# Patent

## Title
Method and apparatus for determining multiscale similarity between NMR measurements and a reference well log

## Abstract

Pulse sequences are applied to a fluid in an earth formation in a static magnetic field and NMR spin echo signals are obtained. The signals are processed to give a distribution of a relaxation time at a plurality of depths. Semblance between the distributions and a log of a formation property are used to identify depths at which portions of the distributions are similar to the formation property logs.

## Background
CROSS-REFERENCES TO RELATED APPLICATIONS
This application is a continuation-in-part of U.S. patent application Ser. No. 12/022,755 filed on Jan. 30, 2008.


FIELD OF THE DISCLOSURE
This disclosure relates to apparatus and techniques for making nuclear magnetic resonance (NMR) measurements in boreholes and to methods for determining magnetic characteristics of formations traversed by a borehole. Specifically, the disclosure relates to design an accurate NMR measurement technique, which is aimed at defining the diffusion properties of the fluid in the pore matrix in the presence of the internal magnetic field gradients.
BACKGROUND OF THE DISCLOSURE
A variety of techniques have been used in determining the presence and in estimating quantities of hydrocarbons (oil and gas) in earth formations. These methods are designed to determine parameters of interest, including among other things, porosity, fluid content, and permeability of the rock formation surrounding the borehole drilled for recovering hydrocarbons. Typically, the tools designed to provide the desired information are used to log the borehole. Much of the logging is done after the boreholes have been drilled. More recently, boreholes have been logged while drilling of the boreholes. This is referred to as measurement-while-drilling (“MWD”) or logging-while-drilling (“LWD”). Measurements have also been made when tripping a drillstring out of a borehole: this is called measurement-while-tripping (“MWT”).
One evolving technique uses nuclear magnetic resonance (NMR) logging tools and methods for determining, among other things porosity, hydrocarbon saturation and permeability of the rock formations. NMR logging tools excite the nuclei of the fluids in the geological formations in the vicinity of the borehole so that certain parameters such as spin density, longitudinal relaxation time (generally referred to in the art as “T1”), and transverse relaxation time (generally referred to as “T2”) of the geological formations can be estimated. From such measurements, formation parameters such as porosity, permeability and hydrocarbon saturation are determined, which provides valuable information about the make-up of the geological formations and the amount of extractable hydrocarbons.
A typical NMR tool generates a static magnetic field B0 in the vicinity of the borehole and an oscillating field B1 in a direction perpendicular to B0. This oscillating field is usually applied in the form of short-duration pulses. The purpose of the B0 field is to polarize the magnetic moments of nuclei parallel to the static field and the purpose of the B1 field is to rotate the magnetic moments by an angle controlled by the width tp and the amplitude B1 of the oscillating pulse. For NMR logging, the most common sequence is the Carr-Purcell-Meiboom-Gill (“CPMG”) sequence that can be expressed as

TW−90−(τ−180−τ−echo)n  (1)

where TW is a wait time, 90 is a 90° tipping pulse, 180 and is a 180° refocusing pulse and 2τ=TE is the interecho spacing.

After being tipped by 90°, the magnetic moment precesses around the static field at a particular frequency known as the Larmor frequency ω, given by ω=γB0, where B0 is the field strength of the static magnetic field and γ is the gyromagnetic ratio. At the same time, the magnetic moments return to the equilibrium direction (i.e., aligned with the static field) according to a decay time known as the “spin-lattice relaxation time” or T1. Inhomogeneities of the B0 field result in dephasing of the magnetic moments and to remedy this, a 180° pulse is included in the sequence to refocus the magnetic moments. This refocusing gives a sequence of n echo signals. These echo sequences are then processed to provide information about the relaxation times. U.S. Pat. No. 6,466,013 to Hawkes et al. and U.S. Pat. No. 6,429,654 to Itskovich et al., both having the same assignee as the present disclosure, teach the use of modified CPMG pulse sequences with reduced power requirements in which the refocusing pulse angle may be less than 180°. These modified pulse sequences may be referred to as optimal rephrasing pulse sequences (ORPS).
Also associated with the spin of molecular nuclei is a second relaxation time, T2, called the transverse or spin-spin relaxation time. At the end of a 90° tipping pulse, all the spins are pointed in a common direction perpendicular, or transverse, to the static field, and they all precess at the Larmor frequency. However, because of small fluctuations in the static field induced by other spins or paramagnetic impurities, the spins precess at slightly different frequencies and the transverse magnetization dephases with a relaxation time T2.
Interpretation of NMR core or log data is often started by inverting the time-domain CPMG (or ORPS) echo decay into a T2 parameter domain distribution. In general, the T2 of fluids in porous rocks depends on the pore-size distribution and the type and number of fluids saturating the pore system. Because of the heterogeneous nature of porous media, T2 decays exhibit a multiexponential behavior. The basic equation describing the transverse relaxation of magnetization in fluid saturated porous media is




  
    
      
        
          M
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            ∫
            
              T
              
                2
                ⁢
                min
              
            
            
              T
              
                2
                ⁢
                
                  
                
                ⁢
                max
              
            
          
          ⁢
          
            
              P
              ⁡
              
                (
                
                  T
                  2
                
                )
              
            
            ⁢
            
              e
              
                
                  -
                  t
                
                /
                
                  T
                  2
                
              
            
            ⁢
            
              
            
            ⁢
            
              d
              
                T
                2
              
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




where M is magnetization and effects of diffusion in the presence of a magnetic field gradient have not been taken into consideration. Eq. (2) is based on the assumption that diffusion effects may be ignored. In a gradient magnetic field, diffusion causes atoms to move from their original positions to new ones which also cause these atoms to acquire different phase shifts compared to atoms that did not move. This contributes to a faster rate of relaxation.

