Relevance of computational rock physics

Jack Dvorkin1, Naum Derzhi2, Elizabeth Diaz2, and Qian Fang2

ABSTRACT

To validate the transport (fluid and electrical) and elasticproperties computed on CT scan pore-scale volumes ofnatural rock, we first contrast these values to physical labora-tory measurements. We find that computational and physicaldata obtained on the same rock material source often differfrom each other. This mismatch, however, does not precludethe validity of either of the data type — it only implies thatexpecting a direct match between the effective properties oftwo volumes of very different sizes taken from the same het-erogeneous material is generally incorrect. To address thissituation, instead of directly comparing data points gener-ated by different methods of measurement, we comparetrends formed by such data points. These trends include per-meability versus porosity; electrical formation factor versusporosity; and elastic moduli (elastic-wave velocity) versusporosity. In the physical laboratory, these trends are gener-ated by measuring a significant number of samples. In con-trast, in the computational laboratory, these trends are oftenhidden inside a very small digital sample and can be derivedby subsampling it. Hence, we base our validation paradigmon the assumption that if these computational trends matchrelevant physical trends and/or theoretical rock physicstransforms, the computational results are correct. We presentexamples of such validation for clastic and carbonate sam-ples, including drill cuttings.

INTRODUCTION

Computational rock physics can be defined as computer simula-tions of processes in natural pore space to obtain the effective per-meability, electrical conductivity, and elastic moduli, as well asother properties. It has become a rapidly (if not widely) evolvingfield with the advent of two crucial elements: robust fine-resolution

3D imaging of rock and computer hardware and software to simu-late processes in such images accurately and in real time. Thisdevelopment has evolved from “do it because we can” (e.g., Boslet al., 1998 and Keehm et al., 2001) to “do it because it’s needed”(e.g., Arns et al., 2002; Oren and Bakke, 2003; Knackstedt et al.,2003; Dvorkin et al., 2008; Dvorkin, 2009; Sharp et al., 2009; andTolke et al., 2010).In computational rock physics, we image rock at the pore level

followed by simulating a physical process in this pore space thatwill yield permeability, electrical conductivity, and the elastic prop-erties. This approach carries an inherent conflict between the reso-lution of an image and its field of view: The finer the former thesmaller the latter, which indicates that the imaged rock samples inwhich the details of the pore space are adequately represented canonly be a few mm in size and often much smaller. Compare this to afew-cm size of samples used in the physical laboratory.Yet, we often judge the validity and quality of computational data

on whether they match laboratory measurements on a sample hostto the microimage. This intuitive perception is incorrect. Becausenatural rock is heterogeneous at all scales, the properties of a smallsubsample used in imaging and computing do not have to match theproperties of its cm-sized host, no more so than the properties of alaboratory sample do not have to match those inferred for a 30.5-m(100-ft)-sized subsurface object from remote sensing. These proper-ties do not have to match.Indeed, we often observe the aforementioned mismatch between

the computational and physical data (Dvorkin and Nur, 2009). Thisobserved difference results from rock’s heterogeneity and does notpreclude the validity of either of the two measurement methods. Yetthe task of verifying computational rock physics results remains.To quality control (QC), these results and make them applicable

to field-scale problems, we invoke the paradigm of a trend formedby pairs of data points (e.g., between porosity and electrical resis-tivity or porosity and the elastic moduli). These trends can be com-puted from a microscopic piece of rock material, such as those usedin pore-scale CT-scan imaging, by selecting multiple subsamples ofthe original image and computing the effective physical propertiesof these subsamples. We propose that if such computationally

Manuscript received by the Editor 30 October 2010; revised manuscript received 2 March 2011; published online 8 November 2011.1Stanford University, Department of Geophysics, Stanford, California, USA; Ingrain, Inc., Houston, Texas, USA. E-mail: dvorkin@stanford.edu.2Ingrain, Inc., Houston, Texas, USA. E-mail: derzhi@ingrainrocks.com; diaz@ingrainrocks.com; fang@ingrainrocks.com.

© 2011 Society of Exploration Geophysicists. All rights reserved.

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derived trends comply with those produced in the physical labora-tory on similar rock material and/or with the trends predicted byrelevant theoretical models, the computational results are validand can be used in the field. Below, we provide such validationexamples in which we compare some computational results to ex-perimental data whereas others are compared to well-establishedtheoretical models.

WHAT IS THE QUESTION?

