Abstract:
The pore structure of rocks and other materials can be determined through microscopy and subjected to digital simulation to determine the properties of fluid flows through the material. To determine a porosity-permeability over an extended range even when working from a small model, some disclosed method embodiments obtain a three-dimensional pore/matrix model of a sample; measure a distribution of porosity-related parameter variation as a function of subvolume size; measure a connectivity-related parameter as a function of subvolume size; derive a reachable porosity range as a function of subvolume size based at least in part on the distribution of porosity-related parameter variation and the connectivity-related parameter; select a subvolume size offering a maximum reachable porosity range; find permeability values associated with the maximum reachable porosity range; and display said permeability values as a function of porosity.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims priority to Provisional U.S. Application Ser. No. 61/692,541, titled “Digital Rock Analysis Systems and Methods that Reliably Predict a Porosity-Permeability Trend” and filed Aug. 23, 2012 by Giuseppe De Prisco, which is incorporated herein by reference. 
     
    
     BACKGROUND 
       [0002]    Microscopy offers scientists and engineers a way to gain a better understanding of the materials with which they work. Under high magnification, it becomes evident that many materials (including rock and bone) have a porous microstructure that permits fluid flows. Such fluid flows are often of great interest, e.g., in subterranean hydrocarbon reservoirs. Accordingly, significant efforts have been expended to characterize materials in terms of their flow-related properties including porosity, permeability, and the relation between the two. 
         [0003]    Scientists typically characterize materials in the laboratory by applying selected fluids with a range of pressure differentials across the sample. Such tests often require weeks and are fraught with difficulties, including requirements for high temperatures, pressures, and fluid volumes, risks of leakage and equipment failure, and imprecise initial conditions. (Flow-related measurements are generally dependent not only on the applied fluids and pressures, but also on the history of the sample. The experiment should begin with the sample in a native state, but this state is difficult to achieve once the sample has been removed from its original environment.) 
         [0004]    Accordingly, industry has turned to digital rock analysis to characterize the flow-related properties of materials in a fast, safe, and repeatable fashion. A digital representation of the material&#39;s pore structure is obtained and used to characterize the flow-related properties of the material. However, the size of the digital representation often proves to be an important factor in that a model that is too small will fail to be representative of the physical material, and a model that is too large will consume a disproportionate amount of computational resources with little or no additional benefit. In some cases the size of the model is determined by equipment limitations and it is necessary to make the best of it. It would be desirable to optimize the size of the digital rock model and maximize the amount of information that can be derived from it. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0005]    Accordingly, there are disclosed herein digital rock analysis systems and methods that measure permeability over an extended porosity range. In the drawings: 
           [0006]      FIG. 1  shows an illustrative high resolution focused ion beam and scanning electron microscope. 
           [0007]      FIG. 2  shows an illustrative high performance computing network. 
           [0008]      FIG. 3A  shows an illustrative volumetric representation of a sample. 
           [0009]      FIG. 3B  shows an illustrative coordinate system for performing sample analysis. 
           [0010]      FIG. 4  shows an illustrative division of a model region into slices. 
           [0011]      FIG. 5A  shows an illustrative distribution of subvolume porosity. 
           [0012]      FIGS. 5B-5C  show illustrative distributions of porosity-related parameter variation. 
           [0013]      FIG. 6  illustrates a dependence of distribution moments on subvolume size. 
           [0014]      FIG. 7A  shows an illustrative porosity-permeability trend derived using a randomly-positioned sub-Darcian subvolume. 
           [0015]      FIG. 7B  shows the illustrative porosity-permeability trend derived by positioning the subvolume with conditional probability. 
           [0016]      FIG. 8  is a flowchart of an illustrative analysis method. 
       
    
    
       [0017]    It should be understood, however, that the specific embodiments given in the drawings and detailed description below do not limit the disclosure. On the contrary, they provide the foundation for one of ordinary skill to discern the alternative forms, equivalents, and other modifications that are encompassed in the scope of the appended claims. 
       DETAILED DESCRIPTION 
       [0018]    For context,  FIG. 1  provides an illustration of a high-resolution focused ion beam and scanning electron microscope  100  having an observation chamber  102  in which a sample of material is placed. A computer  104  is coupled to the observation chamber instrumentation to control the measurement process. Software on the computer  104  interacts with a user via a user interface having one or more input devices  106  (such as a keyboard, mouse, joystick, light pen, touchpad, or touchscreen) and one or more output devices  108  (such as a display or printer). 
