
                               a 
                                
                               ξ 
                             
                             w 
                           
                           ) 
                         
                         2 
                       
                     
                     + 
                     
                       2 
                       
                         
                           ( 
                           
                             
                               a 
                                
                               ξ 
                             
                             w 
                           
                           ) 
                         
                         3 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               γ 
               w 
             
             = 
             
               
                 
                   
                     i 
                      
                     ω 
                   
                   
                     D 
                     W 
                   
                 
                 + 
                 
                   
                     σ 
                     w 
                   
                   
                     
                       ϵ 
                       w 
                     
                      
                     
                       D 
                       w 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               ξ 
               W 
             
             = 
             
               
                 
                   i 
                    
                   ω 
                 
                 
                   D 
                   W 
                 
               
             
           
         
       
     
     where a is characteristic length of inclusion phase; ω is the angular frequency of the electric field; i is square root of −1; λ is surface conductance of nonconductive particle; σ is electrical conductivity; ε is dielectric permittivity; and D is diffusion coefficient of charge carriers. 
     If the nonconductive particle is immersed in non-wetting phase, the surface conductance may be set to be a very small number. It should be noted that the equations for the disclosed model above represent one example embodiment. That is, there can be other alternative forms for the dipolarizability of nonconductive particle (e.g. clay, sand, oil) completely immersed in wetting phase. 
     Inversion Algorithm 
     We applied the Markov Chain Monte Carlo (MCMC) inversion algorithm for the purposes of estimating water saturation, wettability of solid particles, conductivity of water/brine filling the porous material, and clay surface conductance. Implementation of the inversion scheme coupled with the new mechanistic model of wettability effects improves the interpretation and processing of subsurface electromagnetic log. 
     Mechanistic Model Predictions of Multi-Frequency Complex Conductivity 
     Based on some assumption of the properties of solid particles and fluid phases in the fluid-filled porous material and the list of unknown properties to be estimated, the disclosed mechanistic model may be used to identify the frequency range where frequency dispersions in conductivity and/or permittivity will be dominant and measurable for purposes of desired estimations. Electromagnetic (EM) measurements in all the following cases (presented in  FIGS. 17A-D , and  FIGS. 18A-D ) were tuned to be within the frequency range identified using the mechanistic model, such that the number of discrete frequencies as which the measurements were acquired is at least 3 times the number of physical properties to be estimated. 
       FIGS. 17A, 17B, 17C, and 17D  (i.e.,  FIGS. 17A-17D ) show graphs indicating the effect of contact angle on the properties of a nonconductive solid particle. More specifically, the curves  148 ,  150 , and  152  of  FIG. 17A  show the effective conductivity at a contact angle of 30 degrees, 90 degrees, and 150 degrees for a mixture containing water-wet sand and clay particles with a surface conductance of 10 −6  S, partially saturated with brine/water and an oil saturation of 10%. 
     The curves  154 ,  156 , and  158  of  FIG. 17B  show the effective conductivity at a contact angle of 30 degrees, 90 degrees, and 150 degrees for a mixture containing water-wet sand and clay particles with a surface conductance of 10 −6  S, partially saturated with brine/water and an oil saturation of 90%. 
     The curves  160 ,  162 , and  164  of  FIG. 17C  show the effective permittivity at a contact angle of 30 degrees, 90 degrees, and 150 degrees for a mixture containing water-wet sand and clay particles with a surface conductance of 10 −6  S, partially saturated with brine/water and an oil saturation of 10%. 