The question often asked is whether porosity, permeability, elec-trical resistivity, and elastic properties computed by simulating phy-sical processes in a virtual pore-scale rock sample are correct.One part of this question is concerned with the robustness and

repeatability of the elements of the computational rock physicstechnology: (a) 3D imaging; (b) image processing (segmentation)of the raw image to delineate the pores, the mineral matrix, as wellas separate minerals within the matrix and fluids inside the porespace; and (c) the suitability and stability of the computer simula-tion codes. These three pivotal elements of the technology (Dvorkinet al., 2009) can be (and have been) extensively verified by, respec-tively, (a) ridding the raw images of many artifacts related to theacquisition settings and tomographic reconstruction and verifyingthese images against appropriate scanning electron microscope(SEM) slices and optical microscope snapshots; (b) checking seg-mented images against thin sections as well as ensuring that theimaged mineralogy matches X-ray diffraction (XRD) results andthe imaged porosity matches that calculated from weight andvolume; and (c) applying the computer codes to idealized virtualsolids with pores for which analytical solutions are available.The other part of this question implies that the rock properties

computed on a 3D pore-scale image should match those measuredon a 2.5-cm (1-in.)-sized plug or in the well [approximately 30.5 cm(1 ft) scale]. Because in natural rock geometrical heterogeneityof the pore space persists at all scales, we argue that this part ofquestion is ill-posed: We should not expect the effective physical

properties of a rock fragment be the same as of its very smallfraction.Well data demonstrate that these properties can vary appreciably

between two points separated by only several feet. Figure 1 (top)shows log curves for the gamma-ray, porosity, and the compres-sional- and shear-wave versus in a thick wet sand. Present in thisrelatively homogeneous interval is a substantial porosity variationrange, between 0.1 and 0.2. The concurrent P- and S-wavewave velocity variations are from about 4.0 to 4.5 and 2.3 to2.7 km/s, respectively. Each velocity-porosity data point differsfrom the averaged velocity-porosity datum for which we use thearithmetic average for porosity and the Backus average for velocity(Figure 1, bottom).Heterogeneity also persists at the pore scale. In Figure 2 (middle),

we display a segmented CT scan of a mm-sized Berea sandstonesample whose porosity (the ratio of the number of the black voxelsto the total number of voxels) is 0.15. To explore how the propertiesvary with location and scale inside this sample, we divide it intoeight (23) subsamples. The porosity of each of the eight subsamplesis shown in Figure 3. It varies noticeably around the mean value:The minimum porosity is 0.11 and the maximum porosity is 0.18.This porosity heterogeneity is even more pronounced in an oil

sand sample with a large cracklike pore (Figure 2, bottom). Theporosity of the entire sample is 0.26. That of the eight subsamplesvaries between 0.21 and 0.36 (Figure 3).To explore how other properties vary versus position and scale,

we choose absolute permeability and simulate single-phase fluidflow in these two digital samples as well as in their subsamples.For this simulation, we use the Lattice-Boltzmann method(LBM) that computes slow viscous flow in a pore space as imaged,without any idealization or simplification of the apparent pore spacegeometry (e.g., Tolke et al., 2010).The computed permeability of the large Berea sample is 73 mD

(Figure 3). The minimum and maximum values for the eightsubsamples are 16 and 122 mD, respectively. For the oil sand, these

Figure 1. Top: Gamma ray, porosity, and theP- and S-wave velocity versus depth. Bottom:Velocity-porosity cross-plot for the interval high-lighted as black. The black squares rimmed bywhite are for the seismic-scale (Backus-averaged)velocity plotted versus the sand porosity arithme-tically averaged in this interval.

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three numbers are 480, 278, and 1432 mD, respectively. We candefinitely expect similar spatial nonstationarity of porosity and per-meability in a different scale range, such as between laboratorymeasurements (approximately 3 cm) and a reservoir simulation cell(approximately 3 cm).These examples indicate (as expected) that porosity and perme-

ability vary in space even within a fraction of mm and, hence, thesetwo properties lack spatial stationarity within a selected rock vol-ume. These properties may change at the mm scale and, certainly, atthe inch and foot scale. This means that data depend on the scaleand position of measurement. Hence, because a datum will varyalong a short distance, or if measured on differently sized volume,we argue that such variability renders invalid the concept of adata point.To this end, we reformulate the question “Do the physical

properties computed on microscopic pieces of rock match thoseobtained in the physical laboratory, in the well, and/or in thefield?” as “Assuming that the technical components of the digital

rock physics method are correct, how relevant are its results tofield applications?”There is no alternative to using the results of controlled experi-

ments (in the well, physical laboratory, and/or computational la-boratory) conducted at one scale for interpreting data acquired ata very different scale. Hence, perhaps the most important questionof digital rock physics is: “How to use the data computed at a sub-mm scale at the well and field scales?”

VALIDATION PRIMER

In light of heterogeneity of natural rock and because of the scaledifference between physical laboratory (25 to 50 mm) and digitallaboratory (0.1 to 3.0 mm) samples, the latter do not and, generally,should not match the former. Hence, a QC procedure should not befocused on direct number matching across these scales ofmeasurement.The laboratory data set used here to elucidate this point includes

over 50 outcrop calcite carbonate samples (Vanorio et al., 2008).

Figure 2. Top: Subsampling procedure (cartoon).Middle: The original 3D Berea cube (left). The re-solution is 8.6 micron per pixel. The size is 150 by150 by 150 pixels (1.3 by 1.3 by 1.3 mm). Eightsubcubes of the original cube (right). The size ofeach subcube is 75 by 75 by 75 pixels (0.645 by 0.645 by 0. 645 mm). The first subcube has the high-est porosity (0.189) while the third one has thelowest porosity (0.140). Bottom: Same but foran oil sand sample.