         [0019]    For high resolution imaging, the observation chamber  102  is typically evacuated of air and other gases. A beam of electrons or ions can be rastered across the sample&#39;s surface to obtain a high resolution image. Moreover, the ion beam energy can be increased to mill away thin layers of the sample, thereby enabling sample images to be taken at multiple depths. When stacked, these images offer a three-dimensional image of the sample to be acquired. As an illustrative example of the possibilities, some systems enable such imaging of a 40×40×40 micrometer cube at a 10 nanometer resolution. 
         [0020]    The system described above is only one example of the technologies available for imaging a sample. Transmission electron microscopes (TEM) and three-dimensional tomographic x-ray transmission microscopes are two other technologies that can be employed to obtain a digital model of the sample. Regardless of how the images are acquired, the following disclosure applies so long as the resolution is sufficient to reveal the porosity structure of the sample. 
         [0021]    The source of the sample, such as in the instance of a rock formation sample, is not particularly limited. For rock formation samples, for example, the sample can be sidewall cores, whole cores, drill cuttings, outcrop quarrying samples, or other sample sources which can provide suitable samples for analysis using methods according to the present disclosure. 
         [0022]      FIG. 2  is an example of a larger system  200  within which the scanning microscope  100  can be employed. In the larger system  200 , a personal workstation  202  is coupled to the scanning microscope  100  by a local area network (LAN)  204 . The LAN  204  further enables intercommunication between the scanning microscope  100 , personal workstation  202 , one or more high performance computing platforms  206 , and one or more shared storage devices  208  (such as a RAID, NAS, SAN, or the like). The high performance computing platform  206  generally employs multiple processors  212  each coupled to a local memory  214 . An internal bus  216  provides high bandwidth communication between the multiple processors (via the local memories) and a network interface  220 . Parallel processing software resident in the memories  214  enables the multiple processors to cooperatively break down and execute the tasks to be performed in an expedited fashion, accessing the shared storage device  208  as needed to deliver results and/or to obtain the input data and intermediate results. 
         [0023]    Typically, a user would employ a personal workstation  202  (such as a desktop or laptop computer) to interact with the larger system  200 . Software in the memory of the personal workstation  202  causes its one or more processors to interact with the user via a user interface, enabling the user to, e.g., craft and execute software for processing the images acquired by the scanning microscope. For tasks having small computational demands, the software may be executed on the personal workstation  202 , whereas computationally demanding tasks may be preferentially run on the high performance computing platform  206 . 
         [0024]      FIG. 3A  is an illustrative image  302  that might be acquired by the scanning microscope  100 . This three-dimensional image is made up of three-dimensional volume elements (“voxels”) each having a value indicative of the composition of the sample at that point. 
         [0025]      FIG. 3B  provides a coordinate system for a data volume  402 , with the x-, y-, and z-axes intersecting at one corner of the volume. Within the data volume, a subvolume  404  is defined. The illustrated subvolume  404  is a cube having sides of length a, but other subvolume shapes may alternatively be used, e.g., a parallelogram having the same shape as the overall data volume, a sphere, or a tetrahedron. It is desirable, though not necessary, for the chosen subvolume shape to be scalable via a characteristic dimension such as diameter or an edge length. The subvolume  404  can be defined at any position  406  within the data volume  402  using a displacement vector  408  from the origin to a fixed point on the subvolume. Similarly, sub-subvolumes can be defined and positioned within each subvolume. For example,  FIG. 4  shows a subvolume divided into slices  502  perpendicular to the flow direction (in this case, the z-axis). 
         [0026]    One way to characterize the porosity structure of a sample is to determine an overall parameter value, e.g., porosity. The image is processed to categorize each voxel as representing a pore or a portion of the matrix, thereby obtaining a pore/matrix model in which each voxel is represented by a single bit indicating whether the model at that point is matrix material or pore space. The total porosity of the sample can then be determined with a straightforward counting procedure. Following the local porosity theory set forth by Hilfer, (“Transport and relaxation phenomena in porous media” Advances in Chemical Physics, XCII, pp 299-424, 1996, and Biswal, Manwarth and Hilfer “Three-dimensional local porosity analysis of porous media” Physica A, 255, pp 221-241, 1998), when given a subvolume size, the porosity of each possible subvolume in the model may be determined and shown in the form of a histogram (see, e.g.,  FIG. 5A ). Note that the distribution will vary based on subvolume size. While helpful, the distribution of  FIG. 5A  reveals only a limited amount of information about the heterogeneity of the model and does not account for directional anisotropy of the sample. 
