Comput GeosciDOI 10.1007/s10596-015-9536-1

ORIGINAL PAPER

Dimensionally reduced flow models in fractured porousmedia: crossings and boundaries

Nicolas Schwenck1 ·Bernd Flemisch1 ·Rainer Helmig1 ·Barbara I. Wohlmuth2

Received: 7 August 2014 / Accepted: 29 September 2015© Springer International Publishing Switzerland 2015

Abstract For the simulation of fractured porous media, acommon approach is the use of co-dimension one modelsfor the fracture description. In order to simulate correctlythe behavior at fracture crossings, standard models are notsufficient because they either cannot capture all importantflow processes or are computationally inefficient. We pro-pose a new concept to simulate co-dimension one fracturecrossings and show its necessity and accuracy by meansof an example and a comparison to a literature benchmark.From the application point of view, often the pressure isknown only at a limited number of discrete points and aninterpolation is used to define the boundary condition atthe remaining parts of the boundary. The quality of theinterpolation, especially in fracture models, influences theglobal solution significantly. We propose a new method tointerpolate boundary conditions on boundaries that are inter-sected by fractures and show the advantages over standardinterpolation methods.

� Nicolas Schwencknicolas.schwenck@iws.uni-stuttgart.de

Bernd Flemischbernd.flemisch@iws.uni-stuttgart.de

Rainer Helmigrainer.helmig@iws.uni-stuttgart.de

Barbara I. Wohlmuthwohlmuth@ma.tum.de

1 IWS, Department of Hydromechanics and Modellingof Hydrosystems, University of Stuttgart, Pfaffenwaldring 61,70569 Stuttgart, Germany

2 Institute for Numerical Mathematics, Technische UniversitatMunchen, Boltzmannstrasse 3, 85748 Garching, Germany

Keywords Reduced flow models · Fractured porousmedia · Boundary conditions · Fracture crossings

1 Introduction

Characteristic properties for fractured porous-media sys-tems are geometrical anisotropic inclusions and stronglydiscontinuous material coefficients which differ in orders ofmagnitude for the fractures and the rock matrix at the scaleof interest. Such fractured porous-media systems appearin many fields of applications in earth science, such asreservoir engineering, groundwater-resource management,carbon capture and storage (CCS), radioactive-waste repo-sition, coal bed methane migration in mines or landslides[11], and geothermal engineering. The concept of fracturedporous media can also be applied to biomedical engineer-ing applications, e.g., where capillaries can be treated asfractures in the matrix, the capillary bed [14], or technicalapplications, e.g., PEM fuel cells [2], where the gas chan-nels can be treated as fractures in the surrounding porousgas diffusion layers. A more detailed description of frac-tured porous-media systems in earth science can be foundfor example in Bear et al. [6], Berkowitz [7], and Neumann[23].

In this work, a fracture is assumed to be a heterogeneity,where the transversal extension is very small in compari-son to the lateral extension. Areas which are not fracturesare called rock matrix. Fractures limit the representative ele-mentary volume (REV) size [5], and they even can inhibitthe existence of REVs on specific scales. We assume thateverywhere in the domain REVs can be found [7, 10]. Thatallows us to use Darcy’s law to describe the momentumbalance for the flow in the rock matrix as well as in thefractures. There are no free-flow regions. The choice of the

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threshold value which determines if a heterogeneity has tobe treated as a fracture is still somehow arbitrary. Through-out this article, we assume that we have a clearly separablefracture and matrix geometry [25].

Depending on the fracture-rock matrix permeabilityratios, fracture geometry, fracture density, and fracture con-nectivity, different flow regimes can be identified [21]. Theflow regime can be dominated by flow through fractures, byflow through the rock matrix, or by a combination of both.Corresponding to the dominant flow regime, there exists arange of models. Some focus on pure fracture network flow,e.g., [27], where the matrix influence is neglected. Oth-ers assume the fracture to be totally impermeable, so thatthey enter the fracture geometries as boundaries and flowis only modeled in the porous rock matrix. The model pre-sented in this manuscript falls into a third category wherethe flow potentially neither is purely dominated by the frac-tures nor the matrix so that the model has to consider bothparts explicitly. This means that the heterogeneity in perme-ability can lead to highly conductive areas as well as almostimpermeable regions which have in common that they canbe treated as lower dimensional objects.