Figure 3. Left: Porosity (filled symbols) in theeight subsamples of each of the original digitalsamples shown in Figure 2. The horizontal lineis the porosity of the original samples. Gray sym-bols: Berea sandstone. Black symbols: Oil sand.Right: Same but for the decimal logarithm of com-puted permeability. In this graph, 1 stands for10 mD, 2 for 100 mD, and 3 for 1000 mD.

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The measured (ultrasonic pulse transmission) dry-rock elastic com-pressional and shear moduli plotted versus the measured porosityexhibit fairly tight trends (Figure 4, left). Three of these sampleswere selected for micro-CT scanning. The porosity of these digitalmm-size rock samples was computed by simply counting the pore-space voxels and the elastic moduli were computed by using thefinite-element method. The resulting values did not match thosemeasured in the lab on the inch-sized samples (Figure 4, right).Next, each of the three digital samples was numerically divided

into eight subcubes and the porosity and elastic constants were com-puted for each of them. The resulting data set of 24 (3 × 8) digitalsamples formed tight elastic moduli versus porosity trends that es-sentially duplicated the laboratory trends (Figure 4, right).Moreover, the upper and lower elastic (Hashin-Shtrikman)

bounds computed for the selected three lab samples are close toeach other and fall on the same trend if plotted versus the averagedporosity. These bounds can represent data for a scale much largerthan the 1-in. lab scale.Finally, a theoretical model appropriate for this rock type, the

modified upper Hashin-Shtrikman bound or the stiff sand model(Mavko et al., 2009), was used to calculate the elastic moduli versusporosity. The resulting curves are consistent with the laboratory anddigital moduli-porosity trends.This example indicates that:

a) Data pairs (e.g., porosity and velocity) obtained on an inch-sized sample and computed on a mm-sized subsample thereofoften do not match.

b) However, these pairs computed on a few microscopic samplesand their subsamples can form a trend that appears to persist in awide-scale range. This trend can be stationary versus space andscale and, hence, can be used in inferring one unknown property(e.g., porosity) from the other (e.g., velocity) that is measured inthe field.

c) Such a trend may be hidden inside a very small rock sample.Subsampling and subsequent calculations help expand the por-osity and elastic property ranges to form a meaningful trend.

d) Digital rock physics often fails to exactly match physical databut remarkably succeeds in producing meaningful transformsbetween (at least) two rock properties that can be utilized ata field scale.

Hence, our validation paradigm is not to match disparate datapoints but instead to match trends formed by these data points.

HARDWARE AND SOFTWARE

The idea of digital rock physics, “image and compute,” impliestwo principal steps: (a) digitally imaging sediment in 3D to resolveits pore-scale features and then (b) simulating physical processesinside this digital sample to determine its effective properties, suchas permeability (absolute and relative); electrical resistivity at dif-ferent stages of saturation; and the elastic moduli (including the an-isotropy constants if required). To implement this idea is far fromsimple: It calls for high-resolution 3D imaging machines supple-mented by image-reconstruction and processing methods as wellas by accurate process simulation computational engines. Ofcourse, to obtain the desired properties on large digital samplesand in real time, such algorithms have to be configured to runon the modern computer clusters.A computational experiment includes three main steps:

a) 3D imaging. A small rock sample is rotated inside a micro-CT-scan machine and illuminated with X-rays. Then its 3D geo-metry is reconstructed tomographically. The resolution of theimage is mostly determined by the acquisition geometry andnoise level. This resolution (the size of a voxel) can be as smallas tens of nanometers or as large as a millimeter. Because thefile size should not be uncontrollably large, the finer the resolu-tion, the smaller the field of view. Three-dimensional imagesappear in shades of gray (Figure 5). The brightness of a voxelis directly affected by the effective atomic number of thematerial (its density). Pyrite appears brighter than calcite, whichis brighter than quartz, which is brighter than porous clay. Thepore space is darkest where it is filled with air and is brighter

Figure 4. Left: Carbonate laboratory data for thecompressional (top) and shear (bottom) modulusversus porosity. The curves are from the theoreti-cal model explained in the text. Right: Data pairsfor three laboratory samples selected from thisdata set (large gray circles); computed propertiesof digital fragments of these three samples (largegray squares); and computed properties of eightsubsamples of each of these three digital samples(stars). Black bar is for the elastic Hashin-Shtrik-man bounds for the selected three lab data pairs.

