1-s2.0-S0920410503000627-main

8/13/2019 1-s2.0-S0920410503000627-main

1/23

Reconstruction of Berea sandstone and pore-scalemodelling of wettability effects

Pal-Eric ren *, Stig BakkeStatoil Research Centre, N-7005 Trondheim, Norway

Received 29 July 2002; received in revised form 24 January 2003

Abstract

We present an integrated procedure for estimating permeability, conductivity, capillary pressure, and relative permeability of porous media. Although the method is general, we demonstrate its power and versatility on samples of Berea sandstone. Themethod utilizes petrographical information obtained from 2D thin sections to reconstruct 3D porous media. The permeabilityand conductivity are determined by solving numerically the local equations governing the transport. The reconstructedmicrostructure is transformed into a topologically equivalent network that is used directly as input to a network model.Computed two-phase and three-phase relative permeabilities for water wet conditions are in good agreement with experimentaldata. We present a method for characterizing wettability on the pore-scale from measured Amott wettability indices. Simulatedeffects of wettability on waterflood relative permeabilities and oil recovery compare favourably with experimental results.D 2003 Elsevier Science B.V. All rights reserved.

Keywords: Berea; Wettability; Relative permeability; Oil recovery

1. Introduction

The microstructure of a porous medium and the physical characteristics of the solid and the fluids that occupy the pore space determine several macroscopic properties of the medium. These properties include

petrophysical and transport properties of engineeringinterest, such as permeability, electrical conductivity,relaxation times, capillary pressure, and relative per-meabilities. Understanding the relation betweenmicrostructure and physical properties is therefore of great theoretical and practical interest in many fieldsof technology. Important industrial examples arise in

the production of oil and gas from petroleum bearingreservoirs and in pollutant transport and cleanup(Dullien, 1992) .

The prediction of physical properties of porousmedia from their microscopic origins involves threemajor steps:

(i) A quantitative characterization of the micro-structure.

(ii) Characterization of wettability and the relevant pore-scale physics.

(iii) Exact or approximate solutions of the equationsof motion that govern the transport phenomena of interest.

Until the 1990s, attempts to relate physical proper-ties of a rock to its microstructure were mainly

0920-4105/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0920-4105(03)00062-7

* Corresponding author. E-mail address: [email protected] (P.-E. ren).

www.elsevier.com/locate/jpetsciengJournal of Petroleum Science and Engineering 39 (2003) 177199

8/13/2019 1-s2.0-S0920410503000627-main

2/23

hampered by the difficulty of adequately describingthe complex nature of the pore space. Direct measure-ments of a 3D microstructure are now availab le via

synchrotron X-ray computed microtomography (Dun-smoir et al., 1991; Coles et al., 1994, 1996; Spanneet al., 1994; Hazlett, 1995; Coker et al., 1996) . In practice, however, information about the microstruc-ture of porous materials is often limited to 2D thinsection images.

A number of statistical models have been proposedfor reconstructi ng 3D porous media from 2D thinsection images (Joshi, 1974; Quiblier, 1984; Adler et al., 1990, 1992; Adler, 1992; Roberts, 1997; Hazlett,1997; Yeong and Torquato, 1998a,b; Manswart andHilfer, 1998) . In short, it consists of measuringstatistical properties, such as porosity, correlationand lineal path functions on 2D thin section imagesof the sample. Random 3D models are then generatedin such a manner that they match the measuredstatistical properties. Recent quantitative comparisonsof these models with tomographic images of sedi-mentary rocks have shown that statistical reconstruc-tions may differ significantly fr om the original samplein their geometric connectivity (Hazlett, 1997; Biswalet al., 1999; Manswart et al., 2000; ren and Bakke,2002) .

In contrast to statistical models, process-based models try to account for the fact that the porestructure is often the re sult of physical processes.Bakke and ren (1997) developed a process-basedreconstruction procedure which incorporates grainsize distribution and other petrographical dataobtained from 2D thin sections to reconstruct 3Dsandstones. ren and Bakke (2002) applied the pro-cedure to reconstruct Fontainebleau sandstone. Quan-titative comparisons with microtomographic imagesshowed that the reconstructed model adequately

reproduced important geometrical and connectivity properties of the actual sandstone.With the advance in computational fluid dynamics,

it is now possible to compute single-phase propertiessuch as permeability and conductivity directly onquite large 3D microstructure images (Adler et al.,1990, 1992; Schwartz et al., 1994; Ferreol and Roth-man, 1995; Widjajakusuma et al., 1999; ren andBakke, 2002) . This is generally not the case for multiphase flow. Although it is possible to solve themultiphase transport equations in pore spaces of

arbitrary geometry (Rothman, 1990; Gunstensen andRothman, 1993; Hazlett et al., 1998; Adler andThovert, 1998) , the solutions are computationally

very expensive and in practice limited to relativelysmall 3D systems. A commonly applied simplifica-tion is to represent the microstructure by an inter-connected network in which larger pores (pore bodies) are connected by sma ller pores (throats).Since the pioneering work of Fatt (1956) , network models have been used extensively to study different displacement processes in simple or idealized porousmedia.