Dimensionally reduced models are very attractive fromthe point of model complexity. New challenges arise fromthe dimension reduction such as the formulation and imple-mentation of suitable coupling conditions. Such models aredeveloped and analyzed for example for single-phase Darcyflow with continuous pressure in Alboin et al. [3] and withdiscontinuous pressure in Angot et al. [4] and Martin et al.[20]. Despite the advantages of those models, there arisealso difficulties related to the dimension reduction in cer-tain areas (fractures), e.g., a possible pressure jump normalto the averaged fracture thickness. The standard Galerkinfinite-element method, however, cannot handle disconti-nuities in the solution except by resolving them throughthe grid, i.e., the degrees of freedom at the discontinuitiesare doubled and decoupled, e.g., in structural mechanicsand with a Lagrangian approach, the displacement degreesof freedom literally move away from each other. Comingfrom the structural-mechanics problem of evolving cracks,which lead to discontinuities in the solution (displacement,stress, strain), an extension to this standard finite-elementscheme was developed [12, 13, 22] and called “extendedfinite element method” (XFEM). The XFEM uses additionaldegrees of freedom and modified, discontinuous basis andtest functions to capture possible discontinuities at prede-fined locations—in our case the fractures. The discretizationmethod used in this work is the XFEM which allows fornon-matching grids of the lower dimensional fractures andthe matrix and possible discontinuities in the matrix regionacross the lower dimensional fractures. This poses a strongadvantage in the mesh generation process over classicaldiscretization schemes.

After introducing the reduced model in Section 2, thisarticle shows two important novelties for co-dimension onefracture models and their implementation into a numericalsoftware framework. Section 3 discusses the intersection ofseveral fracture branches and how the lower-dimensionalmodel can represent the equi-dimensional pressure distribu-tion correctly in and across the area of intersection. This isa 2d-1d-0d coupling. The second part, Section 4, is aboutthe advantage of Dirichlet boundary condition interpolationand gives two possibilities to set proper Dirichlet values onfractured boundaries. This can be seen as the coupling of a1d-0d problem with a 2d-1d problem.

2 Reduced model

Single-phase, incompressible flow in a porous medium ismodeled with the continuity equation and Darcy’s law on abounded open domain Ω ⊂ R

2 (in general the model canbe extended and applied to R3, but this is not covered in thismanuscript) and suitable boundary conditions to close thesystem. Throughout this article, dimensionless variables areused. In the strong form, we seek a pressure field p such that

∇ · ( −K ∇ p ) = s in Ω, (1)

where the permeability tensorK is assumed to be symmetricand positive definite and s is a source term.

The domain of interest contains heterogeneities, i.e., vari-ations in the permeability tensor K, and the extent of thoseheterogeneities is small in one direction. We call those het-erogeneities fractures. In this section, we present the modelfor one single fracture of co-dimension one. The exten-sion to several, not intersecting fractures, is straightforward.Crossing fractures are discussed in Section 3, which allowsthe handling of complex fracture networks. Complex net-works also possibly contain fractures which end within thedomain. Such situations are not part of this manuscript.A theoretical and numerical evaluation can be found forexample in Angot et al. [4].

The lower-dimensional fracture γ divides the domain Ω

into two partsΩm,1 andΩm,2 such thatΩ\γ = Ωm,1∪Ωm,2

and Ωm,1 ∩ Ωm,2 = ∅, cf. Fig. 1. A thorough and detailedderivation for such a system is carried out in [20]. For easeof notation, we restrict the notation here to inhomogeneousDirichlet boundary conditions, which leads to a primalstrong form with the unknowns pm,i , pf. The subscript mand f indicate the parameters and variables restricted to thematrix and fracture respectively.

