Computational modeling of fluid flow through a fracture in permeable rock

$: Computational modeling of fluid flow through a fracture in permeable rock$
Transp Porous Med (2010) 84:493–510DOI 10.1007/s11242-009-9516-9

Computational Modeling of Fluid Flow througha Fracture in Permeable Rock

Dustin Crandall · Goodarz Ahmadi · Duane H. Smith

Received: 15 April 2009 / Accepted: 2 December 2009 / Published online: 8 January 2010© Springer Science+Business Media B.V. 2010

Abstract Laminar, single-phase, finite-volume solutions to the Navier–Stokes equationsof fluid flow through a fracture within permeable media have been obtained. The fracturegeometry was acquired from computed tomography scans of a fracture in Berea sandstone,capturing the small-scale roughness of these natural fluid conduits. First, the roughness ofthe two-dimensional fracture profiles was analyzed and shown to be similar to Brownianfractal structures. The permeability and tortuosity of each fracture profile was determinedfrom simulations of fluid flow through these geometries with impermeable fracture walls. Asurrounding permeable medium, assumed to obey Darcy’s Law with permeabilities from 0.2to 2,000 millidarcies, was then included in the analysis. A series of simulations for flows infractured permeable rocks was performed, and the results were used to develop a relationshipbetween the flow rate and pressure loss for fractures in porous rocks. The resulting friction-factor, which accounts for the fracture geometric properties, is similar to the cubic law; it hasthe potential to be of use in discrete fracture reservoir-scale simulations of fluid flow throughhighly fractured geologic formations with appreciable matrix permeability. The observedfluid flow from the surrounding permeable medium to the fracture was significant when theresistance within the fracture and the medium were of the same order. An increase in thevolumetric flow rate within the fracture profile increased by more than 5% was observed forflows within high permeability-fractured porous media.

Keywords Fractured porous media · Single-phase flow · Friction-factor · Cubic law ·Navier–Stokes CFD

D. Crandall · G. AhmadiMechanical and Aeronautical Engineering Department, Clarkson University,Potsdam, NY, 13699-5665, USAe-mail: [email protected]

D. Crandall (B) · D. H. SmithNational Energy Technology Laboratory, United States Department of Energy,Morgantown, WV, 26507-0880, USAe-mail: [email protected]

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1 Introduction

Understanding how fluids are transported through fractures is important for assessing andplanning subsurface activities, such as the geologic sequestration of CO2, remediation of sub-surface contaminated zones, geothermal power generation, designs for nuclear waste repos-itories, and the recovery of oil and natural gas (National Research Council 1996; Selrooset al. 2002; Berkowitz 2002; U.S. Department of Energy 2007). Fractures provide prefer-ential pathways for fluid migration through rocks with low permeability, with a lower flowresistance than the surrounding porous medium. Many of the world’s oil and gas reserves arewithin fractured reservoirs (National Research Council 1996) and most geothermal reservoirsare fractured (O’Sullivan et al. 2001). In order to accurately predict fluid transport behaviorthrough highly fractured reservoirs at the kilometer scale, a thorough understanding of theprocesses that affect fluid transport through a single fracture must be known.

Natural fractures in rock have complex geometries. Over the past few decades, numerousmeasurements of fracture-wall roughness have been performed to characterize the geom-etry of fractures. Fracture roughness has been obtained from measurements using stylusprofilometers (Tsang 1984; Poon et al. 1992; Al-Yaarubi et al. 2005), light transmissiontechniques (Brown et al. 1998; Detwiler et al. 1999), confocal laser scanning microscopes(Chae et al. 2004), and X-ray computerized tomography (CT) scanning (Hirono et al. 2003;Rangel-German et al. 2006; Karpyn et al. 2007). All of these methods have shown that thefracture geometry is highly complex, with many showing that the structure of fractures iswell-described by non-Euclidian fractal geometry (Odling 1994; Kulatilake and Um 1999;Lanaro 2000). Self-affine synthetic fracture geometries have been created using this knowl-edge and used for experimental and numerical studies (e.g., Wang et al. 1988; Roux et al. 1998;Mourzenko et al. 2001). The fracture profile geometries used for this study were obtainedfrom sections of a micro-CT-scanned fracture within Berea sandstone (Karpyn et al. 2007),capturing much of the complex geometry of this fracture in a natural porous medium.

