Simulation of Transient Well-TestSignatures for Geologically Realistic Faults

in Sandstone ReservoirsStephan K. Matthai, Swiss Federal Inst. of Technology; Atilla Aydin and David D. Pollard, Stanford U.; and

Stephen G. Roberts, Australian Natl. U.

SummaryFor the design of production strategies one must identify how faultsthat are detected in seismic surveys influence fluid flow in hydro-carbon reservoirs. In this study we use the Entrada sandstoneformation, Arches Natl. Park, Utah, as an outcrop analog of afaulted sandstone reservoir. Field measurements of the geometry,thickness, and layered permeability of the normal faults that cross-cut this sandstone are incorporated in two-dimensional (2D) fluid-flow models, facilitating the simulation of fault signatures intransient well tests with a high-resolution finite-element method.

In our simulations, structure and inhomogeneous permeabilitylead to fault signatures in derivative plots that differ significantlyfrom those of idealized impermeable barriers. Highly permeableslip planes facing the well create the illusion of a nonsealing ornonexisting fault because remote fluids are focused along the slipplanes toward the well. Measured low, fault-normal permeabilityshields fault-bounded blocks of the analog sandstone reservoir fromdrawdown and large pressure differentials build up across faultsduring production.

Thus, test data from a single well are insufficient to assess theflow properties of a nearby fault with an inhomogeneous perme-ability like the normal faults in the Arches Natl. Park. Test re-sponses from multiple wells need to be considered to detect faultsegmentation or fault terminations even if the general fault trend isunderpinned by seismic data. Flow paths in the reservoir duringproduction are complex. Importantly, formation water is likely toflow into the reservoir along the permeable slip planes of the normalfaults.

IntroductionHydrocarbon formations are often cross-cut and offset along nor-mal faults, which can act as both flow barriers1,2 and/or conduitsfor fluids.3-7 Whether a particular fault impedes fault-normal fluidflow during production or whether it connects the reservoir to otherpermeable domains in the sedimentary pile cannot be inferred fromthe seismic or well-log data.

Fluid-flow properties of faults may be estimated from transientwell testing.8-12 Common methods of testing examine the changeof fluid pressure in a well while it is being produced at a constantrate (drawdown test) or shut in after a prolonged production period(buildup test). The well tests are interpreted by comparison withanalytical type curves for a range of reservoir geometries andpermeabilities, or nonlinear least squares estimation techniques(automated type-curve or history matching) are used for theestimation of reservoir parameters from well-test data. Type curvesare commonly plotted in derivative plots that display the dimen-sionless pressure in the wellbore,pD, and the dimensionless rate ofpressure change with time multiplied by time,t(p/t) (seeFig.1).13,14Dimensionless time,tD, is normalized for permeability andis frequently divided by the dimensionless wellbore storage,cD,

such that derivative plots obtained for different reservoirs can becompared. Wellbore storage effects are not addressed in this paper.In consistent units,pD and tD can be defined as

pD 52pkh

qBm~pi 2 pw! , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

tD 5kt

fmctrw2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

and ct 5V

dp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

where pi, pw, k, q, B, m, F, ct, rw, V, and C, are the initialreservoir pressure, pressure at the wellbore, permeability, constantrate of production, formation factor, fluid viscosity, porosity, totalsystem compressibility, wellbore radius, fluid volume, and thestorage capacity of the well, respectively.15 These definitions andnotation will be used throughout this analysis.

If a reservoir is homogeneous and infinite, constant-rate pro-duction is accompanied by an initially rapid change of fluidpressure followed by a period of radial flow and constant-ratepressure decline (Fig. 1). Interaction with an inhomogeneity like afault will bring the constant-rate pressure-decline period to an end.The rate of decline will either increase if the fault has a lowerpermeability than the reservoir (Fig. 1; e.g., Ref. 11) or decrease ifthe fault is a highly permeable fracture.16 Such rate changes arereflected in a distortion of the radial drawdown pattern, whichwould persist in an isotropic, infinite reservoir.

The strategy of estimating fault properties by comparing theirsignatures in derivative plots with existing analytical solutions forlinear reservoir inhomogeneities and combinations thereof iswidely applied (see, for instance, Refs. 8, 11, 12, and 44). Com-monly, however, faults represent composites of several semiplanarzones with different deformation structures and hydrological prop-erties.5,6,17-19This topology implies an inhomogeneous fault per-meability for which we did not find analytical solutions in theliterature.

Copyright 1998 Society of Petroleum Engineers

Original SPE manuscript received for review 27 February 1997. Revised manuscriptreceived 14 November 1997. Paper (SPE 38442) peer approved 12 December 1997.

Permission to copy is restricted to an abstract of no more than 300 words.Illustrations may not be copied. The abstract should contain conspicuous acknowl-edgment of where and by whom the paper was presented or published. WriteLibrarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01214-9529435.

Fig. 1Derivative plot13,14 of a typical drawdown test showingthe normalized cumulative change of dimensionless pressurepD 5 (2 p kh/qBm) (pinitial 2 p(t)well) (stippled curve) and thenormalized rate of pressure change (solid curve) plotted againstdimensionless time, tD (for notation see Nomenclature section).

62 SPE Journal, March 1998

In this paper, we incorporate detailed field measurements of thegeometry, inhomogeneous permeability, and porosity of normalfaults in the Entrada sandstone, Arches Natl. Park, Utah (Fig. 2),into 2D, single-phase flow, finite-element models of sandstonereservoirs. Through numerical simulations, we identify the tran-sient well-test signatures of these realistic normal faults withvarious permeability distributions. These signatures are signifi-cantly different from faults with a homogeneous permeability.

First, we describe how we simulate transient well tests in ourmodels, and then we review the field measurements of Antonelliniand Aydin5,6on the dimensions and fluid-flow properties of normalfaults in sandstones. We then present the results of the simulations,

and we conclude with a discussion of well-test interpretations thatare informed by our new results.