         [0027]    One example of a more sophisticated measure is the standard deviation of porosity along a specific direction. As shown in  FIG. 4 , a volume (or subvolume) can be divided into slices perpendicular to the flow direction. The structure of the pores may cause the porosity to vary from slice to slice, from which a standard deviation of porosity along the flow direction can be determined. While this measure itself provides a useful indication of the pore structure, it can be extended. If the sample volume is divided into subvolumes (see, e.g.,  FIG. 3B ) and the standard deviation of porosity measured (relative to the average porosity of the whole sample and normalized by that same averaged porosity) for each subvolume, it yields a histogram such as that shown in  FIG. 5B . Again, this histogram is a function of the subvolume size. As the subvolume size grows from near zero to the size of a representative elementary volume (“REV”), the histogram converges and becomes nearly Gaussian in shape. (By way of comparison, when the subvolume dimension in a perfectly periodic “ideal” sample has a size that is an integer multiple of the REV size, the histogram is going to have zero mean and zero variance, in other words a Dirac delta function centered at zero.) 
         [0028]    The REV size depends on the statistical measure used to define it. The foregoing approach yields an REV suitable for Darcian analysis, and hence the REV size (e.g., diameter, length, or other dimension) is referred to herein as the “integral scale” or “Darcian scale”. Other length scales may also be important to the analysis. For example, the percolation scale, defined here as the subvolume size at which the average difference between total porosity and the connected porosity (porosity connected in some fashion to the inlet face) falls below a threshold, for example without limitation: 1%. This difference is also termed “disconnected porosity”, and depending on the specific context, may be limited to as little as 1% or as much as 10%, though the upper limit is preferably no more than 2%. Other threshold values may also be suitable, and it is believed that other definitions of percolation length would also be suitable. See, e.g., Hilfer, R. (2002), “Review on scale dependent characterization of the microstructure of porous media”, Transp. Porous Media, 46, 373-390, doi:10.1023/A:1015014302642. We note that the percolation scale can be larger than, or smaller than, the integral scale, so generally speaking the larger of the two should be used to define a truly representative elementary volume. 
         [0029]    Another measure of porosity structure is the standard deviation of surface-to-volume ratio. If the surface area (or in a two-dimensional image, the perimeter) of the pores in each slice  502  ( FIG. 4 ) is divided by the volume (or in 2D, the surface area) of the corresponding pores, the resulting ratio exhibits some variation from slice to slice, which can be measured in terms of the standard deviation. As the standard deviation of the surface-to-volume ratio is determined for each subvolume in a model, a histogram such as that in  FIG. 5B  results. As before, the histogram should converge and approximate a Gaussian distribution when the subvolume size reaches or exceeds the integral scale. 
         [0030]      FIG. 6  compares the moments of both histograms (standard deviation of porosity and standard deviation of surface-to-volume ratio (SVR)) for two different samples as a function of subvolume size. The first four moments (mean, standard deviation, skew, and kurtosis) are shown for subvolumes sizes as measured by edge length of the subvolume (which is a cube) in the range from 60 to 480 units. The first moment for both samples approaches zero, i.e., the center of the standard deviation of porosity and SVR distributions approaches a zero variation with respect to the porosity and SVR of the whole sample, meaning that on average the subvolumes have the same porosity and SVR as the whole sample when the subvolume size reaches about 200 units. The second moment for both samples becomes similarly close to zero at this length scale (200), i.e., the probability of a subvolume having the same standard deviation of porosity and SVR as the whole sample is quite high. The asymmetry of the distribution (as indicated by the skew value) and the kurtosis also become small at and above this threshold, suggesting that the REV size, to define an integral length scale according to Darcy analysis, is no larger than 200 units. As explained in U.S. Provisional Application 61/618,265 titled “An efficient method for selecting representative elementary volume in digital representations of porous media” and filed Mar. 30, 2012 by inventors Giuseppe De Prisco and Jonas Toelke (and continuing applications thereof), either or both of these measures can be employed to determine whether reduced-size portions of the original data volume adequately represent the whole for porosity- and permeability-related analyses. 
         [0031]    Various methods for determining permeability from a pore/matrix model are set forth in the literature including that of Papatzacos “Cellular Automation Model for Fluid Flow in Porous Media”, Complex Systems 3 (1989) 383-405. Any of these permeability measurement methods can be employed in the current process to determine a permeability value for a given subvolume. 