∇ · ( −Km,i ∇ pm,i

) = sm,i inΩm,i , i = 1, 2, (2a)

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Fig. 1 Domain decomposition with lower-dimensional fracture (dash-dotted line) and three pressure values and an aperture associated toevery point on the fracture

Km,i ∇ pm,i · ni = 2kf,nd

(pf − pm,i ) on γ, i = 1, 2,

(2b)

∇t · (−kf,td∇tpf) = sf −2∑

i=1

(Km,i ∇ pm,i · ni ) on γ, (2c)

pm,i = g on ∂Ωm,i , (2d)

pf = g on ∂γ. (2e)

The fracture width (aperture) is denoted by d. The equa-tions for the fracture are already written in the lowerdimensional form. The unit vectors t, n define the tangen-tial and normal direction of the lower dimensional fracture,respectively, ni are the unit outer normal vectors of Ωm,i .In general, the permeabilities in the equi-dimensional modelcan be full tensors. In the lower-dimensional model, how-ever, the fracture permeability is reduced to two scalarvalues, the normal permeability, kf,n , and the tangentialpermeability, kf,t . The gradient and divergence operatorsalong the tangential direction of the fracture are denoted bythe subscript t . Equations 2a to 2e form a bi-directionallycoupled system.

The standard continuity equations and Darcy relationsare given in Eq. 2a for the rock matrix and in Eq. 2c forthe fracture. The coupling from fracture to matrix is real-ized with Robin type boundary conditions in Eq. 2b. In theother coupling direction, the surface flow of the rock matrixenters as source term into the continuity equation for thefracture, Eq. 2c, to account for the exchange flow seen bythe fracture.

Using the definition for a jump and an average andwith n = n1 = −n2, namely, the jump of fluxes is[[ − Km ∇ pm · n ]] = Km,1 ∇ pm,1 · n1|γ + Km,2 ∇ pm,2 ·n2|γ and the average of fluxes is {{−Km ∇ pm · n}} =

0.5(−Km,1 ∇ pm,1 ·n1|γ +Km,2 ∇ pm,2 ·n2|γ ), for the pres-sures respectively [[pm ]] = pm,2|γ − pm,1|γ and {{pm}} =0.5(pm,1|γ + pm,2|γ ), the two coupling conditions (2b) canbe reformulated as

[[ − Km ∇ pm · n ]] = 4kf,nd

(pf − {{pm}}) on γ, (3a)

{{−Km ∇ pm · n}} = −kf,n

d[[pm ]] on γ . (3b)

Introducing the following spaces,

Qg = {q = (qm,1, qm,2, qf)

∈ H1g;∂Ωm,1\γ

(Ωm,1

)× H1g;∂Ωm,2\γ

(Ωm,2

)

× H1g(γ )}H1

g;∂Ωm,i\γ(Ωm,i

)

..={vi ∈ H1(Ωm,i ) : vi |∂Ωm,i\γ g

}

and the obvious meaning for H1g(γ ), the weak formulation

can be derived, cf. (4a).Find p = (pm,1, pm,2, pf) ∈ Qg such that

∑

i

(Km,i ∇ pm,i , ∇ qm,i

)Ωm,i

+(4kf,n

d{{pm}}, {{qm}}

)

γ

+(

kf,n

d[[pm ]], [[ qm ]]

)

γ

=(4kf,n

dpf, {{qm}}

)

γ

(4a)

(kf,t d ∇t pf, ∇t qf

)γ

+(4kf,n

dpf, qf

)

γ

=(4kf,n

d{{pm}}, qf

)

γ

(4b)

for all test functions q ∈ Q0.Both systems are coupled via the interface forces which

appear in the reduced equations as source/sink terms onthe right hand side. The solution to the model presentedabove exhibits discontinuities due to the presence of thefracture. Therefore, the numerical method that is applied tosolve the problem has to be able to handle discontinuities.This renders the standard Galerkin finite-element methodinapplicable. Coming from the structural-mechanics prob-lem of evolving cracks, which lead to discontinuities in thesolution (displacement, stress, strain), an extension to thisstandard finite-element scheme was developed [12, 13, 22],and called XFEM.

