Comput GeosciDOI 10.1007/s10596-009-9148-8

ORIGINAL PAPER

Deformation band populations in fault damagezone—impact on fluid flow

Dmitry Kolyukhin · Sylvie Schueller ·Magne S. Espedal · Haakon Fossen

Received: 1 December 2008 / Accepted: 29 June 2009© Springer Science + Business Media B.V. 2009

Abstract Fault damage zones in highly porous reser-voirs are dominated by deformation bands that gener-ally have permeability-reducing properties. Due to anabsence of sufficiently detailed measurements and theirregular distribution of deformation bands, a statisticalapproach is applied to study their influence on flow.A stochastic model of their distribution is constructed,and band density, distribution, orientation, and flowproperties are chosen based on available field observa-tions. The sensitivity of these different parameters onthe upscaled flow is analyzed. The influence of a het-erogeneous permeability distribution was also studiedby assuming the presence of high permeability holeswithin bands. The fragmentation and position of theseholes affect significantly the block-effective permeabil-ity. Results of local upscaling with a diagonal andfull upscaled permeability tensor are compared, andqualitatively similar results for the flow characteristicsare obtained. Further, the procedure of iterative local–global upscaling is applied to the problem.

D. Kolyukhin (B) · S. Schueller · M. S. Espedal · H. FossenCentre for Integrated Petroleum Research,University of Bergen, Allégaten 41, 5007 Bergen, Norwaye-mail: dmitriy.kolyukhin@cipr.uib.no

M. S. Espedale-mail: magne.espedal@cipr.uib.no

H. Fossene-mail: haakon.fossen@geo.uib.no

Present Address:S. SchuellerIFP, 1&4 avenue de Bois-Préau,92852 Rueil-Malmaison Cedex, Francee-mail: sylvie.schueller@ifp.fr

Keywords Fault damage zone · Deformation band ·Effective permeability · Stochastic models · Upscaling

1 Introduction

In reservoirs, fault zones represent tabular zones ofdeformed rock that introduce significant petrophysicalanisotropy, which potentially perturbs fluid flow duringproduction. Fault zones are typically composed of afault core, which accumulates most of the displacement,and an enveloping damage zone [33]. Even though thefault core represents the most heterogeneous structuralelement in a deformed reservoir rock, the damage zonecan have an additional impact on fluid flow. The dam-age zone is a deformed rock volume that comprisesvarious subsidiary tectonic elements, notably structuraldiscontinuities such as minor faults and slip surfaces,veins, and features known as deformation bands anddeformation band clusters.

Deformation bands are millimeter thick deformationzones, typically with shear offsets that rarely exceed afew centimeters even though their length can exceed100 m (see [14] for a review). They occur in highlyporous reservoirs (∼15% or higher porosity), particu-larly in fault damage zones, and are found not only assingle structures but also as clusters, especially closeto the fault core (Fig. 1). Their formation involvesgrain reorganization by grain sliding, rotation, and/orfracture, most commonly during compaction-assistedshearing. The internal changes in grain arrangementcan modify the porosity and permeability within theband. This is particularly true for cataclastic deforma-tion bands (Fig. 1), in which grain size is reduced bygrain fracturing and permeability reductions of several

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Fig. 1 A Deformations bandsin the Entrada sandstone inUtah [13]. Note thedistribution in clusters as wellas the conjugate arrangement.B Photomicrograph of asingle cataclastic deformationband displaying a reductionof porosity due to graincrushing

orders of magnitude is common (e.g., [13]). Cataclasticdeformation bands in porous siliciclastic sedimentaryrocks may, thus, potentially baffle fluid flow (e.g., [14])at least at reservoir production timescale.

The small thicknesses of deformation bands maketheir real influence on fluid flow in reservoirs difficultto assess. Even the thickest deformation band clustersis invisible on seismic data, and it is, therefore, im-possible to decide whether poor well performance orpoor communication between wells is to be explainedby deformation bands, subseismic faults, or both. It isalso difficult to separate their effect from that of theassociated fault core. Calculations and field observa-tions presented by Fossen and Bale [13] indicate thatdeformation bands only have limited effects on reser-voir production unless the bands show unusually highpermeability reductions (higher than four orders ofmagnitude) or occur in very high numbers in the dam-age zone. The results of their study are in part basedon averaging the permeability over a distance of 500 mbetween an injector and a producer (including a 50 mwide damage zone). The model is one-dimensionaland, hence, does not capture the more complex flowpathways that could be generated at a smaller scale.However, Odling et al. [24] suggest that even if thefault core dominates the bulk permeability of the wholefault zone, the contribution of the fault damage zone tothe fault sealing capacity cannot be neglected. Sternlofet al. [31] study the flow and transport effects of com-paction bands in sandstone. They simulate flow in two-dimensional domains with linear dimensions of severalhundred meters, using a discrete-feature model, basedon finite-volume approximation. The numerical resultsshow that the presence of deformation bands has anessential impact on the flow. The impact depends partlyon the band orientation. These contrasting views showthat large uncertainty is still associated with the roleof damage zone structures in reservoirs under produc-tion. Moreover, in reservoir-scale flow models, wheredamage zones cannot be modeled explicitly and wheremajor fault zones often are represented by only onegrid block, upscaling of the fine scale flow patterns is

still a challenge. The ratio between the thickness ofthe deformation bands (millimeter to centimeter scale)and the computational scale (1- to 10-m scale) may bethree to four orders of magnitude. Therefore, upscalingmethods are needed [11].

The objective of this work is to develop and study thesensitivity of parameter data on a computational scale,suited for the simulation of flow in fault damage zones.More specifically, we seek to characterize the influenceof some properties of deformation band populationson fluid flow in a coarse block (with a size of 1 ×1 m) and understand how the effective permeability isaffected. Focus is on characteristics observed in nat-ural damage zones represented by deformation bandpopulations, notably the orientation of the bands (syn-thetic or antithetic to the fault core), their clustering,the heterogeneous distribution of permeability alongbands, and the variable density of deformation bandsin a damage zone. Results are presented in order toillustrate the effect of these different deformation bandconfigurations on the upscaled bulk permeability.

For some parameters, the uncertainty in the geo-logical field measurements is large, leading to a largemodeling error. The modeling error is approachedthrough statistical models based on distributions gen-erated from databases and from data given in the liter-ature. Thus, both modeling error and error originatingfrom loss of subgrid information in the upscaling proce-dure might be evaluated.

2 Deformation band population characteristicsin the damage zone

This study focuses on deformation band populations infault damage zones, and any additional secondary faultsor slip surfaces that may exist in the damage zone arenot taken into account. We are aware that if such sub-sidiary faults and slip surfaces are open, permeabilitywithin the fracture is enhanced [3, 12, 20], and theycould strongly enhance fluid flow in the damage zoneby connecting domains that are otherwise separated

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by a low permeability zone (such as clusters of defor-mation bands). These fractures often contribute to fluidflow across the stratigraphic layering. At the same time,slip surfaces are mostly contained in deformation bandclusters [19], and the slip planes themselves have thincataclastic wall rocks of very low permeability. Hence,layer-parallel fluid flow can be expected to be retardedby both deformation band clusters, and these low per-meability fracture wall rocks. Moreover, slip planes aregenerally sparse and unconnected within fault dam-age zones in highly porous sandstones and only formconnected networks in the fault core [24]. We, thus,emphasize our study on the effect of the deformationbands solely.