     The curves  166 ,  168 , and  170  of  FIG. 17D  show the effective permittivity at a contact angle of 30 degrees, 90 degrees, and 150 degrees for a mixture containing water-wet sand and clay particles with a surface conductance of 10 −6  S, partially saturated with brine/water and an oil saturation of 90%. 
     In this example, the frequency dispersion of effective conductivity is relatively negligible for frequencies lower than 10 MHz, and the frequency dispersion for effective permittivity is relatively negligible for frequencies lower than 10 KHz. 
     As contact angle increases, i.e. the surface-charge bearing nonconductive particle becomes oil wet, the frequency dispersion of permittivity reduces. This is because the clay surface is covered more by oil, which has much less charge carriers than water and impedes the interfacial polarization in the fluid phase. The conductivity increases as contact angle decreases, because clay surface conductance will assist charge transport. At low frequency, the charge carriers quickly reach the equilibrium distribution around the surface-charge-bearing nonconductive particles&#39; interface, so that the particles act as insulators, which lead to lower σ eff  and higher ε r,eff . This model prediction shows that the EM measurements and log responses may be acquired at low frequencies and high frequencies to capture the frequency dispersions in both permittivity and conductivity, respectively. Moreover, the effect of wettability on conductivity is higher at higher oil saturation. 
       FIGS. 18A, 18B, 18C, and 18D  (i.e.,  FIGS. 18A-18D ) show graphs indicating the effect of oil saturation on the properties of a nonconductive solid particle. More specifically, the curves  172 ,  174 , and  176  of  FIG. 18A  show the effective conductivity for a mixture containing water-wet sand and clay particles partially saturated with brine/water and an oil saturation of 10%, 50%, and 90% for a with a surface conductance of 10 −6  S with a contact angle of 30 degrees. 
     The curves  178 ,  180 , and  182  of  FIG. 18B  show the effective conductivity for a mixture containing water-wet sand and clay particles partially saturated with brine/water and an oil saturation of 10%, 50%, and 90% for a with a surface conductance of 10 −6  S with a contact angle of 150 degrees. 
     The curves  184 ,  186 , and  188  of  FIG. 18C  show the effective permittivity for a mixture containing water-wet sand and clay particles partially saturated with brine/water and an oil saturation of 10%, 50%, and 90% for a with a surface conductance of 10 −6  S with a contact angle of 30 degrees. 
     The curves  190 ,  192 , and  194  of  FIG. 18D  show the effective permittivity for a mixture containing water-wet sand and clay particles partially saturated with brine/water and an oil saturation of 10%, 50%, and 90% for a with a surface conductance of 10 −6  S with a contact angle of 150 degrees. 
     In this example, when comparing the different curves in  FIGS. 19A  and C or  FIGS. 19B  and C, there is negligible frequency dispersion of effective conductivity at frequency range of 100 Hz to 1 GHz, while effective permittivity shows some dispersion phenomena at frequency range of 100 Hz to 10 kHz. As oil saturation increases, both σ eff  and ε r,eff  will reduce, which has the similar trend as predicted by Archie&#39;s law and CRI model. 
       FIGS. 19A, 19B, 19C, and 19D  (i.e.,  FIGS. 19A-19D ) show graphs indicating the effect of surface conductance on the properties of a nonconductive solid particle. More specifically, the curves  196 ,  198 ,  200  of  FIG. 19A  show the effective conductivity for a mixture containing water-wet sand and clay particles partially saturated with brine/water and an oil saturation of 10% with a surface conductance of 10 −5  S, 5×10 −6  S, and 10 −6 , respectively, and with a contact angle of 30 degrees. 
     The curves  202 ,  204 , and  206  of  FIG. 18B  show the effective conductivity for a mixture containing water-wet sand and clay particles partially saturated with brine/water and an oil saturation of 10%, with a surface conductance of 10 −5  S, 5×10 −6  S, and 10 −6 , respectively, and with a contact angle of 150 degrees. 
     The curves  208 ,  210 , and  212  of  FIG. 18C  show the effective conductivity for a mixture containing water-wet sand and clay particles partially saturated with brine/water and an oil saturation of 10%, with a surface conductance of 10 −5  S, 5×10 −6  S, and 10 −6 , respectively, and with a contact angle of 30 degrees. 
     The curves  214 ,  216 , and  218  of  FIG. 18D  show the effective conductivity for a mixture containing water-wet sand and clay particles partially saturated with brine/water and an oil saturation of 10%, with a surface conductance of 10 −5  S, 5×10 −6  S, and 10 −6 , respectively, and with a contact angle of 150 degrees. 
     When comparing the different curves in  FIGS. 19A-D , it may be observed that for the smaller the surface conductance of clay, the less obvious the conductivity dispersion will be, and permittivity exhibits dispersion. As surface conductance decrease, the effective conductivity and permittivity also decrease. At low values of contact angles, i.e., water-wet state, an increase in the surface conductance of clay leads to a drastic change in both conductivity and permittivity. 