Numerous numerical models have been used to describe the flow through a single fracture.Often the fracture is described using a fairly simple geometry, such as a composite set ofperpendicular channels (Ingham et al. 2006) or a penny-shaped crack used to analyticallyevaluate the propagation of cracks due to high-pressure fluid flow (Detournay 2004). Whenthe area of study is on the reservoir scale, a high level of approximation is required due to theinherent computational cost of modeling flow through thousands of fractures (McKoy andSams 1997). Discrete fracture simulators often assume fractures as having closely spacedparallel walls. The accuracy of such simulations may be improved by inclusion of phenome-nological equations for flow through real fractures. For single-fracture studies that focus ondescribing fluid flow within a fractured piece of rock, a geometry that captures the compli-cated structure of naturally occurring fractures is desired.

Similar to the treatment of flow through porous media, various simplified mathemati-cal relationships have been examined to describe flow through fractures. The well-knownDarcy’s equation, phenomenologically determined over a century ago, has been used exten-sively to describe the volumetric flow Q [L3/T], through various porous media under animposed pressure head ∇ P[M/L2T2],

Q = κ

(A

µ

)∇ P. (1)

Here, µ [M/LT] is the fluid dynamic viscosity, A [L2] is the cross-sectional area perpendicularto the mean flow, and κ[L2] is the permeability of the medium.

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Computational Modeling of Fluid Flow 495

The cubic-law is the solution of the Navier–Stokes equations for viscous flow throughvery wide, closely spaced parallel plates (with no-slip walls) caused by an imposed pressurehead (White 1999). If a fracture has a large width w [L], and a small, constant aperture h [L],flow through the individual fracture can be described using the cubic-law,

Q = −(

wh3

12µ

)∇ P = −

(h2

12

) (A

µ

)∇ P. (2)

Here, the influence of gravity on the flow has been disregarded for simplicity. The similar-ity of the cubic-law to Darcy’s Law is shown by the expression on the right of Eq. 2, whereκ = (h2/12). This relationship has been shown to be quite useful in predicting fluid transportthrough highly fractured reservoirs (McKoy and Sams 1997), but does not account for thesmall-scale roughness of the fracture or the possibility that flow adjacent to the fracture wallsexist due to the rock permeability. A relationship with the simplicity of the cubic-law, whileincorporating the complexity of the fracture geometry and including flow contributions fromthe surrounding permeable medium, would have obvious benefits for use in reservoir-scalediscrete fracture simulators.

Numerical studies using complex geometries include the study of Piri and Karpyn (Karpynand Piri 2007; Piri and Karpyn 2007) where an invasion percolation pore-throat model wasused to model multiphase immiscible flow through a CT-scanned fracture. This modeling wasshown to compare favorably to multiphase experiments (Piri and Karpyn 2007). Tracer dis-persion was modeled using lattice-Boltzmann techniques by Drazer and Koplik (2001) withina two-dimensional (2D) rough-walled fracture. Their results showed that the dispersion ofparticles at small Péclet numbers ducts with small wall roughness is well-described by Taylordispersion. Numerical simulations of flow through a natural fracture in sandstone were per-formed by Al-Yaarubi et al. (2005) using the Navier–Stokes equations and the finite elementcode FLUIDITY. These simulations were shown to compare well to experiments performedthrough similar 2 cm square fracture geometries, and the deviations from the Forchheimerequations at low flow rates were examined. The goal of this study was to perform simulationssimilar to these studies and to obtain a relationship that is appropriate to describe laminar,single-phase fluid flow in a fracture for use in a reservoir-scale, discrete fracture simulator.

While fractures often play a crucial role in determining the flow distribution though frac-tured rock, fluid can also move through the surrounding permeable media (Rangel-Germanet al. 2006). Flow through non-fractured porous media occurs at the micro-scale throughinterstitial pores, but has been described macroscopically using the fluid resistance parame-ter of κ with great success. When the resistance of a fracture to fluid flow is similar to theκ of the surrounding rock, we hypothesize that the flow through the porous matrix will bequite important for describing how fluid is transported through the fracture.

An analytical solution of Darcy’s law with a low permeability “fracture” within it (Mityu-shev and Adler 2006) has shown that it is possible to develop relationships for flow throughfractured permeable media, albeit with an idealized fracture geometry. The development ofa useful relationship for flow through fractured porous media, similar to Darcy’s Law, whichaccounts for some of the complexity of the fracture roughness has yet to be determined.