Mathematical Model

We use an algebraic multigrid finite-element method20,21 to sim-ulate single-phase fluid flow to a well that is produced at a constantrate, as in a drawdown test (e.g., Ref. 15). To quantify the fluiddiffusivity in the reservoir we need to quantify fluid viscosity,m,permeability,k, and storage capacity,S (storativity) of the heter-ogeneous rock. The storativity,S, describes the change with fluidpressure,p, of the fluid volume,V, stored in the porosity,F, of a

Fig. 2Map and cross-section of the Cache Valley showing the Entrada Sandstone and normal faults and joints, Arches Natl. Park,Utah (adapted from Antonellini and Aydin5,6). (a) Map of normal faults with segmented slip planes and salt anticlines in the CacheValley area. (b) Cross section along the profile A A* through the Cache Valley. The graphs below the profile show measured andinferred permeabilities of joints and faults.

63SPE Journal, March 1998

unit volume of reservoir rock. We calculateSfrom the sand matrixand fluid compressibilities,a and b, respectively, for a constantmatrix mass:

~Vf!

p5 @~1 2 f!a 1 fb# 5 S. . . . . . . . . . . . . . . . . . . . . . . (4)

This constitutive equation is similar to the definition of storativityfor an aquifer in Freeze and Cherry,22 and it represents a simpli-fication of the formulation of Braceet al.,23which also incorporatesthe compressibilities of mineral grains. Because the present-day

compressibility of the Entrada sandstone may have been modifiedduring exhumation and because compressibility varies with con-fining stress (see, for instance, Ref. 24), we use compressibilitymeasurements under in-situ conditions. Dvorkin and Nur25 mea-sured thea of poorly cemented Troll sandstone, which was sampledfrom hydrocarbon reservoirs in the North sea. At a porosity of 25%and an effective pressure of 30 MPa, the matrix compressibilityaof the Troll sandstone is approximately 25 GPa, corresponding toa storativity of 3.13 10210 m3/Pa, for a light oil (API5 45) asa pore fluid.

The finite-element code calculates the spatial and temporalchange in fluid pressure from the pressure diffusion equation with

Fig. 3Spatially variably refined triangular finite-element mesh used in the transient fluid flow simulations: largest element with50-m edge length; smallest element with 0.01-m edge length.

Fig. 4Typical structure of a small normal fault in the Entradaformation (after Antonellini and Aydin5).

Fig. 5Permeability and porosity profile across a deformationband in the Moab member of the Entrada sandstone at DelicateArch viewpoint (from Antonellini and Aydin5).

64 SPE Journal, March 1998

Darcys law

Sp

t5

k

Sm 2p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

We limit this analysis to single-phase flow, and we thereforeconsider a petroleum liquid above the bubblepoint. For commonreservoir conditions such as a fluid pressure of 25 MPa, a temper-ature of 50C, a gas/oil ratio of 0.1, and an oil density of 800kg/m23, an oil viscosity of about 4 cp seems appropriate.26 Thecompressibility of this oil at reservoir conditions is about 1.131029 Pa21.27

To implement the reservoir models, the faulted sandstone unit,in planview, is represented by a 2D finite-element mesh of trian-gular elements (Fig. 3). This mesh is constructed with a specialoctree-based scheme implemented in the commercial mesh-gener-ation tool ICEM TETRA. Spatially variable refinement facilitated

the representation of centimeter-scale features in the kilometer-scale models. Calculated flow properties and an initially uniformfluid pressure are assigned to this mesh. A circular well with aradius of 9 cm [0.3 ft] is represented by eight triangular elementsforming an octagon such that a perimeter correction of 0.97 relativeto a perfectly circular well is applied. Constant-rate production issimulated by specifying fluid sink terms on the well elements. Thepressure at the wellbore is measured at the boundary nodes of thewell region. Temporal and spatial variations in fluid pressure arecomputed with exponentially increasing timestep size(exponent51.2) ranging from 1 second to 3 days. For the largest timestep we stillobtain a signal to numerical noise ratio of 10.9 This treat-ment of reservoir fluid flow relies on the following assumptions:

1. Porosities and permeabilities measured in an outcrop aresimilar to in-situ values.

2. Three-dimensional reservoir flow can be approximated with atwo-dimensional model.

Fig. 6Summary of permeability measurements for zones of deformation bands, Arches Natl. Park and San Raphael desert. khr,kp, and kn are the permeability of the host rock, the deformation-band-parallel permeability, and the normal permeability,respectively.

Fig. 7Permeability of the wall rock of slip planes at Arches Natl. Park. The lower limit of permeability perpendicular to slip planesis unconstrained because of the detection limit of the minipermeameter. khr, kp, and kn are the permeability of the host rock, thedeformation-band-parallel permeability, and the normal permeability, respectively. *

65SPE Journal, March 1998

3. Porosity and permeability of the sandstone are nearly uniformisotropic (f 5 25%, 1000 md, and field variations#1.5 orders ofmagnitude).

4. An effective permeability28 can be applied to compensate forover-representation of slip plane thickness in the model (10 cminstead of 0.5 mm to 3 cm). This correction reduces the perme-ability by one order of magnitude.

5. The fluid can be modeled as slightly compressible single-phase petroleum liquid above the bubblepoint.

6. Compaction of the sandstone during the transient well test isneglected.

Regarding assumption 1, porosity and permeability measured inan outcrop at Arches Natl. Park are similar to in-situ measurementson reservoir sandstones of comparable origin (e.g., Horne15). Theslip planes of the normal faults frequently contain open-spacefillings of carbonate and other minerals that were precipitated andsubsequently deformed while the analog reservoir was submersedand the faults were active, respectively. Thin-sections of the Moabsandstone do not indicate a significant generation of secondaryporosity during uplift and erosion.5,6All these observations suggestthat assumption 1 is permissible and that the simulated faultsignatures should be applicable to in-situ faulted reservoirs.