         [0032]    As mentioned in the background, the size of the model may be constrained by various factors including physical sample size, the microscope&#39;s field of view, or simply by what has been made available by another party. It may be that a model is subjected to the foregoing analysis and shown to have an integral scale and/or a percolation scale that is substantially same as the size of the model. In such a case, the size of the representative elementary volume should be set based on the larger of the integral scale or the percolation scale, and there are a sharply limited number of subvolumes in the model having this size. 
         [0033]      FIG. 7A  is a graph of permeability (on a logarithmic scale) as a function of porosity for a sample of Fountain Blue rock. The solid curve is derived from the published literature on this rock facies and is shown here solely for comparative purposes. It is desired to apply the disclosed methods to samples for which this curve may be unknown, so that the curve can be determined or at least estimated. Alternatively, the disclosed methods may be employed to verify that the sample actually exhibits the expected relationship. 
         [0034]    The size of the whole sample in this case is larger than the REV size. When taken as a whole, the model yields only one permeability vs. porosity data point which appears in  FIG. 7A  as a diamond. The diamond is on the curve, indicating that the measurement is accurate. Due to the limited amount of information this single point provides, it is desired to extend the range of data points insofar as it is possible with the given model, so as to predict the permeability values for different hypothetical members of this rock family having larger or smaller porosities. 
         [0035]    One solution is to employ REV-sized subvolumes and independent measure their porosity-permeability values. This approach is only feasible if the whole sample is much larger than the REV size. This is often not the case, so the measured porosity-permeability values are likely to be limited to a very small range close the measurement for the sample as a whole. 
         [0036]    Since the model exhibits a range of subvolume porosities (see, e.g.,  FIG. 5A ) that increases as the subvolume size shrinks, the user may be tempted to reduce the subvolume size below the REV size, making a greater number of subvolume positions available for exploring the poro-perm trend, each subvolume potentially yielding a different porosity-permeability measurement. The triangles in  FIG. 7A  show the resulting measurements from a randomly-chosen set of subvolume positions, with the subvolume having an edge length of about half that of the REV. Notably, these data points suggest a substantially incorrect relationship between porosity and permeability. Hence, a random-choice strategy appears destined to fail. 
         [0037]    Accordingly, we propose a strategy using reduced-size subvolumes with conditionally-selected positions rather than randomly chosen positions.  FIG. 7B  is a graph similar to  FIG. 7A , but this time with conditionally-chosen subvolume positions as provided further below. Note that the substantial majority of measurements are now aligned with the solid curve, indicating that it is possible to extract a porosity-permeability relationship over an extended porosity range even when using a subvolume size below the REV size. The porosity ranges  702 ,  704 , and  706  are referenced further below, but we note here that  702  indicates the porosity range that is reachable in this sample with a subvolume dimension about half that of the REV size, variation below a threshold of 40% of the mode of the standard deviation of porosity distribution, and disconnected porosity of no more than 1% of the subvolume&#39;s total porosity. Porosity range  704  indicates the region where the average disconnected porosity lies between 1% and 10%, and porosity range  706  indicates the region where the average disconnected porosity is on the order of 20%. 
         [0038]    Given the foregoing principles and practices, we turn now to a discussion of certain methods that enable determination of a porosity-permeability relationship over an extended porosity range when analyzing a small digital rock model.  FIG. 8  is an illustrative flowchart to support this discussion. 
         [0039]    The illustrative workflow begins in block  802 , where the system obtains one or more images of the sample, e.g., with a scanning microscope or tomographic x-ray transmission microscope. Of course the images can be alternatively supplied as data files on an information storage medium. In block  804 , the system processes the images to derive a pore/matrix model. Such processing can involve sophisticated filtering as set forth in existing literature to classify each image voxel as representing a pore or a portion of the matrix. 
         [0040]    In block  806 , the system determines a flow axis. This determination may be based on external factors (e.g., the orientation of the material sample relative to the well, formation pressure gradients, client specifications). When the axis is not based on external factors, it may be selected based on an analysis of the pore/matrix model (e.g., choosing the axis with the highest absolute permeability, or the axis having the lowest standard deviation of porosity). 
         [0041]    In block  808 , the system finds the distributions of standard deviation of porosity (e.g.,  FIG. 5B ) at different subvolume sizes, distributions of standard deviation of surface to volume ratio (e.g.,  FIG. 5C ), and/or other distributions of porosity-related parameter variation as a function of subvolume size. From these distributions, the system determines a (Darcian) integral scale. Note that for a useful porosity-permeability trend to be extractable, we believe that the dimensions of the original model should be at least four or five times the dimensions of the REV. 