     With the foregoing in mind,  FIG. 20  illustrates an example process  220  that may be employed by the data processing system  28  to determine properties of a geological formation comprising certain types of solid particles (e.g., conductive and nonconductive) that may be used for certain oil and gas decisions, in accordance with embodiments described herein. The steps of the process  220  may be stored in the memory  32 . Before proceeding, it should be noted that the process  220  is described as being performed by the processor  30  of the data processing system  28 , but the process  220  may be performed by other suitable computing devices. Although described in a particular order, which represents a particular embodiment, it should be noted that the process  220  may be performed in any suitable order. Additionally, embodiments of the process  220  may omit process blocks and/or include additional process blocks. 
     At block  222 , the processor  30  may identify a type of solid particle within the geological formation. In general, the processor  30  identifying the type of solid particle (e.g., particles) within the geological formation based on an input specifying the type of solid particles. For example, an individual may provide an input specifying that the geological formation includes nonconductive particles (e.g., clay, calcite, and quartz) or conductive particles (e.g., graphite and pyrite). In some embodiments, the processor  30  may identify the type of solid particle based on data associated with well logging measurements received by the processor. For example, the processor  30  may receive elemental data from a well logging measurement that indicates a relative percentage of certain elements. The processor  30  may compare the relative percentages to reference elemental data that indicates types of solid particles (e.g., stored in the memory  32 ). As such, the processor  30  may identify a type of solid particle when the received elemental data matches a particular reference elemental data for a type of solid particle. In some embodiments, the processor may 
     At block  224 , the processor  30  may receive electromagnetic measurements at a set of frequencies. In general, block  222  may occur in a general similar manner as block  52  of the process  50  of  FIG. 3 . For example, the processor  30  may identify a set of frequencies to perform an electromagnetic measurement based on the identified type of solid particle. That is, and as discussed herein, the material and subsurface characterization model  46  may be utilized by the processor  30  of the data processing system  28  to determine a set of frequencies that the electromagnetic well-logging tool  12  may operate to acquire electromagnetic measurements. 
     In some embodiments, the processor  30  may determine a type of material and subsurface characterization model  46  to use to identify the set of frequencies based on the identified type of solid particles within the geological formation. For example, the processor  30  may determine the type of material and subsurface characterization model  46  to use based on a relative conductivity (e.g., conductive, nonconductive, above or below a conductivity threshold) of regions of the geological formation, particles identified in the geological formation, or particles suspected of being in the geological formation. That is, if the identified type of solid particle corresponds to a conductive type of solid particle (e.g., graphite and pyrite), the material and subsurface characterization model  46  may be based upon the effective complex conductivity of a porous fluid-filled geomaterial containing conductive particles of any wettability (e.g., graphite particle) and fully wetted surface-charge-bearing nonconductive particles (e.g., water-wet sand and clay particles), as discussed herein. Additionally or alternatively, if the identified type of solid particle corresponds to a nonconductive type of solid particle, the material and subsurface characterization model  46  may be based upon the effective complex conductivity of a porous fluid-filled geomaterial containing surface-charge-bearing nonconductive particles (e.g., water-wet sand and mixed-wet clay particles) at any saturation of the wetting phase (e.g., water) also discussed herein. That is, the processor  30  may select one of the models described herein to determine the set of frequencies for the electromagnetic well-logging tool  12 . In some embodiments, the memory  32  of the data processing system  28  may store both models (e.g., a first model based on the nonconductive particles and a second model based on the conductive particles). As such, when a received input, determination by the processor  30 , or other indication specifies that the processor  30  should utilized the first model or the second model, the processor  30  may retrieve the model. 