We will not describe the XFEM in detail but rathergive a short literature overview. From the more theoreticalpoint, Nitsche’s method, intentionally developed to han-dle Dirichlet constraints, evolved to a new possibility totreat interface problems [18]. XFEM and Nitsche’s methodapplied to interface problems are in this case essentiallythe same approach. An overview of recent problems whereXFEM methods are investigated is given in [1]. To be able

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to represent the discontinuities in the solution, i.e., the jumpand the average term, we use the extended finite elementmethod as it is described for example by Dolbow et al. [13],Hansbo and Hansbo [17], and Mohammadi [22]. XFEM isused in the context of fractured porous media for exam-ple in D’Angelo and Scotti [9], Fumagalli and Scotti [16]for lower dimensional fractures introducing a discontinu-ous solution in the matrix, for lower dimensional fracturenetworks having different permeabilities in the networkand therefore also discontinuities [8], as well as for thinheterogeneities (equi-dimensional) which are not resolveddirectly with the grid but rather with the XFEM, [19]. Thesystem is completely assembled and solved with a directsolver. Iterative solvers or problem specific pre-conditionersremain future work. To keep the computational costs as lowas possible, a structured quadrilateral grid is used for thediscretization of the rock matrix.

3 Fracture crossings

In lower dimensional fracture models, every fracture isdefined by a center line with a (possibly varying) thick-ness and associated properties such as normal and tangentialpermeabilities. Such fractures of co-dimension one, γj ⊂Rn−1, define a network of fractures as a set of discrete lines

γ = ∪j γj .In a crossing, however, the different properties of every

fracture branch can overlap and a unique association ofproperties is not always possible, so that, in general, newproperties have to be defined especially for the crossingarea. This is in fact also a requirement from the physi-cal point of view. If there is a crossing of fractures withvery different permeabilities, one fracture always dominatesa crossing from a geological point of view. For example,if there exists a highly permeable fracture which becomesintersected over time by an almost impermeable fracture,the crossing’s permeability is more likely to be almostimpermeable than highly conductive or averaged. It is thenneither a good idea to always average the permeabilitiesin a crossing nor to neglect the connection between differ-ent fractures. The information how the crossing affects theflow has to be provided to the modeler, as the other soilparameters, by an expert.

The flow through such a crossing of fractures, with possi-bly very different physical parameters in a global coordinatesystem, is not at all trivial and classical models are notalways able to capture the important flow features correctlyor lack numerical stability, e.g., the standard model in For-maggia et al. [15] or Section 6.2 in Neunhauserer [24]. InFormaggia et al. [15], a model for two straight intersectingfractures is introduced. This article proposes a new methodto handle arbitrary numbers (>2) of fracture branches which

Fig. 2 Example 4-crossing with intersection geometries and locationof pressure unknowns in the equidimensional model

end in the same point. Starting at an equi-dimensional cross-ing with the equi-dimensional fracture branches �i , pressurerelations in the lower-dimensional model are derived forall fracture branches γi and the crossing. An example ofa crossing is shown in Fig. 2. The equidimensional modeldomain can now be decomposed into three different domain

types: Matrix, fracture, and crossing, Ω =(

∪iΩi

)∪

(∪i�i

)∪ ϒ . We define the crossing area ϒ with bound-

aries to the fractures (∂ϒ)f,i ..= ∂�i ∩ ∂ϒ (solid red lines)and boundaries to the rock matrix (∂ϒ)m ..= ∂ϒ\(∪

i∂�i)

(dashed red lines). The intersection of the fracture centerlines is denoted by υ.

With the crossing area defined, Fig. 2, we introduce anadditional degree of freedom, the pressure at the intersec-tion of the fracture center lines, pυ . Introducing the greenboundaries in Fig. 3 by connecting the appropriate cornersin our crossing region, we get a closed control area forwhich we can write down the mass conservation equationwhich gives a relation between the crossing area pressureand the adjacent fracture pressures pi .

