# Patent

## Title
Method for determining permeability of rock formation using computer tomograpic images thereof

## Abstract

A method for estimating permeability properties of a porous material from a sample thereof includes making a three dimensional tomographic image of the sample of the material. The image is segmented into pixels each representing pore space or rock grains. The image is divided into sub-volumes. A porosity is estimated for each sub-volume. The components of a permeability tensor are determined by conducting simulations. Principal permeability values and directions are then calculated. The permeability properties are stored in a computer or displayed.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
Not applicable.
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
Not applicable.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates generally to the field of estimating material properties of porous media. More specifically, the invention relates to methods for estimating such properties using computer tomographic (CT) images of porous media such as subsurface rock formation.
2. Background Art
Estimating material properties such as effective elastic and shear moduli, electrical resistivity and fluid transport properties of porous media, for example, mobility of hydrocarbon in subsurface rock formations, has substantial economic significance. Methods known in the art for identifying the existence of subsurface hydrocarbon reservoirs, including seismic surveying and well log analysis, need to be supplemented with reliable methods for estimating how fluids disposed in the pore spaces of reservoir rock formations will flow over time in order to characterize the economic value of such reservoir rock formations.
One method known in the art for estimating fluid transport properties is described in U.S. Pat. No. 6,516,080 issued to Nur. The method described in the Nur patent includes preparing a “thin section” from a specimen of rock formation. The preparation typically includes filling the pore spaces with a dyed epoxy resin. A color micrograph of the section is digitized and converted to an n-ary index image, for example a binary index image. Statistical functions are derived from the two-dimensional image and such functions are used to generate three-dimensional representations of the rock formation. Boundaries can be unconditional or conditioned to the two-dimensional n-ary index image. Desired physical property values are estimated by performing numerical simulations on the three-dimensional representations. For example, permeability is estimated by using a Lattice-Boltzmann flow simulation. Typically, multiple, equiprobable three-dimensional representations are generated for each n-ary index image, and the multiple estimated physical property values are averaged to provide a result.
In performing the method described in the Nur patent, it is necessary to obtain samples of the rock formation and to prepare, as explained above, a section to digitize as a color image. Economic considerations make it desirable to obtain input to fluid transport analysis more quickly than can be obtained using prepared sections. Recently, devices for generating CT images of samples such as drill cuttings have become available. Such CT image generating devices (CT scanners) typically produce three-dimensional gray scale images of the samples analyzed in the scanner. Such gray scale images can be used, for example, essentially contemporaneously as drill cuttings are generated during the drilling of a wellbore through subsurface rock formations.
Using images of samples of rock formations, it is possible to obtain estimates of petrophysical parameters of the imaged rock sample, for example, porosity, permeability, shear and bulk moduli, and formation resistivity factor. The foregoing parameters are typically distributed within ranges in each rock formation, and there may be determinable relationships between such parameters such that determining one parameter value can enable determining one or more of the other parameters. One way to establish such relationship is to determine one or more rock physics transforms. A rock physics transform is a mathematical formula that relates one property of a rock formation to another. Such transforms can be based on an idealized mathematical model of rock, such as the differential effective medium that models rock as a solid with ideal-shape inclusions or the Hertz-Mindlin model that models rock as a composite made of perfect elastic spheres. Such transforms can also be based on a sufficient number of experimental data (e.g., well log measurements or laboratory measurements) using a statistically fit expression that approximates such data. An example of the latter is the Raymer transform between porosity φ and the compressional wave (P-wave) velocity of the rock (Vp). The transform is the expression Vp=(1−φ)2Vps+φVpf, where Vps is the P-wave velocity in the mineral (matrix or solid) phase of the rock (e.g., quartz) and Vpf is the P-wave velocity in the pore fluid (e.g., water). The elastic (compressional) wave velocity is directly related to the bulk K and shear G moduli by the expression Vp=√{square root over ((K+4G/3)/ρ)}, where ρ is the bulk density of the rock. The foregoing moduli can be obtained by laboratory measurement, and can also be obtained by calculations made from an image of a rock sample. Another example is the relationship between the absolute permeability k and the porosity φ of a rock formation called the Kozeny-Carman relation, represented by the expression k=d2φ3/[72τ2(1−φ)2], where d is the mean rock grain size and τ is the pore tortuosity (typically represented by a number between 1 and 5). Yet another example is Humble's relationship between the electrical resistivity formation factor F and the porosity φ, represented by the expression F=a/φm, where a and m are constants that are determined experimentally. As in the P-wave velocity example, the parameters that enter these two equations, one for permeability and the other for the formation factor, can be obtained by laboratory measurement and also by calculations based on an image of a rock sample. Instead of using the permeability (k) and formation factor (F) equation examples above, one may conduct a large number of laboratory tests on samples that represent the formation under examination. Alternatively, such data can be obtained by digital calculations on a digitally imaged rock sample.
Obtaining and calibrating permeability measurements using physical samples and using measurements made on actual rock samples require extensive laboratory and/or well measurements. There exists a need to use images such as the foregoing CT scan images to determine permeability characteristics without the need for extensive laboratory or well measurements.
SUMMARY OF THE INVENTION
A method according to one aspect of the invention for estimating permeability properties of a porous material from a sample thereof includes making a three dimensional tomographic image of the sample of the material. The image is segmented into pixels each representing pore space or rock grains to form a segmented image. The components of a permeability tensor are determined. The principal permeability values are calculated using the components of the permeability tensor. The permeability directions are calculated using the components of the permeability tensor. The relationship and the modeled petrophysical property for each sub-volume are stored in a computer or displayed.
Other aspects and advantages of the invention will be apparent from the following description and the appended claims.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 shows an example of obtaining cuttings during drilling of a wellbore and analysis thereof during the drilling.
FIG. 2 shows a flow chart of an example process for CT image segmentation.
FIG. 3 shows a flow chart of an example analysis procedure according to one embodiment of the invention.
FIG. 4 shows directional axes and the relationship between those axes.
FIG. 5 shows a graphical representation of a re-imaging technique of one example of the present invention.
FIG. 6 shows a simplified image of a re-imaging technique of one example of the present invention.