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where liquid is present. To resolve microcrystalline calcite orshale structures, a nano-CT scanner has to be used. However,even its resolution is not fine enough to resolve clay structures.For this purpose, focused ion beam combined with scanningelectronic microscope (FIB/SEM) is engaged. The beam gradu-ally shaves off very thin layers of the sample, leaving essentiallyunaltered flat surfaces that are captures by SEM. A sequence ofsuch slices constitutes a 3D image at resolution as fine as 5 to10 nm (Figure 6).

b) Image processing and segmentation. Various artifacts that maybe present in the raw image, such as concentric shadows in CTscan and stripes in FIB/SEM images are removed. Followingthis, the gray scale is reduced to a small number of integers,first to separate the pore space from the mineral matrix(Figure 5) and then to delineate minerals within the matrixand fluids inside the pores. Simple brightness thresholding isoften not adequate for this purpose and, in this case, advancedimage processing, such as noise reduction and multibandthresholding (e.g., Russ, 2007) needs to be utilized. Becausethis procedure defines the pores where fluids flow and solidsresponsible for elastic stress distribution, proper segmentationis key to obtaining meaningful results from process simulators.

c) Property simulation. Physical processes are simulated in thesegmented pore space and mineral matrix. Specifically, absoluteand relative permeability come from simulating single- andmultiphase flow by using LBM (e.g., Tolke et al., 2010)whereas the electrical and elastic simulations use the finite-element method (Garboczi and Day, 1995). It is essential inall simulations that appropriate local physical constants are as-signed to each voxel (the mineral bulk and shear moduli, con-ductivity, viscosity, wettability angles, and interfacial tension).In flow simulations, these parameters must come from labora-tory tests or computed for relevant reservoir conditions fromtables and equations available. On the one hand, this require-ment appears restrictive. On the other hand, this flexibilitycan be considered advantageous since we can rapidly computeand address various scenarios relevant to the life span of a re-servoir, which is extremely difficult if not impossible in the phy-sical laboratory.In electrical flow simulations, assigning local conductivities arefairly straightforward for rock filled with conductive brine andmade of virtually nonconductive minerals, such as quartz or cal-cite. It becomes challenging but manageable where conductiveclay, pyrite, or a conductive microporous element (such as mi-crite) are present.

Most importantly, all computational experiments are staged onthe same shared object which can be stored for as long asneeded and revisited as new questions and demands appear.

TESTING COMPUTATIONAL ENGINES

The accuracy of these digital engines is validated by computingthe rock properties on analytically constructed idealized geometriesfor which theoretical solutions exist. For slow viscous flow, such asolution is available for cylindrical pipes of elliptical, triangular, andrectangular cross sections. We constructed several synthetic digital3D rock samples to compare exact analytical solutions with digitalresults in a wide permeability range. These results (Figure 7) vali-date the high accuracy of our LBM computational engine.The electrical current engine was tested on a number of simple

layered models as well as on packs of identical dielectric spheres(the so-called Finney pack) for which laboratory data exist. Ourcomputations accurately match analytical and experimental resultsfor the electrical formation factor which is defined as the ratio of theresistivity of the sample to that of the conductive phase in the porespace (Figure 8).The elastic finite-element method (FEM) engine is designed to

only account for very small deformation, on the order of 10−7 to10−6, such as induced by a seismic wave. This engine is basedon linear elasticity, meaning that it produces the so-called

Figure 5. 2D slices of a 3D CT-scan of a carbonate sample. Left:Gray-scale image. Right: The same image segmented. The pores areblack while the mineral matrix is white. The sample is about 0.8 mmacross with the voxel size 2 microns. Courtesy Ingrain.

Figure 6. 3D FIB-SEM image of shale sliced attwo progressing horizontal planes. The field ofview is about 2 microns while the voxel size is7 nanometers. Courtesy Ingrain.

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“dynamic” elastic moduli (K for bulk and G for shear) that are ob-tained from seismic P- and S-wave velocity (Vp and Vs, respec-tively) and bulk density (ρb) as K ¼ ρb½V2

p − ð4∕3ÞV2s � and

G ¼ ρbV2s . By no means the moduli thus obtained should be used

in rock mechanics applications where the so-called “static” or large-deformation moduli are required.In our tests, this FEM engine was used on synthetic elastic ob-

jects with elliptical and cylindrical inclusions of varying aspect ratio(the ratio of the short to the long axis). These computational results

were compared to the theoretical differential effective medium(DEM) solutions in a range of inclusion aspect ratio and concentra-tion. The larger the inclusion concentration (porosity) the smallerthe modulus, and the smaller the aspect ratio the smaller the mod-ulus. The digital and theoretical results match even for very thin(aspect ratio 1∕20 ¼ 0.05) inclusions, which confirms the accuracyof this FEM computational engine (Figure 9).

REAL ROCK TESTS

Before proceeding with real-rock validation examples, let us ar-gue that permeability is the most important product of digital rockphysics. The reason is that both the elastic and electrical properties,as well as the bulk density, can be measured in the well whereaspermeability cannot.Fortunately, permeability appears to be the most reliable product

of digital rock physics laboratory. The reason is that in most rocks,fluid flow is governed by relatively large pores (fluid flux is ap-proximately proportional to the pore size to the fourth power) easilyresolved in CT-scan images. Hence, even if very thin cracks or poresare not resolved in the image, the computed permeability is stillreasonably accurate. The electrical current nature in rock is morecomplex because it is approximately proportional to the pore sizesquare and, hence, is influenced by these thin cracks stronger thanpermeability. Also, conducting components in rock are, in additionto the pores filled with brine, microporous minerals, such as claysand micrite, as well as conductive minerals (pyrite). These elementshave extremely low or zero permeability but finite conductivity.The situation with the elastic properties is the opposite: Very thin

defects in the solid matrix (cracks) as well as uncemented, verycompliant, grain contacts often act to drastically soften the rockand, hence, reduce the elastic moduli. This means that to correctlycompute the elastic properties of rock, we need to accurately imageand resolve these fine features. As shown below, our imaging andimage processing techniques meet this challenge.