The 2D representation of the reservoir (2) is adequate in view ofthe planar geometry of the sandstone unit sandwiched betweenimpermeable shale layers, and cross-cut by the faults. The simu-lation of a homogeneous isotropic reservoir sandstone (3) maxi-mizes the sensitivity of our forward simulations with regard todetecting boundaries, because the effect of small-scale inhomoge-neities in the sandstone would be to obscure and blur the fault signalover time. In this sense, faults should be detected with greater easein the simulated well tests rather than in the real tests.

Field Measurements of Sandstone and Normal-FaultFlow PropertiesThe Moab and Slickrock sandstones in the Entrada formation,Arches Natl. Park, Utah (Fig. 2), typify many reservoir sandstones,and their structure, porosity, and permeability have been docu-mented in detail by the field and laboratory work of Antonellini andAydin.5,6 In the Delicate Arch area (Fig. 2), the fluid-flow prop-erties and the geometric attributes of normal faults that cross-cut theEntrada sandstone also have been documented by Antonellini andAydin.5,6Therefore, this area is ideally suited as an analog reservoirfor our simulations.

The Moab and Slickrock sandstones are crossbedded, well-rounded, and well-sorted dune deposits.29 Their porosity variesbetween 4 and 28%, with an average value of about 22%, and theircombined thickness varies between 80 and 200 m. The thickness ofthe Moab sandstone ranges between 20 to 40 m in the Delicate Archarea (typically 30 m). As determined by Antonellini and Aydins5

minipermeameter measurements, the permeability of the unde-formed Moab and Slickrock sandstones ranges over three orders ofmagnitude between 10 and 10 000 md. In most of the Delicate Archarea, this range is narrower, between 100 and 10 000 md. In oursimulations, we treat the porosity and permeability of the unde-formed reservoir sandstone as uniform (25% at 1000 md).

As is typical for the stratigraphy of many sedimentary basins, theMoab and Slickrock sandstones occur between the impermeableDewey Bridge and the Morrison mudstones. These units bound theanalog sandstone reservoir.

Normal Faults as Composite Zones of DeformationBands, Slip Planes, and JointsJoints and deformation bands form the basic structural elements thataccount for small-scale inhomogeneities in the permeability of theMoab/Slickrock sandstones. Herein, the term joint refers to discretefractures with small shear displacements. A more general discus-sion of joints is given in Pollard and Aydin.30 Deformation bands(seeFig. 4) are 1-mm-wide planar zones of crushed grains andreduced permeability and porosity relative to undeformed sand-stone in the reservoir.31,32

Ensembles of tens to hundreds of deformation bands associatedwith slip planes, which tend to occur on the hanging wall side ofthe ensemble, constitute the kilometer-scale normal faults at ArchesNatl. Park (Figs. 2 and 4). At the terminations of these faults, at thepoint where the slip decreases to less than 6 m, fault-parallel jointsare observed in the hanging wall. Along the strike, slip planesterminate into zones of deformation bands and often are associatedwith joints subperpendicular to the slip plane. These two different

TABLE 1MODEL PARAMETERIZATION

model length (X-dimension) 2400 m

model width (Y-dimension) 2400 m

Sandstone (reservoir matrix)

permeability: 1.00E212 m2 (1000 md)

compressibility: 4.16E211 Pa21

porosity: 0.25

Total System Compressibility (Eq. 4)

ct 5 S: 3.10E210 m3/Pa21

Low-Permeability Part of Fault

thickness: 1 m

permeability: 1.00E218 m2 (0.001 md)

S: 1.24E211 m3 Pa21

Slip Plane (if present)

thickness: 0.1 m

(effective) conductivity 2.08E204 m/s21

S: 1.11E209 m3/Pa21

Fluid Properties (at 25MPa, 50C)

API 45

GOR 0.1

density 800 kg/m3

dynamic viscosity 4 cp

compressibility 1.11E209 Pa21

Well Characteristics. Implementation: octagon, 8 triangularfinite elements

wellbore radius 0.09 m

placement, center (50 m from fault)

X-coordinate 1200 m

Y-coordinate 1200 m

well perimeter correction 0.9744953

Model M1

with fault (see left column)

Model M2. (with segmented slip plane)

slip-plane segment length 290 m

slip-plane segment separation 20 m

Model M3. (slip planes and fault as in Model M2)

zone of d.b.s thickness 1 m

zone of d.b.s continuouslength

100 m

gaps in zones of d.b.s 20 m

separation from slip plane 1 m

zone of d.b. permeability 1.00E215 m2 (1 md)

Model M4. (with slip planes continuous along fault segments)

fault segment length 200 m

fault segment overlap 20 m

fault segment separation 20 m

fault permeability 1.00E218 m2 (0.001 md)

Model M5. (with slip planes continuous along fault segments)

fault segment length 290 m

fault segment separation 20 m

fault permeability 1.00E218 m2 (0.001 md)

66 SPE Journal, March 1998

types of joints are expected to increase the permeability parallel andnormal to the zone of deformation bands, respectively.

Permeability of Zones of Deformation Bands.Antonellini andAydin5 define the thickness of the normal fault zones in theMoab/Slickrock sandstones from the extent, normal to the slipplane, of densely clustered, anastomosing zones of deformationbands. This thickness correlates with fault displacement and variesalong fault strike from tens of centimeters to 23 m.

The deformation bands are thin shearbands (millimeters to sev-eral centimeters thick), along which the porosity is reduced and, inmost cases, grains were crushed during brittle shearing.31,32 An-tonellini and Aydin5 measured porosities of 1% to 3% in the centerof the deformation bands accompanied by permeability reductionsof up to three orders of magnitude (Fig. 5). Where zones ofdeformation bands in the Moab sandstone are filled with calcitecement, permeabilities down to the detection limit of the miniper-meameter (1023 md) were measured (Fig. 6).