A friction-factor correlation for flow through rock fractures has been proposed by Nazri-doust et al. (2006), where the fracture tortuosity (defined in the next section) and somefracture-wall roughness parameters were used to obtain a reasonable fit to simulated flows.Their simulations were performed through 2D fracture profiles within Berea sandstonewhere the fracture walls were assumed to be well-described by no-slip boundary conditions,i.e., that the surrounding rock was of sufficiently low κ that the fluid within the fracture wouldnot flow through the rock matrix. This friction-factor correlation, which accounts for the nat-

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Fig. 1 Fracture profiles used to generate computational domains

ural roughness of a fracture, is extended to describe flows in rock fractures in permeablemedia for the present study.

1.1 Rationale and Approach

In this study, single-phase numerical simulations of fluid flow through fracture profiles inpermeable media were performed. The fracture geometries used were obtained from slicesof a micro-CT-scanned fracture in Berea sandstone by Karpyn et al. (2007). Nazridoust et al.(2006) transformed the original data into four separate, 2D, open-fracture sections, whichcaptured the complexity of the original fracture geometry. An analysis of the roughness andthe fractal properties of these 2D profiles is first presented. A review of the friction-factorcorrelation proposed by Nazridoust et al. (2006) and a new formulation to account for con-tributions to flow, due to a surrounding porous medium, comprise the following section. Thenumerical simulations, which solved the Navier–Stokes equations using the finite-volumecode FLUENT, are then described. Flow was modeled in the region surrounding the fractureprofiles, and the κ of the surrounding medium was varied from 0.2 to 2,000 millidarcies(md) to assess its impact on the flow. The flow through the fracture and surrounding rockmatrix was solved with a constant pressure gradient across the length of the domain. Differ-ent aspects of the flow solutions, including the transport of fluid between the fracture andmedium, as well as the applicability of the new friction-factor, are presented in Sect. 5. Thisnew friction-factor may be useful in simulations of highly fractured reservoirs where theformation permeability is appreciable.

2 Fracture Properties

The fracture used in this study was created by Karpyn et al. (2007) using the Braziliancompression method on a 2.54 cm in diameter, 10.15 cm in long core of Berea sandstone.Micro-CT scanning was performed on this fracture, embedded within the rock, with a voxelresolution of 27.4µm. Information about the micro-scanning techniques and detailed three-dimensional (3D) fracture properties can be found in Karpyn et al. (2007). The fracturegeometries, described in this article, were obtained by transforming the 3D data obtainedfrom the CT-scans into four separate, completely open, 2D profiles by Nazridoust et al.(2006) for flow simulations in fractured impermeable rock. These profiles were used to cre-ate the current fracture models, and are shown in Fig. 1. The length of the fracture profiles,10.15 cm, is shown in Fig. 1 as well.

The fracture profiles were created by applying a threshold to the original gray-scale CTnumbers and retaining the open voxels at a size of 240 µm (Nazridoust et al. 2006). Within

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Fig. 2 Aperture frequencies of the four fracture profiles

Table 1 Fracture profile properties

havg (µm) σ (µm) Area( mm2) Df (–) κfrac (md) θ (%)

Profile #1 606 302 61.5 1.62 449 3.39Profile #2 573 295 58.2 1.57 335 3.87Profile #3 586 303 59.5 1.55 362 3.95Profile #4 594 293 65.2 1.66 406 4.62

each profile, apertures were measured at the perpendicular distance between the top and thebottom fracture walls. Due to the voxel size, the aperture height varied by increments of240 µm within the fracture profiles. The frequency distributions of these fracture aperturesfrom 240 to 1, 680 µm are shown in Fig. 2. As can be seen, these profiles are completelyopen (no zero apertures), and there are no apertures greater than 1.44 mm.

Statistical properties of the fracture profiles, such as the average profile aperture, havg,and the standard deviation, σ , of aperture values were obtained by Nazridoust et al. (2006),and are shown in Table 1. As can be seen, the havg for all the profiles is approximately600 µm, and the σ is approximately 300 µm. The total open area of each slice of the fracturewas measured, and as shown in Table 1, is around 60 mm2. The fractal dimension Df [−],the fracture profile permeabilities κfrac, and the tortuosity θ [%] of the fracture profiles werealso measured and are listed in Table 1. The method of determination, comparison to otherrecorded values, and significance of the Df , κfrac, and θ are further examined as follows.