REAL ROCK: PERMEABILITY

Fontainebleau sandstone permeability versus porosity trend is aclassical test-bench for new data. Figure 10a displays this physicallaboratory data set. Added is a Kozeny-Carman equation curve thattheoretically links the absolute permeability k to porosity ϕ as

k ¼ d2

72τ2ðϕ − ϕpÞ3

ð1 − ϕþ ϕpÞ2; (1)

Figure 7. Flow simulation in square and rectangular pipes. Exam-ples of computed versus theoretical permeability. The axes are thedecimal logarithm of permeability (e.g., number 0 corresponds to1 mD and 2 to 100 mD). The black line is a diagonal.

Figure 8. Electrical current simulation in layered (gray circles)models and sphere packs (black squares). Examples of computedversus theoretical formation factor. The axes are the decimal loga-rithm of the formation factor. The black line is a diagonal.

Figure 9. Elasticity of pure-quartz solid withempty elliptical and cylindrical inclusions. Com-puted bulk (a) and shear (b) elastic moduli versusthose theoretically predicted by DEM. Squares:Aspect ratio 1.00. Gray circles: Aspect ratio0.10. Gray stars: Aspect ratio 0.05. The black lineis a diagonal.

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where d is the average grain size; τ is the tortuosity; and ϕp is thepercolation porosity at which the pore space becomes disconnected,and as a result, the permeability becomes zero (Mavko and Nur,1997). These theoretical curves are computed for a porosity rangebetween 0.02 and 0.30 to approximately encompass the data. Spe-cifically, here we used ϕp ¼ 0.02; τ ¼ 2.5; and d ¼ 0.20, 0.25,and 0.4 mm.Figure 10b displays the porosity and permeability computed on

CT-scan images of small fragments of the same rock material (voxelsize 4 microns). The same Kozeny-Carman curves are plotted forreference. The digital data fall on the same trend as the laboratorydata and also follow the theoretical trend.Laboratory permeability-porosity data in micritic carbonate

(Vanorio et al., 2008) are plotted in Figure 11. In the same figurewe display the permeability computed from small CT-scanned testsof the same samples (voxel size 2 microns). There is hardly apermeability-porosity trend in these data, especially at low porosity.The reason is the complex nature of these carbonates which contain

large calcite grains, large pores, as well as regions filled withmicroporous micrite (Figure 12). Yet, in spite of this complexity,the permeability computed on mm-sized rock tests is consistentwith that measured on inch-sized samples (Figure 11b).Permeability versus porosity computed for the two digital

rock samples displayed in Figure 2, Berea sandstone andoil sand, as well as their subsamples are shown in Figure 13a.These digital data form a trend that is supported by a Kozeny-Carman curve.Moreover, by further subdividing the original Berea cube into 27

subsamples and calculating their porosity and permeability, weobtain a set of data pairs falling approximately on the sametrend (Figure 13b). A Berea physical laboratory permeability-porosity point lies on the same trend.Permeability computed on the images of carbonate cuttings is

plotted versus their computed porosity in Figure 14. Naturally,no physical data accompanied these cutting. The digital rock phy-sics is, arguably, the only means of quantifying the properties of

Figure 10. Permeability versus porosity in Fontai-nebleau sandstone. (a) Physical laboratory data.Small circles: Bourbie and Zinszner (1985). Largesquares: Gomez (2009). (b) Digital rock physicsdata (stars). The curves in both plots are the sameand computed using the Kozeny-Carman equationas explained in the text.

Figure 11. (a) Permeability versus porosity inmicritic carbonate samples. Laboratory valuesare black while computed values are gray.(b) Computed versus measured permeability inthe same samples. The black line is a diagonal.

Figure 12. SEM images of micritic carbonatesamples with large calcite grains and microcrystalfilled domains (micrite) are discernable. CourtesyTiziana Vanorio.

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such material. The computed permeability-porosity trend is reasonably tight and is consis-tent with laboratory data from a different set ofcarbonate samples (Vanorio et al., 2008) as wellas with relevant Kozeny-Carman curves.Porosity and permeability computed on

high-porosity unconsolidated sands from anoffshore oil field are displayed in Figure 15.No laboratory data were available for compar-ison. Nevertheless, we are confident in the ac-curacy of these digital data: (a) the newlyobtained permeability-porosity trend continuesthat of Fontainebleau sandstone and links itto the sorted Ottawa sand data by C.-A. Estes,personal communication (1994); (b) these digi-tal data fall close to the North Sea Troll fielddata (S. Strandenes, personal communication,1992), in which the sediment has a very simi-lar character; and (c) these digital data are con-sistent with the Kozeny-Carman curves.