Slip-Plane Permeability. In the normal faults the slip planes occurat the interface with the hanging wall sandstone. The country rockadjacent to the slip planes shows varying degrees of cataclasis andcarbonate cementation. The two surfaces of a slip plane are smoothand polished with thin lineations (slickensides). Perpendicular tothe slip direction, the surfaces are wavy with a wavelength of 2 to15 cm and an amplitude of 1 cm or less. The map pattern of slipplanes is segmented on a scale ranging from tens of meters to 1 km(Fig. 2). Curvilinear slip planes with a strike length of up to a fewhundred meters also occur. The slip planes are straight in cross-section, and their present-day aperture varies from less than 1 mmto a few centimeters. Local in-situ dilatation of slip planes up to 15cm is indicated by open space fillings of calcite, which also maybe considered evidence for preferential fluid circulation in thefaults. Dilatation normal to the slip plane is expected because thisis the direction of the least principal compressive stress duringfaulting.

In view of these constraints it appears reasonable to assign anaverage in-situ aperture of 1 cm to slip planes in models of largernormal faults at Arches Natl. Park (Fig. 2). This aperture falls in themidrange of Cruishanket al.s33 aperture estimates for shearedjoints in the Entrada sandstone. It implies a high fault-parallelpermeability.34,35 However, because the slip planes tend to besegmented, they should represent discontinuous fault-parallel flowconduits.

Fluid flow in the slip plane is treated as flow between parallelplates36,37 leading to an intrinsic permeabilityk

k 5d2

12, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

whered is the mean fracture aperture.38 Brown39 compared thepermeability predicted by this approximation with solutions to the

Reynolds equation for fluid flow through rough-walled fractureswith contacting asperities. His results indicate that fracture flow isapproximated by the parallel-plate law within a factor of 2. Forwider and open fractures (d $ 100 mm) deviations further dimin-ish.39Renshaw40demonstrated the validity of the parallel-plate lawfor fractures with an average critical aperture greater than 1025 m

Fig. 82D, transient fluid-flow Model M1 of faulted Moab sand-stone in a horizontal cross-section through the analog reservoir.The fault is 15 m wide, straight, with a uniform low permeabilityof 1023 md (10218 m2). The well is located at a horizontal distanceof 50 m from the fault. The permeability of the sandstone, 1000md (10212 m2), is six orders of magnitude higher than that of thefault representing a zone of altered deformation bands withouta slip plane

Fig. 9Derivative plots for constant-rate production of the wellin Model M1. (a) The upper curve (diamonds) shows the nor-malized cumulative change in wellhead pressure, and the lowercurve (square data points) shows the normalized rate of pres-sure change. These characteristic curves are superposed ontocurves computed for a homogeneous isotropic model (solidlines), such that deviations after t 5 3 minutes define the faultsignature. The reference curves also indicate when interactionwith the model boundaries begins to show in the well response.This interaction is indicated by the rise of the rate of pressurechange after 1.4 days. (b) Comparison of the derivative curvefrom Model M1 with analytical solutions for finite conductivityand sealing faults (modified Fig. 9, p. 594, Yaxley11). Here aA isthe specific transmissibility ratio between the fault and theformation with respect to the distance, b, between the well andthe fault (aA 5 (kfh/lfm)/(kh/bm)). In contrast with the derivativecurve from Model M1 (Fig. 9), the analytical solutions are derivedfor a fault of infinite length. Therefore, the derivative curve forModel M1 reaches a lower maximum value and decays morerapidly than the analytical solutions after tDA 5 10 and beforeinteraction with the model boundary begins.

67SPE Journal, March 1998

and a typical surface roughness. He argued that in fractures withcontacting asperities, only a small percentage of the fracture planeis in contact. Otherwise one should not be able to observe thefrequently described lognormal or fractal distributions of fractureaperture (e.g., Ref. 41). In our models, we treat slip planes as if theirsurfaces were in contact only over a small percentage of their area.This treatment leads to slip-plane conductivities for the laminarflow of oil, ranging from 0.001 to 10 m/s21 for apertures of 0.1 and1 cm, respectively.

We represent each slip plane by a mesh of triangular finiteelements with fixed width. For 1-cm-wide slip planes, the width ofthe slip plane in the model is overrepresented by a factor of 10. We

compensate for this overrepresentation by a reduction of the slip-plane normal permeability by one order of magnitude using thetreatment recommended by Deutsch.28

Permeability of Wall Rock Adjacent to Slip Planes. In theSlickrock and Moab sandstones the permeability of slip-planewallrock is approximately 1 md and less than 1023 md, respectively(seeFig. 7). The latter value is an upper bound, because it justreflects the detection limit of the minipermeameter. The porosity inthe wallrock adjacent to well-developed slip planes along faultzones in sandstone is very low (%1%) because of pronounced graincrushing, recrystallization, and calcite precipitation in the pores of

Fig. 10Spatial variation in fluid pressure 5 minutes (a), 3 hours (b), and 1.4 days (c) after the onset of production from Model M1.Fluid pressure is displayed in a banded gray scale and juxtaposed pressure contours. (a) Fault-induced perturbation of the radialdrawdown when a noticeable increase in the rate of pressure change occurs (see derivative plot, Fig. 9). (b) Spatial variation of fluidpressure at time t 5 3 hours when a fault-induced increase in the second derivative of pressure approaches its maximum. (c) Asignificant pressure differential has built up across the fault when interaction with the no-flow boundaries leads to an acceleratedpressure decrease in the well.

68 SPE Journal, March 1998

the sandstone in some cases. A correspondingly lower storativity of1.24 3 10211 m3/Pa21 as calculated from Eq. 1 is thereforeassigned to regions representing slip-plane wallrock.