The effect of field gradients is given by an equation of the form




  
    
      
        
          1
          
            T
            2
          
        
        =
        
          
            1
            
              T
              
                2
                ⁢
                bulk
              
            
          
          +
          
            1
            
              T
              
                2
                ⁢
                surface
              
            
          
          +
          
            1
            
              T
              
                2
                ⁢
                diffusion
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




where the first two terms on the right hand side are related to bulk relaxation and surface relaxation while the third term is related to the field gradient G by an equation of the form





  
    
      
        
          T
          
            2
            ⁢
            diffusion
          
        
        =
        
          C
          
            
              TE
              2
            
            ·
            
              G
              2
            
            ·
            D
          
        
      
    
    
      
        (
        4
        )
      
    
  




where TE is the interecho spacing, C is a constant and D is the diffusivity of the fluid.

Some prior art teaches the use of a gradient-based, multiple-frequency NMR logging tool to extract signal components characteristic for each type of hydrocarbon. Measurements at different frequencies are interleaved to obtain, in a single logging pass, multiple data streams corresponding to different recovery times and/or diffusivity for the same spot in the formation.
One of the main difficulties in defining self-diffusion parameters of the fluid in the pore matrix is related to the fact that different fluids having the same relaxation times and different diffusion coefficients cannot be effectively separated. Due to the practical limitation of the signal-to-noise ratio, none of the existing inversion techniques allow an effective and stable reconstruction of both the relaxation and diffusion spectra.
Another difficulty in relaxation and diffusion spectra reconstruction is caused by internal magnetic gradients. Typically, the values of the internal gradients are unknown. Thus, the diffusion parameters cannot be correctly defined if the internal gradients are not considered in both the measurement and interpretation scheme. These have been addressed, for example, in U.S. Pat. No. 6,597,171 to Hurlimann et al., and in U.S. Pat. No. 5,698,979 to Taicher et al. having the same assignee as the present application and the contents of which are incorporated herein by reference, and U.S. Pat. No. 7,049,815 to Itskovich et al., having the same assignee as the present application and the contents of which are incorporated herein by reference. A common aspect of all the prior art methods discussed above is that they do not consider measurements by other logs in the processing of the NMR data, and measurements at one depth are processed substantially independently of measurements and other depths.
SUMMARY OF THE DISCLOSURE
One embodiment of the disclosure is a method of evaluating an earth formation. The method includes: conveying a logging tool into a borehole; obtaining nuclear magnetic resonance (NMR) signals at a plurality of depths in the borehole; processing the NMR signals to obtain a distribution of a relaxation time at each of the plurality of depths, each of the distributions comprising at least one of (i) a plurality of bins, and (ii) a plurality of components; determining a semblance of a first log indicative of a property of the formation with the obtained distribution; and using the semblance to identify a portion of at least one of the distributions characteristic of the property of the formation.
Another embodiment of the disclosure is an apparatus configured to evaluate an earth formation. The apparatus includes: a logging tool configured to be conveyed into a borehole; a sensor arrangement on the logging tool configured to obtain nuclear magnetic resonance (NMR) signals at a plurality of depths in the borehole; at least one processor configured to: (i) process the NMR signals to obtain a distribution of a relaxation time at each of the plurality of depths, each of the distributions comprising at least one of (I) a plurality of bins, and (II) a plurality of components; (ii) determine a semblance of a first log indicative of a property of the formation with the obtained distribution; and (iii) use the semblance to identify a portion of at least one of the distributions characteristic of the property of the formation.
Another embodiment of the disclosure is a computer-readable medium product having stored thereon instructions that when read by a processor cause the processor to execute a method. The method includes: processing NMR signals at a plurality of depths obtained by a logging tool conveyed in a borehole for: estimating a distribution of a relaxation time at each of the plurality of depths, each of the distributions comprising at least one of (i) a plurality of bins, and (ii) a plurality of components; determining a semblance of a first log indicative of a property of the formation with the obtained distribution; and using the semblance to identify a portion of at least one of the distributions characteristic of the property of the formation.



BRIEF DESCRIPTION OF THE DRAWINGS
The file of this patent contains at least one drawing executed in color: Copies of this patent with color drawing(s) will be provided by the Patent and Trademark Office upon request and payment of the necessary fee. The present disclosure is best understood with reference to the following figures in which like numerals refer to like elements, and in which:
FIG. 1 depicts diagrammatically an NMR logging tool in a borehole;
FIG. 2 (prior art) shows an exemplary configuration of magnets, antenna and shield suitable for use with the present disclosure;
FIG. 3 shows a flow chart illustrating some of the important features of the present disclosure;
FIG. 4 (prior art) shows an exemplary display of results obtained using a prior art method for processing of NMR data;
FIG. 5 shows an exemplary plot of T2 distributions obtained by processing of the data in FIG. 4 at a plurality of depths;
FIG. 6 shows plots of resistivity, gamma ray, clay bound water, bound volume irreducible and movable water over a portion of the depth interval shown in FIGS. 4 and 5;
FIG. 7 shows plots of resistivity, gamma ray and selected T2 bin logs over the depth interval of FIG. 6;
FIG. 8(a)-(d) illustrate exemplary cross-correlations between loges measured in different runs;
FIG. 9(a) shows an exemplary T2 distribution over a depth interval that includes oil and FIG. 9(b) shows different measures of similarity of the T2 distribution with gamma ray and resistivity logs;
FIG. 10 (in color) shows (a) a 2-D distribution of the data in the T2-diffusivity plane, (b) the distribution in 1-D with respect to the T2-axis, (c) the distribution in 1-D with respect to the diffusivity axis, and (d) a portion of a graphic display interface;
FIG. 11 (in color) shows a 2-D distribution of data in the T2-diffusivity plane for a depth interval that includes oil;
FIG. 12(a)-(b) show the Pearson correlation coefficient and the p-value for data in FIG. 11;
FIG. 13 (in color) shows the T2 distribution for the data of FIG. 11 color coded according to a measure of similarity;
FIG. 14 (in color) shows a 2-D distribution of data in the T2-diffusivity plane for a depth interval that is water-bearing;
FIG. 15(a)-(b) show the Pearson correlation coefficient and the p-value for data in FIG. 14;
FIG. 16 shows petrofacies and the corresponding T2 distribution at 100% and irreducible water saturation;
FIG. 17 shows simulation results for 100% water saturation;
FIG. 18 shows simulation results for 100% oil saturation;
FIG. 19 is a mobility map for a reservoir;
FIG. 20 (in color) shows (a) a reference log, (b) the continuous wavelet transform (CWT) of the reference log, (c) a second log, (d) the CWT of the second log, and (e) the coherence between the CWTs of the reference log and the second log;
FIG. 21 (in color) shows (a) another reference log, (b) the continuous wavelet transform (CWT) of the another reference log, (c) another second log, (d) the CWT of the another second log, and (e) the coherence between the CWTs of the another reference log and the another second log;
FIG. 22 (in color) shows a representation of an exemplary T2 spectrum by 28 bins;
FIG. 23 (in color) shows binned NMR logs (track 5) with gamma ray logs in track; and
FIG. 24 shows exemplary spectral distributions obtained by fitting NMR data by split Gaussian curves.