     At block  226 , the processor  30  may determine one or more physical properties of the geological formation using the received electromagnetic measurements as generally described with respect to block  54  of the process  50  of  FIG. 4 . For example, a Markov-Chain Monte-Carlo may be applied to the EM measurements received in process block  224  to determine properties such as the contact angles and other physical properties as described herein. 
     As one nonlimiting example of how the above-described techniques may be applied, the processor  30  may use the mechanistic model to identify the range of operating frequency within which the EM measurements and logs may be acquired for purposes of reliably estimating the desired properties of the fluid-filled porous material. According to the identified range of operating frequencies, an EM tool/equipment may be tuned to acquire the multi-frequency electromagnetic measurements and log responses. Following that, an inversion scheme coupled with a mechanistic model processes the multi-frequency electromagnetic (EM) measurements or log responses of fluid-filled porous materials to estimate the desired properties of the fluid-filled porous material. The mechanistic model is coupled with a Markov-Chain Monte Carlo (MCMC) inversion scheme to simultaneously estimate the water saturation, clay surface conductance, brine/pore-filling-fluid salinity/conductivity, and the contact angle of the particles giving rise to interfacial polarization phenomena. 
     Accordingly, aspects of the present disclosure provide techniques to quantify the multi-frequency complex conductivity and/or complex permittivity of fluid-filled porous materials so as to account the effects of contact angle or wettability of conductive or surface-charge-bearing nonconductive particles (or other types of solid particles that give rise to interfacial polarization) on the conductivity and permittivity and their frequency dispersions (i.e., frequency-dependent behavior). Estimate the contact angle (wettability) of conductive particles (e.g. graphite and pyrite) and surface-charge-bearing nonconductive particles (e.g. clay, calcite, and quartz) in fluid-filled porous geomaterials (in subsurface or on surface) or other fluid-filled porous materials. The disclosed techniques may be used to simultaneously estimate fluid saturations, contact angle of conductive particles, contact angle of surface-charge-bearing particles, fluid conductivity/salinity, surface conductance of solid particles, diffusion coefficients of charge carriers in various material constituents, and volume fractions of fluid and solid components in the material. Further, the disclosure techniques may enable for simultaneously estimation wettability (i.e., depends on contact angle) and oil saturation (i.e., depends on water saturation). Further still, the disclosed techniques may be used to estimate contact angle of solid particles/grains that can give rise to interfacial polarization when surrounded by fluid phases/components for various wettability scenarios. Even further, the disclosed techniques may be used to quantify the effects of contact angle or wettability of solid grains/particles (i.e., conductive or surface-charge-bearing nonconductive particle) on the net charge transport and net charge accumulation as a function of the frequency of the external electromagnetic field. The net charge transport determines the conductivity and net charge accumulation determines the permittivity that govern the electromagnetic measurements and log responses of the fluid-filled porous material. Additionally, the disclosed techniques may be used to quantify the multi-frequency complex conductivity and/or complex permittivity of fluid-filled porous materials so as to account the effects of contact angle or wettability of conductive or surface-charge-bearing nonconductive particles (e.g., other types of solid particles that give rise to interfacial polarization) on the conductivity and permittivity and their frequency dispersions (i.e., frequency-dependent behavior). 
     The techniques presented and claimed herein are referenced and applied to material objects and concrete examples of a practical nature that demonstrably improve the present technical field and, as such, are not abstract, intangible or purely theoretical. Further, if any claims appended to the end of this specification contain one or more elements designated as “means for [perform]ing [a function] . . . ” or “step for [perform]ing [a function] . . . ”, it is intended that such elements are to be interpreted under 35 U.S.C. 112(f). However, for any claims containing elements designated in any other manner, it is intended that such elements are not to be interpreted under 35 U.S.C. 112(f). 
     The specific embodiments described above have been shown by way of example, and it should be understood that these embodiments may be susceptible to various modifications and alternative forms. It should be further understood that the claims are not intended to be limited to the particular forms disclosed, but rather to cover all modifications, equivalents, and alternatives falling within the spirit and scope of this disclosure.