         [0042]    In block  810 , the system finds a measure of pore connectivity at different subvolume sizes. The connectivity can be characterized in a variety of ways, including finding a percentage or normalized volume of the subvolume pore space connected to an inlet face of the pore/matrix model. In accordance with the Hilfer&#39;s Local Porosity theory, a histogram of the subvolume connectivities will exhibit a mean and a mode, either of which could be used as a connectivity measure. As an alternative, the percentage difference between the total subvolume pore space and the subvolume pore space connected to an inlet face (hereafter termed the disconnected pore percentage) can be used. From the relationship between the connectivity measure and subvolume size, the system determines a percolation scale, e.g., as one step above the largest subvolume size with a disconnected pore percentage below a threshold. 
         [0043]    In block  811 , the system determines whether the whole model size is substantially (four or five times) bigger than the larger of the percolation scale or the integral scale. If the model is substantially bigger than the larger of these scales, an adequate porosity range should be reachable using a subvolume size equal to the larger of these two scales. Hence the system proceeds with a systematic (potentially a pseudo-random), unconditional selection of subvolumes and in block  816  determines the porosity-permeability value associated with each subvolume. A plot of these data points, with an optional matching curve determined in block  817 , should reveal the porosity-permeability relationship for the sample. 
         [0044]    Otherwise, when the whole model size is approximately the same as the larger of the percolation or integral scales, it is desirable to derive this relationship using smaller (sub-REV) subvolumes. (This might not be possible if the larger of the two scales is the percolation scale.) In block  812 , the system determines a reachable porosity, either for a given subvolume size or for each of multiple subvolume sizes. As previously mentioned, for a given size, each subvolume position yields a different potential measurement on the porosity-permeability space. However, when operating in the sub-REV domain, not all subvolumes are used to extract the poro-perm trend. Rather, the system performs a conditional probability analysis, placing one or more conditions on which subvolumes can be employed in this analysis. One condition is that the standard deviation of porosity of the subvolume should not vary too much from the standard deviation of porosity of the full model. Accordingly, the system requires that the subvolume have a standard deviation of porosity below a given threshold (hereafter termed the “variation threshold”). In one embodiment, this threshold is set at the 20 th  percentile, meaning that the subvolume should have a standard deviation in the bottom fifth of the distribution (in  FIG. 5B , this would be below about 0.16). In another embodiment, the variation threshold is set at the 40 th  percentile (about 0.19 in  FIG. 5B ). Some system embodiments may employ a similar threshold on the standard deviation of SVR as an additional or alternative condition. 
         [0045]    Another condition that may be imposed by the system is that the subvolume have only a disconnected porosity that is below a given threshold (hereafter termed the “connectivity threshold”). In one embodiment, the system requires that the subvolume have no more than 1% disconnected porosity. In another embodiment, the system allows up to 2% disconnected porosity. In a preferred system embodiment, the subvolume must be below both the variation threshold and the connectivity threshold. The range of porosities possessed by subvolumes satisfying these conditions is the “reachable porosity”. In  FIG. 7B , range  702  is the reachable porosity for subvolumes having an edge size of about 50% that of the REV below the 40 th  percentile threshold with no more than 1% disconnected porosity. The system optionally adjusts the thresholds to determine the effect on the reachable porosity range and the data point scatter. For example, it may be the case that only a small sacrifice in range is made by reducing the connectivity threshold from 10% to 4%, but a significant reduction in scatter may be achievable by this reduction. Conversely, the system may determine that a variation threshold of the 20 th  percentile yields an insufficient range of reachable porosity and accordingly increases the variation threshold to the 40 th  percentile to achieve the desired range. The system may vary the thresholds to extract this information, and in block  814  select the threshold values that yield the best results (i.e., maximum reachable porosity that still yields with acceptable data scatter). 
         [0046]    In block  816 , the system determines the porosity-permeability measurements for the selected subvolumes satisfying the screening conditions. In block  817 , the system optionally fits a curve to the data points and/or provides a comparison curve derived from the literature. The resulting relationship can then be displayed to a user, e.g., in a form similar to  FIG. 7B . 
         [0047]    For explanatory purposes, the operations of the foregoing method have been described as occurring in an ordered, sequential manner, but it should be understood that at least some of the operations can occur in a different order, in parallel, and/or in an asynchronous manner. 
         [0048]    Numerous variations and modifications will become apparent to those skilled in the art once the above disclosure is fully appreciated. For example, the foregoing disclosure describes illustrative statistics for determining an REV size, but other suitable statistics exist and can be employed. It is intended that the following claims be interpreted to embrace all such variations and modifications.