DETAILED DESCRIPTION OF THE DISCLOSURE
FIG. 1 depicts a borehole 10 drilled in a typical fashion into a subsurface geological formation 12 to be investigated for potential hydrocarbon producing reservoirs. An NMR logging tool 14 has been lowered into the hole 10 by means of a cable 16 and appropriate surface equipment (represented diagrammatically by a reel 18) and is being raised through the formation 12 comprising a plurality of layers 12a through 12g of differing composition, to log one or more of the formation's characteristics. The NMR logging tool may be provided with bowsprings 22 to maintain the tool in an eccentric position within the borehole with one side of the tool in proximity to the borehole wall. The permanent magnets 23 provide the static magnetic field. Signals generated by the tool 14 are passed to the surface through the cable 16 and from the cable 16 through another line 19 to appropriate surface equipment 20 for processing, recording, display and/or for transmission to another site for processing, recording and/or display. Alternatively, the processor may be located at a suitable position (not shown) downhole, e.g., in the logging tool 14.
FIG. 2 (prior art) schematically illustrates an exemplary embodiment of an apparatus (sensor arrangement) suitable for use with the method of the present disclosure. This is discussed in detail in U.S. Pat. No. 6,348,792 of Beard et al., having the same assignee as the present disclosure and the contents of which are fully incorporated herein by reference. The tool cross-sectional view in FIG. 2 illustrates a main magnet 217, a second magnet 218 and a transceiver antenna comprising wires 219 and core material 210. The arrows depict the polarization (e.g., from the South pole to the North pole) of the main magnet 217 and the secondary magnet 218. A noteworthy feature of the arrangement shown in FIG. 2 is that the polarization of the magnets providing the static field is towards the side of the tool, rather than towards the front of the tool (the right side of FIG. 2).
The second magnet 218 is positioned to augment the shape of the static magnetic field by adding a second magnetic dipole in close proximity to the RF dipole defined by the wires 219 and the soft magnetic core 210. This positioning moves the center of the effective static dipole closer to the RF dipole, thereby increasing the azimuthal extent of the region of examination. The second magnet 218 also reduces the shunting effect of the high permeability magnetic core 210 on the main magnet 217. In the absence of the second magnet, the DC field would be effectively shorted by the core 210. Thus, the second magnet, besides acting as a shaping magnet for shaping the static field to the front of the tool (the side of the main magnet) also acts as a bucking magnet with respect to the static field in the core 210. Those versed in the art will recognize that the bucking function and a limited shaping could be accomplished simply by having a gap in the core; however, since some kind of field shaping is required on the front side of the tool, in one embodiment of the disclosure, the second magnet serves both for field shaping and for bucking. If the static field in the core 210 is close to zero, then the magnetostrictive ringing from the core is substantially eliminated.
Within the region of investigation, the static field gradient is substantially uniform and the static field strength lies within predetermined limits to give a substantially uniform Larmor frequency. Those versed in the art will recognize that the combination of field shaping and bucking could be accomplished by other magnet configurations than those shown in FIG. 2.
NMR spin echo signals are obtained using the apparatus of FIG. 2. Optionally, additional measurements may be made using an external gradient field as discussed in Reiderman. Measurements made with the gradient field enable the determination of diffusivity. The manner in which these measurements are used is discussed next.
Shown in FIG. 3 is a flow chart outlining some of the steps of the present disclosure. NMR measurements are obtained 303. In addition, gamma ray logs indicative of the shale content of the formation, and resistivity logs are obtained 301. The measurements of resistivity and gamma ray logs may be done simultaneously with the acquisition of the NMR data or, as is more common, the resistivity and/or gamma ray logs may be obtained in a different logging run than the NMR logs.
The NMR measurements are inverted using prior art methods to obtain a T2 distribution at each of a plurality of depths 305. The T2 distribution is characterized by its values in a plurality of bins. FIG. 5 shows an exemplary plot of the T2 distribution at a plurality of depths. In the example shown, approximately 30 bins were used with a total time of greater than 1024 ms.
Digressing briefly, FIG. 4 shows an exemplary display of prior art processing. Track 1 401 shows the T2 distribution from a well over a depth range of around 600 ft. Track 2 403 shows the bound volume irreducible 411, the bound water moveable 415 and the clay bound water 417. Track 3 405 shows the determined permeability and track 4 407 shows the misfit in the determination of permeability and porosities. The total porosity includes clay-bound water (CBW), capillary bound water (also known as Bulk Volume Irreducible or BVI), movable water and hydrocarbons. See U.S. Pat. No. 6,972,564 to Chen.
An important feature of the present disclosure is that the NMR data are sorted to produce an NMR bin log for each of the bins of the distribution 307. The method of the present disclosure is based upon identifying and using similarities between the NMR bin logs and the resistivity and gamma ray logs. FIG. 6 shows a display of the resistivity 601 log over an 8 ft. (2.44 m) interval, the gamma ray log 607, a log of clay-bound water (from FIG. 4), a plot of BVI 605 and a plot of bound water movable 609. FIG. 7 shows the gamma ray and resistivity plots, and in addition, shows a log of NMR bins 2-8 over the same depth interval 701, 703, 705, 707, 709, and 711.
The present method is based on two principles. First, an increase in the resistivity log is commonly due to an increase in the oil in the formation (provided the gamma ray indicates a small amount of shale). Accordingly, in depths in which the resistivity is high, the bin that characterizes oil in the formation will tend to have a larger value. Thus, referring back to FIG. 3, a depth log is obtained for each bin of the T2 distribution 307. Still referring to FIG. 3, a measure of similarity is determined between the T2 bin logs and the resistivity log and the gamma ray log 309. The measure of similarity is discussed below.
The similarity test can be performed using any linear or non-linear correlation, depending on the case. However, tests results suggest using cross correlation or only the correlation matrix if there is no depth shift between the logs. Once the oil peak has been found, the correlation to the viscosity can be established using known models. Furthermore, by removing the effects of the oil exemplified by the oil peak 311 from the original T2 distribution, the water T2 distribution can be obtained and hence the water saturation, Sw. This is discussed further below after the discussion of FIG. 14. In the special case of heavy oil, and probably in more general cases, but depending on the oil viscosity and degree of water saturation, the T2 bin logs with negative correlation with the resistivity logs and their gradients are more likely to represent the movable water, to be drained by the oil when the oil saturation increases. By this way it is possible to trace back the T2 distribution of the water saturation using a Gaussian curve fitting or an approach like that in U.S. Pat. No. 7,363,161 to Georgi et al. which also generates a scale for correlating the T2 bin with the grain size. When a model for the water T2 distribution is obtained, a model for the capillary pressure curve and rock quality (P. Romero, SPWLA 2004) and for the relative permeability (Corey-Burdine) can be established 313. Furthermore, a cross plot of the hydrocarbon T2 distribution vs. T2 distribution for the 100% water saturation can be built, which indicates the zones of different productivity indexes. This is discussed below with reference to FIGS. 16-19.
Those versed in the art would recognize that logs made at different times in the same borehole may not be perfectly registered. This is illustrated in FIG. 8. Shown in FIG. 8(a) is the cross-correlation between the resistivity and the gamma ray logs. As expected, there is a negative correlation between the two and a time shift of zero because the logs were measured in the same logging run. FIG. 8(b) shows the cross-correlation between the resistivity and Clay bound water log. Since the latter is derived from NMR measurements made in a separate logging run, there is a depth shift of approximately 10 ft. (2.44 m) between the logs. A similar shift is noted in FIG. 8(c) of the bound volume irreducible and in FIG. 8(d) for the movable water. Such a cross-correlation step, while not illustrated in FIG. 3, may be necessary before determination of similarity 309. As an alternative to using a peak of the cross-correlation technique for different time shifts, the mutual entropy between two shifted traces may be used. The entropy is defined as H(x)=−E{x log(P(x))} where H(.) is the entropy, E{.} is the expected value and P(.) is the probability density function.
We next discuss measures of similarity that are used in the present disclosure between the different logs. These are for exemplary purposes only and other measures of similarity could be used with the present method.
The first concept is that of covariance. This is a measure of how much the variations of the variables are interrelated. Consider a random vector