2.1 Deformation band distribution around the faultcore

Fault frequency profiles across fault damage zonesshow an overall decrease in deformation band densityaway from the fault core [2, 4, 15, 17, 21, 30]. Based ona comprehensive database consisting of damage zonedata from Utah and Sinai (CIPR database), this de-crease can be modeled as a logarithmic decline in morethan 80% of the cases (Fig. 2). The frequency of de-formation bands can, thus, vary from 1 to 200 bands/m(largest extreme value observed in Utah) along a singledamage zone, with the maximum density being closeto the fault core. If we consider a homogeneous dis-tribution of deformation bands in the damage zone,

Fig. 2 Decrease of the frequency of deformation bands permeter (Y) in a damage zone in siliciclastic reservoirs. The highestdensities are found close to the fault core; however, significantlylarge cluster of deformation bands also exist at a certain distanceof the fault core. X is the distance from the fault core, and R2 isthe fitting factor of the logarithmic decrease to the data

the average density of deformation band is around 10–15 bands/m, regardless of the fault throw.

2.2 Spatial distribution of the band in the damage zone

Modeling deformation band density reduction awayfrom the fault core by a logarithmic function is asimplification that does not reflect the tendency of de-formation bands to form cluster. The clustering of de-formation bands has been characterized by Du Bernardet al. [9] by means of a correlation analysis. The corre-lation analysis consists of calculating the correlation in-tegral based on the discretized equation of Grassbergerand Procaccia [16]. The equation used was adapted as:

C(r) = 2

N × (N − 1)

∑

i< j

�(r − ∣∣xi − x j

∣∣) ≈ rDc (1)

where N is the total number of deformation bands inthe scan lines, r is the distance, and x is the position ofthe deformation band in the scan line (distance fromthe fault core). � is the Heaviside function, which isdefined as �(x) = 1 if x > 0, and as �(x) = 0 ifx < 0.

The correlation function corresponds to the relativenumber of pairs of deformation bands that have spacinglower than a certain distance r. This analysis requiresknowledge of the exact position of each deformationband along a scan line in the damage zone. If thedistribution of the deformation band is fractal, then thefunction appears linear in a log–log diagram (Fig. 3A),and its slope corresponds to the correlation dimensionDc. The degree of clustering is, thus, characterized byDc. For 1D fractal organizations, Dc can vary between0 (complete clustering) and 1 (no clustering; Fig. 3B).Theoretically, a Dc value of 0 implies all of the defor-mation bands to be at the same position, and a corre-lation dimension of 1 corresponds to a homogeneousdistribution of deformation bands in the damage zone.Observations of several damage zones of normal faultsin sandstones (using the CIPR database) have shownthat the average correlation dimension is 0.84 with astandard deviation of 0.06, implying that deformationbands are fractally organized in the damage zone.

2.3 Orientation of the deformation bands

Deformation bands in damage zones generally trendsubparallel to the major fault with synthetic and anti-thetic dips [2, 19, 29], although some may trend obliqueto the major fault [12, 20]. Some studies indicate thatsynthetic and antithetic structures appear to be geolog-ically coeval and in equal proportions for faults with

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Fig. 3 Correlation analysis:A Correlation function as afunction of the distance. Theslope of this function in alog–log plot corresponds tothe correlation dimension.B Examples of values of Dcand their correspondingfrequency graphs. Theorganization the closest tothe one observed in nature isDc = 0.85

displacement larger than around 30 m whereas smallerfaults show a dominance of synthetic structures [2, 17,24, 29]. Within the damage zone, the abundance of eachset can also vary with location. Within a cluster, thedeformation bands have roughly the same dip direction.Considering the correlation of dip direction betweentwo successive deformation bands indicates that twodeformation bands have more than 70% chance ofhaving the same dip direction (synthetic or antithetic)if their spacing is less than 10 cm (analysis performedon four scan lines using the CIPR database).

2.4 Petrophysical properties of deformation bands

Cataclastic bands are a common type of deformationbands in fault damage zones, particularly in cases wheredeformation occurred at depths in excess of 1 km; onlythis type of bands will be considered in this study. Evenif these structures have been widely studied, the precisedescription of their geometry (length distribution, ori-entation, spatial continuity) and particularly the varia-tion of their petrophysical properties are still not wellknown. The thickness of the deformation bands, whichtypically is around 1 mm [14], varies along many bands.In this study, an average width of 1 mm was chosen. Sig-nificant variations in porosity and permeability alongselected deformation bands have been documented byTorabi and Fossen [32], but more extensive work is

needed to map the frequency and understand the mech-anisms responsible for these changes.

3 Numerical method

The solver used in this study is based on a finite volumemethod and was developed for the numerical solutionof the filtration problem. The main computational dif-ficulty of the solution deals with the extremely irregu-lar spatial distribution of the permeability field causedby the existence of complex networks of deformationbands crossing the media. It is possible to assume thatthe computing complexity of the problem will growwith an increase in number of deformation bands, per-meability contrast, or decrease in band’s width.

We consider a steady flow through a saturatedporous medium. For a stationary 2D flow, we solve thefollowing Darcy law and continuity equation:

q = − 1

μK ∇ p (2)

∇ · q = 0 (3)

where q is the Darcy velocity, p is the pressure, μ thedynamic viscosity (constant in all the simulations), andK the permeability.

Because of discontinuity of the permeability K, thefinite volume method was used with a rectangular grid

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for solving the problem. Integrating Eqs. 2 and 3 overthe small volume surrounding each node point on amesh and applying Gauss’s theorem, we get the follow-ing equation:∫

Vk

∫div

[K grad p

]dVk

=∫

Sk=Gk1∪Gk2∪Gk3∪Gk4

K∂ p∂ n

dSk = 0 (4)

where Vk is the volume of the cell k, Sk is the surface ofthe cell k, and Gk1, Gk2, Gk3, and Gk4 represent the cellboundaries (Fig. 4).

The derivative on boundaries Gki in Eq. 3 was ap-proximated using the work of Marchuk [23]. For exam-ple, the flow rate through Gk4:

Qk−1/2 =∫

Gk4

Kdpdn

dSk4 ≈ pk − pk−1∫

[Xk−1,Xk]

1K dx

A, (5)

where A is the length of Gk4,[Xk−1, Xk

] ⊥Gk4

(Fig. 4), pk is the pressure in the middle of the cell k,and Qk−1/2 is the flow rate through Gk4.

Thus, we have a so-called dual-grid structure. Thefirst grid consists of a set of Sk boundaries of controlvolumes Vk. The second grid consists of a set of Xk—centers of control volumes Vk. For correct flow simula-tion the second grid Xk should be chosen so that eachbody included in its cell should intersect at least oneboundary of this cell.

We solve Eqs. 2 and 3 by using the approximation(5) in the domain D = {

0 ≤ x ≤ Lx, 0 ≤ y ≤ Ly}. The

host rock has a permeability of 1 darcy. Each upscaledblock has a size of 1 × 1 m. By default, a rectangular

Fig. 4 Method of calculation based on the finite volumes method.Sketch of a cell k with center point Xk, volume Vk, surface Sk,and pressure pk at Xk. Gk1, Gk2, Gk3, andGk4 are the boundariesof the cell k

grid contains 80 × 80 nodes for one upscaled block,and each deformation band corresponds to a 1 mmthick band with a default permeability of 10−2 darcy.The representation of thin deformation bands in anumerical model is a rather difficult problem. There-fore, the method of integral identities [23] was appliedto solve the problem with discontinuous permeabilitycoefficient. In our model, the integral on the right sideof Eq. 5 is computed analytically because permeabilityis a piecewise function with known borders. Therefore,the solution of Eq. 5 is exact in 1D even if the widthof the bands is much smaller than the grid size. In 2D,some errors are introduced in the solution because thebands are not parallel to the grid. The accuracy of thesolution was verified by comparing analytical resultswith results obtained on highly resolved models. Thenumerical tests presented in Fig. 5 show that the resultsof the calculations do not change significantly whenthe numerical resolution is increased. In these tests,the flow rate is computed by using 40 × 40, 80 × 80and 260 × 260 grid nodes for 20 random realizationsof deformation band distributions (Fig. 5). Here, Dc =0.8, Iθ = 0, and the number of deformation bands isN = 10 (see Section 5 for the meaning of the differentparameters). On two opposite boundaries, we set fixedpressures P1 = 2 Pa and P2 = 1 Pa, whereas no flowboundary conditions apply to the other borders (seeFig. 7 for model setup). The relative difference betweenthe curves corresponding to 80 × 80 and 260 × 260nodes is less than 2%. We consider, thus, that thethin deformation bands are accurately resolved in thechosen grid (80 × 80 nodes).