Notice a large permeability spread in these oil-bearing sands:Two digital samples with porosity 0.25–0.28 have permeabilityover 104 mD whereas another digital sample with porosity 0.30has permeability an order of magnitude smaller. The reason isrelatively large and well-sorted sand grains present in the firsttwo samples and poor grain-size sorting apparent in the thirdone (Figure 16).The next example is for porosity and permeability computed

from cuttings in medium-to-high porosity competent sandstone.These data pass our QC test because the data pairs fall on a trendformed by similar sandstone from the North Sea as well as by theFontainebleau sandstone (Figure 17).Figure 18 displays computed permeability versus porosity on two

sets of high-porosity oil sands. No laboratory data were available forthese samples: Conducting such physical tests on sands filled withbitumen is virtually impossible. By comparing these digital data tophysical data from lithologically and texturally similar samples, wedeem these computational results valid.Our final example demonstrates the power of digital rock physics

in understanding rock and, by so doing, addressing practical fielddevelopment issues. The problem faced by a producer was that awell with apparently significant porosity did not produce gas.The question was whether to abandon this well or keep trying tobring it on line. To address this question, we have imaged rock sam-ples from the interval in question (Figure 19). The high-resolutionFIB-SEM images show that the pore space is predominantly filledwith practically immovable organic matter and whatever gas is pre-sent in this sediment is trapped inside the organic material in such away that the gas-filled pores are disconnected from each other. As aresult, the permeability for gas is zero.

REAL ROCK: ELECTRICAL PROPERTIES

In this section, we present the electrical formation factor (FF)calculated on a number of digital rock samples filled withconductive fluid. By definition, FF is the ratio of the measured(or computed) resistivity of rock (Rt) fully saturated with conductivefluid to the resistivity of this fluid (Rw):

Figure 13. (a) Permeability versus porosity in Berea (light square) and oil sand (darksquare). Smaller round symbols of the same shade of gray are for the eight subsamplesof these two samples. (b) Berea sandstone with its eight and twenty seven (small blackcircles) subsamples. The open star symbol is from a physical measurement. The Fon-tainebleau trend (small light circles) is given for reference as well as a Kozeny-Carmancurve (black, same in both plots).

Figure 14. Permeability versus porosity in carbonate cuttings(black). The gray symbols are a laboratory carbonate data set(Vanorio et al., 2008). The curves are from the Kozeny-Carmanequation in a range of the average grain size.

Figure 15. Permeability versus porosity computed for high-porosity oil-bearing sand (black circles). The gray symbols are fromlaboratory Fontainebleau sandstone data set. Light squares are la-boratory data for sorted Ottawa sand. Light stars are laboratory datafor high-porosity unconsolidated sand from the North SeaTroll field. The curves are the same Kozeny-Carman curves asin Figure 10.

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FF ¼ Rt∕Rw: (2)

Our first example is for sintered glass beads.The digital rock was produced from the Finney(1970) pack which is an analytical representationof a physical pack of identical rigid spheres withthe coordinates of the centers of these spheresrecorded. This is what is called a dense randompack of spheres whose porosity is about 0.36.Porosity reduction was analytically simulatedby uniformly expanding the radii of the originalspheres. The results displayed in Figure 20 showthat our calculations match the experimentaldata for a similar artificial rock — sintered glassbeads (Sen et al., 1981) — in the medium-to-high porosity range and slightly overestimatethe physical data at low porosity. A reason forthis mismatch may be that disconnected poresstart to appear in the digital pore space createdby uniformly expanding the Finney pack parti-cles as the porosity decreases, unlike in the phys-ically sintered pack where the accumulation of the material occursmostly at the contacts.Figure 21 displays FF versus porosity computed on carbonate

cuttings, the same material that was used to compute permeabilitydisplayed in Figure 14. No laboratory measurements were availablefor this material. Our QC was, hence, based on a commonly usedArchie’s analytical expression

FF ¼ a∕ϕm; (3)

that links the formation factor to porosity ϕ. A modification of thisequation accounting for the percolation porosity ϕp (similar toequation 4 for permeability) is

FF ¼ a∕ðϕ − ϕpÞm: (4)

The cementation exponentm is between 1 and 2 in sandstone andcan be much larger in carbonates, especially with a variety of poreshapes (Figure 21b). To validate these digital data, we used a ¼ 1

and m ¼ 2.0; 2.5; and 3.0. The analytical curves thus producedbound the digital data with the m ¼ 2.5 curve falling betweenthe bounds.The results of computations conducted on Fontainebleau sand-

stone digital samples are compared to laboratory measurements(Gomez, 2009) in Figure 22. This plot contains the results for largedigital samples as well as eight subsamples of each of them. Thecomputed and laboratory formation factor versus porosity trendsmatch. These results are bounded by the two equation 4 curves withm ¼ 1.0, 1.5, and 2.0 and ϕp ¼ 0.025.The formation factor in micritic carbonate (the same samples that

were used in Figures 4 and 11) is plotted versus porosity inFigure 23. It shows a good match between the computed and mea-sured (Gomez, 2009) values. The cementation exponent values usedto produce the equation 4 curves to trace this trend are the same asused for the Fontainebleau data set (Figure 22). The reason it worksin carbonates is that these samples have a granular sandstonelikefabric.Figure 24a illustrates the power of subsampling for trend genera-

tion. Here we use just two mm-sized carbonate tests (no laboratory

Figure 17. Permeability versus porosity computed from cuttingsfrommedium-to-high-porosity competent sandstone (large open cir-cles). Small gray circles are from laboratory Fontainebleau sandstonedata set. Black circles symbols are laboratory data for high-porositycontact-cemented sand from the Oseberg field in the North Sea(S. Strandenes, personal communication, 1992). Light squares are la-boratory data for high-porosity unconsolidated sand from the NorthSea Troll field (S. Strandenes, personal communication, 1992).