{right arrow over (X)}=(X1,X2, . . . Xn)  (5.).

For any pair of components, the covariance is defined by the expectation

cov(Xi,Xk)=Σik=E{(Xi−μi)(Xk−μk)}  (6),

Where μi and μk are the mean values of Xi and Xk. The covariance is then given by the covariance matrix:





  
    
      
        
          Σ
          →
        
        =
        
          
            [
            
              
                
                  
                    Σ
                    
                      1
                      ,
                      1
                    
                  
                
                
                  
                    Σ
                    
                      1
                      ,
                      2
                    
                  
                
                
                  
                    Σ
                    
                      1
                      ,
                      3
                    
                  
                
                
                  ...
                
                
                  
                    Σ
                    
                      1
                      ,
                      n
                    
                  
                
              
              
                
                  
                    Σ
                    
                      2
                      ,
                      1
                    
                  
                
                
                  
                    Σ
                    
                      2
                      ,
                      2
                    
                  
                
                
                  
                    Σ
                    
                      2
                      ,
                      3
                    
                  
                
                
                  ...
                
                
                  
                    Σ
                    
                      1
                      ,
                      n
                    
                  
                
              
              
                
                  
                    Σ
                    
                      3
                      ,
                      1
                    
                  
                
                
                  
                    Σ
                    
                      3
                      ,
                      2
                    
                  
                
                
                  
                    Σ
                    
                      3
                      ,
                      3
                    
                  
                
                
                  ...
                
                
                  
                    Σ
                    
                      3
                      ,
                      n
                    
                  
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  
                    Σ
                    
                      n
                      ,
                      1
                    
                  
                
                
                  
                    Σ
                    
                      n
                      ,
                      2
                    
                  
                
                
                  
                    Σ
                    
                      n
                      ,
                      3
                    
                  
                
                
                  ...
                
                
                  
                    Σ
                    
                      n
                      ,
                      n
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        7
        )
      
    
  




The second concept is that of correlation. Correlation is a dimension less than measure of linear dependence of random variables. Pearson's Correlation Coefficient r is the best estimate of correlation of normally distributed variables:




  
    
      
        
          
            ρ
            ik
          
          =
          
            
              cov
              ⁡
              
                (
                
                  
                    x
                    i
                  
                  ,
                  
                    x
                    k
                  
                
                )
              
            
            
              
                σ
                i
              
              ⁢
              
                σ
                k
              
            
          
        
        ,
      
    
    
      
        (
        8
        )
      
    
  




where σi and σk are the standard deviations of Xi and Xk. The correlation matrix is:





  
    
      
        
          ρ
          →
        
        =
        
          
            [
            
              
                
                  
                    ρ
                    
                      1
                      ,
                      1
                    
                  
                
                
                  
                    ρ
                    
                      1
                      ,
                      2
                    
                  
                
                
                  
                    ρ
                    
                      1
                      ,
                      3
                    
                  
                
                
                  ...
                
                
                  
                    ρ
                    
                      1
                      ,
                      n
                    
                  
                
              
              
                
                  
                    ρ
                    
                      2
                      ,
                      1
                    
                  
                
                
                  
                    ρ
                    
                      2
                      ,
                      2
                    
                  
                
                
                  
                    ρ
                    
                      2
                      ,
                      3
                    
                  
                
                
                  ...
                