Furthermore, a statistical approach is used to studythe flow in the damage zone. In this work, we constructa model describing a deformation band’s populationin the damage zone. Due to strongly irregular bandstructures, we assume that several geometrical parame-ters of this model are random functions. Therefore, thepermeability field is also considered as a random fielddescribed completely by the probability distributionof random parameters. Then, any flow characteristicξ (flow rate, velocity, effective permeability, etc.) alsobecomes a random function. This approach allows us tocompute averaged flow characteristics only. Having theensemble of the random parameters’ realizations sam-pled accordingly with the correspondent distribution,we can calculate the value of flow characteristic ξ i foreach realization as well as the effective properties byusing the following statistical averaging:

Eξ ≈ 〈ξ〉 = 1

NS

Ns∑

i=1

ξi (6)

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Fig. 5 Flow rate computed by using 40 × 40, 80 × 80, and260 × 260 grid nodes for 20 random realizations of deformationband population, N = 10 deformation bands; Dc = 0.8, Iθ =0, dip =π/3, KDB/KHR = 0.01. On two opposite boundaries,

we set the fixed pressures P1 = 2 Pa and P2 = 1 Pa, whereas noflow boundary conditions apply to the other borders. The relativedifference between the two curves “80 × 80” and “260 × 260nodes” is less than 2%

with NS as the number of realizations. Here, 〈 〉 meansthe ensemble averaging.

In Monte Carlo methods, the evaluation of the sta-tistical error ν(ξ) is essential. The standard estimationhas the following form [27]:

ν(ξ) = α(β)σξ√NS

(7)

where α is a coefficient depending on the confidentcoefficient β. For example, α(0.997) = 3 and α(0.95) =1.96. σξ is the standard deviation of random value ξ .The number of realizations used in all our Monte Carlosimulations is NS = 1,000.

4 Upscaling

As noted previously, geological models frequently in-corporate heterogeneities on the scale of a few cen-timeters and can contain in the order of 107 grid cells(e.g., [24, 25]). However, current flow simulators aretechnically limited to 104 to 105 cells, with grid cellstypically being a few hundred meters horizontally anda few tens of meters vertically. Upscaling methods are,thus, required for flow modeling. The aim of upscalingis to reproduce the global behavior of the reservoir butstill capture the local behavior, and the challenge is totransfer the geological information to the coarser cells

without losing crucial information existing at the finestscale.

A large number of papers have been devoted to dif-ferent upscaling techniques. A good survey of differentmethods is presented in [11] and reviews are found in[22, 24, 26, 28]. In general, these different approachescan be divided in two main groups: local approaches[5, 10, 25] and global ones [6, 18]. In the local methods,the upscaled permeability of the coarse blocks onlydepends on the real media permeability inside eachblock (pure local upscaling) and on the permeabilityof the surrounding area (extended local upscaling). Bycontrast, global and quasi-global methods utilize thefine-scale permeability on the whole domain for thecalculation of the upscaled permeability for each coarseblock.

The advantage of global upscaling is that the infor-mation about the fine-scale permeability in the entiredomain is used for flow computation. However, themain shortcoming of the global upscaling is an ex-tremely large computational cost. Therefore, for thenumerical solution of the flow problem in whole do-main, we apply the adaptive local–global upscaling(ALG) [7]. The ALG procedure is briefly described inpart 6.3, and an example of numerical computations isgiven.

For the sake of simplicity, the effect of deformationband populations on fluid flow will be mainly studiedfor a diagonal upscaled permeability tensor. On two

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opposite boundaries, the pressures are fixed to constantvalues P1 and P2, whereas no flow boundary conditionsapply to the other borders (example in Fig. 7). Aflow calculated numerically allows us to estimate theeffective permeability Keff of a coarse upscaled blockD from equation [26]:

〈q〉 = − 1

μKeff 〈∇ p〉 . (8)

1,000 realizations of deformation bands configurationare sampled according to the procedure described be-low. Then, the mean values for the flow rate and pres-sure gradient are estimated by using a local upscalingmethod. Finally, Keff is calculated. In this upscalingprocedure, the averaged upscaled permeability Kups

is a positive definite diagonal tensor, where diagonalelements are equal to the effective permeability Keff

in x and y directions. Strictly speaking, we use in thispaper an extended local upscaling. The linear size of theextended domain between boundaries with Dirichletboundary conditions is equal to the size of the coarseblock plus one grid cell size.

Diagonal Kups may not give accurate quantitativeresults, but we expect that the results are well suitedfor the sensitivity analysis. Some of these calculationsare repeated in a 2D upscaling framework where Kups

is a full positive definite tensor. We aim at showingthat the trends obtained with diagonal Kups for theeffect of some characteristics of the deformation banddistribution are qualitatively similar in the upscalingmethod with full Kups. Finally, we deal with the up-scaling problem in the global domain and emphasizethe advantage of the adaptive local–global upscaling incomparison with the pure local upscaling.

5 Parameters tested

One of the problems considered in this study is theinfluence of deformation band distribution on flow inthe damage zone. Based on the modeling presented inthe previous section, we will upscale the effects on flowin the damage zone based on the following parameters:

– Spatial distribution of the bands (i.e., clustering)– Orientation of the deformation bands– Heterogeneity of the petrophysical properties of the

deformation bands– Variation in band density.

The sensitivity of these different parameters is of spe-cial interest in defining the effective permeability re-lated to damage zones in porous sandstones.

The clustering of the deformation bands is charac-terized by the correlation dimension Dc defined byEq. 1. Since only a 1D fractal organization is taken intoaccount in these simulations, the correlation dimensionDc is varied between 0 and 1. The following values areparticularly tested: 0.2, 0.5, 0.8, and 1. Furthermore,in considering upscaling in the domain D, we assumethat all deformation bands intersect the segment Zconnecting the points with coordinates (0.5Lx, 0) and(0.5Lx, Ly). Using the procedure described below, aset of bands’ centers {Ci} ∈ Z is sampled according tochosen Dc under the condition

∣∣Ci − C j∣∣ > d, ∀ i, j, and

d being the width of each deformation band. In oursimulations, d is equal to 1 mm.

In this work, for simulating the set of points with agiven correlation dimension Dc in 1D, we use multi-plicative cascade processes [16]. In 1D, the recursive al-gorithm described in Darcel et al. [8] is used. As a resultof this algorithm, we get a subdivision Z = {Z1, Z2, . . . ,Z M} for the entire domain (Z = Z1 ∪ Z2 ∪ . . . ∪ Z M)

and the corresponding set of probabilities P = {p1, p2,. . . , pM} such that (p1 + p2 + . . . + pM = 1). When Zand P are constructed, we distribute each deformationband center Ci randomly and mutually independentlyin one of the subdomains Z j according to the prob-abilities {pj} by using the Monte Carlo method. Oneach iteration, each Zj is divided in two equal parts.The iteration procedure can be repeated infinitely, butin our simulations, we used a number of iterations Tbetween 5 and 7 producing M subdomains (M = 2T).