DETAILED DESCRIPTION
The description of this invention is made with reference to using drill cuttings obtained during the drilling of a wellbore through subsurface formations. It should be clearly understood that drill cuttings is only one example of samples of rock formation that may be used with the present invention. Any other source of a rock formation sample, e.g., whole cores, sidewall cores, outcrop quarrying, etc. may provide suitable samples for analysis using methods according to the invention. Consequently, the invention is not limited in scope to analysis of drill cuttings.
An example of drilling a wellbore to obtain samples of rock formations for evaluation by examples of a method according to the invention will be explained with reference to FIG. 1. A drilling unit or “rig” 10 is disposed at the Earth's surface. The rig 10 includes lifting equipment (not shown separately) for raising and lowering one of several types of device used to rotate a drill string 14. The device, shown at 18 in the present example may be a top drive, although the use of a tope drive is not a limit on the scope of the invention. The drill string 14 is assembled by threadedly coupling segments of drill pipe end to end. A drill bit 16 is disposed at the lower end of the drill string 14 and cuts through subsurface rock formations 11 to form a wellbore 12. During the drilling of the wellbore 12, the rig 10 is operated to cause some of the axial load (weight) of the drill string 14 to be applied to the drill bit 16. The top drive 18 rotates the drill string 14 and the drill bit 16 at the lower end thereof. The combination of axial load and rotation causes the drill bit 16 to cut through the formations 11.
The rig 10 includes a tank or pit 22 having drilling fluid (“mud”) 20 stored therein. A pump 24 lifts the mud 20 and discharges it through suitable flow lines 26 so that the mud 20 passes through an internal passage in the drill string 14, whereupon it is discharged through suitable orifices or courses in the drill bit 16. The discharged mud 20 cools and lubricates the drill bit 16 and lifts the cuttings generated by the bit 16 to the Earth's surface. The cuttings and mud thus lifted enter separation and cleaning devices, shown generally at 28 and including, for example, devices known as “degassers” and “shale shakers” to remove the cuttings and contamination from the mud 20. The mud after such cleaning is returned to the pit 22 for subsequent use in drilling the wellbore 12.
In the present example, the cuttings removed from the separation and cleaning device 28 may be transported to a computer tomographic (“CT”) scanner 30, which may use x-rays for analysis of internal structure of the cuttings, for generation of three dimensional (3D) images of the cuttings. The images so generated may be in numerical form and their content will be further explained below. After CT scanning, the cuttings may be saved for further analysis or may be suitably discarded. An example of a suitable CT scanner for making images usable with methods according to the invention is sold under model designation MicroXCT Series 3D tomographic x-ray transmission microscope by Xradia, Inc., 5052 Commercial Circle, Concord, Calif. 94520.
In some examples, an analysis of the cuttings from the CT scan images may provide, substantially in real time during the drilling of the wellbore, an estimate of certain properties of the subsurface formations being drilled, for example fluid mobility of one or more constituent fluids in the pore spaces of the rock formations 11. In the present example, images generated by the CT scanner 30 may be transferred to a computer 32 having program instructions for carrying out image analysis and subsequent formation property modeling as described below.
It should also be understood that drill cuttings are only one type of rock sample that may be analyzed according to the invention. In other examples, the drill bit 16 may be an annular type configured to drill whole cores of the rock formations 11. In other examples, percussion sidewall core samples may be obtained during drilling or when the drill string 14 is withdrawn from the wellbore 12 such as for “wireline” well evaluation techniques. Accordingly, the scope of the invention is not limited to analysis of drill cuttings. As explained above, the invention is also not limited to use with rock samples obtained from a wellbore drilled through subsurface rock formations.
CT scan imaging of a porous material sample (e.g., a sample of rock formation) is used in the invention to produce a numerical object that represents the material sample digitally stored in the computer 32 for subsequent numerical simulations of various physical processes, such as viscous fluid flow (for permeability estimation); stress loading (for the effective elastic moduli); electrical current flow (for resistivity); and pore size distribution for nuclear magnetic resonance relaxation time properties, including distribution of relaxation time. In some examples, such analysis can be performed while drilling operations are underway, substantially in real time.
The CT scan image produced by the CT scanner 30 may be a 3D numerical object consisting of a plurality of 2D sections of the imaged sample. Each 2D section consists of a grid of values each corresponding to a small region of space defined within the plane of the grid. Each such small region of space is referred to as a “pixel” and has assigned thereto a number representing the image darkness (or for example the density of the material) determined by the CT scan procedure. The value ascribed to each pixel of the 2D sections is typically an integer that may vary between zero and 255 where 0 is, e.g., pure white, and 255 is pure black. Such integer is typically referred to as a “gray scale” value. 0 to 255 is associated with eight digital bits in a digital word representing the gray scale value in each pixel. Other gray scale ranges may be associated with longer or shorter digital words in other implementations, and the range of 0 to 255 is not intended to limit the scope of the invention. For the purpose of simulating a physical process using such a numerical object (the gray scale), however, the numerical object is preferably processed so that all the pixels allocated to the void space in the rock formation (pore space) are represented by a common numerical value, e.g., by only 255s, and all the pixels associated with the rock matrix (or rock grains) are represented by a different numerical value, for example, zeroes. The foregoing process is called image segmentation. Subsequently, the resulting numerical object can be normalized so that the pore spaces are represented by, for example, ones and the rock grains are represented by zeroes. The foregoing may be described as converting the image into a binary index. In other examples, the image may be converted into an index having any selected number, n, of indices. It has been determined that sufficiently accurate modeling of rock properties may be obtained using a binary index, in which one value represents pore space and another single value represents rock grains.