                
                  
                    ρ
                    
                      1
                      ,
                      n
                    
                  
                
              
              
                
                  
                    ρ
                    
                      3
                      ,
                      1
                    
                  
                
                
                  
                    ρ
                    
                      3
                      ,
                      2
                    
                  
                
                
                  
                    ρ
                    
                      3
                      ,
                      3
                    
                  
                
                
                  ...
                
                
                  
                    ρ
                    
                      3
                      ,
                      n
                    
                  
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  
                    ρ
                    
                      n
                      ,
                      1
                    
                  
                
                
                  
                    ρ
                    
                      n
                      ,
                      2
                    
                  
                
                
                  
                    ρ
                    
                      n
                      ,
                      3
                    
                  
                
                
                  ...
                
                
                  
                    ρ
                    
                      n
                      ,
                      n
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        9
        )
      
    
  




Corresponding to the correlation matrix is a correlation/p-value matrix:





  
    
      
        
          ρ
          →
        
        =
        
          
            [
            
              
                
                  
                    
                  
                
                
                  
                    ρ
                    
                      1
                      ,
                      2
                    
                  
                
                
                  
                    ρ
                    
                      1
                      ,
                      3
                    
                  
                
                
                  ...
                
                
                  
                    ρ
                    
                      1
                      ,
                      n
                    
                  
                
              
              
                
                  
                    ρ
                    
                      2
                      ,
                      1
                    
                  
                
                
                  
                    
                  
                
                
                  
                    ρ
                    
                      2
                      ,
                      3
                    
                  
                
                
                  ...
                
                
                  
                    ρ
                    
                      1
                      ,
                      n
                    
                  
                
              
              
                
                  
                    ρ
                    
                      3
                      ,
                      1
                    
                  
                
                
                  
                    ρ
                    
                      3
                      ,
                      2
                    
                  
                
                
                  
                    
                  
                
                
                  ...
                
                
                  
                    ρ
                    
                      3
                      ,
                      n
                    
                  
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  
                    ρ
                    
                      n
                      ,
                      1
                    
                  
                
                
                  
                    ρ
                    
                      n
                      ,
                      2
                    
                  
                
                
                  
                    ρ
                    
                      n
                      ,
                      3
                    
                  
                
                
                  ...
                
                
                  
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        10
        )
      
    
  




In simple terms, a p-value is the probability of obtaining a finding as a result of chance alone.

Turning now to FIG. 9, shown in FIG. 9(a) is an exemplary averaged T2 distribution 911 derived over a depth interval from measurements made in a well. The plot shows the averaged T2 as a function of bin number. 913 is the Pearson correlation coefficient between the NMR bin logs and the gamma ray log, while 915 is the Pearson correlation coefficient between the NMR bin logs and the resistivity log. Denoted by 919 is the p-value of resistivity, while 917 is the p-value of the gamma ray. It should be emphasized that the values for 913, 915, 917 and 919 for any particular bin are obtained using the NMR T2 bin log for that particular bin. It can be seen that except for the largest bin numbers, the p-values are close to zero, meaning that there is a very low probability that the observed correlations are a result of chance.
A similar processing may be carried out with measurements made with a field gradient to give plots as a function of diffusivity. FIG. 10(c) shows the same T2 distribution 911 from FIG. 9(a). FIG. 10(a) is a 2-D plot in the T2-diffusivity plane and FIG. 10(b) is a plot of the Pearson correlation coefficient as a function of T2 distribution and diffusivity. The line 1001 shows the expected position for a water-wet rock while 1003 shows the expected trend for oil-saturated rock. The depth interval corresponding to this plot was a water saturated interval. A point to note is that examples given later using 1-D T2 distributions can also be done with 1-D Diffusivity distributions and 2-D distributions in the T2-diffusivity plane.
FIG. 11 shows a Pearson correlation 2-D plot of another example of field data in which there is considerable amount of oil present as indicated by the high values near the oil line 1003. 1201 in FIG. 12(a) shows the averaged NMR T2 distribution for this interval, 1203 shows the Pearson correlation coefficient between the gamma ray log and the averaged T2 bin log, 1205 shows the Pearson correlation coefficient between the resistivity log and the averaged T2 bin log. 1207 and 1209 show the p-values corresponding to 1203 and 1205 respectively. Shown in FIG. 12(b) is the averaged diffusivity 1251, the Pearson correlations of the diffusivity bin logs with gamma ray (1253) and resistivity (1255) and the corresponding p-values 1257, 1259.
FIG. 13 shows the averaged T2 distribution that has been color coded to show the correlation of the T2 bin log for a particular bin with the resistivity log. The large correlations in bins 8-10 are indicative of a significant amount of oil in the interval.
FIG. 14 shows the Pearson correlation as a function of T2 and diffusivity for a water wet interval. Note the high values near the water line 1001. 1501 in FIG. 15(a) shows the averaged NMR T2 distribution for this interval, 1403 shows the Pearson correlation coefficient between the gamma ray log and the averaged T2 bin log, 1505 shows the Pearson correlation coefficient between the resistivity log the T2 bin log. The p-values are not plotted as they are close to zero. Shown in FIG. 15(b) is the averaged diffusivity 1551, the Pearson correlations of the diffusivity bin logs with gamma ray (1553) and resistivity (1555).
Returning now to FIG. 11, in one embodiment of the invention, a 2-D filtering is applied to the data to attenuate portions of the data near the oil line 1003. After this filtering is done, projecting the filtered data onto the T2 axis will give an estimate of the T2 distribution of water in the formation. Alternatively, filtering may be done to attenuate data that is not near the oil line 1003, projecting the filtered data onto the T2 axis, and subtracting the projected distribution from the original T2 distribution to give an estimate of the water T2 distribution.
FIG. 16 shows a plot of a number of rock samples of the permeability in mD (ordinate) against the porosity (ordinate). Also indicated on the plot are curves of constant pore throat sizes: for 0.1 μm (1601), 0.5 μm (1603), 2.0 μm (1605), 10 μm (1607) and 40 μm (1609). Also shown in the side-panel of FIG. 16 are plots of the T2 distribution for the four groups of rocks of different grain sizes. In each of the plots in the side-panel, the dominant curve is for 100% water saturation while the light colored curve is for irreducible water saturation. The curves on the plot are the limits of the facies (families or clusters) found in the sample set. These facies are characterized by the pore throat size and have a characteristic T1 or T2 distributions. In case that the real T1 or T2 distributions are not known, one can take a characteristic T1 or T2 distribution for the facies (permeability vs. porosity plot) from a facies-NMR-look-up table.
For very well sorted grain sizes (not necessarily the case for the data of FIG. 16), the T2 distribution of petrofacies (with grain size as parameter) can be simulated as shown in the FIG. 17. In this case of simulation using the methods of Georgi is basically uni-modal, where the position of the T2 logarithmic mean value can be correlated with the permeability, following e.g. SDR perm-model. The T2 distribution for three different grain sizes for 100% water saturation is shown by 1701, 1703, 1705. The bin numbers correspond to T2 values ranging logarithmically from 0.25 ms to 16381.37 ms. Real rocks may include a combination of unimodal distributions, giving rise to a multimodal distribution.
In FIG. 18, the simulation of different oil types in absence of any field gradient (this means is equal to T2 intrinsic—no diffusion effect. When increasing the oil API gravity from around 12 (heavy oil) up to 40 API (light) the FIG. 18 shows that, in general, the lighter the oil and hence the smaller its viscosity, the T2 or T1 distribution shifts to the right on the x-axes (ms or # of Bin). The T1 or T2 distribution of oil can be obtained using Diffusivity-T2 maps (DT2 maps) as discussed above. Gas, which has very low viscosity, would lie at the right end of the scale.
As the location of the distribution along the T2 axes for 100% water saturated sample, not only corresponds to grain size and hence facies, but also to permeability (SDR equation KSDR=a′φm′ (T2LM)N′ with a: constant; m, n: exponents parameters, φ: porosity and T2LM: T2 log mean of the distribution), and the location of the distribution for oil samples along the same T2 axes corresponds to the oil API gravity—and much better to viscosity—, it is possible to generate a mobility (permeability over viscosity) map. Up to some geometrical variables, the mobility shows the proportionality between flow rate and differential pressure (Darcy's law):