The computing rule of P is illustrated briefly in Fig. 6.Each element of P has the form

ql1 qT−l

2 , 0 ≤ l ≤ T.

The set P involves ClT elements. Here, q1 and q2 are

calculated from [8]:

q1 + q2 = 1 andq2

1 + q22

(1/2)Dc = 1.

If we assume an isotropic cascade process, the pairof probabilities corresponding to the two parts of thedivided subdomains can be randomly permuted. Thus,for each element of the ordered set Z , we constructthe corresponding element in the ordered set P. Thedistribution of deformation bands along a 1D scan linewill, thus, reproduce a clustered or fractal organizationcharacterized by the correlation dimension Dc.

Each band is characterized by its dip (angle be-tween the horizontal and the band). Only the senseof dip (toward the left or the right) is varied in thisstudy. Furthermore, each deformation band crosscutsthe whole model. The distribution of the orientations

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Fig. 6 Hierarchical construction of a fractal set points. Z (i)j is

the length of the jth subdomain of the domain Z at the ithiteration. q1 and q2 are the probabilities of finding one point inone subdomain so that q1 + q2 = 1

can be random and correlated with a specific correla-tion coefficient, described hereafter.

In some models, the angle between the deformationbands and the boundaries with fixed pressure is θ i =±π /6 (complementary angle of the dip). We assumethat {θ i} is the set of random variables ordered by they coordinate. Let the sign of θ1 be chosen randomlyaccording to the equal probability p = 0.5 for π /6 and−π /6. The orientation of the other deformation bandsis then defined recursively. If two neighboring bandshave their centers in the positions Yi−1 and Yi and theorientation of the first band is known, then the secondband has a same orientation with a probability of:

p(r) = p(Yi−1,Yi) = 1

2(1 + exp (−α × r))

with r = |Yi−1 − Yi|The correlation coefficient (radii) is defined as:

ρθ(θi−1, θi) = E(θi−1θi)√Eθ2

i−1

√Eθ2

i

= exp(−αr)

where Eθ i−1 = Eθ i = 0 and Iθ = 1/α. Iθ is the correlationlength of the random field θ .

The correlation length Iθ was varied between 0 and0.2 m. The correlation length characterizes the distancebelow which two bands will have a correlated orienta-tion (same dip direction). In the case where Iθ = 0, theorientations of the deformation bands are statisticallyindependent.

As observed in nature, the permeability inside a de-formation band is not necessarily constant and homoge-neous. In some simulations, holes are introduced in thebands. It means that a certain length of the deformationband presents an increased permeability in compari-son with the rest of the band. In our simulations, anaverage permeability contrast between the band andthe host-rock of around 10−2 is fixed. Hence, if weconsider a deformation band with a permeability KDB

of 0.001 darcy, the holes having a permeability Khole of0.091 darcy and the total length of the holes in eachdeformation band being 10 cm, then the perpendicularaverage permeability of the band KDBaverage is calcu-lated as the arithmetic average (L−h)×KDB+h×Khole

L (withL, the length of the deformation band—here 1 m—andh the total length of the holes) and equals 0.01 darcy.Four configurations are tested: a single hole having thesame position in each deformation band, a single holewith a random position, eight uniformly spaced holeswith a total length of 9 cm, and eight holes randomlyspaced (Fig. 10). In all the simulations, the host rockpermeability KHR was fixed to 1 darcy.

The deformation band frequency is not constant inthe damage zone but decreases away from the faultcore. Six densities were tested: 5, 10, 15, 25, 30, and 60deformation bands per meter. The default frequency is10 deformation bands in each upscaled block.

6 Numerical results

6.1 Local upscaling: diagonal permeability tensor

6.1.1 Reference bulk-effective permeability

If the deformation bands are oriented perpendicular tothe flow direction and if they present a constant per-meability, the flow along a grid element perpendicularto the bands can be calculated analytically [13, 24].In this case, the bulk permeability corresponds to theharmonic average of the permeabilities crossed by theflow:

Kref =

N∑i=1

li

N∑i=1

liki

(9)

where li is the thickness of the different domainscrossed by the flow, ki the corresponding permeability,and N is the number of existing domains. This value isindependent of the clustering of the deformation bandsin the media and was used as a reference value.

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6.1.2 Role of the orientation of the deformations bandsand their clustering

In the simulations presented in Fig. 7, the bands presenta homogeneous constant permeability of 10−2 darcy.Ten bands are considered in the upscaled block. Thedeformation bands are dipping 60◦ either toward theleft or toward the right. We consider here a variablecorrelation coefficient Iθ between the orientations ofthe dips as described above (parameters tested).

Figure 7 presents the bulk-effective permeabilityKeff in the upscaled coarse block as a function ofthe correlation dimension Dc and various correlationlengths for the orientation of the band dips. The grayline corresponds to the analytical average bulk per-meability Kref for bands perpendicular to the flowdirection.

For a specific clustering, increasing the correlationlength increases the bulk-effective permeability. Vary-ing the clustering of the bands has a negligible effectwhen the orientations of the bands are mutually inde-pendent (Iθ = 0 m). If the correlation length betweenthe orientations of the bands is strong (i.e., over acertain length, the bands have the same orientation),increasing the clustering of the bands (by decreasingDc) leads to a slight increase in the bulk permeability.The correlation between the dip orientations has how-ever not a strong effect on the effective permeabilityin the system: even for a strong clustering (low Dc)and for Iθ varying between 0 and 0.2 m, the variationof the effective permeability is less than 0.03 (∼5%).

Fig. 7 Effective permeability Keff in the upscaled coarse block asa function of the correlation dimension Dc for various correlationlengths Iθ of the dip orientations. Kref is the reference permeabil-ity calculated for continuous deformation bands perpendicular tothe flow direction

Fig. 8 Effective permeability Keff in the upscaled coarse blockas a function of the dip of the deformation bands

(dip = π/2 − θ

).

Iθ = 0 (random orientation of the dip), number of struc-tures = 10; permeability contrast between the bands and thehost rock: 10−2 and Dc = 0.8. Kref is the reference permeabilitycalculated for continuous deformation bands perpendicular tothe flow direction, corresponding to θ = 0 or a dip equal to 90◦

Actually, the difference between the curves Iθ = 0.1 mand Iθ = 0.2 m is comparable with the statistical errorsof the numerical calculations.

Varying the dip of the deformation bands also influ-ences the effective permeability in the block (Fig. 8):The more parallel to the flow direction the bands are,the higher the effective permeability. As expected, flowalong rather than across the deformation bands will befavored.

Decreasing the deformation band permeability(KDB) decreases the effective permeability in the sys-tem. Moreover, we note that the effect of clustering andcorrelation of dip orientation becomes stronger whenthe permeability contrast increases (Fig. 9). For clus-tering characterized by large correlation dimensions(Dc = 0.8–1), the effect of the clustering has only asmall effect on the effective permeability in the up-scaled block.