The superscript (·)∗ denotes values which are inside ϒ ,i.e., d∗

i = ∣∣∂�∗i

∣∣ is the length of the interface (green) for

Fig. 3 Example 4-crossing with definition of geometrical parametersinside the crossing area

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fracture i within ϒ , t∗i is the unit inward normal vector

of fracture i on ∂�∗i . The distance between the crossing

point υ and the center of ∂�i for every fracture is denotedby �∗

i . The classical mass conservation without sourcesor sinks for a steady-state situation for the green regioninside the crossing area

∫∂ϒ∗ u∗ · n∗ ds = 0 can be writ-

ten as the sum of the normal fluxes across each fractureboundary to the inner crossing area (green lines in Fig. 3),∑nf

i=1 f ∗i = 0. Here, nf is the number of fracture branches

connected to this crossing. The normal flux is defined asf ∗

i = ∫∂�i

u∗i,ϒ t∗

i,ϒ ds for every fracture i. Using the Darcyclosure, u∗

i = −Kϒ(∇ pi)ϒ , to eliminate the velocitiesleads to∑

i

[(−Kϒ(∇ pi)ϒ) t∗

i

]d∗i = 0 . (5)

Keep in mind that the permeability of the crossing, Kϒ

is, as all the other permeabilities in our model, given as inputparameter to the modeler. To incorporate this continuousmass conservation equation into our flow fracture model,we need to discretize it. Because we want to get an explicitexpression for the pressures, we need to approximate thegradient.

The rotation matrix R(θ) =(cos(θ) − sin(θ)

sin(θ) cos(θ)

)is

defined as the transformation (rotation) from local fracturecoordinates into the reference coordinate system where thetangential vector is the first axis (x) and points towards thecrossings center. The pressure gradient in the fracture, how-ever, is in our lower dimensional model defined as the finitedifference along the tangential fracture direction, becausewe only know one average pressure and not a variation ofpressure in fracture normal direction. That means

(∇ pi)ϒ = R(θi) [(∇ pi)ϒ ]f,i = R(θi)

[(pυ−pi

�∗i

0

)]

f,i

(6)

approximates the gradient in global coordinates dependenton the local fracture gradient and the orientation θ of thefracture. We can now write the compatibility equation as

∑

i

{(

−Kϒ

(

R(θi)

[(pυ−pi

�∗i

0

)]

f,i

))

t∗i

}

d∗i = 0 . (7)

This approximation of the pressure gradient is exact (inthe context of the fracture normal averaged pressure) insidethe respective fracture branch. Inside ϒ the global pressuregradient possibly varies in space and is dependent on alladjacent averaged fracture pressures. Thus, our approach isan approximation of this complex dependency.

Remark 1 In the case of four fractures with the same aper-ture intersecting at 90◦, Fig. 4 left, the red and green

Fig. 4 Two special crossing cases. Left four fractures intersecting 90◦,which is not a problem for the proposed method. Right two paral-lel fractures with changing aperture, which cannot be handled by theproposed method

lines fall together. This does not pose any problem for thisapproach.

Remark 2 The case of two parallel fractures with differ-ent aperture intersecting, Fig. 4 right, which effectivelydescribes a single fracture with changing cross section, can-not be handled with this approach because the crossing areaϒ shrinks to a line and all three pressures p1, p2, pυ arelocated at the same geometrical position in the equidimen-sional domain.

3.1 Example 1: different crossing area permeabilities

This example shows a small fracture network consisting ofthree fractures as shown in Fig. 5. It includes an X-crossingand a Y-crossing, highly conductive and almost imperme-able fractures. The matrix permeability is the unit tensorKm = I. All fractures have the same aperture d = 10−4.The horizontal fracture is highly conductive kf,n = kf,t =104, whereas the vertical fractures are almost impermeable

Fig. 5 Simulation domain (−1, 1) × (−1, 1) with fractures andboundary conditions

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Fig. 6 Coarse grid matrix pressure solution and location of the lineplot

kf,t = kf,n = 10−4. The permeabilities in the crossings areKϒ = 10−4I. All given values here are normalized.

The same type of boundary condition is applied on eachside, i.e., for fracture and matrix. Dirichlet boundary condi-tions are set on the left and right boundary with a pressuregradient from left to right. The bottom boundary is closed(Neumann no flow) and an injection (Neumann inflow) isapplied on the top boundary.