The power and usefulness of network models arein their ability to relate macroscopic behaviour directly to the underlying pore-scale physics. Untilthe 1990s, most network model studies focused ontwo-phase flow phenomena. In recent years, our physical understanding of three-phas e flow at the pore level has increased significantly (Kantzas et al.,1988; ren and Pinczewski, 1992, 1995; Blunt et al.,1995; Keller et al., 1997) . This has led to d evelopment of network models for three-phase flow (ren et al.,1994; Pereira et al., 1996; Fenwick and Blunt,1998a,b; Mani and Mohanty, 1998; Lerdahl et al.,2000). It is crucial, however, that network modelsare given proper pore structure information as input

if they are to provide realistic predictions of transport properties.

Despite strong evid ence that spatial correlationsexist at the pore scale (Wardlaw et al., 1987; Bryant et al., 1993a,b; Knackstedt et al., 1998) , most net-work models assume that the pore structure is ran-dom. There have been few attempts to construct porenetworks that replicate the true microstructure of themedium. With the availability of tomographic or reconstructed 3D images, it is now possible to extract the exact network representation of the pore space and

use it directly in network models. Sok et al. (2002)generated network replica of four tomographic imagesof Fontainebleau sandstone. Simulated two-phaseflow properties were compared with those computedon equivalent stochastic networks. In all cases, thesimulations on the stochastic networks provided a poor representation of the results from the direct network replica. ren et al. (1998) reconstructedBentheimer sandstone and extracted network replicaof the microstructure. Predicted drainage and imbibi-tion relative permeabilities for water wet conditions

P.-E. ren, S. Bakke / Journal of Petroleum Science and Engineering 39 (2003) 177199178

8/13/2019 1-s2.0-S0920410503000627-main

3/23

were found to be in good agreement with experimen-tal data.

The wettability characteristics of a reservoir rock

plays a significant role in determining transport prop-erties such as relative permeability, capillary pressure,and oil recovery. Predictive modelling of reservoir rocks thus requires an accurate characterization of wettability. Unfortunately, there is little experimentalevidence of wettability and the appropriate fluidconfigura tions on the pore s cale in nonwater wet samples. Kovscek et al. (1993) proposed a theoretical pore level model for wettability alteration and fluidconfigurations that appears to capture much of theimportant physics of nonwater wet displacements. Themodel has been used to explore the effects of wett-ability on relative permeability and oil recovery (Dixit et al., 1999, 2000; Blu nt, 1998; ren et al., 1998) .Dixit et al. (1998, 1999) introduced the regime theorythat explained experimental trends in oil recovery interms of wettability characterized by contact anglesfor the oil wet regions and the fraction of pores that become oil wet. Unfortunately, there is no a priori wayof determining these parameters for reservoir rocks.The hope is that macroscopic measurements, such asAmott wettability indices, will be sufficient to esti-mate them.

In the present work, we reconstruct 3D Bereasandstone. To assess the quality of the reconstructionwe compare quantitatively the geometry and theconnectivity of the simulated microstructure withtwo other models of Berea sandstone. One of themodels is a microtomographic image of the actualsandstone and the other one is a statistical reconstruc-tion of the same rock. The permeability and formationfactors of the models are derived by solving numeri-cally the local equations governing the transport.Topological and geometrical analyses are then used

to build network replica of the process-based recon-structed microstructure. Predicted two-phase andthree-phase relative permeabilities for water wet con-ditions are compared with the experimental results of Oak (1990) . Finally, we describe a physically basedscenario for wettability alteration on the pore leveland show how we can utilize measured Amott wett-ability indices to characterize wettability on the pore scale. We investigate how wettability affectswaterflood relative permeabilities and oil recoveries.Computed oil recoveries for a range of wettability

conditions are compared with the exp erimental resultsof Jadhunandan and Morrow (1995) .

2. Reconstruction of Berea sandstone

2.1. Experimental sample

Berea sandstone has been used by the petroleumindustry for many years as a standard material in coreanalysis research and in laboratory core floodingexperiments. This is because the rock is relativelyhomogenous, well characterized and, until recently,readily available. It is made up of well-sorted andwell-rounded predominately quartz grains, but it alsocontai ns minor amounts of feldspar, dolomite, andclays (Churcher et al., 1991) .

The particular Berea sample that we study has a porosity of 23% and an absolute air permeability of 1100 mD (Hazlett, 1995) . High resolution X-raycomputer tomography was performed on a cylindricalmicroplug that was drilled from the larger originalcore. Details about the experimental apparat us and theimage acquisition procedure are given by Jasti et al.(1993) . Each voxel of this 3D image has a linear dimension of 10 Am and takes on the value 0 (solid

and clays) or 1 (pore). The size of the reconstructedimage is 128 3 voxels and the porosity / p = 0.1778which is to be compared with the experimental valueof / = 0.23. The discrepancy in porosity is mainlyattributed to the presence of clay microporosity belowthe 10-Am resolution of the thresholded image.