6.1.3 Effect of heterogeneous petrophysical propertiesalong the bands combined with variation inclustering

In these simulations (Fig. 10), the deformation bandsare perpendicular to the flow direction. A heteroge-neous distribution of permeability inside each band isnow considered. In order to simplify the configurationof the simulations, we first consider a 10-cm “hole”per band having permeability Kholes of 0.091 darcy; thepermeability in the rest of the band KDB being equal to

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Fig. 9 A Effective permeability Keff calculated in the upscaledblock and reference permeability Kref if the bands are perpen-dicular to the fluid flow as a function of the correlation dimensionDc. Different permeability contrasts KDB/KHR are tested; KDBis the permeability in the deformation band, and KHR is thepermeability in the host rock. Number of deformation bands =10; Iθ = 0.1. B Effective permeability Keff as a function of thepermeability contrast KDB/KHR for various correlation length ofdip orientation Iθ and various correlation dimensions Dc

10−3 darcy. The average permeability across the bandis equal to 10−2 darcy by considering a flow strictlyperpendicular to the band, which does not capturethe 2D nature of the flow. In this case, two referencevalues were calculated: the one without holes “Kref forKDB/KHR = 0.001” and the one taking into account theholes “Kref for KDB/KHR = 0.01.”

The first configuration tested is the case where theholes have exactly the same position in all the bands,implying that the holes are aligned parallel to the flowdirection (Fig. 10, 1h-uniform). In this case, increasingthe clustering of the bands increases the bulk perme-ability in the band (Fig. 10, curve 1h-uniform).

In nature, the probability of having holes alignedparallel to the flow direction is rather low, unless theholes are controlled by stratigraphic heterogeneitieswithin the layer. Therefore, a second configuration wastested in order to randomly position the hole on eachdeformation band (Fig. 10, 1h-random). This randomconfiguration reduces the general bulk permeability inthe block in comparison with the uniform one. More-over, increasing the clustering (by decreasing Dc) inthat case leads to an additional decrease of the bulk-effective permeability, contrary to the uniform case.

Because deformation band heterogeneities can occurrepeatedly along bands at the microscale [13], a thirdconfiguration was explored, where the single deforma-tion band hole was replaced by eight smaller holes ofthe same total length (10 cm). In this configuration,the hole length (1.25 cm) is larger that the grid step(1.23 cm). The results indicate that this fragmentationof the permeability heterogeneity globally increases thebulk-effective permeability in the block. If the holesare randomly distributed on each band (Fig. 10, curve8h-random), then increasing the clustering decreasesthe bulk-effective permeability. On the contrary, if theholes are uniformly positioned, then increasing the clus-tering (by decreasing Dc) increases the bulk-effectivepermeability Keff .

Comparing the difference in bulk-effective perme-ability Keff as a function of the clustering for the casesabove (uniform and random positions of the holes andfor only one large heterogeneity and for a fragmentedheterogeneity) indicates that clustering has a lower ef-fect on bulk-effective permeability than the distributionof permeability anomalies in the deformation bands.It is also interesting to note that assigning an averageconstant permeability to the band such as “Kref forKDB/KHR = 0.01” leads in that case to an overestima-tion of the effective permeability in the coarse block.

6.1.4 Effect of deformation band density

Deformation band density is varying within the damagezone and is globally decreasing when moving awayfrom the fault core. In Fig. 11, the effect of variabledensity is tested in combination with the effects ofclustering and orientation. As expected, the effectivepermeability decreases with increasing deformationsband density in each coarse block. The results indicatethat for densities higher than 10 deformation bands permeter, the effective permeability in the coarse blockis reduced at least by 50% compared to the host rockpermeability, if we consider deformation bands havinga constant permeability of 0.01 darcy and being per-pendicular to the flow direction (curve Kref in Fig. 11).

Comput Geosci

Fig. 10 Effects of permeability discontinuities along the defor-mation bands (10 deformation bands perpendicular to the flowdirection). KDB = 0.001 and Kholes = 0.1. The total length ofholes in each deformation band is equal to 9 cm in all thesimulations. The different configurations tested are: 1h-uniformonly one hole placed in a similar position along the bands in allthe bands; 1h-random only one hole placed randomly along the

band; 8h-uniform eight holes placed similarly in all the bands;8h-random eight holes placed randomly along the bands. Thetwo gray curves correspond to the analytical solution for tenbands with a uniform permeability: the lower curve correspondsto bands having a permeability of 0.001, and the upper one tobands having a permeability of 0.01 (which is average across bandpermeability of the band with the holes)

If we consider deformation bands with ±60◦ dip (andDc = 0.8, KDB = 0.01 darcy), a density of 15 deforma-tion bands is required to reduce the effective perme-ability by 50%. The correlation of the dip orientationshas no real influence on the effective permeability if the

Fig. 11 Effective permeability Keff in an upscaled block as afunction of the density of deformation bands (number of defor-mation bands in block of 1 m long; KDB = 0.01) . The referencepermeability Kref corresponds to deformations bands being per-pendicular to the flow direction and having a constant permeabil-ity. Two types of clustering have been tested characterized by acorrelation dimension Dc equal to 0.2 (strong clustering) and 0.8(weak clustering), respectively

correlation dimension Dc is large (0.8–1). For lower Dc,the influence of the correlation of dip orientation has tobe taken into account.

6.2 Local upscaling: full permeability tensor

In this paragraph, we deal with 2D extended localupscaling. Here, for each geological realization, wesolve two problems with fixed pressure on two oppositeboundaries and linear pressure variation on the otherboundaries parallel to the main flow direction [11]. Theresulting upscaled permeability is a positive definite fulltensor. The upscaled coarse block here has a size of 1× 1 m and the total number of the inner nodes in thecoarse block is 80 × 80. The linear size of the extendeddomain is equal to the size of the coarse block plus onegrid cell size.

In Table 1, we study the significance of off-diagonalterms of the upscaled permeability tensor. In thefirst row, the relations

∣∣Kupsxy

∣∣/Kupsxx and

∣∣Kupsxy

∣∣/Kupsyy are

shown for N = 10, Dc = 0.8, |θ | = π/

6 and Iθ =0 m. Twenty thousand realizations of the distributionof the deformation bands were sampled. In the nextrows, we stepwise change one of these parameters tocompare the results. Here, we present the percentage ofrealizations for which these ratios were greater or equalto 5%, 10%, 15%, and 20%, respectively. Note thatthe relative value of the off-diagonal terms grows with

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Table 1 Influence of different parameters of the model on the relation between off-diagonal and diagonal terms of the upscaledpermeability tensor

∣∣∣Kupsxy

∣∣∣Kups

xx,

∣∣∣Kupsxy

∣∣∣Kups

yy≥ 0.05

∣∣∣Kupsxy

∣∣∣Kups

xx,

∣∣∣Kupsxy

∣∣∣Kups

yy≥ 0.1

∣∣∣Kupsxy

∣∣∣Kups

xx,

∣∣∣Kupsxy

∣∣∣Kups

yy≥ 0.15

∣∣∣Kupsxy

∣∣∣Kups

xx,

∣∣∣Kupsxy

∣∣∣Kups

yy≥ 0.2

47.2%, 59.1% 15%, 27.3% 2.7%, 9.6% 0.2%, 2.3%θ = ±π/4 60.5%, 60.6% 29.3%, 29.6% 10.7%, 11% 3%, 3.1%N = 20 44.8%, 60.4% 12.4%, 29.9% 2%, 11.5% 0.17%, 3.5%Dc = 0.2 36.4%, 49% 8.1%, 18% 1%, 5.1% 0.07%, 1.1%Iθ= 0.1 73%, 78.7% 46.7%, 57.8% 24.3%, 38.6% 7.9%, 22.1%

20,000 realizations of deformation band distributions were sampled. The reference model (first row) has the following parameters: N= 10, Dc = 0.8, |θ | = π/6, Iθ = 0 m

increasing |θ |. The ratios grow also with increasing per-meability contrast and correlation length of band orien-tation Iθ . On the other hand, the clustering (decreasingof Dc) decreases the relative value of the off-diagonalterms. At the same time, it seems that increasing theband density is not important for the significance ofthe off-diagonal terms: the difference between N = 10(number of bands) and N = 20 does not exceed thestatistical error.