Figure 18. Permeability versus porosity computed from two groupsof oil sand samples (large light and darker squares). Black circlesare laboratory data for high-porosity contact-cemented sand fromthe Oseberg field in the North Sea (S. Strandenes, personal com-munication, 1992). Gray circles are the Fontainebleau data.

Figure 16. Examples of 2D slices of 3D CT-scan images on high-permeability (a) andlower permeability (b) digital oil sand samples used in Figure 15. Images courtesy In-grain, Inc.

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data are available). Their segmented CT-scanimages were broken into eight subsamples each.The two original samples had porosity 0.22 and0.37. Subsampling helped extend and evenlycover this porosity range (0.12 to 0.49). Allthe subsampled digital data fall approximatelyon the same trend represented by equation 4 withthe percolation porosity 0.025 and cementationexponent 2.0.Subsampling the third carbonate sample (por-

osity approximately 0.13) left us with more ques-tions than answers (Figure 24b). The porosityrange covered by the eight subsamples extendedfrom 0.05 to almost 0.50. However, the samplingcoverage of this range appeared extremely unevenwith most of the subsamples clustered within the0.05 to 0.10 porosity range. In the modeling ac-cording to equation 4, in this case, we used thepercolation porosity 0.016— the value for the dis-connected porosity computed on the main digital

sample. The cementation exponent required to match the data byusing this equation varied from 1.5 for the high-porosity subsampleto 3.0 for the main sample to almost 5.0 for the low-porosity subsam-ple. In spite of the apparent diversity in these data, we feel that there isan important practical message delivered by these results: More di-gital experiments and concurrent physical laboratory measurementsare needed to fully comprehend this diversity in FF.Computations performed on fragments of oil sands from several

geographical areas are presented in Figure 25. No laboratory datawere available for these samples: Removing bitumen from this sandfor laboratory measurements destroys the original rock fabric: Com-putational rock physics is perhaps the only feasible option for mea-suring the properties of oil sands. The computed formation factorversus porosity trend is fairly tight and, as porosity decreases,moves from the equation 4 curve computed form ¼ 1.5 to that com-puted for 2.0.Our final example is for tight gas sandstones (Figure 26). No la-

boratory data were available for these samples. For reference, in thesame graph, we plot laboratory data for comparable rock samples.Some of the computed formation factor values appear to exceed theexpected range (light-gray symbols in Figure 26). Still, such valuesare plausible because they can be matched by an equation 4 curvewith the cementation exponent 2.5. Such high resistivity may be due

Figure 20. Formation factor in sphere packs. Gray circles are la-boratory data (Sen et al., 1981). Black squares are our calculationson the original and altered Finney pack.

Figure 19. Tight rock expected to produce gas. (a) An FIB slice showing mineral frame(gray) and pores (dark) this image is about 50 micron across. (b) Zoom on a pore. Theorganic matter (dark) is surrounded by mineral (gray) and occupies essentially the entirepore space. The small disconnected gas pockets are visible as black inclusions inside theorganic material. Gas is trapped inside and cannot flow to the well. Images courtesyIngrain, Inc.

Figure 21. (a) Formation factor versus porosity incarbonate cuttings (gray squares). The curves areproduced using equation 4 with the percolationporosity 0.025 and cementation exponent 2.0,2.5, and 3.0 (from bottom to top). (b) A 2D sliceof a 3D CT-scan image of a carbonate cutting usedin this example. This image is about 1 mm wide.

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Figure 22. Formation factor versus porosity in Fontainebleau sand-stone as computed (gray squares) and measured (black circles). Thecurves are produced using equation 4 with the percolation porosity0.025 and cementation exponent 1.0, 1.5, and 2.0 (from bottom totop).

Figure 25. Computed formation factor versus porosity in oil sand.Different shades and symbols corresponds to different geographicalareas. The curves are produced using equation 4 with the percola-tion porosity 0.025 and cementation exponent 1.0, 1.5, and 2.0(from bottom to top).

Figure 26. Computed formation factor versus porosity in tight gassandstone (light and dark gray squares). Black circles are from la-boratory measurements on a different set of tight gas sandstones.The curves are produced by equation 4 with the percolation porosity0.025 and cementation exponent 1.5, 2.0, and 2.5 (from bottomto top).