2.2. Process-based model

2.2.1. Thin section analysisThe input parameters for the process-based recon-

struction algorithm are obtained from back-scatteredelectron (BSE) images of standard 2D petrographicthin sections. The BSE images are digitized and re- presented by an array of grey-scale values (0255)according to the differences in atomic mass density. Byappropriately thresholding the images, we can distin-guish between different phases such as pore, quartz,clays, and feldspar. Proper thresholding is obtained byvisual comparison of the original and thresholdedimages. Fig. 1 shows a BSE image of a thin sectionof Berea sandstone.

P.-E. ren, S. Bakke / Journal of Petroleum Science and Engineering 39 (2003) 177199 179

8/13/2019 1-s2.0-S0920410503000627-main

4/23

To describe each phase i in the image, we introducethe binary phase function

Z ix 1 if x a phase i

0 otherwise8

8/13/2019 1-s2.0-S0920410503000627-main

5/23

grain centre towards the centre xy-plane of the modelaccording to the formula

z 0:5k z z max z min z o1 k z e z 2

where z is the new vertical position, z o is the originalone, k z is the compaction factor, and e z is a randomvariance that mimics grain rearrangement. Further details ab out the compaction modelling are givenelsewhere (Bakke and ren, 1997; ren and Bakke,2002) . We modelled Berea sandstone using k z = 0.05and e z randomly distributed in the interval [ 0.02,0.02].

2.2.4. DiagenesisOnly a subset of known diagenetic processes are

modelled, namely quartz cement overgrowth, disso-lution and metasomatosis of feldspars, and authi-genic clay growth. We model quartz cement over-growth by an algorithm similar to that described bySchwartz and Kimminau (1987) . If Ro(r ) denotesthe radius of the originally deposited grain, its newradius along the direction r from the grain centre isgiven by

Rr Ror mina l r c

; l r 3

where l (r ) is the distance between the surface of theoriginal grain and the surface of its Voronoi poly-hedron along the direction r . The constant a con-trols the amount of cement growth (i.e. the poros-ity). Positive values of a result in quartz cement growth while negative values mimic quartz disso-lution or overpressured sandstones with abnormallylarge intergranular porosity.

The exponent c controls the direction of cement

growth. Positive values of c favour growth of quartzcement in the directions of large l (r ) (i.e. pore bodies)while negative values favour growth of cement in thedirections of small l (r ) (i.e. pore throats). If c = 0,quartz cement grows equally in all directions asconcentric overgrowth. The Berea sample was mod-elled using c = 0.5.

Pore lining clays such as chlorite typically formcrystals that grow radially outward from the surfaceof the detrital grains. We model such clays byrandomly precipitating clay particles on the surfaces

of the grains or the quartz cement. Pore filling clayssuch as pseudo-hexagonal booklets of kaolinite aremodelled using a clustering algorithm which causes

new clay particles to precipitate preferentially inrandomly selected pore bodies that already containclay. The modelling of pore bridging clays, such asillite, is more complex and we ref er the interestedreader to ren and Bakke (2002) . For the Bereasample, the volume fraction of clay / c was 0.09.Approximately 90% of the clays were deposited as pore filling while the rest was precipitated as porelining.

Dissolution of feldspars may result in a variety of secondary porosity textures. Feldspar may be parti-ally or completely dissolved and replaced by either void or microporous clay. The dissolution patternsometimes reflects the crystal structure of the feld-spar while clay may show feldspar pseudomorphism.For the Berea sample, we model these processes byrandomly converting 20% of the feldspar grains intomicroporous clay and 10% into void. The main input parameters for the r econstru ction of Berea sandstoneare summarized in Table 1 .

2.2.5. Gridding For practical purposes, we construct the medium

in a discrete manner. The reconstruction is performedon a 3D grid of size M x M y M z with latticespacing a. The discretized medium is composed of small cubes, each of size a, that are filled either withvoid or with solid. The solid cubes or voxels arecharacterized in terms of their mineralogy (quartz,feldspar, or clay). The Berea sandstone was con-

Table 1Input parameters for the process based reconstruction of Berea

sandstone/ p 0.18 intergranular porosity/ c 0.09 volume fraction of clays/ f 0.01 volume fraction of feldspar d 1 80330 Am grain diameter f ell 0.20 fraction ellipsoidal grainsd 2/ d 1 [1, 1.6] grain aspect ratiok z 0.05 compaction factor e z [ 0.02, 0.02] compaction variancec 0.5 quartz cement exponent p f 0.90 fraction pore filling clays p l 0.10 fraction pore lining clays


8/13/2019 1-s2.0-S0920410503000627-main

6/23

structed on a cubic grid of size 300 3 voxels. Thelattice constant a = 10 Am.