A technique known in the art for segmenting a gray-scale object is called “thresholding”, where all pixels having a gray scale value below a selected threshold value (e.g., a gray scale value of 150 on a scale of 0 to 255) are identified as grains, while all other pixels are identified as pore space. The foregoing approach is often not satisfactory, however, because, due to numerical clutter in an unprocessed CT scan image, some pixels physically located inside a grain may have the gray level of the pore space and vice versa. In one embodiment of the invention, a type of image segmentation known as “region growing” can be used. Region growing may be described as follows. Consider a 2D section of a CT scan image made of a porous rock formation such as sandstone, which has primarily quartz rock grains. A substantial number of “seeds” (each seed consists of one or more pixels having a similar pixel gray scale level, e.g., 250±5) is placed within the image. All pixels within a seed are assigned the same gray scale level which may be an average (e.g., arithmetic) of the gray levels of all the pixels within the seed. The seeds in the image frame do not overlap spatially. Next, two or more adjacent seeds are merged and are identified as a “region” if the gray scale levels of the adjacent seeds have gray scale values within a selected difference threshold of each other. Each identified region is assigned a uniform (fixed) gray level, which can be a weighted average of the gray scale values of all the seeds that have been merged into the identified region. The foregoing process continues for all regions thus formed in the image frame. As a result, the unprocessed CT image is transformed into internally uniform regions plus unclassified pixels that were not be assigned to any of the identified regions (because such pixels included gray scale values outside the allocation threshold criteria). Each of such unclassified pixels can be assigned to an adjacent region with the closest gray scale level. If the resulting number of regions is greater than two, however, the foregoing method simply fails to allocate the CT image correctly into grains and pores.
To address the foregoing problem with extending (“growing”) seeds into regions, in the invention, instead of using seeds having different gray scale values, only two classes of seeds are used: all pixels having a gray scale value below a selected initial limit for the gray scale level of rock grains (e.g., 60) are classified as rock grains; and all pixels in which the gray scale level is larger than a selected initial limit for pore spaces (e.g., 130) are classified as pore space. One simple way of specifying these initial limits is by selecting the gray scale levels corresponding to the peaks of a gray level histogram. In many subsurface formations, such a histogram will be bimodal, wherein one mode value will correspond to the gray scale level of pores, and another mode value will correspond to the gray scale level of rock grains.
The next element in image classification according to the invention is to grow each of the two initially formed seeds by allocating to such seeds all adjacent pixels having gray scale levels within a selected tolerance, e.g., 130−5 for pore spaces and 60+5 for rock grains. The foregoing process can continue by incrementally increasing the gray scale lower limit for rock grains and incrementally reducing the gray scale upper limit for pore spaces until the two limits meet. The result is that all pixels will be allocated to either pore space or to rock grains, thus providing a fully segmented image.
A possible advantage of the foregoing procedure is that instead of forming multiple regions, the foregoing technique grows only two distinctive regions from start to end, thus avoiding the situation where multiple distinctive regions appear and then have to be reclassified into either pores or grains. If the resulting segmented image appears noisy (cluttered), it can be smoothed by any of conventional filters.
A schematic outline of the foregoing procedure follows. First is to preprocess the original image using the median or 2D Gaussian kernel filter. The size of the filter is provided by the user and should depend on, among other factors, the quality of the image (level of noise). It should be noted that the image segmenting procedure that follows has been demonstrated to be sufficiently noise resistant as to make the preprocessing frequently unnecessary.
Next, two user-selected thresholds, t1 and t2, are selected to determine initial regions for pore space and rock grains, respectively. The initial thresholds may be selected, for example, by analysis of a histogram of the gray scale values in the CT image. For every pixel pi having a gray scale level represented by B(pi):
if B(pi)>t1 then pi is identified as pore space; and
if B(pi)<t2 then pi is identified as rock grain.
If there are two or more contiguous pixels in any subset of the image frame that are classified according to the threshold procedure above, such contiguous pixels may be referred to as “clusters.” All of the pixels allocated as explained above then become the image seeds from which region growing proceeds.
Finally, for each pixel classified as a pore, its eight neighbors (spatially contiguous pixels) in the 2D image plane are interrogated. If any of the interrogated neighbor pixels is not already identified as pore or rock grain, and the gray scale level of such pixel is within a preselected tolerance level of (or initially selected different between) the gray scale level assigned to the “pore” seed (as in Step 2 above), the interrogated neighbor pixel is then classified as a pore and is allocated to the “pore” cluster.
The foregoing contiguous pixel interrogation is also performed for pixels classified as rock grain. Contiguous, previously unallocated pixels having gray scale level within a preselected tolerance of the gray scale level of the rock grain seed are allocated to the rock grain cluster.