Q=(k/μ)*Δp:  (11)

Where

Q is the flow rate.
K: permeability.
μ: viscosity.
Δp: differential pressure.
As, in general, the inverse of the oil viscosity is proportional to T2oil the mobility (k/μ) is therefore proportional to k*T2oil. Being k proportional to T2water, we obtain the mobility map by multiplying T2water*T2oil. The map is shown in the FIG. 19.
The abscissa of this mobility plot is the viscosity indicator derived from T1 or T2 as discussed above. The ordinate is T1 or T2 distribution of water in a porous medium. 1901, 1903, 1905 and 1907 refer to the distributions for microporosity, mesoporosity, macroporosity and megaporosity respectively. C.f. FIG. 16. 1911, 1913, 1915 and 1917 refer to extra-heavy oil, heavy oil, medium oil and light oil respectively.
In general, when for a given reservoir or oil bearing layer, the spot on the map falls on the right up corner the fluid mobility is high 1921 (the best case in terms of production); however, if the spot falls on the lower left corner, 1923 the opposite situation occurs. In situations in which a strict classification of rock quality (facies) and oil viscosity (or API gravity) in terms of T1 or T2 windows, does not hold in every situation, the maps has a universal validity as it represents the Darcy's Law, and can be use to show tendencies in a reservoir evaluation. When spots have a tendency to be above the diagonal with positive slope, it can be understood as that the rock quality overweights the fluid quality.
Those versed in the art and having benefit of the present disclosure would recognize that the NMR relaxation spectra of T1, T2 and of Diffusivity reflect the presence of water and hydrocarbon, as individual one-dimensional spectra (1D), as their combined two dimensional representations (e.g. 2D−(T1/T2 vs. T2), D vs. T2, Diff vs. T1 or in three dimensions as typically used in Diff vs. T2 vs Ti/T2). These representations, even if they appear to be continuous, are discrete and set up of bins or points in the corresponding dimensionality. As these bins reflect the fluids inside the formation, it is possible to retrieve information about the type of fluids and its volumetric (saturations of fluids in a reservoir of a given fluid-filled porosity).
Fluids are also known to affect some other well logs, e.g. resistivity and saturation logs. As an example: in a conventional reservoir, an increase of resistivity can be associated to the presence of hydrocarbon or fresh water, where a decrease of resistivity can be associated to the presence of saline water. Fluids may also be indirectly associated to geological characteristics of the formations as to the lithology e.g. Clay-Bound-Water (CBW) in shale-sand sequences. This CBW, detected by NMR, can be directly compared with the shale volume typically calculated using Gamma Ray, Neutron and Density or Formation Lithology/Chemical logs, that reflect the mineralogy of the formation. Another example is the NMR-derived porosity; it can be compared with the porosity obtained from other conventional logs as Density, Neutron and Acoustic based on different physical phenomena. In very specific cases, when compared with e.g. Density porosity or core-plug derived porosity; it is possible to associate a lower NMR-derived porosity to the presence of gas or very heavy-solid-like-oil.
The examples shown above are aimed to extracting more information from NMR data. In this context, it is very important to detect the degree of similarity between the NMR bin-logs and a given reference log, which is understood to be driven by a particular effect or fluid that is also expressed or represented in the NMR data. Treated as time series data, well logs can be similar but in not all scales or frequency contents. This means that as an example, the resistivity curve may be very similar to the hydrocarbon T2 component up to some constant value; this means that there is an offset most likely in the resistivity curve, a constant resistivity value (zero frequency component, lower scale of data representation) that is not found in the NMR bin, however beyond this the curves may be very similar. In this typical example, if we calculate the similarity of these two curves, we may get a low degree of similarity, because the curves are not similar in the zero frequency scale. On the other hand, at higher frequencies, a higher similarity may be found. For example, if the first derivative of the two curves is taken for comparison, the degree of similarity may be higher, when using a standard criterion as Pearson Correlation. To address this problem, in the present disclosure, analysis of the similarity of curves in many scales of representations is done.
This multi-scale analysis is done using Wavelet transforms and determining the semblance between the transformed logs. Once the degree of similarity is found it is possible to associate a given NMR component to a characteristic reservoir fluid (water-hydrocarbon-mud filtrated) or eventually to a geological characteristic (shale-sandstone, reservoir-nonreservoir).
The wavelet transform has certain desirable properties.