The numerical results presented in Fig. 12 are similarto the curves in Fig. 7. The resulting average upscaledpermeability is an average of permeabilities computedby extended local upscaling for Ns = 1,000 realiza-tions. The curves presented in Fig. 12A are higherthan the corresponding curves in Fig. 7. The highervalue observed for the permeability is caused by thedifferent boundary conditions. However, for Iθ = 0 and0.2 m, the relative difference does not exceed 5.5% and12.5%, respectively, even for Dc = 0.2. The nondiag-onal tensor element Kups

xy can change sign. Therefore,the averaged value <Kups

xy > is small in comparison with<Kups

xx > and <Kupsyy > and does not exceed 0.006 darcy.

Thus, the averaged upscaled permeability tensor ispractically diagonal for our model, as the diagonalterms are of the order of unity.

6.3 Adaptive local–global upscaling

We have mentioned above the procedure of theadaptive local–global upscaling. Following Chen andDurlofsky [7], we prefer the transmissibility upscaling.The basic idea of this iterative procedure is to use thepressure obtained from a global coarse scale solutionfor the calculation of the local properties. At the firststep, the coarse flow transmissibility is calculated byusing an extended local upscaling. Furthermore, theglobal coarse scale pressure is calculated and used forthe determination of local boundary conditions. Then,these boundary conditions are used for calculating lo-cal properties on the next iteration. The convergence

Fig. 12 Simulations using 2D extended local upscaling: influenceof the correlation length Iθ on the dip orientations. Number ofbands = 10; the permeability in the deformation bands KDB is0.01, the permeability in the host rock KHR = 1. A <Kups

yy > (aver-aged upscaled permeability in the coarse block along the y axis—at 60◦ of the deformation band) as a function of the correlationdimension Dc. B <Kups

xx > (averaged upscaled permeability in thecoarse block along the y axis—at 30◦ of the deformation band) asa function of the correlation dimension Dc

Comput Geosci

criterion was chosen as a measure of the differences inpressure and flow rate from one iteration to the next.

We consider a simple model, where the square rec-tangular domain D consists in 4 × 4 equal coarse blocksand an extended area (Fig. 13). Each block has linearsizes equal to 1 × 1 m, Lx = Ly = 5 m. There are 50deformation bands having a width of 0.001 m distrib-uted in D according to Dc = 0.8 and Iθ = 0 m. Thepressures in the corners are P(0,0 m) = 1 Pa, P(5,0 m)= P(0,5 m) = 2 Pa, P(5,5 m) = 3 Pa. The pressureson the boundaries are fixed and are defined as a linearfunction between the pressures in the correspondingcorners.

In Fig. 14, we present the result for accuracy of theupscaled characteristics. The coarse grid pressure Pc iscalculated in blocks centers Xk, and Qx, Qy are the flowrates in x and y directions calculated on the correspond-ing inner boundaries. The measure of accuracy is basedon a comparison of the coarse block results with theresults of fine scale simulations

(Pf

c, Qfx, and Qf

y

)and

is defined as ‖Pc−Pfc‖

0.5(‖Pc‖+‖Pfc‖) . The method converges after

a few iterations. The relative error for Pc, Qx, Qy areabout 0.5%, 21%, 29%. For comparison, on 0th itera-tion, the accuracies of the extended local upscaling are0.9%, 54%, 67%, respectively. These results indicate asexpected that some types of multiscale method shouldbe used to obtain sufficient accuracy in specific applica-tions. For comparison in Fig. 14, we present the resultsof local–global upscaling. We address the same model

Fig. 13 Test model for the 2D local–global upscaling. The globalboundary conditions are given by the pressures at the corners ofthe extended area and linear function in between. Each coarseblock has a size of 1 × 1 m. The flow rates Qx and Qy arecalculated across the block boundaries, and the coarse blockpressure Pc is calculated in the middle of the block

as given above, but we divide the computational regioninto 10 × 10 coarse grid blocks. The total number of finegrid nodes is the same. It is natural that relative errordecreases to 0.25%, 16.5%, and 24%, respectively.

6.4 Probability density of the upscaled permeability

The influence of the model parameters has been studiedpreviously only for average upscaled permeabilities.But it is also interesting to estimate some more complexstatistical characteristics like variance or correlationfunction. It allows us to assess an error of the globalupscaling when the effective permeability is used ascoarse block permeability. Also, it allows to check theaccuracy of statistical averaging by using Eq. 7.

In Fig. 15, 3,000 realizations of deformation bandnetworks have been sampled in order to estimate theprobability density of the y component of diagonalupscaled permeability tensor Kups in a coarse block ofone meter by 1 m. The following parameters have beenchosen: a density of 10 deformation bands per meter,KDB/KHR= 0.01, Dc = 0.8, and θ = {π/6, − π/6}. ForIθ = 0, 0.1, and 0.2 m, the effective permeability (aver-aged upscaled permeability) <Kups

yy > is equal to 0.5806,0.5875, and 0.5872; the mean deviation σ

(Kups

yy)

beingequal to 0.026, 0.030, and 0.032, respectively. Figure 15presents the density of the random value Kups

yy . It pointsout that the distribution is essentially non-Gaussian, butthe results for Iθ = 0.1 and 0.2 m are comparable apartfrom the statistical error.

7 Discussion—concluding remarks

7.1 Computational method

A 2D statistical model of fault damage zone architec-ture, incorporating the characteristics of deformationband populations observed in natural examples, hasbeen built in order to investigate fluid flow and per-meability upscaling in fault damage zones containingdeformation bands. The finite volume method devel-oped for the numerical flow computation allows touse a fine mesh with mesh size larger than the thinwidth of deformation bands. This method was usedfor simple 2D rectangular domains with the main flowbeing parallel to the borders. Numerical tests showthat our method keeps the conservation law with highaccuracy, but the upscaling process applied in practicemay represent a computational problem with complexgeometries. Therefore, more accurate numerical meth-ods and tools are needed to reduce the computationalerror. Advanced multipoint approximation methods [1]

Comput Geosci

Fig. 14 Accuracy of theupscaled characteristics:a coarse grid pressure Pc;b flow rates Q in x and ydirections. The measure ofaccuracy is made bycomparing the results (X)with the ones obtained at finescale (Xf): relative error =2|X−Xf||X|+|Xf|

and nonregular grids may be useful in order to pro-duce reliable results. Further development in multiscalemethods may also improve the upscaling methods. But,in many cases, the model error may be the domi-nating one.

7.2 Results

Our results show that the effect of deformation bandson permeability in the fault damage zone depends ontheir spatial distribution (clustering), distribution oforientation, and distribution of permeability anomaliesor “holes” in the bands. While damage zone flow isoverestimated when deformation bands are neglect-ed, quasi 1D models assuming continuous bands with

homogeneous permeability oriented perpendicular tothe flow direction, as done in the reference model, leadsto an underestimation of flow. Figure 16 summarizesthe parameters affecting the permeability in the dam-age zone and illustrates how changes in these parame-ters can make the effective permeability in the mediadiffer from the permeability calculated analytically inthe reference model. More complex and realistic mod-els have been considered through two cases. The firstone considers the effect of adding an orientation to thedeformation bands (horizontal complexity evolution ofthe reference model, Fig. 16). The second one takes intoaccount heterogeneous petrophysical properties alongthe bands. The effect of the different parameters isevaluated by analyzing the relative difference of the

Comput Geosci

Fig. 15 Probability density of the upscaled permeability. Acoarse block of 1 × 1 m is considered. The band density is 10deformation bands per meter, KDB/KHR = 0.01, Dc = 0.8, θ ={π/6, − π/6}; 3,000 realizations of deformation band networkswere sampled to estimate the probability density

effective permeability in the more complex models withthe permeability defined analytically in the referencemodel ((Keff − Kref)/Kref).