Fractures have been shown to be well-described by fractals, since seminal the study ofMandelbrot et al. (1984) and their examination of fractures in metals. Numerous studies havemeasured the Df of fracture profiles in concrete (Dougan et al. 2000; Issa et al. 2003) ornatural rock (Poon et al. 1992; Odling 1994; Brown 1995). The Hurst exponent H [−] is thequantity of interest reported in most of these studies, and is related to the Df with,

H = 2 − Df . (3)

The H values of the fracture profiles used in this study were determined using the variablebandwidth method, which is described in detail by Dougan et al. (2000). As they explain, awindow of length s[L] is placed over the profile, and the standard deviation of the windowend displacement σs[L] is calculated by

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Fig. 3 Logarithmic plot of variable bandwidth measurements of fracture profiles

σs =√√√√(

1

N − s

) N−s∑t=1

(BH(x) − BH(x+s)

)2 (4)

where BH(x)[L] is the height of the profile at point x[L] and N[–] is the total number of datapoints. From this, a logarithmic plot of σs versus s can be constructed and, if a linear rela-tionship is observed, the profile is well-described by fractals, and the slope of this line is theHurst exponent (Dougan et al. 2000). From this and Eq. 3, the Df was calculated. Figure 3 isthe logarithmic plot of σs versus s for the four fracture profiles, along with the R2 goodnessfit of the linear lines used to calculate the slopes. The fracture profile BH(x)’s were calculatedfor both the top and bottom fracture-wall surfaces, and the average of the two was plotted inFig. 3, for each s. The high values of R2 for the power-law fittings of the logarithmic plots inFig. 3 indicate that the profiles are well-described by this fractal analysis technique.

As is shown in both Fig. 3 and Table 1, the Df of the fracture profiles vary from 1.55 to1.66, or H from 0.45 to 0.34. These Hurst exponents are lower than the average H observedby Dougan et al. (2000) in concrete fractures (H = 0.77), and the H = 0.8 observed in fourdifferent rock fractures by Poon et al. (1992), and just below the range of H observed in natu-ral fracture profiles by Odling (1994) , where 0.5 < H < 1.0. The Df of our fracture profilegeometries are similar to those of a Brownian fractal profile, with H = 0.5. The self-affineBrownian fractal structure has been used as a fracture geometry analogy in numerous studiesof flow through fractures (e.g., Wang et al. 1988; Roux et al. 1998; Mourzenko et al. 2001),and we believe our fracture modeling will be applicable to fractal fractures that have similarroughness characteristics.

The results of the single-phase flow simulations through fractures with impermeable wallspresented by Nazridoust et al. (2006) were used to determine the κfrac of each of the profiles.The pressure losses across the fractures were reported for different flow rates of air and water.With this information, and Eq. 1, the κfrac were determined and are reported in Table 1. It isimportant to note that using Eq. 2, and assuming the aperture h, equal to the average fractureprofile aperture havg, over-predicts the κfrac (as h2

avg/12) by five orders of magnitude. Thisdiscrepancy highlights the importance of performing computational fluid dynamics (CFD)

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simulations on real fracture geometries; the small-scale roughness and aperture variabilityhas a large influence on the flow through fractures.

The tortuosity θ through a fracture has been defined as the percentage increase in distancetraveled by fluids flowing through the fracture due to the presence of wall roughness andasperities (Tsang 1984; Nazridoust et al. 2006). This can be written as

θ = Le

L− 1 (5)

where Le [L] is the length that the fluid travels, and L [L] is the length of the fracture andsurrounding matrix (L = 10.15 cm for the four profiles). The value of tortuosity is difficultto measure in physical situations, therefore we used CFD to directly measure the length offlow path lines along the fracture length for low velocity flows within the fracture profiles.The details of the computational methods used to solve the fluid flow with the fracture aregiven in the next section. For these θ measurements, the working fluid was water with a setinlet velocity of 0.1 cm/s. In order to model the fluid particle trajectories, a minimum of fourparticles with a diameter of 10−8 m (10 nm) were tracked from the inlet to the exit of thefracture using the discrete phase modeling technique (with Brownian motion effects turnedoff). For these low velocity flows, the fracture walls were modeled as no-slip boundaries[unlike in the following sections discussing flow through permeable matrix discussion, butsimilar to the models discussed by Nazridoust et al. (2006)]. The length of these particletrajectories were evaluated and averaged to determine the length that a particle travels withinthe fracture. This averaged valued was used as an estimate of Le. The values of θ determinedin this fashion varied from 3.4 to 4.6%, as shown in Table 1. The particles tended to meanderin large aperture locations and constrict into a smaller area within small aperture regions.

In summary, the fracture geometries used in this study were obtained from the thres-holding of micro-CT-scanned data of a real fracture in Berea sandstone reported by Karpynet al. (2007). Four individual open-fracture profiles, spanning a length of 10.15 cm, wereconstructed by Nazridoust et al. 2006 from this original scan. The fractal roughness of theseprofiles has been examined and shown to be similar to other reported natural fractures inrock. Through the use of CFD, with no-slip fracture walls and discrete phase flow modeling,we were able to obtain a reasonable estimate of the θ for the flow through these profiles aswell as an estimate of the individual fracture profile permeabilities.