2.3. Statistical model

We follow the reconstruction algorithm proposed by Yeong and Torquato (1998b) for reconstruction of a two-phase porous medium. The main idea is togenerate a 3D configuration that matches certainstochastic functions of a reference medium. In our case, the reference medium is the microtomographyimage. The reconstruction is carried out in a discretemanner by use of a sim ulated annealing algorithm(Kirkpatrick et al., 1983) . The target or energy func-tion is defined as

E t X J

k 1x k Xx A f k t x f

k ref x A

24

where f ref k is the k th function of a set of J stochastic

functions to be reconstructed, f t k is the value of this

function at iteration step t , and x k is the weight of thek th function.

Starting from a random configuration with poros-ity / p, two voxels of different phases (i.e. matrix and

pore) are chosen randomly and swapped. The poros-ity is thus preserved during the reconstruction proc-ess. The new energy is computed according to Eq.(4) and the new configuration is accepted with probability

p 1 if D E V 0

e D E =T if D E > 08

8/13/2019 1-s2.0-S0920410503000627-main

7/23

where m is the total number of placements of the segment of length u. We use a cut-off lengthuc = 50 voxels for both G 2 and PL and impose periodic

boundary conditions on the sample boundaries. Dur-ing the reconstruction, we update only those terms of G 2 and PL that have been affected by the exchange of voxels. This was accomplished by storage of line-by-line contributions of the functions. Both functionshad the same weight x 1 = x 2 = 0.5.

3. Reconstructed Berea models

Fig. 2 shows the 3D pore space in a 128 3 voxelssubregion of the different Berea models. The micro-tomographic image is denoted CT, the process-based

model PM and the statistical model SM. Visualinspection suggests that none of the reconstructedmicrostructures closely resemble that of the micro-

tomography image. This is especially the case for sample SM which appears to be more tortuous.Visually, the process-based reconstruction provides a better representation of the true sandstone microstruc-ture. However, it is important to realize that average properties of different media may agree closely even if visual appearances suggest otherwise.

3.1. Local porosity distributions

Local porosity distributions (Hilfer, 1991, 1996)are obtained from the indicator Z (x) in the followingway: Let / (x, L) = h Z (x)i M denote the porosity of a

Fig. 2. The pore space in a 128 3 voxels subregion of samples CT (left), PM (right), and SM (bottom). The resolution of all the images is 10 A m.


8/13/2019 1-s2.0-S0920410503000627-main

8/23

cubic measurement cell M (x, L) of side length Lcentred at position x. The local porosity distribution,l (/ , L) is then defined as

l / ; L 1mXx d/ / x ; L 10

with the indicator

d/ / x ; L 1 if A/ / x ; LAV D /

0 otherwise118>:l (/ , L) is the probability density that a givenmeasurement cell of side length L has a porosity inthe range [ / , / F D / ]. In Eq. (10), m denotes thetotal number of measurement cells. Ideally, themeasurement cells should not overlap, but in practicethis cannot be done be cause of poor statistics(Biswal et al., 1998, 1999) . The present results wereobtained by placing M (x, L) on all voxels which areat least a distance L/2 from the boundaries of thesample.

The local porosity distri butions l (/ , L*) of the

three samples are shown in Fig. 3. The characteristiclength L*=150 Am. Although differences do exist,the local porosity distributions of the three models arefairly similar. Unfortunately, characterization of thesamples in terms of l (/ , L*) does not allow us toquantify the connectivity of the samples. Transport

properties, of course, depend critically on the con-nectivity of the pore space.

3.2. Fraction percolating cells

A measurement cell M is said to percolate in allthree directions if each face in the cell is connected tothe opposite face by a continuous path that liesentirely inside the pore space. The total fraction of percolating cells, P ( L), is given by

P L 1mXx Kx ; L 12

where K(x, L) is an indicator of percolation

Kx ; L 1 if M x ; L percolates

0 otherwise8

8/13/2019 1-s2.0-S0920410503000627-main

9/23

This does not appear to be the case for the statisticalreconstruction. Sample SM shows a drasticallyreduced connectivity compared to the other samples

particularly at large L.

3.3. Transport properties

3.3.1. Formation factor The directional formation factor, F i, is defined as

the inverse of the macroscopic electrical conductivity,r i, of a porous medium in a given i direction:

F i r wr i

; 14

where r w is the bulk electrical conductivity of thefluid that fills the pore space. We compute r i for thethree orthogonal directions from a linear relation between the total electrical flux, Qi, and the applied potential gradient, ( UI UO )/ L:

Q i r iUI UO

L A 15

The potential UI is applied to an inlet face, XI, of area A, and the potential UO to an outlet face, XO ,separated from XI by a distance L. To obtain theelectrical flux, we solve the Laplace equation

j r wj U 0 16

subject to the boundary condition

j U n 0 on S p 17

where n is the unit vector normal to the pore wall S p.The microscopic fluid conductivity r w is given by

r w x; y; z 1 if x; y; z is inside the pore

0 otherwise188>:

The electrical field is calculated from U byE = r ij U, and the total electrical flux throughXI is

Q i Z XI nE dXI 19

Details of the solution strategy are described byren and Bakke (2002) . The average formation factor is defined as the harmonic mean of the directional

dependent formation factors, i.e.,

1 F

13

1 F x

1 F y

1 F z 20

3.3.2. PermeabilityThe directional dependent absolute permeability,

k i, is defined by Darcys law

Qi k il

pI pO L

A 21

where Qi is the macroscopic flux obtained fromapplying a macroscopic pressure gradient ( pI pO )/ L in a direction i. The low Reynolds number flow of an incompressible Newtonian fluid is governed by theusual Stokes equations

l j 2v j p 22

j v 0 23

v 0 on S p 24

where v , p, and l are the velocity, pressure, andfluid viscosity, respectiv ely. We employ the artific ialcompressibility scheme (Peyret and Taylor, 1983) tosolve this partial differential equation on the digi-tized samples. From the resulting velocity field, themacroscopic flux Qi is obtained by integrating v over the inlet area. A directionally averaged perme-ability is defined as the arithmetic mean k =(k x + k y +

k z )/3.The computed transport properties for the differ-ent samples are summarized in Table 2 . As expected,sample SM has a significantly lower conductivityand permeability than sample CT. This shows that the pore space of the statistical reconstruction is sig-nificantly more tortuous and less connected than that of the microtomography data. The directionally aver-aged formation factor of the process-based recon-struction is similar to that of sample CT (36.9 versus41.6) while the permeability is roughly 20% higher.


8/13/2019 1-s2.0-S0920410503000627-main

10/23

In general, we find the agreement between samplesPM and CT quite encouraging.

4. Network representation

4.1. Topology

The topology or connectivity of the reconstructedmedium is characterized by the skeleton which can beviewed as a line representation of the pore space, analo-gous to a capillary network. The skeleton of sandstonesthat contain microporosity and secondary porosity can be very complex. To simplify matters, we extract the

skeleton from the intergranular pore space that is de-fined by the sedimentary grains and the quartz cement.

We do this by an ultimate dilation of the grains(Bakke and ren, 1997) . The vertices of the resulting

Voronoi polyhedra are defined by the voxels that haveneighbouring voxels from four or more different grains and can be viewed as the nodes of the network.

The edges of the polyhedra consist of voxels that haveneighbouring voxels from three different grains anddefine the flow paths (links) between the nodes.

Many characteristic network model parameters can be obtained directly from the skeleton provided that some care is exercised. In particular, dead ends must be removed and certain nodes must be fused into asingle node. In general, neighbouring nodes are fusedinto a single node if their inscribed sphere radiioverlap. After the fusing operations, the remainingnodes and links are in a one-to-one correspondencewith the percolating pore bodies and throats of theintergranular pore space.

The co-ordination number of a node is the number of links connected to it. This is an important con-nectivity characteris tic that can be obtained directlyfrom the skeleton. Fig. 4 shows the distribution of measured co-ordination numbers. The co-ordinationnumber ranges from 1 to 16 with an average value of 4.45. This is in agreement with values for Bereasandstone reported in the literature (Jerauld and Salter,1990) .

4.2. Pore body and throat sizes

The nodes (vertices) of the skeleton are used todetermine the pore body sizes. Briefly, for each

Fig. 4. Measured distributions of (left) co-ordination numbers and (right) channel lengths for the process-based reconstruction of Berea.

Table 2Computed transport properties for the samples (experimental sampledenoted EX)

Property CT PM SM EX

F x 40.4 33.1 81.9 F y 38.5 36.3 78.8 F z 46.8 42.6 103.0 F 41.6 36.9 86.9k x (mD) 861 1040 243k y (mD) 891 965 254k z (mD) 622 822 167k (mD) 791 942 221 1100


8/13/2019 1-s2.0-S0920410503000627-main

11/23

node, the pore walls are mapped using a rotatingradius vector in a plane that is centred on the nodevoxel. This plane is rotated 18 times in increments of

10j

and all the distances from the node voxel to thequartz/clay surfaces are recorded. Abnormally longdistances are assumed to have been measured into pore throats and are neglected. The highest distancetransform voxel next to the node defines the radiusof an inscribed sphere. We define this radius as thesize of the pore body. Fig. 5 shows the distributionof pore body sizes. The average pore body size is27 Am.

Pore throat sizes are measured similarly. For eachskeleton voxel, the pore walls between two connect-ing nodes are mapped in a plane that is perpendicular to the local direction of the throat. The throat radiusin each plane is measured using a radius vector that is rotated 36 times in increments of 10 j . The in-scribed radius for each plane is recorded. The small-est radius corresponds to the throat constriction andit is defined as the effective size of the pore throat.Fig. 5 shows the measured distribution of throat sizes for the reconstructed Berea. The average throat size is 14 Am.