The foregoing cluster allocation and region growing process continues for both pore space and rock grain until all the pixels in the 2D image frame are interrogated. If any of the pixels is not classified as pore space or rock grain, the foregoing tolerance value for each of the pore space and the rock grain may be increased by a selected increment (for example five gray scale numbers), and the contiguous pixel interrogation and classification may be repeated. The foregoing tolerance increase and repeated adjacent pixel interrogation may be repeated until all or substantially all the pixels in the 2D image frame are allocated to either rock grain or pore space.
The foregoing region growing procedure is then repeated for each 2D image frame in the 3D CT scan image. The result is a three dimensional characterization of the pore structure of the rock samples on which CT imaging has been performed.
An example implementation of the above process for image segmentation is shown in a flow chart in FIG. 2. At 40, a 2D image frame of a CT scan image is selected. The image frame may be subjected to histogram analysis, at 42 to determine possible mode values of gray scale for pore spaces and for rock grains. At 44, the possible modes of the histogram may be used to set initial values for the image segmentation thresholds t1 and t2. At 46, using the initial segmentation thresholds, all pixels in the image frame are interrogated may be are allocated to pore space or to rock grains, depending on whether the gray scale value in each pixel exceeds the respective segmentation threshold. The allocated pixels are then segmented into seeds where two or more contiguous pixels are allocated to either pore space or rock grain. At 48, pixels adjacent to the each of the seeds are interrogated. Previously unallocated pixels having a gray scale value falling within an initially selected threshold difference (or tolerance) of the adjacent cluster pixel gray scale value are allocated to the seed at 50. At 54, the image frame is interrogated to determine if all or substantially all the image frame pixels have been allocated to either pore space or rock grain. At 54, the number of allocated pixels is counted and at 60 if all or substantially all the pixels in the image frame have been allocated, a new 2D image frame can be selected, at 58, and the above process repeated. Typically the next 2D image frame will be adjacent to the most recently analyzed 2D image frame. The above process can be repeated until all available 2D image frames have been analyzed. If all pixels in the image frame have not been allocated, at 52, the tolerance or difference threshold values used at 50 may be increased and the interrogation of pixels adjacent to the existing seeds can be repeated, at 48, and the remainder of the process can be repeated.
The result of the foregoing procedure is a segmented 3D image of the rock sample including image elements for rock grain and for pore space. Such image can be stored or displayed in a computer and can be used as input to one or more rock property characterization models.
The foregoing implementation of image segmentation may prove advantageous when implementing analysis techniques according to the invention, which require a segmented image as input for further analysis. Such techniques may be explained as follows. In many cases, a relatively large range for each input parameter (e.g., porosity) can be obtained from a single sample of rock formation, for example drill cuttings (FIG. 1), sidewall cores, outcrop quarry or whole drill cores. Referring to FIG. 3, the sample of rock formation, obtained at 60, should be imaged, such as by CT-scanning, to obtain a high-resolution 3D image of the pore space and rock grains (matrix). The foregoing is shown at 62. Next, at 64, the image should be segmented to allocate image portions to the rock matrix and the pore space. Image segmentation may be performed as explained above with reference to FIG. 2. By counting the pixels in the segmented image that are allocated to the pore space and dividing their number by the total number of pixels in the image, an estimate of the porosity φ may be obtained, at 66.
Once porosity is obtained, it is possible to model permeability. At 68, a simulation of a physical experiment may be conducted on the sample image. In the present example, the Lattice-Boltzmann method can be used to numerically solve Navier-Stokes equations for flow simulation for permeability modeling. Such solution may be used to calculate permeability of simulated 3D volumes. The Lattice-Boltzmann method is a robust tool for flow simulation, particularly in media with complex pore geometry. See, for example, Ladd, Numerical Simulations of Particulate Suspensions via a discretized Boltzmann Equation, Part 1: Theoretical Foundation, J. Fluid Mech., v271, 1994, pp. 285-309; Gunstensen et al., “Lattice Boltzmann Model of Immiscible Fluids, Phys. Rev. A., v.43, no. 8, Apr. 15, 1991, pp. 4320-4327; Olsen et al., Two-fluid Flow in Sedimentary Rock: Simulation, Transport and Complexity, J. Fluid Mechanics, Vol. 341, 1997, pp. 343-370; and Gustensen et al., Lattice-Boltzmann Studies of Immiscible Two-Phase Flow Through Porous Media,” J. of Geophysical Research, V. 98, No. B4, Apr. 10, 1993, pp. 6431-6441).
The Lattice-Boltzmann method simulates fluid motion as collisions of imaginary particles, which are much larger than actual fluid molecules, but wherein such particles show almost the same behavior at a macroscopic scale. The algorithm used in the Lattice-Boltzmann method repeats collisions of these imaginary particles until steady state is reached, and provides a distribution of local mass flux. In accordance with the present example, the Lattice-Boltzmann method is applied successfully for many pore structures, including cylindrical tubes, random densely packed spheres, and 3D rock samples digitized by CT scanning as explained above. See, for example, U.S. Pat. No. 6,516,080 issued to Nur.
Because of the naturally layered nature of many rock formations, their permeability is anisotropic. This means that a particular pressure gradient will drive a smaller flux of fluid in the direction of smaller permeability, compared to the direction of larger permeability.
One method to quantify permeability anisotropy is through use of a permeability tensor k. In 70, the permeability tensor is determined. The permeability tensor k can be represented by:




  
    
      k
      
        _
        _
      
    
    =
    
      (
      
        
          
            
              k
              xx
            
          
          
            
              k
              xy
            
          
          
            
              k
              xz
            
          
        
        
          
            
              k
              yx
            
          
          
            
              k
              yy
            
          
          
            
              k
              yz
            
          
        
        
          
            
              k
              zx
            
          
          
            
              k
              zy
            
          
          
            
              k
              zz
            
          
        
      
      )
    
  
  ,




which is further described by the following three equations:





  
    
      V
      x
    
    =
    
      
        -
        
          1
          μ
        
      
      ⁢
      
        (
        
          
            
              k
              xx
            
            ⁢
            
              
                ∂
                P
              
              
                ∂
                x
              
            
          
          +
          
            
              k
              xy
            
            ⁢
            
              
                ∂
                P
              
              
                ∂
                y
              
            
          
          +
          
            
              k
              xz
            
            ⁢
            
              
                ∂
                P
              
              
                ∂
                z
              
            
          
        
        )
      
    
  
  ,
  
    

  
  ⁢
  
    
      V
      y
    
    =
    
      
        -
        
          1
          μ
        
      
      ⁢
      
        (
        
          
            
              k
              yx
            
            ⁢
            
              
                ∂
                P
              
              
                ∂
                x
              
            
          
          +
          
            
              k
              yy
            
            ⁢
            
              
                ∂
                P
              
              
                ∂
                y
              
            
          