3 Fracture Friction-Factor

Recently, Nazridoust et al. (2006) have proposed using a friction-factor to describe the laminarflow of a fluid within fractures. Friction-factors, f [–], are commonly used for analyzing fluidlosses in pipes and ducts, and may be a reasonable way to describe the relationship betweenfluid velocity and pressure losses within rough fractures. By starting with the Darcy–Weisbachequation for laminar flow in a pipe (White 1999), and assuming that changes in gravitationalpotential energy are negligible, one can define a friction-factor for flow in a pipe as,

f = 2�Pd

ρLV 2 . (6)

Here, V [L/T] is the average flow velocity, ρ [M/L3] is the fluid density, P [M/LT2] is thepressure, and d [L] is the controlling length scale (diameter for pipe flow). Equation 6 relatesthe resistance to fluid motion to the geometric parameters of the confining conduit, the fluidproperties, and the associated pressure drop. For fractures in impervious rocks, Nazridoustet al. (2006) suggested

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f = 123ReH̄

(1 + 0.12Re0.687

H̄

)when ReH̄ ≤ 10 . (7)

Here, the Reynolds number was defined as

ReH̄ = ρV H̄

µ= ρQf

µ(8)

where H̄ [L] is the effective fracture aperture and Qf [L2/T] is the flow rate through the frac-ture per unit width. Note that, due to large pressure drops in small apertures, Nazridoustet al. (2006) suggested that H̄ = havg − σ instead of havg as the appropriate length scale todescribe the flow geometry of the fracture. Equations 6, 7, and 8 may be used for evaluatingthe pressure drop along single fractures with known statistical properties in impervious rocks.That is,

�P = 61.5 µL(1 + θ)Qf

H̄3

(1 + 0.12Re0.687

H̄

), (9)

where L(1 + θ) is the length of the fracture that the fluid “sees”, accounting for tortuosity ofits path. When fluid flows through adjacent permeable rocks the flow rate the total flow rateper unit width through the domain may be given as

Qt = Qf + Qm, (10)

where Qm[L2/T] is the flow rate through the adjacent permeable matrix. Assuming Darcy’slaw for the surrounding permeable matrix,

Qm = κhm

µL�P (11)

where hm[L] is the formation height. The total flow rate then is given as,

Qt =⎛⎝ H̄3

61.5(1 + θ)(

1 + 0.12Re0.687H̄

) + κhm

⎞⎠ �P

µL. (12)

For fractured geologic formations with a low-permeability matrix commonly encounteredin the field, flow rates through fractures are typically much higher than the matrix. For thiscondition, the expression for the friction-factor f of the fracture within a permeable mediumbecomes,

f = 123

ReH̄

(1 + 0.12Re0.687

H̄

)

1 + 61.5(1 + θ)(

1 + 0.12Re0.687H̄

)κhm/H̄3

. (13)

This modified f accounts for the roughness of the fracture, fluid properties, and the κ ofthe surrounding media to relate the fluid flow through a natural fracture due to an imposedpressure gradient. In the limit of impermeable matrix, with κ = 0, this new correlationreduces to the f developed by Nazridoust et al. (2006). In the next section, we will discuss theCFD modeling techniques that were used to solve the flow through the fracture embedded inpermeable media to test the applicability of the friction-factor.

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4 Numerical Modeling

In order to model the steady-state, incompressible, laminar flow within the fractured porousmedia the conservation of momentum,

ρ (ui · ∇) ui = −∇ P + µ∇2ui + Fi , (14)

and the conservation of mass,

∇ · ui = 0, (15)

were solved within the computational domain. Here, ui [L/T] is the velocity vector, andFi [M/LT2] is a momentum source term to account for flow through a homogenous porousmedium. Fi was obtained by rearranging Eq. 1 (Darcy’s Law for volumetric flow through amedium of cross-sectional area A) to solve for the gradient of pressure in terms of ui = Qi /A,

∇ P = − µ

κiui = Fi . (16)

Here, the permeability κi is given as a vector, but in the modeling, we applied a homogenousresistance, i.e., κi = κ j .