Slip-Plane Storativity. The slip-plane storativity is important be-cause it influences how much fluid the slip plane can supply orconsume during pressure changes, independently from the countryrock. Thus, in our simulations, we implement discrete slip-planestorativities calculated from the fluid compressibility,Z, and ana-lytical solutions for the compressibility of straight open fractureswithout contacting asperities.42From these solutions, the slip-planestorativity,Ss, is derived as

SsVsp

1 Vsbdp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

Vs 52 pa3~v 2 1!~p 1 syy!

mm, . . . . . . . . . . . . . . . . . . . . . . . (8)

and

Vsp

52 pa3~v 2 1!

mm, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

whereVs, n, mm, andsyy are the slip-plane volume, Poissons ratio,shear modulus, and the slip-plane normal stress, respectively;a isthe half-length of the fracture. We use a shear modulus of 15 GPaand a Poissons ratio of 0.25, as measured on the Troll sandstonefrom the North Sea.25

Because the effect of contacting asperities on fracture compress-ibility is ignored, this treatment overemphasizes slip-plane storat-ivity. We find, however, that the slip-plane storativities calculatedfrom Eq. 4 to Eq. 6 exceed the storativity of the sandstone only if

the slip planes extend for more than 100 m along a strike. In ourmodels we calculate slip-plane storativity according to the strikelength of the slip planes.

The composite normal faults are implemented with a fixed widthof 1.1 m. Within this zone we resolve a discrete slip plane (0.1 m)and an altered-wallrock/zones-of-deformation-band assembly (1m). These two fault components are assigned effective permeabili-ties, as calculated from the parallel-plate approximation for fractureflow and the permeability measurements of Antonellini and Ay-din5,6 on deformation bands and the undeformed sandstone, re-spectively. The effective permeabilities are calculated with thepower-averaging method of Deutsch.28

ResultsSimulations are presented for four rectangular models that aremodifications of a reference model for comparison with the ana-lytical solution for a fault as an impervious reservoir boundary(Table 1, Fig. 2). In the reference model, M1 (24003 2400 m)shown in Figs. 4 and8, the fault is a continuous 1-m-wide zonealong which the permeability is reduced by six orders of magnituderelative to the reservoir sandstone. The four other models, M2through M5, cover the variability in geometry and permeability ofthe normal faults in the Arches Natl. Park (see Fig. 2). These modelshave the same size and central well placement at a distance of 50 mfrom the fault as Model M1. The faulted reservoir is represented inplanview, and in the illustrations only the central part is shown(Observation Area, see Figs. 8, 11, 14, 17, and 20). A uniforminitial fluid pressure and no-flow boundary conditions are assigned.

Because no flow is permitted across the outer model boundaries,the rate of pressure change increases when this boundary is en-countered by the pressure perturbation spreading from the well. Forthe chosen model dimensions and well placement this interactionbegins after 1.4 days of drainage from the well when the signal ofthe fault has disappeared or ceases in the derivative plots. This timespan does, however, vary among the models due to the presence orabsence of slip planes in the faults, as will become evident by meansof the illustrations of drawdown pressure.

As discussed earlier, constant-rate fluid sinks are specified on theelements forming the circular well. Time-dependent fluid pressureat the well in Models M1 through M5 is visualized in derivativeplots (Fig. 1). For reference, all of these graphs also show theresponse of an isotropic, uniform, homogeneous model in solidlines without datapoints (see, for instance,Fig. 9a).

These lines identify when the interaction with model boundariesbegins and when the presence of the fault creates a significantdeviation from the behavior of a homogeneous isotropic reservoir.

Model M1: Low-Permeability Fault. The curves for Model M1compare well with the analytical solutions for partially communi-cating faults,11 as is shown in Fig. 9b. This good fit demonstratesthat our numerical model is suited for the simulation of transientwell tests and indicates that the permeability contrast betweenundeformed and altered/deformed Moab sandstone in the normalfaults suffices to compartmentalize fluid pressure in the sandstonereservoir (Fig. 10c). For the 1-m-wide fault with a permeability of10218m2 in Model M1, the magnitude of the normalized derivativedeviation (0.9) in Fig. 9b is close to that of a completely imper-meable boundary (derivative5 1.0). A sensitivity analysis showsthat an effect of the fault on the derivative of pressure is visible inthe simulated tests up to a minimum permeability contrast of oneorder of magnitude between the fault and the country rock.

In Model M1, after 1.4 days of production, the fluid pressure inthe rock on the far side of the fault has changed by less than12%of the pressure difference between the unperturbed reservoir pres-sure and the pressure at the wellhead. This compartmentalization isinstrumentalized solely by the cataclastic deformation of the sand-stone and/or the diagenetic carbonate cementation. It can thereforeoccur in faults cutting sandstone reservoirs, independently fromother physical mechanisms, such as the smearing of shale along thefault from shale layers above and below the sandstone layer. Thespatial variation of fluid pressure in Models M1 to M5, as induced

Fig. 11Two dimensional, transient fluid-flow Model M2 offaulted Moab sandstone with a 1-m-wide, straight fault with auniform low permeability of 1023 md and a 10-cm-wide perme-ability equivalent of a 1-cm-wide slip plane with an effectiveconductivity of 2.08 3 1024 m/s21. The slip plane has a 20-m-wide gap.

Fig. 12Derivative plot for a simulated drawdown test inModel M2.

69SPE Journal, March 1998

by fluid withdrawal from the well, is displayed in reservoir snap-shots at 4.5 minutes (directly after the onset of fault interaction, e.g.,Fig. 10a), after 3 hours of production (when the rate of pressurechange accelerates or reaches a maximum, e.g., Fig. 10b), and after1.4 days (before the interaction with the model boundaries becomesappreciable, e.g., Fig. 10c).

Except for Model M4, for which a simple gray scale is used, thepressure plots use a banded gray scale scheme to emphasize smallspatial variations in fluid pressure. The bands are comparable toisobars in fluid pressure, and some of the plots contain additionalcontour and vector overlays illustrating absolute changes in fluidpressure and the directions of fluid flow in the reservoir, respectively.