    
    
        A transform which localizes a function both in space and scaling and has some desirable properties compared to the Fourier transform. The transform is based on a wavelet matrix, which can be computed more quickly than the analogous Fourier matrix.
        Weisstein, Eric W. “Wavelet Transform.” From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/WaveletTransform.html
    
    


For the purposes of the present disclosure, the commonly used definition of semblance is adopted:

    
    
        A measure of multichannel coherence. If fij is the jth sample of the ith trace, then the semblance coefficient Sc is
    
    






  
    
      
        
          
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        where M channels are summed; the coefficient is evaluated for a window of width N centered at k. It is basically the energy of the stack normalized by the mean energy of the components of the stack. This is equivalent to the zero-lag value of the unnormalized autocorrelation of the sum trace divided by the mean of the zero-lag values of the autocorrelations of the component traces. Perfect agreement yields a value of unity.
    
    


Encyclopedic Dictionary of Applied Geophysics, Fourth Edition.
Going back to FIG. 12(a), it is noted that the Pearson coefficient is displayed as a function of the T2 bin for an averaged T2 distribution over a depth interval that has oil in it. The Pearson correlation is calculated with the gamma ray log and the resistivity log. Similarly, FIG. 12(b) shows the Pearson correlation of the average diffusivity over a depth interval with gamma ray and resistivity logs. It is convenient to think of the Pearson coefficient as a special case of coherence as explained next.
The Pearson coefficient is a measure of similarity between two curves. Examples of this are shown in FIG. 12(a) where the curve 1203 is the Pearson coefficient between a first curve 1201 that happens to be the averaged T2 distribution and a second curve (not shown) which is the gamma ray curve. The data are from an interval known to have oil. The objective is to generalize this method to the actual T2 values (instead of an averaged value) over a range of depths that may have a variety of hydrocarbon and water saturations. This is accomplished by taking the continuous wavelet transform (CWT) of the binned T2 distributions, such as those shown in FIG. 7, with the CWT of a selected log, such as the gamma ray log. The coherence (given by eqn. 12) then gives the similarity between a particular range of T2 with the selected log at each depth. To illustrate the concept, a simple example is given.
FIG. 20(a) shows a reference log. The abscissa is depth and the ordinate is the log value in arbitrary units. FIG. 20(b) shows the CWT of the reference log in FIG. 20(a). The abscissa for the CWT is depth while the ordinate is a “wavelength”. FIG. 20(c) shows a second log while FIG. 20(d) shows the CWT of the second log. The semblance according to eqn. (12) between the CWT in FIG. 20(b) and the CWT in FIG. 20(d) is shown in FIG. 20(e). The semblance is zero. This may have been expected by those versed in the art as the two logs in FIG. 20(a) and FIG. 20(c) are seen to be anti-correlated.
FIG. 21(a) shows another reference log. The abscissa is depth and the ordinate is the log value in arbitrary units. FIG. 21(b) shows the CWT of the another reference log in FIG. 21(a). FIG. 21(c) shows another second log while FIG. 21(d) shows the CWT of the another second log. The semblance according to eqn. (12) between the CWT in FIG. 21(b) and the CWT in FIG. 21(d) is shown in FIG. 20(e). As can be seen, there are certain combinations of depth interval and wavelength where the semblance is high and there are other combinations of depth intervals and wavelength where the semblance is low. Such displays make it possible to highlight depths (such as 100 and 150) where the second log has a high semblance to the reference log for particular values or ranges of values of the reference log. This is in contrast to the Pearson coefficient which gives a single value over a depth range for all values of the reference log.
The procedure discussed above is used in one embodiment of the disclosure with one modification. Instead of a second log, the binned NMR T2 distributions discussed above with reference to FIG. 7 are used. When this is done, the semblance highlights depth ranges where specific ranges of the T2 distribution have a high semblance to the reference log in specific depth intervals. An example of a representation of a T2 spectrum by 28 bins is shown in FIG. 22. The abscissa is the T2 value, also partitioned into 28 bins. The ordinate is the incremental porosity. The value in the middle of the bin is used to represent the bin. This bin-related information reflects partial porosity, because the whole area under the Spectrum is related to total porosity. The sum of the bin amplitude is equal to the total area under the spectrum curve. This is shown in track 5 in the log display of FIG. 23. The similarity between the gamma ray log at the depth indicated by 2301 with the binned NMR log for the 1024 ms bin indicated by 2303 is indicative of a sand with free fluid. It should be noted that similar displays can be prepared for T1 spectra and for diffusivity spectra.
As an alternative to displaying the binned NMR logs as in track 5 of FIG. 23 the spectra can be shown in terms of spectral components. An example of the spectral components is shown in FIG. 24 where the abscissa is the spectral component and the ordinate is the depth. The area below each of the curves represents the total porosity at a particular depth (vertical axis). The spectra can be obtained by fitting or inversion procedures. A method of fitting NMR distributions using split Gaussian curves has been described, for example, in Romero (2009) “Fitting NMR Spectra for retrieving fluid distributions”.
The semblance of these individual logs (bins or components) to a log characteristic of a particular property of the formation can then be analyzed using semblance of the logs to a identify a portion of at least one of the distributions characteristic of a property of a formation such as porosity, resistivity, water saturation, clay bound water, bound water irreducible, moveable water and/or permeability.
The method of the present disclosure is described above with reference to a wireline-conveyed NMR logging tool. The method may also be used on logging tools conveyed on coiled tubing in near horizontal boreholes. The method may also be used on NMR sensors conveyed on a drilling tubular, such as a drillstring or coiled tubing for Measurement-While-Drilling (MWD) applications. As is standard practice in well-logging, the results of the processing are recorded on a suitable medium. Implicit in the processing of the data is the use of a computer program implemented on a suitable machine readable medium that enables the processor to perform the control and processing. The machine readable medium may include ROMs, EPROMs, EAROMs, Flash Memories and Optical disks.
While the foregoing disclosure is directed to the specific embodiments of the disclosure, various modifications will be apparent to those skilled in the art. It is intended that all variations within the scope and spirit of the appended claims be embraced by the foregoing disclosure.