          +
          
            
              k
              yz
            
            ⁢
            
              
                ∂
                P
              
              
                ∂
                z
              
            
          
        
        )
      
    
  
  ,
  
    

  
  ⁢
  
    
      V
      z
    
    =
    
      
        -
        
          1
          μ
        
      
      ⁢
      
        (
        
          
            
              k
              zx
            
            ⁢
            
              
                ∂
                P
              
              
                ∂
                x
              
            
          
          +
          
            
              k
              zy
            
            ⁢
            
              
                ∂
                P
              
              
                ∂
                y
              
            
          
          +
          
            
              k
              zz
            
            ⁢
            
              
                ∂
                P
              
              
                ∂
                z
              
            
          
        
        )
      
    
  
  ,




where the V terms represent the ratio of the volumetric flux Q to the corresponding cross-sectional area A:

Vx=Qx/Ax, Vy=Qy/Ay, Vz=Qz/Az, 

By measuring the flux components along the x, y, and z directions for a fixed orientation of pressure gradient (in one of these directions), it is possible to determine the components of the permeability tensor k.
One method of determining the permeability tensor k is to conduct three experiments with the following conditions
Experiment 1—∂P/∂x≠0, ∂P/∂y=∂P/∂z=0;
Experiments 2—∂P/∂x=0, ∂P/∂y≠0, ∂P/∂z=0; and
Experiment 3—∂P/∂x=∂P/∂y=0, ∂P/∂z=0.
In each of the foregoing experiments, V is measured in three directions:
Experiment 1—Vxa, Vya, and Vza;
Experiment 2—Vxb, Vyb and Vzb; and
Experiment 3—Vxc, Vyc, and Vzc.
The combination of the three experiments yields the components of the permeability tensor k from the following volumetric flux equations:



  
    
      
        V
        xa
      
      =
      
        
          -
          
            1
            μ
          
        
        ⁢
        
          k
          xx
        
        ⁢
        
          
            ∂
            P
          
          
            ∂
            x
          
        
      
    
    ,
    
      
        V
        ya
      
      =
      
        
          -
          
            1
            μ
          
        
        ⁢
        
          k
          yx
        
        ⁢
        
          
            ∂
            P
          
          
            ∂
            x
          
        
      
    
    ,
    
      
        
          V
          za
        
        =
        
          
            -
            
              1
              μ
            
          
          ⁢
          
            k
            zx
          
          ⁢
          
            
              ∂
              P
            
            
              ∂
              x
            
          
        
      
      ;
    
  




  
    
      
        V
        xb
      
      =
      
        
          -
          
            1
            μ
          
        
        ⁢
        
          k
          xy
        
        ⁢
        
          
            ∂
            P
          
          
            ∂
            y
          
        
      
    
    ,
    
      
        V
        yb
      
      =
      
        
          -
          
            1
            μ
          
        
        ⁢
        
          k
          yy
        
        ⁢
        
          
            ∂
            P
          
          
            ∂
            y
          
        
      
    
    ,
    
      
        
          V
          zb
        
        =
        
          
            -
            
              1
              μ
            
          
          ⁢
          
            k
            zy
          
          ⁢
          
            
              ∂
              P
            
            
              ∂
              y
            
          
        
      
      ;
    
  




  
    
      
        V
        xc
      
      =
      
        
          -
          
            1
            μ
          
        
        ⁢
        
          k
          xz
        
        ⁢
        
          
            ∂
            P
          
          
            ∂
            z
          
        
      
    
    ,
    
      
        V
        yc
      
      =
      
        
          -
          
            1
            μ
          
        
        ⁢
        
          k
          yz
        
        ⁢
        
          
            ∂
            P
          
          
            ∂
            z
          
        
      
    
    ,
    
      
        V
        zc
      
      =
      
        
          -
          
            1
            μ
          
        
        ⁢
        
          k
          zz
        
        ⁢
        
          
            
              ∂
              P
            
            
              ∂
              z
            
          
          .
        
      
    
  



V may be measured either physically through the use of a permeameter or may be simulated, e.g., numerically through the use of a flow simulation technique such as the LBM model described above.
The components of the permeability tensor k may be determined by digitally or numerically subdividing the original segmented image at different angles in the Cartesian coordinate system to obtain subsamples of different orientation and performing fluid transport simulations on those subsamples. As shown in FIG. 4, the direction of the pressure gradient can be represented by unit vector n 100, whose components are cos α 102, cos β 104, and cos γ 106. Directional axes are depicted in FIG. 4 as x-axis 108, y-axis 110 and z-axis 112. Permeability kn is measured along the same direction as the unit vector n 100 and is related to the permeability tensor k by the expression




  
    k
    n
  
  =
  
    
      (
      
        
          