Figure 24. Formation factor versus porosity.(a) Two carbonate samples (large squares) andtheir subsamples (smaller circles of the corre-sponding shade). The black curve was producedusing equation 4 with the percolation porosity0.025 and cementation exponent 2.0. (b) Anothercarbonate sample (square) and its subsamples. Thecurves are according to equation 4 with the perco-lation porosity 0.016 and the cementation expo-nent increasing from 1.0 (bottom) to 5.0 (top)with increment 0.5.

Figure 23. Formation factor versus porosity in micritic carbonate ascomputed (gray squares) and measured (black circles). The curvesare produced using equation 4 with the percolation porosity 0.025and cementation exponent 1.0, 1.5, and 2.0 (from bottom to top).

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to the presence of disconnected pores in the mineral matrix (as wasconfirmed by the images of these samples). Yet, at this point wecannot accept or refute these values with certainty.

REAL ROCK: ELASTIC PROPERTIES

As mentioned earlier, computing the elastic properties is oftenmore challenging than computing permeability and electrical resis-tivity. The reason is that the compliant cracks and grain contacts thatare key to the effective elastic moduli are not easy to image. If thesesoft elements are absent in digital rock, the computed elastic modulibecome unrealistically high. Knackstedt et al. (2009) mitigate thisproblem by replacing the actual mineral material at grain contactsby a hypothetical softer material. We use a different approach that

makes the computational experiment predictiveinstead of making it match an a priori availableresult: Our special image processing (describedpreviously in the text) targets the compliantelements of the mineral frame and accuratelyresolves them to arrive at realistic elastic proper-ties. Results displayed in Figure 4 validate thisapproach and, moreover, show how to create adefendable elastic moduli versus porosity trendby using just three mm-sized digital samples. Be-low, we proceed with more such examples.Figure 27 displays the elastic moduli versus

porosity for the same set of carbonate cuttingsthat were used in Figures 14 and 21. This digitaldata trend is matched by laboratory measure-ments conducted on comparable carbonate sam-

ples. This trend is also matched by the stiff sand model (Mavkoet al., 2009) relevant for this fast well-consolidated rock.The Fontainebleau sandstone digital velocity data are compared

to the laboratory measurements in Figure 28. The digital trend isalso supported by the stiff sand model. A remarkable feature of thisresult is that the number of the digital data points by far exceeds thatfrom the laboratory. Once a digital image is obtained, the porosityrange can be dramatically widened by subsampling and the corre-sponding velocity-porosity trends produced. To further emphasizethis point, consider Figure 29 in which a digital trend matching alaboratory trend was produced just from a single mm-sized carbo-nate sample.Our final example is for the sediment whose elastic properties are

difficult to correctly compute: high-porosity unconsolidated sand.Producing such data requires accurate imaging of the verycompliant contacts between the grains. The results shown inFigure 30 confirm our ability to handle this challenge.In addition to the results shown here, computational rock physics

also allows us to simulate changes, such as diagenesis, or, in other

Figure 27. Velocity versus porosity computed on carbonate cut-tings (light squares). Relevant laboratory data are black circles.The stiff sand model is represented by black curves.

Figure 28. Velocity versus porosity computed on Fontainebleau digital samples andtheir subsamples (squares). Laboratory measurements are shown as gray circles. Curvesare from the stiff sand model.

Figure 29. Velocity versus porosity computed on a digital carbo-nate sample and its subsamples (black squares). Laboratory mea-surements on about a dozen samples from this formation (twodifferent locations) are shown as filled circles. Different shademeans a different location.

Figure 30. Velocity versus porosity computed ondigital unconsolidated sand samples (squares). La-boratory measurements on similar sand samplesare shown as black circles.

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words, move rock through geologic time and space by applying re-levant mathematical transforms to a digital image of a sample takentoday and at a known location (Figure 31).

CONCLUSION

The advent of computational rock physics has brought to light theoften ignored question: How applicable are controlled-experimentdata acquired at one scale to interpreting measurements obtained ata different scale? An answer is not to use a single data point or evena few data points but rather find a trend that links two or more rockproperties to each other in a selected formation type. We show thatsuch a trend, retrieved from a small sample of rock, may be applic-able in a range of scales.Computational rock physics is uniquely tooled for finding such

trends: Although it is virtually impossible to subsample a physicalsample and consistently conduct the same laboratory experimentson each of the subsamples, it is straightforward to accomplish thistask in the computer. Following this, computational and analyticaltechniques can be used to ensure the utility of such trends in a rangeof scales. To paraphrase the saying, we may be able “to see the rockin a grain of sand.”The paradigm of computational rock physics executed by using

modern imaging and computing techniques opens a new avenue toquantifying and understanding rock. It generates massive data andin real time on fragments that cannot be handled otherwise. The fullspan of applications of this paradigm is evolving. Three things arealready clear: (a) computational rock physics has become a power-ful technique of the present and will evolve as such into the future;(b) computationally produced data can pass validation tests if thecriteria are correctly formulated; and (c) it will immensely contri-bute to both applied technology and fundamental science.

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Figure 31. Original sample of sandstone (a) withpore space eroded to simulate cementation (b) andmineral matrix eroded to simulate early sedimen-tation (c). Porosity is shown as numbers below theimages.

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