Model M2: Continuous Normal Fault With Slip Plane Facingthe Well. Model M2 is identical to M1 except that it contains the10-cm-wide permeability equivalent of a 1-cm-open, segmented

slip plane in the hanging wall of the fault (seeFig. 11). The twoslip-plane segments have a strike length of 290 m and are separatedby a 20-m gap located at the shortest distance between the well andthe fault (Table 1).

In contrast to Model M1, the derivative plot for Model M2 inFig.12 indicates a short-lived, but pronounced, decrease in the rate ofpressure decline ($0.1 at 25 minutes) as the well interacts withthe fault.

The decrease in the pressure derivative accompanies rapidspreading of the pressure perturbation along the slip plane in thehanging wall of the fault (Figs. 13a and 13b). The fluid pressurein the slip plane next to the well stays nearly constant.

The fault behaves similarly to a constant pressure boundary.Twenty-five minutes after the onset of production (Fig. 13b), thepressure pattern near the well has developed, reflecting the seg-mentation of the slip plane. Although the earlier flow was focused

Fig. 13Spatial variation in fluid pressure in Model M2. (a) 6.5 minutes, (b) 25 minutes, and (c) 1.4 days after the onset of production.Fluid pressure is displayed in a banded gray scale, and the vectors indicate the direction of fluid flow. The snapshots of the spatialvariation in fluid pressure correspond to when (a) a pressure perturbation induced by drawdown reaches the fault; (b) a fault-induceddecrease in the second derivative of pressure reaches a maximum; and (c) the drawdown begins to interact with model boundaries.

70 SPE Journal, March 1998

radially into the well, the slip plane is now underpressured relativeto the country rock along its strike such that fluid in the sandstonereservoir at a greater distance to the well first flows into the slipplane and then is focused toward the well. After 25 minutes,semiradial drawdown patterns develop at the far tips of the slipplanes around the fault terminations (Fig. 13b). As drainage con-tinues, interaction with the lateral no-flow boundaries of the modelbegins earlier than in Model M1, leading to a final increase in therate of pressure decline. As in Model M1, within the first day ofdrainage there is little change in the fluid pressure on the far sideof the well (Fig. 13c). The fault compartmentalizes fluid pressurein the reservoir.

Importantly, we can infer with certainty that the changes in thepressure derivative are related to rapid flow along the slip plane ofthe fault, only because structure and fluid-flow properties of ModelM2 are known. A comparison of this result with the derivative plotfor Model M1 in Fig. 9 illustrates that single-well transient testingonly provides nondirectional (scalar type) information on the flowproperties of the reservoir. Therefore, given the derivative plot ofa single well, we cannot discriminate between a fault that permitscrossflow and a sealed fault that has a moderately, but not highly,permeable slip plane facing the well. Such discrimination is,however, vital for the design of well placement and reservoirexploitation strategies. Also, because highly permeable slip planesare likely to provide flow connections to the stratigraphy above andbelow the reservoir unit, they represent a high risk for water coningduring reservoir recovery. Such potential fluid supply from thestrata above and below is not considered in Model M2.

Model M3: Continuous Normal Fault With Shielded SlipPlane. Model M3 is similar to Model M2, but the slip plane iscontinuous and a 1-m-wide zone of deformation bands in the

hanging wall of the fault intermittently shields the slip plane againstthe undeformed reservoir sandstone (Fig. 13). There are four20-m-wide gaps in the shield over the length of the fault. In thesegaps the highly permeable slip plane is in contact with the unde-formed sandstone. This arrangement of structures is frequentlyobserved near fault terminations or overlapping segments of normalfaults in the Arches Natl. Park5 (Fig. 2), and it is comparable to whatis known as the skin effect in petroleum engineering. Theeffective permeability of the hanging-wall zone of deformationbands is simulated as a reduction by three orders of magnituderelative to the undeformed sandstone (see Fig. 6).

The derivative plot inFig. 15shows that the shielding of the slipplane leads to a fault signature that shows a rise somewhat similarin shape to Model M1 (Fig. 9). However, the increase in the rateof pressure change is much less pronounced (#0.15 log units at 6.1hours). In the idealized model this increase is distinct, but it isquestionable whether it would be identified as a fault signature ina field situation where other reservoir inhomogeneities are present.

Compared with Model M2, significantly less fluid is supplied tothe well, because the amount of fluid in the slip plane itself is smallcompared to the reservoir storage. Thus, a pronounced slip-planeeffect is seen only when the slip plane can draw fluid over a largearea from highly permeable reservoir rock along the fault strike.Because of the shielding, the slip plane in Model M3 can draw fluidfrom the reservoir sandstone only at the gaps in the shield along thefault strike and at the fault terminations. At these terminations aradial drawdown pattern develops (Fig. 16).

Model M4: Segmented, Stepping Normal Fault.Model M4represents a right-stepping normal fault with slip planes in thehanging wall (Fig. 17). The slip planes are continuous over thethree, 200-m-long fault segments, which are separated by 10-m-wide rock bridges providing connections to the sandstone on thefar-side of the fault. The distance of the well to the central faultsegment is 50 m, as in the other flow models.

If the highly permeable slip planes are omitted from Model M4,the segmented fault has a signature shown in the derivative plot inFig. 18(circle symbols) that is similar to a continuous uniform faultwith a low permeability (normalized derivative maximum 0.82 at1.1 hours; see Figs. 2 and 10).