            
              cos
              ⁢
              
                
              
              ⁢
              α
            
          
          
            
              cos
              ⁢
              
                
              
              ⁢
              β
            
          
          
            
              cos
              ⁢
              
                
              
              ⁢
              γ
            
          
        
      
      )
    
    ·
    
      (
      
        
          
            
              k
              xx
            
          
          
            
              k
              xy
            
          
          
            
              k
              xz
            
          
        
        
          
            
              k
              yx
            
          
          
            
              k
              yy
            
          
          
            
              k
              yz
            
          
        
        
          
            
              k
              zx
            
          
          
            
              k
              zy
            
          
          
            
              k
              zz
            
          
        
      
      )
    
    ·
    
      
        (
        
          
            
              
                cos
                ⁢
                
                  
                
                ⁢
                α
              
            
          
          
            
              
                cos
                ⁢
                
                  
                
                ⁢
                β
              
            
          
          
            
              
                cos
                ⁢
                
                  
                
                ⁢
                γ
              
            
          
        
        )
      
      .
    
  




Therefore,

kx=(kxx+kxy+kxz)cos2α+(kyx+kyy+kyz)cos2β+(kzxkzy+kzz)cos2γ.

It will be appreciated by those skilled in the art that if k is symmetrical, six independent experiments or simulations may be used to determine the six components of k. These simulations or experiments may be defined by the following equations:

α=0, β=γ=90°; kn1=kxx+kxy+kxz.

β=0, α=γ=90°; kn2=kxy+kyy+kyz.

γ=0, α=β=90°; kn3=kxz+kyz+kzz.

α=β=45°, γ=90°; kn4=(kxx+kyy+kxz+kyz)/2+kxy.

α=γ=45°, β=90°; kn5=(kxx+kzz+kxy+kyz)/2+kxz.

β=γ=45°, α=90°; kn6=(kyy+kzz+kxy+kxz)/2+kyz.

In one example of the present invention, six numerical permeability simulations may be conducted to determine the six permeability values kni (i=1 to 6). One non-limiting example of conducting such simulations is to render the original segmented sample image to create a new rectangular digital sample image out of the original sample image. For instance, as shown in FIGS. 5 and 6, the new rectangular digital sample image may be oriented at a desired angle, such as for instance, 45° with respect to the selected direction of the previous sample image.
Typically, in order to conduct a digital simulation on the new digital sample image, it may be required to re-index the original sample image. When re-indexing the image, the elements of the original digital sample image may be related to the elements to the new digital sample image as shown in FIG. 5. Such elements may be any elements necessary or useful to define the digital sample image. FIG. 5 depicts elements i 120, j 122 and k 124 of the original sample image, corresponding to elements in the x, y and z directions respectively. These may be related to elements I 130, J 132, and K 134 of the new digital sample image. For example, where the new rectangular sample is oriented at 45° to the original sample, I 130=i 120 and:




  {
  
    
      
        
          
            
              k
              =
              
                
                  (
                  
                    K
                    +
                    J
                  
                  )
                
                /
                
                  2
                
              
            
            ,
          
        
      
      
        
          
            
              j
              =
              
                
                  
                    (
                    
                      J
                      -
                      K
                    
                    )
                  
                  /
                  
                    2
                  
                
                +
                
                  n
                  /
                  2
                
              
            
            ,
          
        
      
    
     
  




in which n is the length of original image side.
An example of such re-indexing operation is shown in FIG. 6. As shown in FIG. 6, original sample image 150 rendered to new sample image 160. New sample image 160 is selected from within original sample 150 and is oriented at a 45° degree angle from the original sample. During re-imaging, the elements (I 130, J 132, and K 134) of the new digital sample may be determined. Based on these determinations, it is then possible to run the six digital experiments referenced above.
Once the sample has been re-imaged and the digital experiments run, the resulting kni(i=1 to 6) values may be used to resolve the linear system of the following equations

kxx+kxy+kxz=kn1,

kxy+kyy+kyz=kn2,

kxz+kyz+kzz=kn3,

(kxx+kyy+kxz+kyz)/2+kxy=kn4,

(kxx+kzz+kxy+kyz)/2+kxz=kn5,

(kyy+kzz+kxy+kxz)/2+kyz=kn6.

to find kxx, kyy, kzz, kxy, kxz and kyz.