The cross-sectional shape of every pore body andthroat is described in terms of a dimensionless shape

factor G = A/ P 2 where A and P are the average areaand perimeter length, respectively, of the planes usedin the mapping of the pore body or throat. The shape

factor approximates the irregular shape of a pore bodyor throat by an equi valent irregular tria ngular, cubic,or cylindrical shape (ren et al., 1998) . Fig. 6 shows

the distribution of shape factors for the pore bodiesand throats.

4.3. Lengths and volumes

We define the pore channel length, l ij , as the length between two connecting nodes i and j on the skeleton.The distribution of channel le ngths for the recon-structed Berea is shown in Fig. 4. For network modelling purposes, we also define an effective pore body length l i and an effective throat length l t accord-ing to

l i l t i 1 0:5 r t r i 25

l j l t j 1 0:5 r t r j 26

l t l ij l i l j 27

where l i( j )t is the distance between node i( j ) and the

channel constriction, r t , is the radius at the constric-tion, and r i( j ) is the radius of pore body i( j ). Theintergranular volume of each pore body, V p, is deter-

Fig. 5. Measured pore body (left) and throat (right) size distributions for the reconstructed Berea.


8/13/2019 1-s2.0-S0920410503000627-main

12/23

mined by measuring the volume out to each of theconnecting throats. Similarly, the intergranular vol-ume of each throat, V t , is measured by counting thenumber of pore voxels inside the domain bounded by planes located at l i and l j . We note that l i and l j , andhence V t , depend on the pore body-to-throat aspect ratio (i.e. r i( j )/ r t ) and that V t decreases as the aspect

ratio increases. The clay associated volume V c of every pore body or throat is determined by countingthe number of clay voxels inside the pore body or throat domain. The total pore volume of the model isgiven as

PV Xntot

i1V p;i / Ac V c;i X

ltot

l 1V t ;l / Ac V c;l 28

where ntot and ltot are the total number of pore

bodies and throats, respectively, and / A c is the clay porosity. We model the porous clay as a bundle of parallel pores with fractal cross section. The fractalmodel is constructed by an iterative process whichcommences by dividing the half perimeter of acircular pore of radius r 0 in n parts and replacingeach part by half a circle (Vizika and Lenormand,1991) . The radius r 0 =1/2a / A c where a is the reso-lution of the reconstructed model. The linear fractaldimension is taken to be D L = 1.6, which is typical of sandstones.

5. Effects of wettability

The predictive power of network modelling tech-niques f or strongly water wet systems is well estab-lished (Bryant and Blunt, 1992; Bakke and ren,1997; ren et al., 1998; Lerdahl et al., 2000 ).However, few, if any, oil reservoirs are strongly

water wet. Predictive modelling of reservoir rocksthus requires an accurate characterization of wett-ability. Below, we describe what has recently becomethe standard model for describing wettability effectsin network models. Although the model has not beenvalidated through quantitative predictions for non-water wet rocks, it appears to capture much of the pertinent physics.

Kovscek et al. (1993) proposed a physically based pore level scenario of wettability change and fluiddistribution. Initially, we consider the reservoir rock to

be fully saturated with water and strongly water wet.When oil invades the pore space during oil migration,a water film protects the pore surface from beingcontacted by oil. The stability of the water film depends on the prevailing capillary pressure and on theshape of the disjoining pressure isotherm. At a criticalcapillary pressure, the water film ruptures, oil contactsthe pore surface, and its wettability changes. Regionsof the pore space where the thick water film coats the pore surface remain water wet as do the water-filledcorners of oil invaded pores, as shown in Fig. 7a. The

Fig. 6. Measured distributions of pore and throat shape factors.


8/13/2019 1-s2.0-S0920410503000627-main

13/23

degree of wettability alteration depends on the com- position of the oil and brine and on the mineralogy of the pore surface.

Within a single pore, different parts of the poresurface have different wettability. The corners arewater wet while the centre of the pore is oil wet. This

results in a number of different fluid configurationsduring waterflooding. If the bulk portion of a pore isoil wet, water invades the pore as a nonwetting phaseand resides in the centre of the pore. In this case, afilm of oil may be sandwiched in between the water inthe corner and the water in the centre, as shown in Fig.7b. These films may significantly increase the con-nectivity of the oil and allow oil mobility down to lowoil saturations.

Kovscek et al. (1993) examined the different fluidconfigurations of this wettability model in pores with

a grain boundary shape. Blunt (1997) extended theanalysis to square pores while ren et al. (1998) andPatzek (2001) examined the fluid configurations intriangular pores and provided mathematical details of entry capillary pressures. Using this pore level sce-nario for wettability change, the effects of wettabilityon relative permeability and oil recovery have beenexplored (Blunt, 1998; Dixit et al., 1999, 2000; renet al., 1998) .