Adding different orientations to the deformationbands will change the effective permeability of thesystem. Decreasing the dip angle of the deformationbands will increase the permeability relatively to thatof the reference models where the dip angle is 90◦(Fig. 8). If the orientations of the dip angles (antitheticor synthetic) are correlated, by imposing a correlationlength Iθ , the permeability will be even higher thanwith a random distribution of orientations (Fig. 7).However, results indicate that the influence of thedip angle value remains higher than having correlatedorientations. The discrepancy between the effectivepermeability and reference permeability can be highlyincreased if the density of deformation bands and thepermeabilities are changed. Increasing the deformationband density (number of deformation bands per meter)leads to a substantial increase in the relative differencebetween the effective and the reference permeability.Likewise, decreasing the permeability contrast betweenhost rock and deformation bands KDB/KHR causes highrelative permeability differences with the referencemodel. Moreover, strong clustering of the deformationbands will also increase the relative permeability dif-ference regardless of the other parameters. The roleof all these parameters (KDB/KHR, density, dip angle,correlation of the dip orientation, and clustering) areinterconnected, and the quantification of the effect ofone parameter depends on the values of the other

parameters; that is why we only give a qualitative andrelative influence of all the parameters in Fig. 16.

Another important factor to take into considerationis the heterogeneity of the petrophysical properties.The vertical axis of the graph in Fig. 16 illustrates howvariations in permeability along deformation bands canaffect the effective permeability of the system and de-tach it from the reference permeability. The hetero-geneity of the deformation band permeability is mod-eled by introducing “holes” in the deformation bands.Two cases can be pointed out: the first correspondingto a weak clustering of the bands (Dc = 0.8) andthe second to a strong clustering (Dc = 0.2). If theclustering is weak, the most important parameter isthe degree of fragmentation of the holes. The morefragmented the holes are, the larger the deviation inpermeability from the reference model without hole. Asecondary parameter to consider is the position of theholes. A uniform position (holes aligned in the model)will favor a larger relative difference. On the contrary,if the clustering is strong, the position of the holesalong the bands is the most important parameter, thefragmentation having a minor effect. In damage zonesin porous sandstones, the clustering of the deformationbands has been characterized by a correlation dimen-sion close to 0.8. Moreover, observations indicate thatthe permeability heterogeneity of the bands is morelikely to occur like numerous small holes rather thana large singularity. The reference model is, thus, goingto give a rather unrealistic estimate of the permeability.If both the orientation, the position of the bands, andthe variability in the petrophysical properties of thebands are all taken into consideration (complex modelin Fig. 14), then the reference model can give significanterrors in permeability estimations.

For example, taking 10 deformation bands per meter(average value observed in normal fault damage zonesin sandstones), a permeability contrast of 0.01 and aclustering of 0.8, adding a dip of 60◦ to the bandsand a correlation length Iθ for the dip orientationsof 0.1 m, the relative error made on the permeabilitymeasurement is equal to (Keff − Kref)/Kref = 17%. In-creasing the density to 30 deformation bands, whichis commonly observed in a damage zone close to thefault core, will lead to a relative error of 29%. Theserelative errors are quite realistic according to observa-tions made in the field. If the clustering was stronger,then the relative difference with the reference modelwould be even larger (say, 52% if Dc = 0.2 in oursimulations). These results were obtained using localupscaling with diagonal upscaled permeability tensor.However, tests using upscaling methods with a fulltensor indicate similar trends. The previous examples

Comput Geosci

Fig. 16 Summary of the parameters that affect the permeabil-ity defined by using the reference model. The reference modelcorresponds to deformation bands with no orientation (perpen-dicular to fluid flow) and homogeneous petrophysical proper-ties. Adding an orientation (dip) to the bands will increase theeffective flow in the media compared to the reference model.The difference with the permeability calculated for the referencemodel will be even larger if the permeability contrast betweenthe deformation band and the host rock (KDB/KHR) is small, ifthe density of deformation bands (number of bands per meter)is large, if the dip angle is low, if the orientations of the dipangle are correlated (Iθ �= 0) and if the deformation bands are

clustered (small Dc). Considering heterogeneous petrophysicalproperties along the bands by adding “holes” will also increasethe relative permeability difference with the reference model. Forweak clustering, hole fragmentation is the parameter increasingthe most the effective permeability in the system, whereas forstrong clustering, the position of the holes along the bands isthe most important factor. The complex model corresponds todeformation bands with variable orientations and position in thesystem and presenting heterogeneous petrophysical properties.This is the model presenting the largest relative difference withthe reference model

are given in order to illustrate the underestimation oroverestimation of the effective permeability done ifgeological properties such as variable orientations orheterogeneous petrophysical properties are omitted inthe models.

The question how an uncertainty in the coarse blockpermeability can influence the uncertainty of the pro-ductivity of the whole damage zone is still an openquestion. A simple quasi 1D model is examined inorder to get some insights into the problem. The globaldomain consists in 1 × M coarse blocks (Lx = 1 m,Ly=M m, M = 10–40). We assume no flow boundaryconditions on the borders parallel to the y directionand fixed pressures P1, P2 on the borders parallel to xdirection. Let the block permeability be Ki = αKref +

σKξi, where the coefficient α characterizes the relativedifference between Kref and the effective permeability(α =Keff/Kref, < K> = Keff), σ K the mean deviationof the random variable K, and ξ i the random variablessampled with standard normal distribution. For the ex-ample, the following parameters are taken: the densityis 10 deformation bands per meter, KDB/KHR = 0.01,Dc = 0.8, θ = {π/6, − π/6} and Iθ = 0.1 m. In thiscase, the numerical calculation give α = 1.17, σ K =0.03 darcy. The resulting flux q can be calculated byusing Eqs. 2 and 9. We compare an average flux <q>

with qref—flux through the global domain, where eachcoarse block has a permeability Kref. It is clear that forsmall σK, <q>/qref should be approximately equal α.Numerical calculations show that this ratio decreases

Comput Geosci

slightly with an increase in σ K. For 1.1 ≤ α ≤ 1.5 andσ K ≤ 0.2 Kref, the relative difference |<q>/qref − α|/αdoes not exceed 3.5%.

To estimate the average relative difference< ‖p − pref‖>/‖pref‖, the equations for pressurewith the original and perturbed permeability derivedfrom correspondent finite-volume approximation areused:

Ap = b(A + δA)(p + δp) = b .

Then, it is possible to estimate the relative pressureperturbation:

∥∥A−1∥∥ ‖δA‖ ≥ ‖δp‖

‖p‖ ≥ ∥∥δA−1∥∥−1 ‖A‖−1 . (10)

In practice, Eq. 10 gives a very coarse estimation. Nu-merical calculations show that this relative differenceincreases linearly with an increase in σ K. But, for thesame values of α and σ K, it does not exceed 1.6%.However, we should also expect that the relative errorincreases with increasing coarse block scale and the sizeof global domain.

From a geological point of view, by considering theclustering of Dc = 0.8 found for the damage zone insandstones, the petrophysical properties of the bandsand their distribution have a relatively larger effect thanthe orientation variations of the bands. Future worksshould, thus, focus on better characterizing the petro-physical properties of the faults and more specificallytheir distribution along the deformation bands.