## Claims

1. A method of evaluating an earth formation, the method comprising:
conveying a logging tool into a borehole;
using the logging tool for obtaining nuclear magnetic resonance (NMR) signals at a plurality of depths in the borehole;
processing the NMR signals to obtain a distribution of a relaxation time at each of the plurality of depths, each of the distributions comprising at least one of (i) a plurality of bins, and (ii) a plurality of components;
determining a semblance of a first log indicative of a property of the formation with the obtained distribution; and
using the semblance to identify a portion of at least one of the distributions characteristic of the property of the formation.

2. The method of claim 1 wherein the distribution comprises a plurality of components, the method further comprising using at least one of: (i) a fitting procedure, and (ii) an inversion procedure to obtain the components.

3. The method of claim 1 wherein the NMR signals comprise spin-echo signals and the relaxation time comprises a transverse relaxation time T2.

4. The method of claim 1 wherein the first log is at least one of: (i) a gamma ray log, and (ii) a resistivity log.

5. The method of claim 1 wherein determining the semblance further comprises using a relation of the form:




  
    
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6. The method of claim 2 further comprising:
(i) obtaining additional spin echo signals at each of the plurality of depths while applying an external magnetic field gradient;
(ii) processing the spin echo signals and the additional spin echo signals to obtain a distribution of a diffusion coefficient (D) at each of the plurality of depths, each of the distribution of diffusion coefficients comprising at least one of: (i) a plurality of diffusion bins, and (ii) a plurality of components;
(iii) determining a semblance of a first log indicative of the property of the earth formation with each of the obtained distribution of diffusion coefficients; and
(iv) using the semblance to identify portion of at least one of the distributions of diffusion coefficients characteristic of the property of the formation.

7. The method of claim 6 further comprising preparing a mobility map and using the mobility map for establishing reservoir quality.

8. The method of claim 2 further comprising using the identified subset of the logs and a T2 distribution at least one of the plurality of depths to obtain a modified T2 distribution responsive primarily to water in the formation.

9. The method of claim 8 further comprising estimating from the modified distribution at least one of: (i) water saturation, (ii) clay bound water, (iii) bound water irreducible, (iv) moveable water, and (v) permeability.

10. The method of claim 1 further comprising conveying the logging tool into the borehole on a conveyance device selected from (i) a wireline, and (ii) a drilling tubular.

11. An apparatus configured to evaluate an earth formation, the apparatus comprising:
a logging tool configured to be conveyed into a borehole;
a sensor arrangement on the logging tool configured to obtain nuclear magnetic resonance (NMR) signals at a plurality of depths in the borehole;
at least one processor configured to:
(i) process the NMR signals to obtain a distribution of a relaxation time at each of the plurality of depths, each of the distributions comprising at least one of (I) a plurality of bins, and (II) a plurality of components;
(ii) determine a semblance of a first log indicative of a property of the formation with the obtained distribution; and
(iii) use the semblance to identify a portion of at least one of the distributions characteristic of the property of the formation.

12. The apparatus of claim 11 wherein the NMR signals comprise spin-echo signals and the relaxation time comprises a transverse relaxation time (T2).

13. The apparatus of claim 11 further comprising at least one of a gamma ray logging tool and a resistivity logging tool and wherein the first log is at least one of: (i) a gamma ray log, and (ii) a resistivity log.

14. The apparatus of claim 11 wherein the at least one processor is configured to determine the semblance by using a relation of the form:




  
    
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15. The apparatus of claim 12 wherein the logging tool is further configured to obtain additional spin echo signals at each of the plurality of depths while applying an external magnetic field gradient; and
wherein the processor is further configured to:
(i) process the spin echo signals and the additional spin echo signals to obtain a distribution of a diffusion coefficient at each of the plurality of depths, each of the distribution of diffusion coefficients comprising at least one of: (I) a plurality of diffusion bins, and (II) a plurality of components;
(ii) determine a semblance of a first log indicative of the property of the earth formation with each of the obtained distribution of diffusion coefficients; and
(iii) use the semblance to identify portion of at least one of the distributions of diffusion coefficients characteristic of the property of the formation.

16. The apparatus of claim 12 wherein the processor is further configured to use the identified subset of the logs and a T2 distribution at least one of the plurality of depths to obtain a modified T2 distribution responsive primarily to water in the formation.

17. The apparatus of claim 16 wherein the processor is further configured to estimate from the modified distribution at least one of: (i) water saturation, (ii) clay bound water, (iii) bound water irreducible, (iv) moveable water, and (v) permeability.

18. The apparatus of claim 11 further comprising a conveyance device configured to convey the logging tool into the borehole, the conveyance device selected from (i) a wireline, and (ii) a drilling tubular.

19. A non-transitory computer readable medium product having stored thereon instructions that when read by a processor cause the processor to execute a method, the method comprising:
processing NMR signals at a plurality of depths obtained by a logging tool conveyed in a borehole for:
estimating a distribution of a relaxation time at each of the plurality of depths, each of the distributions comprising at least one of (i) a plurality of bins, and (ii) a plurality of components;
determining a semblance of a first log indicative of a property of the formation with the obtained distribution; and
using the semblance to identify a portion of at least one of the distributions characteristic of the property of the formation.

20. The non-transitory computer readable medium product of claim 19 further comprising at least one of: (i) a ROM, (ii) an EPROM, (iii) an EAROM, (iv) a flash memory, and (v) an optical disk.