The key parameters in this mixed wettabilitymodel are the fraction a1 of oil-invaded pores that

become oil wet, the spatial distribution of the oil wet pores, and the contact angles. In the presence of contact angle hysteresis, the advancing contact angle

ha is greater than the receding angle hr . Consequently,some of the pores that are oil wet during water-flooding ( ha > 90

j ) will be water wet during secon-dary drainage ( hr < 90

j ). We thus need to estimate thefraction a 2 that are oil wet during secondary drainage.Unfortunately, there is no a priori way of determiningthese parameters. The hope is that macroscopicmeasurements, such as Amott wettability indices,would be sufficient to estimate the parameters andallow quantitativ e predictions for no nwater wet samples. Recently, Dixit et al. (2000) derived limitinganalytical relationships between Amott wettabilityindices and a1 for a bundle of capillary tubes. Herewe attempt to extend their analysis by includingaccessibility effects, phase trapping and contact anglehysteresis.

5.1. Amott wettability indices

The Amott test is one of the most widely usedempirical wettability measurements for reservoir cores. The method combines two spontaneous imbibition measurement s and two forced displacement

measurements (see Fig. 8). The Amott water index I w = D S 1 /(D S 1 + D S 2 ), where D S 1 is the change in

Fig. 8. Schematic definition of the Amott indices to water ( I w) andoil ( I o) in terms of capillary pressure curves. I w = (S 2 S 1)/(S 4 S 1), I o = (S 4 S 3)/(S 4 S 1).

Fig. 7. (a) Oil and water in a triangular pore or throat after primarydrainage. The areas contacted by oil have altered wettability whilethe water-filled corners remain water wet. (b) Formation of oil filmsin the corners during waterflooding.


8/13/2019 1-s2.0-S0920410503000627-main

14/23

water saturation due to spontaneous imbibition andD S 2 is the additional change in water saturation due toforced displacement. Similarly, the Amott oil index

I o = D S 3/(D S 3 + D S 4). The two indices are often com- bined to give the AmottHarvey index I wo = I w I o . I wo varies between + 1 (strongly water wet) and 1(strongly oil wet).

Consider first the process of spontaneous water imbibition. If the water saturation at the start of theimbibition is S wi , the saturation change due to water imbibition can be expressed as

D S 1 1 S wi S ow1 a1 29

where S ow1 is the volume fraction of oil wet pores(ha > 90

j ) and a1 is a trapping function for the water wet regions defined as

a1 1 S or ;ww

1 S wi S ow130

where S or,ww is the residual oil saturation in water wet pores. The volume fraction of oil wet poresS ow1 = a1 f s1 where f s1 is a saturation conversionfunction that depends on the distribution of the oilwet pores. If the oil wet pores are randomly distrib-

uted, wettability is uncorrelated with pore size and f s1 = 1 S wi . If wettability is correlated with pore size, f s1 depends on the pore size distribution.

Similarly, the additional change in water saturationdue to forced displacement can be expressed as

D S 2 a1 f s1a2 31

where a 2 is a trapping function for the oil wet regionsdefined as

a2 1 S or ;ow1S ow1 32

where S or,ow1 is the residual oil saturation in the oilwet pores.

By combining Eqs. (29) and (31), the fraction of oil wet pores can be expressed in terms of I w as

a1 1 S wi

f s11

a2a 1

I w1 I w

1

33

If there is no contact angle hysteresis, pores that are oil wet during waterflooding are also oil wet during secondary drainage (i.e. a1 = a2). In a more

realistic scenario, hr is different from ha and a2 < a1 .This is always true for mixed wet systems where hr may be much less than ha . The change in water saturation due to spontaneous oil imbibition can beexpressed as

D S 3 S ow2 a3 a2 f s2a3 34

where the accessibility function a3 is given by

a3 1 S or ;ow2 S imbw;ow2

S ow2

! 35

where S or,ow2 is the residual oil saturation in oil wet pores (hr > 90

j ) at the end of the forced water displacement and S w,ow2

imb is the volume fraction of oil wet pores that are filled with water at the end of the oil imbibition. Furthermore, if we assume that there is little or no water trapping during secondarydrainage, the change in water saturation due to oilimbibition can be written as

DS 3

g DS 2

I o1 I w 36

By combining Eqs. (31), (34), and (36), the frac-tion of oil wet pores during secondary drainage isgiven by

a2 a1 f s1a2 f s2a3

I o1 I w

37

We note that if ha for the oil wet pores arerandomly distributed between some upper and lower

values, f s2 equals f s1 evaluated at a1 .

5.2. Trapping and accessibility functions

To determine a1 from a given I w, we need toestimate the three functions a1 , a2 , and f s1 . Asmentioned before, f s1 depends on the particular distribution of oil wet pores. We shall consider three possible distributions: (i) oil wet pore bodies arerandomly distributed, denoted MWR, (ii) the largest pore bodies preferentially become oil wet, denoted


8/13/2019 1-s2.0-S0920410503000627-main

15/23

MWL, and (iii) clay-rich pore bodies preferentially become oil wet, denoted MWC. For all three dis-tributions we assume that the throat contact angle iscorrelated to the contact angles in the two connecting pore bodies.