A schematic of the computational domain for flow through the fractured permeablemedium with fracture Profile #4 is shown in Fig. 4a. The domain is square with the open-fracture profile across the entire length. The left side of the fracture is centered at the middleof the domain height. As is indicated in Fig. 4a, the left and right sides of this region wereset to constant pressure conditions, with the higher pressure at the inlet side forcing the flowthrough the permeable matrix and the fracture. The upper and lower boundaries were as-signed symmetry boundary conditions, which is appropriate to model zero shear-slip wallsfar from the open fracture. Flows adjacent to these symmetry boundaries were observed tobe sufficiently far away from the high-flow rate region within the fracture to have no verticalflow component. The regions above and below the open fracture were modeled as permeablemedia with the momentum sink term of Eq. 16.

The interface between the fracture and the medium is shown in Fig. 4b, along with thegridding for a small section of the domain. As is shown in Fig. 4b, an unstructured triangularmesh was used within and surrounding the open fracture, which enabled a high grid densityto be generated locally. Farther from the fracture, a square mesh of larger cells was appliedto reduce the overall mesh size. The total computational cell count was greater than 4.5(106)for each of the four geometries. These meshes were obtained by refining the grid until nosubstantial variation in the flow solutions was noted. Between the open fracture and the per-meable matrix interface boundaries were applied, which enabled different flow conditions tobe specified within the individual regions, while allowing unhindered flow across the inter-face. This interface boundary has been thickened in Fig. 4 for illustrative purposes. All fourmeshes were constructed using GAMBIT� 2.1.13 and refined using FLUENT� 6.3.26, bothdistributed by Ansys, Inc. (Canonsburg, PA).

Numerical solutions of the flow field were obtained with the finite-volume CFD packageFLUENT 6.3.26. The flow was solved using the implicit, double-precision, pressure basedsolver. Momentum equation was solved for using a second-order upwind scheme, pressureterm was solved using the pressure staggering option scheme, and the pressure–velocitycoupling was performed using the pressure-implicit with the splitting of operators scheme.Solution gradients used in these solvers were obtained by averaging the node values alongeach cell face to obtain flow parameter solutions at the center of each cell.

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Fig. 4 a Schematic of flow domain of fracture Profile #4 and surrounding matrix, with computational grid.Note that pressure boundary conditions are uniform along the entire domain height. b Magnification of boxedregion in (a) of fracture Profile #4 and surrounding matrix with grid. The interface between the open fractureand permeable matrix has been thickened for illustrative purposes

Flow of water with ρ = 998 kg/m3 and µ = 10−3 kg/m s was analyzed. The matrix wasassumed to homogeneous and isotopic, with a porosity of 20% and the permeability variedfrom 1.97(10−16) to 1.97(10−12) m2 (0.2–2,000 md). The surrounding permeable mediumwas also assumed to be non-deforming, which is acceptable for the low pressure gradientsmodeled, from 5 to 1,000 Pa across the 10.15 cm domain length. The steady-state solutionswere assumed to be converged when the normalized velocity and continuity residual ratioswere less than 5(10−10). Less stringent convergence criterion resulted in solutions with sim-ilar flow into and out of the entire domain, but the mass flux between the matrix and thefracture was observed to vary until the residual ratios were less than 5(10−10).

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Fig. 5 Pressure distributions through permeable domains, with fracture Profile #2. a Pressure contourswith �P = 5 Pa and κ = 2 md. b Pressure across (a) within the fracture, 0.5, 2.0, and 4.0 cmaway from fracture. c Pressure contours with �P = 1, 000 Pa and κ = 2,000 md d Pressure across(c) within the fracture, 0.5, 2.0, and 4.0 cm away from fracture

5 Results

A constant pressure gradient across the computational domain induced the fluid flow. Farfrom the fracture profiles, the pressure was observed to decrease linearly along the flow path,obeying Darcy’s Law. Near to the fracture profiles, the pressure was observed to fluctuatesignificantly due to changes in the fracture aperture. The pressure contours in Fig. 5a showthis variation surrounding the fracture profile. This is further illustrated by the pressure plotsin Fig. 5b. Similar behavior is shown in Fig. 5c and d for flow under a higher pressure gradientand with a larger permeability. The deviations from the predicted linear pressure decline ofDarcy’s Law is greatest with κ = 2,000 md and lessen as the permeability decreases, whichcan be seen by comparing the computed pressure distributions within the fractures in Fig. 5band d.

As was expected, the fluid flow through the computational domain was focused withinthe open fractures. This is illustrated in Fig. 6 with velocity contours and vectors of waterflow through fracture Profile #3 under an imposed pressure difference of 100 Pa and with aκ of 200 md. The contours demonstrate that lower-velocity flow occurs in the surroundingpermeable media than within the open fracture. The uniform length velocity vectors shown inFig. 6b are at the same location in the computational domain. From this image, the complexinteraction of fluid flow between the surrounding matrix and the open fracture is apparent.While the fluid velocity is much less in the surrounding matrix, the fluid does move throughthe interface between fracture and the matrix.