The presence of the highly permeable slip planes in Model M4,however, masks the low permeability of the fault segments. In thiscase there is a slight negative pressure-derivative response (#0.3log units) within the time window of observation (Fig. 18, squaresymbols). The spatial variation in fluid pressure with time indicatesthe development of enhanced fluid pressure gradients across thefault-facing rock bridges between the slip-plane segments (Figs.19a and 19b). In these zones the flow is defocused, leaving theslip-plane segments distant to the well and entering the slip-planesegment close to the well. This is illustrated in a pressure-gradientvector overlay on an image of the spatial pressure variation in theleft segment stepover of the fault (Fig. 19c). The pressure gradientvectors in the overlay point in the direction of fluid flow. Thesevectors also indicate the focusing toward the well of fluid from thefar side of the fault segments.

Model M5: Segmented, Normal Fault with Slip Plane on FarSide of Well. Simulations for a segmented fault with the slip planefacing away from the well (Fig. 20) show that the presence of theslip plane enhances the coupling with the reservoir on the far sideof the fault. This leads to a strong attenuation (derivative change#0.1 vs. 0.9 in Model M1) of the normalized rise in the derivativeof fluid pressure shown inFig. 21. It should, therefore, be impos-sible or more difficult to identify segmented normal faults intransient well tests on single wells when a slip plane facing awayfrom the well is present. Also, in Model M5, the drainage regiongrows rapidly along the fault strike on the side of the slip plane (Fig.22). This implies a risk of inflow of formation water into thereservoir, especially if oil is not present on the hanging-wall sideof the fault.

Fig. 14Two-dimensional, transient fluid-flow Model M3 offaulted Moab sandstone with a 1-m-wide, straight fault, with apermeability of 1023 md, a 10-cm-wide permeability equivalentof a 1-cm-wide slip plane, and a 1-m-wide discontinuous zoneof deformation bands with a reduced permeability of 1 mdshielding the slip plane.

Fig. 15Derivative plot for simulated drawdown test in Model M3.

71SPE Journal, March 1998

DiscussionThe fluid-flow numerical Models M2 to M5 contain idealized butrealistic representations of the normal faults with a variety ofcomplex internal architectures, as established by the previousstudies of fault geometry and flow properties in Arches Natl.Park5,6 (Fig. 2). Although our finite-element method reproducesanalytically determined drawdown curves for a homogeneous iso-tropic reservoir and impermeable43 and low-permeability bound-aries11,12(Fig. 9), we observe previously not described signaturesfor the realistic model representations of the normal faults withdiscontinuous slip planes and with slip planes shielded by zones ofdeformation bands.

These new signatures are based on permeability-porosity mea-surements obtained after uplift and erosion. However, these valuesclosely match in-situ measurements on highly permeable sandstoneformations (e.g., Ref. 15). The corrugations, fibrous crystal coat-ings, and open space-filling carbonate precipitates in the slip planes

of the normal faults indicate that these were open fractures whenthe faults were active and that the Moab sandstone was submersed.Our simulation results should, therefore, apply to producing faultedsandstone reservoirs.

Our analysis further indicates that transient pressure data from asingle well are insufficient to assess the flow properties of faultsthat have an inhomogeneous permeability. This is due to thenondirectional (scalar) nature of any information on flow propertiesthat can be extracted from test data derived from a single well. Thus,

Fig. 16Spatial variation in fluid pressure in Model M3. (a) 4.5 minutes and (b) 1.4 days after the onset of drawdown. The snapshotsof the spatial variation in fluid pressure correspond to when (a) pressure perturbation reaches the fault and (b) the fault-inducedpressure increase reaches a maximum.

Fig. 17Two-dimensional, transient fluid-flow Model M4 ofMoab sandstone with a 1 m-wide, right-stepping, segmentedfault with a uniform permeability of 1023 md and 10-cm-widepermeability equivalent of 1-cm-wide slip planes facing the well.

Fig. 18Derivative plot for simulated drawdown test in ModelM4. Two sets of curves are shown for Model M4 with and withouthighly permeable slip planes in the hanging wall of the seg-mented fault, respectively.

72 SPE Journal, March 1998

we would estimate that the normal faults at Arches Natl. Park arehighly permeable for crossflow if the test well was placed on thehanging-wall side of a segmented fault, and we would use theanalytical method of Yaxley11 to determine fault permeability fromthe magnitude of the derivative deviation from radial flow. Only bymeans of interference tests of wells on either side of the fault shouldwell testing be able to identify whether a flow connection existsbetween adjacent fault blocks. This applies, however, only if the

well test or the production curve spans a sufficiently long period oftime. If the possibility of fault crossflow has been discounted andif the fault gives no sealing response, pressure data from wells alongthe fault strike have to be examined to evaluate whether the pressureperturbation induced in the well test is transmitted rapidly along aslip plane like in Model M2.

In our 2D Model M3 (Figs. 14 to 16), the partial shielding of theslip plane by zones of deformation bands, comparable to a skin

Fig. 19Spatial variation in fluid pressure in Model M4. (a) 1.1 hours, (b) 1.4 days, and (c) 1.4 days after the onset of drawdown atthe left stepover region along the segmented fault. The snapshots of the spatial variation in fluid pressure correspond to when (a)a fault-induced decrease in second derivative of pressure occurs in the presence of the slip plane and (b, c) the pressure derivativereaches a minimum.

73SPE Journal, March 1998

Fig. 20Two-dimensional, transient fluid-flow Model M5 ofMoab sandstone with a 15-m-wide, segmented fault with auniform permeability of 1023 md and a 10-cm-wide permeabilityequivalent of a 1-cm-wide slip plane facing away from the well.

Fig. 21Derivative plot for simulated drawdown test in Model M5.

Fig. 22Spatial variation in fluid pressure in Model M5. (a) 5 minutes, (b) 25 minutes, and (c) 1.4 days after the onset of production.Note that the pressure gradients are largest across the fault that forms a quasi-impermeable barrier. In (c), after 1.4 days of drainage,significant amounts of fluid are focused through the gap in the fault. Most of this fluid is derived from along the slip plane.