In addition, improved databases for the most sensi-tive parameters have to be developed. As noted earlier,the local upscaling procedure seems to be accurateenough to give the trends caused by the variation ofdifferent parameters. But for more accurate upscalingrequired in specific applications or for more complexmodels, more accurate upscaling tools like the ALG areneeded. In this paper, we have discussed the upscalingproblem in the global domain and shown the advantageof the ALG upscaling in comparison with the pure localupscaling. However, the numerical results of ALG arepresented for only one realization of the bands popula-tion, sampled with given probability distribution. Stud-ies based on ALG or other global upscaling methodsshould be the topic for further research, with a strongerfocus on the quantitative parameters describing themodels.

Acknowledgements The research was carried out as part ofthe Fault Facies Project at Centre for Integrated PetroleumResearch (CIPR), University of Bergen, Norway. The authors

are grateful for financial support from VISTA, a research coop-eration between the Norwegian Academy of Science and Lettersand StatoilHydro; and from The Research Council of Norway,StatoilHydro and ConocoPhillips.

References

1. Aavatsmark, I.: Multipoint flux approximation methods forquadrilateral grids. In: The 9th International Forum onReservoir Simulation, Abu Dhabi, 9–13 December 2007

2. Antonellini, M.A., Aydin, A.: Effect of faulting on fluid flowin porous sandstones: petrophysical properties. AAPG Bull.78(3), 355–377 (1994)

3. Antonellini, M.A., Aydin, A.: Effect of faulting on fluidflow in porous sandstones: geometry and spatial distribution.AAPG Bull. 79(5), 642–671 (1995)

4. Beach, A., Welbon, A.I., Brockbank, P.J., McCallum, J.E.:Reservoir damage around faults: outcrop examples from theSuez rift. Pet. Geosci. 5, 109–116 (1999)

5. Boe, O.: Analysis of the upscaling method based on conser-vation of dissipation. Transp. Porous Media 17, 77–86 (1994)

6. Chen, Y., Durlofsky, L.J., Gerritsen, M., Wen, X.H.: A cou-pled local–global upscaling approach for simulating flow inhighly heterogeneous formations. Adv. Water Resour. 26,1041–1060 (2003)

7. Chen, Y., Durlofsky, L.J.: Adaptative local–global upscal-ing for general flow scenarious in heterogeneous formations.Transp. Porous Media 62, 157–185 (2006)

8. Darcel, C., Bour, O., Davy, P., de Dreuzy, J.R.: Connectiv-ity properties of two-dimensional fracture network with sto-chastic fractal correlation. Water Resour. Res. 39(10), 1272(2003). doi:10.1029/2002WR001628

9. Du Bernard, X., Labaume, P., Darcel, C., Davy, P., Bour, O.:Cataclastic slip band distribution in normal fault damagezones, Nubian sandstones, Suez rift. J. Geophys. 107(B7),2141 (2002)

10. Durlofsky, L.J.: Numerical calculation of equivalent gridblock permeability tensors for heterogeneous porous media.Water Resour. Res. 27, 699–708 (1991)

11. Durlofsky, L.J.: Upscaling and griding of fine scale geologicalmodels for flow simulation. In: The 8th International Forumon Reservoir Simulation, Iles Borromees, Stresa, Italy, 20–24June 2005

12. Flodin, E.A., Aydin, A., Durloksky, L.J., Yeten, B.: Repre-sentation of fault zone permeability in reservoir flow models.In: SPE 71617. SPE, Houston (2001)

13. Fossen, H., Bale, A.: Deformation bands and their influenceon fluid flow. AAPG Bull. 91(12), 1685–1700 (2007)

14. Fossen, H., Schultz, R.A., Shipton, Z.K., Mair, K.: Deforma-tion bands in sandstone: a review. J. Geol. Soc. 164, 755–769(2007)

15. Fowles, J., Burley, S.: Textural and permeability characteris-tics of faulted, high porosity sandstones. Mar. Pet. Geol. 11,608–623 (1994)

16. Grassberger, P., Procaccia, I.: Measuring the strangeness ofstrange attractors. Physica, D 9, 189–208 (1983)

17. Hesthammer, J., Johansen, T.E.S., Watts, L.: Spatial relation-ships within fault damage zones in sandstone. Mar. Pet. Geol.17, 873–893 (2000)

18. Holden, L., Nielsen, B.F.: Global upscaling of permeabilityin heterogeneous reservoirs: the Output Least Square (LOS)method. Transp. Porous Media 40, 115–143 (2000)

Comput Geosci

19. Johansen, T.E.S., Fossen, H.: Internal geometry of fault dam-age zones in interbeded siliciclastic sediments. In: Wibberley,C.A.J., Kurz, W., Imber, J., Holdsworth, R.E., Collettini, C.(eds.) The Internal Structure of Fault Zones: Implicationsfor Mechanical and Fluid-Flow Properties, pp. 35–56. TheGeological Society of London, London (2008)

20. Jourde, H., Flodin, E.A., Aydin, A., Durlofsky, L.J., Wen,X.H.: Computing permeability of fault zones in eolian sand-stones from outcrop measurements. AAPG Bull. 86, 1187–1200 (2002)

21. Knott, S.D., Beach, A., Brockbank, P.J., Lawson Brown, J.,McCallum, J.E., Welbon, A.I.: Spatial and mechanical con-trols on normal fault populations. J. Struct. Geol. 18(2–3),359–372 (1996)

22. Kumar, A., Farmer, C.L., Jerauld, G.R., Li, D.: Efficientupscaling from cores to simulation models. In: SPE 38744Annual Technical Conference and Exhibition. SPE, SanAntonio (1997)

23. Marchuk, G.I.: Methods of Numerical Mathematics, p. 510.Springer, New York (1982)

24. Odling, N.E., Harris, S.D., Knipe, R.J.: Permeability scalingproperties of fault damage zones in siliclastic rocks. J. Struct.Geol. 26(9), 1727–1747 (2004)

25. Pickup, G.E., Ringrose, P.S., Corbett, P.W.M., Jensen, J.L.,Sorbie, K.S.: Geology, geometry and effective flow. Pet.Geosci. 1(1), 37–42 (1995)

26. Renard, P., de Marsily, G.: Calculating effective permeability:a review. Adv. Water Resour. 20, 253–278 (1997)

27. Rubinstein, R.Y.: Simulation and the Monte Carlo Method.Wiley, New York (1981)

28. Sanchez-Vila, X., Grirardi, J., Carrera, J.: A synthesis of ap-proaches to upscaling of hydraulic conductivities. Water Re-sour. Res. 31(4), 867–882 (1995)

29. Shipton, Z.K., Cowie, P.A.: Damage zone and slip-surfaceevolution over [mu]m to km scales in high-porosityNavajo sandstone, Utah. J. Struct. Geol. 23(12), 1825–1844(2001)

30. Shipton, Z.K., James, J.P., Robeson, K.R., Forster, C.B.,Snelgrove, S.: Structural heterogeneity and permeability infaulted eolian sandstones, implications for subsurface mod-elling of faults. AAPG Bull. 86, 863–883 (2002)

31. Sternlof, K.R., Karimi-Fard, M., Pollard, D.D., Durlof-sky, L.J.: Flow and transport effects of compaction bandsin sandstone at scales relevant to aquifer and reser-voir management. Water Resour. Res. 42, W07425 (2006).doi:10.1029/2005WR004664

32. Torabi, A., Fossen, H.: Spatial variation of microstructureand petrophysical properties along deformation bands inreservoir sandstones. AAPG Bull. (2009, in press)

33. Wallace, R.E., Morris, H.T.: Characteristics of faults andshear zones in deep mines. Pure Appl. Geophys. 124(1, 2),107–125 (1986)