The previous study of Nazridoust et al. (2006) has shown the effective aperture that dic-tates fluid movement within rough fractures is inversely proportional to the smallest apertureswithin a fracture. The current modeling effort confirms this finding, as shown by the pressure

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Fig. 6 a Contours of velocity magnitude in Profile #3, with �P = 100 Pa and κ = 200 md. b Uniform lengthvelocity vectors in sub-section of Profile #3 under the same flow conditions

drop versus flow rate plot shown in Fig. 7. The calculated pressure drops for assigned flowrates for the four fracture profiles, with no surrounding permeable media are also shown inFig. 7. A consistent trend of higher flows in Profile #2 was observed. The flow rate for the samepressure drops were always in the order, from largest to smallest, of #2 > #3 > #4 > #1.This is the inverse of the effective aperture and havg values, as given in Table 1, where#1havg > #4havg > #3havg > #2havg. This same order of largest to smallest flow rates wasreported by Nazridoust et al. (2006).

Unlike the study of Nazridoust et al. (2006), a permeable medium surrounding the openfracture was included in this analysis. Rather than a no-slip fracture-wall boundary, an inter-face between the medium and the fracture was modeled. An increase of approximately 10%was observed when the interface boundary condition was used, as compared to the no-slipboundary, regardless of the medium κ . in order to explain this difference velocity profilesof water flowing due to a 5 Pa pressure drop along the fracture at a 240µm aperture regionof the fracture Profile #2 are shown in Fig. 8. The velocity magnitude profiles are shownfor the cases of no permeable medium (no-slip walls), κ = 0.2 md, and κ = 2,000 md. Thepeak velocity within this constricted region is nearly identical for all three cases (4.5 cm/s).A standard parabolic velocity profile for flow in a narrow channel is shown by the solution offlow with no-slip boundaries, as would be expected. The profiles calculated with the perme-able medium and interface boundaries are broader and sharply drop to a near-zero value atthe interface. A number of tests were performed increasing the grid density at the interfaces,

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Fig. 7 Variation of flow rate through the fracture profiles for imposed pressure gradients. Inset shows thevariation of flow rate through fracture Profile #1 with different matrix permeabilities

Fig. 8 Magnitude of velocity across a 240µm opening in fracture Profile #2 with a different κ values, com-pared to a no-slip fracture walls model. Fluid motion due to �P = 5 Pa. The profile location is shown in thesub picture of velocity vectors

and this behavior was consistently observed. ImageJ (National Institutes of Health 2009) wasused to measure the area under each of these profiles, effectively measuring the flow rateat this location. The flow through fracture with a surrounding medium of κ = 0.2 md was9.7% greater than the no-slip condition; flow through the 2,000 md fractured medium was14.3% greater. This approximately 10% increase in flow through the fracture profiles withinpermeable media was consistent across all the studied fracture profiles. By not enforcing the

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Fig. 9 Percent of flow through fracture profiles in permeable media

no-slip wall condition at the fracture-medium interface, the calculated flow rates were largerthan in previous studies.

Flow through the fractures varied with the permeability of the surrounding matrix. Theinset in Fig. 7 shows pressure drop versus flow rate, for Profile #1 and for a small segmentof the pressure drops modeled. The flow rate through the fracture was essentially the samewhen κ = 0.2, 2, and 20 md. When the surrounding matrix permeability approached the κfrac

(449 md; Table1), the flow within the fracture increased. This is shown in the Fig. 7 insetfor the κ = 200 and 2,000 md cases. This result confirms our initial hypothesis that whenthe flow resistance within the fracture and the permeable matrix are similar the fluid motionbetween the two bodies will influence the flow rate observed within the fracture.

As the permeability of the surrounding matrix was increased a smaller percentage of thetotal flow was observed to move through the fracture. For all permeabilities less than 200 md99.9% or more of the total fluid flowed through the fracture. This decreased to a little morethan 99% for κ = 200 md and 93–95% for surrounding media with κ = 2,000 md. Theseresults are shown in Fig. 9 for all four profiles. The smaller κ cases are not shown in Fig. 9,because they were nearly identical to the κ = 20 md case. The total percent area of the flowdomain that was modeled as an open fracture was small, from 0.565 to 0.633%. This indicatesthat for cases when the permeable medium κ is less than the κfrac, it is appropriate to onlymodel the flow through fractures. However, when κ is greater than the κfrac, the percentage of

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Fig. 10 Friction-factor variations with different surrounding media permeabilities

flow through the surrounding permeable matrix is appreciable and may need to be accountedfor.