One of ordinary skill in the art will recognize that if the permeability tensor is symmetrical, there are three principal permeability directions in a rectangular coordinate system attached to a porous material: the direction of maximum permeability; the direction of minimum permeability; and, normal to both of these directions, the direction of intermediate permeability. Principal permeability values correspond to these three principal directions.
The principal permeability values and directions may be obtained as the eigenvalues and directions of matrix k.
In 72, the eigenvalues, i.e., principal permeability values, are determined. One method of determining the principal permeability values is by calculating the three roots, k1, k2, and k3 of the cubic equation:




  
    
    
      
        
          
            
              k
              xx
            
            -
            k
          
        
        
          
            k
            xy
          
        
        
          
            k
            xz
          
        
      
      
        
          
            k
            yx
          
        
        
          
            
              k
              yy
            
            -
            k
          
        
        
          
            k
            yz
          
        
      
      
        
          
            k
            zx
          
        
        
          
            k
            zy
          
        
        
          
            
              k
              zz
            
            -
            k
          
        
      
    
    
  
  =
  0.




The roots k1, k2, and k3 correspond to the principal permeability values.
In addition to the values of the principal permeabilities, it is also possible to determine their direction. In 76, these directions are determined. One method of determining the eigenvector or directional vector corresponding to, for instance, the eigenvalue ki is the (xi yi zi) vector, which is the solution of matrix equation




  
    
      (
      
        
          
            
              
                k
                xx
              
              -
              
                k
                i
              
            
          
          
            
              k
              xy
            
          
          
            
              k
              xz
            
          
        
        
          
            
              k
              yx
            
          
          
            
              
                k
                yy
              
              -
              
                k
                i
              
            
          
          
            
              k
              yz
            
          
        
        
          
            
              k
              zx
            
          
          
            
              k
              zy
            
          
          
            
              
                k
                zz
              
              -
              
                k
                i
              
            
          
        
      
      )
    
    ·
    
      (
      
        
          
            
              x
              i
            
          
        
        
          
            
              y
              i
            
          
        
        
          
            
              z
              i
            
          
        
      
      )
    
  
  =
  0.




This may be repeated for the other two roots. Thus it is possible to determine all three principal permeability values and their corresponding directions.
The permeability values and permeability directions may be stored or displayed on a computer.
Methods according to the invention may enable determining fluid transport properties of anisotropic rock formations using tomographic images of samples of the rock formations without the need for expensive and time consuming laboratory experiments on physical samples of the formations.
While the invention has been described with respect to a limited number of embodiments, those skilled in the art, having benefit of this disclosure, will appreciate that other embodiments can be devised which do not depart from the scope of the invention as disclosed herein. Accordingly, the scope of the invention should be limited only by the attached claims.

## Claims

1. A method for estimating permeability properties of a porous material from a sample thereof, comprising:
making a three dimensional tomographic image of the sample of the material;
segmenting the image into pixels each representing pore space or rock grain to form a segmented image;
determining components of a permeability tensor through use of the segmented image;
calculating principal permeability values using the components of the permeability tensor;
calculating permeability directions using the components of the permeability tensor;
and at least one of storing and displaying the permeability values and permeability directions,
wherein the tomographic image comprises a value of gray scale allocated to each of a plurality of pixels in the image.

2. The method of claim 1 wherein the segmenting comprises: (a) determining an initial gray scale threshold for each of pore space and rock grain; (b) allocating each pixel in the image to pore space or rock grain for each pixel meeting threshold criteria for each of the pore space and rock grain thresholds, respectively;
(c) interrogating pixels adjacent to each seed;
(d) allocating the interrogated adjacent pixels previously not allocated to a seed to the pore space seed or the rock grain seed based on threshold criteria; and
(e) repeating (c) and (d) until substantially all pixels in the image are allocated to the rock grain or the pore space.

3. The method of claim 2 wherein the determining initial gray scale thresholds comprises histogram analysis of the tomographic image.

4. The method of claim 2 wherein the allocating interrogated adjacent pixels comprises determining a difference between a gray scale value and a gray scale value of the adjacent pixel in the seed, and allocating the interrogated pixel to the seed if the difference falls below a selected threshold.

5. The method of claim 4 further comprising:
determining whether unallocated pixels exist in the image;
increasing the selected difference threshold; and
repeating the interrogating adjacent and allocating the interrogate pixels having gray scale values wherein the difference is below the increased difference threshold.

6. The method of claim 1 wherein the porous material comprises a rock formation.

7. The method of claim 1 wherein the step of determining components of a permeability tensor further comprises conducting a numerical simulation.

8. The method of claim 7 wherein the step of determining components of a permeability tensor further comprises evaluating the results of the numerical simulation using a volumetric flux equation.

9. The method of claim 7 wherein the step of determining components of a permeability tensor further comprises
digitally subdividing the original segmented image at an angle to obtain a subsample of different orientation; and
performing simulations on the subsample.

10. The method of claim 9 wherein the step of determining components of a permeability tensor further comprises re-indexing the sample image.