74 SPE Journal, March 1998

around a well, partly restored the characteristic response of animpermeable fault. However, fluid from within the slip plane wasfocused toward the well. In a layered sedimentary sequence, thisbehavior may imply that fluid invades the reservoir from permeablestrata above and/or below the sandstone unit. This fluid is likely tobe formation water.

Our models are idealized as they do not include potential het-erogeneities in the properties of the sandstone or small-scalestructural inhomogeneities such as joints and deformation bands.Such inhomogeneities are expected to distort the radial drawdownpattern before fault interaction and thereby to obscure or add noiseto fault signatures.21 In a real reservoir, the pressure response willnot be unique, for instance, if open joints are present in thesandstone near the well. These will induce derivative variationssimilar to the slip plane in Model M2.16 This highlights theimportance of a close geological characterization of the geometryand the subseismic inhomogeneities in producing reservoirs.

ConclusionsNormal faults in the Entrada sandstone, Arches Natl. Park, Utah,represent ensembles of zones, meter- to tens-of-meter-wide zones,of reduced permeability and porosity (1 to#1023 md at 1 to 5%vs. 1000 md at 25%) on the footwall side of the fault and segmentedbut highly permeable slip planes forming the interface to thehanging wall. This inhomogeneous permeability leads to typecurves that differ strongly from theoretically inferred fault re-sponses.

1. If highly permeable slip planes face the well and are notshielded by 1-m-wide zones of deformation bands, the increase inthe pressure derivative that is characteristic of low-permeabilitybarriers disappears or is even inverted.

2. Rapid flow along the slip plane can create the illusion that thefault is highly permeable, whereas the sandstone reservoir on thefar side of the fault is not drained because it is shielded by thelow-permeability zone in the footwall of the fault.

3. If the fault is discontinuous or stepping, even slip planes facingaway from the well affect the type curves. They improve thecoupling between the tested reservoir block and the reservoir blockon the opposite side of the fault such that the fault-associatedincrease in the pressure derivative is smaller or absent. Slip planesgenerally act as rapid conductors of drawdown-induced pressureperturbations. This makes them prone to focus remote fluids pro-moting water coning.

4. Provided that the normal faults are continuous,$1 m thick,and that the slip planes face away from the well, measured negativefault-permeability deviations of greater than or equal to three ordersof magnitude relative to the reservoir sandstone produce a responseof an impermeable barrier in the simulated drawdown tests. Dis-continuous, low-permeability faults with gaps still show detect-able, but weaker responses than continuous faults without slipplanes.

5. Fault zones with permeabilities reduced by four to six ordersof magnitude due to footwall cementation, recrystallization, andcalcite precipitation compartmentalize fluid pressure in the analogsandstone reservoir and prevent or postpone production of adjacentfault blocks.

Nomenclaturea 5 matrix compressibilityb 5 fluid compressibilitym 5 fluid viscosity

mm 5 shear modulusn 5 Poissons ratio

syy 5 slip plane-normal stressF 5 porositya 5 half length of fracture (slip plane)B 5 formation factorc 5 wellbore storagect 5 total system compressibilityh 5 formation thicknessk 5 permeability

p 5 fluid pressurepi 5 initial reservoir pressure

pw 5 pressure at the wellboreq 5 constant rate of production

rw 5 wellbore radiusS 5 storage capacityt 5 time

V 5 volume

SubscriptsD 5 dimensionlessf 5 fluids 5 slip plane

AcknowledgmentsThis research was supported by the Rock Fracture Project, StanfordU. The authors thank Fikri Kuchuk and Roland Horne for theirhelpful comments and review of this work. Special thanks are dueto Albrecht Honecker from ICEM CFD Engineering Ltd. for thegeneration of the highly refined triangular finite-element meshes.

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SI Metric Conversion Factorscp 3 1.0* E203 5 Pa z sft 3 3.048* E201 5 m

ft3 3 2.831 685 E202 5m3

F (F232)/1.8 5 Cin. 3 2.54* E100 5 cm

lbm 3 4.535 924 E201 5 kgmd 3 9.869 223 E204 5 mm2

psi 3 6.894 757 E100 5 kPa

*Conversion factor is exact SPEJ

Stephan K. Matthai is a research associate at the Dept. of EarthSciences, Swiss Federal Inst. of Technology in Zurich, Switzer-land. He previously conducted postdoctoral research at Cor-nell U., Ithaca, New York, and as a member of the Rock FractureProject at Stanford U., Stanford, California. Matthai holds a Dip.degree in structural geology from the Eberhardt Karls U., Tub-ingen, Germany, and a PhD degree from the Australian Natl. U.,Canberra. Atilla Aydin is Professor of geomechanics and struc-tural engineering, Codirector of the Rock Fracture Project, andDirector of the Shale Smear Project at Stanford U. His researchinterests include fluid flow through fractures and faults withapplications to hydrocarbon migration and recovery; waste-containment problems; physical processes of faulting andfault-seal potential; and the characterization of naturally frac-tured reservoirs and aquifers. Aydin holds a BS degree in geo-logical engineering from Istanbul Technical U. and MS and PhDdegrees in geology from Stanford U. David D. Pollard is Professorin the Geological and Environmental Sciences Dept., School ofEarth Sciences, Stanford U., and Codirector of the StanfordRock Fracture Project. He supervises research focusing on un-derstanding rock fracturing and faulting with applications tofluid flow in structurally heterogeneous reservoirs and aquifers.Pollard holds a BA degree in geology from Pomona College,and MS degree in structural geology and rock mechanics fromImperial College and a PhD degree in geology from StanfordU. Stephen G. Roberts is a lecturer in the Mathematics Dept. atAustralian Natl. U., with research interests in efficient numericalsolution of partial-differential equations associated with fluid-flow problems and geometric-evolution equations. Robertsholds a PhD degree in mathematics from the U. of California,Berkeley. Author photographs are unavailable.

76 SPE Journal, March 1998