In Fig. 9, simulated values of the three functionsare plotted versus a1 for the MWL and MWC cases.The results for MWR were similar. In all thesimulations, ha was distributed randomly between20 j and 90 j for the water wet pores and 90 j and160 j for the oil wet pores. The saturation function,

f s1 , varies almost linearly with a1 for the MWLand MWC cases. We also note that f s1

MWL > f s1MWC >

f s1MWR = 1 S wi .

The oil wet trapping function a2 is zero until a1reaches a critical value a1c . For a1 < a1c , no spanningoil wet cluster exist and oil can only be displaced at positive capillary pressures (i.e. I w =1). If a1>a1c ,spanning oil wet clusters exist and oil can bedisplaced from oil wet pores by forced displacement.The exact value of a1c depends on the distributionof oil wet pores. For the Berea sample considered

here, a1c (S wi = 0.25) = 0.15, 0.19, and 0.25 for theMWL, MWC, and MWR, respectively. For a1>a1c ,the water wet trapping function a1 is fairly close to1. Since we are mainly interested in cases wherea1>a1c (i.e. I w < 1), we make the simplification that a1g 1.

The trapping function a2 is mainly determined bythe collapse of oil films in the corners of oil wet pores. This in turn depends on details such as thecorner half angle, the contact angle, and the max-imum capillary pressure reached during primary

drainage (i.e. S wi ) (ren et al., 1998) . It is thereforealmost impossible to a priori predict a2 for realisticcases. To come up with an approximate expressionfor a2 , we performed a number of waterflood simu-lations commencing at different S wi . The computed(1 a 2) values show a strong power law behaviour when plotted versus S ow1c / S ow1 (Fig. 10) where S ow1cis the volume fraction of oil wet pores at a1c . We propose to estimate a2 by the following empiricalexpression

a2 1 1:2 S ow1c

S ow1 2:1

38

The simu lated accessibility function a3 is plottedversus a2 in Fig. 11 . It is similar for the MWL, MWC,

Fig. 10. Scaling of the oil wet trapping function a2 . In thesimulations, the initial water saturation was varied between 0.25and 0.4.

Fig. 9. The functions a 1, a 2 , and f s1 for MWL (left) and MWC (right) versus a 1 . The initial water saturation S wi = 0.25.


8/13/2019 1-s2.0-S0920410503000627-main

16/23

and MWR cases and increases rapidly towards anasymptotic value for a2>a 2c . The asymptotic value is1 S or * where S or * is the residual oil saturation for the

strongly oil wet case ( a 2 = 1). For a given rock-fluidsystem, S or * depends principally on S wi and decreasesas S wi is reduced. For low S wi , the initial capillary pressure is larger than for high S wi . This improves thestability of the oil films and allows more oil to berecovered by film drainage. Similar to a 2 , we observea strong power law behaviour when we plot (1 a3)versus S ow2c / S ow2 (Fig. 11) and we propose thefollowing empirical expression

a3 1 S ow2c

S ow2 2 Z

39

where Z is the average co-ordination number of the pore network.

5.3. Contact angle hysteresis

It is well known that the effective advancingangle during waterflooding is always higher thanthe effective receding angle during drainage. Thisapparent hysteresis is due to pore surface roughness,

adsorption of surface active materials, and poregeometr y effects (i.e. diverging/convergi ng geome-tries). Morrow and McCaffery (1975) measuredadvancing and receding angles of many chemicalsin air/liquid/teflon systems. With sufficient rough-ening of the Teflon core, they obtained fairly con-sistent results that were independent of pore size andfurther roughening. Their results (referred to as Class

III) are shown in Fig. 12 where the advancing andreceding angles are plotted versus the equilibriumangle he measured on a at surface.

It is of course unclear how these results for low-energy solids can be used to predict the wetting behaviour of porous media formed from high-energy solids such as quartz. Since many crudeoil/brine/rock systems are known to dis play highcontact angle hysteresis (Buckley, 1993) , we sus- pect that the predicted hysteresis may be too small.However, in the absence of direct information, wewill assume that contact angle hysteresis in our Berea sample can be described by the Class IIItype.

Since ha and hr can be characterized in terms of he,the problem of distributing advancing and recedingangles is reduced to determine the distribution of equilibrium contact angles. We define he1 to be theequilibrium angle corresponding to ha = 90

j and he2 to be the equilibrium angle at which hr = 90

j . Unless thesample is strongly water wet ( I w = 1), we simplyassume he for the water wet pores to be randomlydistributed in the range [20 j he1 ].

Similarly, for the oil wet pores, we assume that he is randomly distributed in the range [ he,min

Documents

1-s2.0-S0920410503000627-main