As was previously shown, a new friction-factor has been developed to describe laminarflow in fractured rocks with permeable media. This friction-factor f accounts for the fractureroughness, the surrounding medium κ , and the fluid properties (Eq. 13). The f values of thefour fracture profiles were calculated from numerical simulations performed, and are shownin Fig. 12 for the cases with a surrounding matrix κ of 0.2, 20, 200, and 2,000 md, as a functionof the ReH. As can be seen, the κ = 0.2 and 20 md cases are similar, with a lower f for boththe κ = 200 and 2,000 md cases. In each of the plots, as shown in Fig. 10, all four fractureprofile results are shown together, illustrating that the friction-factor taken into account thedifferent roughness properties of each profile. This is further shown in Fig. 11 where the fvalues of all four fractures, for all four of the κ numerical cases are compared to Eq. 13. Thelower f predicted by incorporating the surrounding matrix κ into the friction-factor derivationis shown to match the simulated flow. Once again, the lowest f is shown to occur in the cases,where the surrounding κ is greater than κfrac.

Two reasons have been identified in this study for why the inclusion of the permeability ofthe medium surrounding an open fracture will increase the flow. The first was the “slippage”along the fracture walls that may occur if fluid is allowed to travel within the surroundingmedium, as was shown in Fig. 8. The other physical mechanism affecting the amount of flowwithin a fracture is the flow between the permeable medium and the open fracture, as wasshown with vectors in Fig. 6. We measured the flow from the permeable media to the fracturesto evaluate how this varied with different fracture profiles and κ . The results are presented

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Fig. 11 Comparison of friction-factor values from numerical simulations to the predictions of Eq. 13

Fig. 12 Flow from surrounding matrix to fracture profile

in Fig. 12, where the increased amount of flow observed exiting the open-fracture profilesis shown for three different κ and all four profiles. An increase in flow to the fracture dueto larger pressure gradients is observed for all profiles, as well as an increase due to highersurrounding matrix κ . The most surprising result obtained from this analysis is the differencein how these increases manifest in the different profiles. A linear increase is observed in bothProfiles #2 and #3, while Profiles #1 and #4 deviate from this simple relationship at higherpressure drops. Profiles #2 and #3 are less rough than the other two profiles, as shown by thelower Df (Fig. 3; Table 1). The rougher fracture profiles show more variation in flux to theprofiles, possibly due to the wider range of apertures and more abrupt changes in the fractureprofile. This characterization of how fracture roughness affects the flow between the matrixand the fracture is a possible avenue of future study.

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6 Conclusions

Numerical simulations of laminar, single-phase fluid flow through four fracture profilesembedded within permeable media with different permeabilities were conducted. The small-scale roughness of the fractures was shown to affect the motion and overall flow throughthese fracture geometries, similar to behavior observed in previous studies. Using the currentsimulation results, a correlation for evaluating the fraction factor for fractures surrounded bya permeable matrix was developed. It was shown that in the limit of impermeable matrix, thenew correlation reduces to those developed in the earlier studies. Inclusion of the permeabil-ity of the matrix around the fracture in the simulation led to an understanding of a numberof features of flow through fractured media. Important conclusions of this study are:

– When the resistance to fluid motion within a fracture and the matrix surrounding thefracture are similar, the flow through the fracture noticeably increases.

– The largest pressure drops were observed through the regions with the smallest aperturesin a fracture.

– Imposing no-slip wall conditions at the interface between fractures and porous matrixleads to a decrease the simulation flow rate by about 10%.

– The new fracture friction-factor presented may find application in predicting transport offluids in fractured media when the matrix has fairly large permeability.

– The amount of flow from the porous medium to the fracture was observed to be linearlyrelated to the permeability of the medium.

– The small-scale roughness of the fracture profile affects the motion of fluid from the matrixto the fracture, with flows through rougher fractures deviating from a linear increase athigher flow rates.

– Even though fractures may comprise a small percentage of the area within a formation,they will dramatically increase the overall flow rate through region.

Acknowledgements A portion of this research was performed, while Dustin Crandall held a NationalResearch Council Research Associateship Award at the National Energy Technology Laboratory in Mor-gantown, WV USA.

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Computational modeling of fluid flow through a fracture in permeable rock

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