Grid convergence is shown by plotting the matrix pres-sure over a line. Figure 6 shows the pressure distribution inthe domain for the coarse grid solution, the fractures and thelocation of the line plot (from [−1.0, −0.2] to [1.0, 0.9]).The solution for three different refinement levels (base grid:21 × 21 elements, levels 1–3 are 41 × 41, 81 × 81 and161 × 161 respectively) plotted over this line is shown in

Fig. 7 Matrix pressures for coarse grid and three refinement levelsalong a line

Fig. 8 Matrix pressures for coarse grid and three refinement levelsalong a part of the line show grid convergence

Fig. 7 and a zoom is shown in Fig. 8. Convergence can easilybe seen.

Furthermore, the influence of different permeabilities inthe crossing regions is considered. A second simulationis carried out with a different set of permeabilities in thetwo crossing regions. The X-crossing is highly permeable,Kϒ = 1.04I, and the Y-crossing has a mixed permeability,

Kϒ =(10−4 00 104

).

The vertical fractures, which are almost impermeable,lead to a pressure jump. In the crossing areas, this dis-continuity depends on the corresponding permeabilities. InFig. 9, the crossing area has almost the same permeabil-ity as the fractures itself, so that the discontinuity persistsover the whole vertical fracture distance. The second case,

Fig. 9 Pressure distribution in the matrix with the same permeabilityin both crossings: Kϒ = 10−4I

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Fig. 10 Pressure distribution in the matrix with a highly permeablecrossing area in the X-crossing:Kϒ = 1.04I, and a mixed permeabilityin the Y-crossing: Kϒ = (10−4 00 104

)

Fig. 10, has two different permeabilities for the two cross-ing regions. The X-crossing is highly conductive in x- andy-direction and therefore forces also the matrix pressure tobe almost continuous. The Y-crossing however, is highlyconductive in y-direction and almost impermeable in x-direction. The discontinuity across the fracture is preservedand the permeability change along this fracture across thecrossing region has almost no influence.

3.2 Example 1: hydrocoin benchmark

Within the international Hydrocoin project [26], a bench-mark for heterogeneous groundwater flow problems waspresented. The domain setup is shown in Fig. 11 with theexact coordinates given in Table 1.

The boundary conditions are Dirichlet piezometric headon the top boundary and Neumann no flow on the other threeboundaries. The hydraulic conductivity in the fracture zonesis 10−6 m/s and in the rock matrix 10−8 m/s, respectively.The inclination of the fracture zones has no influence on the

Fig. 11 Geometry of the modeled domain of the Hydrocoin test case2, [26]. Boundary conditions are hydraulic head on top and Neumannno-flow on the other three sides of the domain

Table 1 Coordinates of the numbered points in the modeled region ofthe problem depicted in Fig. 11, [26]

Point x (m) z (m)

1 0 150

2 10 150

3 395 100

4 405 100

5 800 150

6 1192.5 100

7 1207.5 100

8 1590 150

9 1600 150

10 1600 −1000

11 1505 −1000

12 1495 −1000

13 1007.5 −1000

14 992.5 −1000

15 0 −1000

16 1071.35 −566.35

17 1084.04 −579.04

18 1082.5 −587.5

19 1069.81 −574.81

permeability tensor and in lower dimensional models, thenormal and tangential permeabilities are exactly the samescalars.

The original benchmark shows the piezometric head dis-tribution along five horizontal lines through the modeleddomain. Here, only the plot at a depth of −800 m is shownand compared to the range of results from the Hydrocoinsummary report [26], in Fig. 12. The dashed-dotted lineshows the pressure head for a relatively coarse grid (303

Fig. 12 Hydraulic head for constant y =−800 m, the grey shaded areamarks the range of different simulator results from the Hydrocoin sum-mary report [26], the dashed-dotted line and the dashed line show theXFEM solution for a coarse (341 unknowns) and a refined grid (1049unknowns), and the solid line represents an equidimensional referencesolution computed on a very fine grid with an MFD

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Fig. 13 Hydraulic head contour plot for the XFEM solution on a finegrid (1049 unknowns)

unknowns, 17 × 13 standard and 82 enriched dofs) andthe solid line for a uniformly refined grid (979 unknowns,33 × 25 standard and 154 enriched dofs). The fracture ele-ment size is chosen in the same order of magnitude as thematrix element size and therefore also uniformly refined forthe second case (38 and 70 dofs respectively). It is clearlyvisible that the solution is already very good close to thefractures (and the crossing) on the coarse grid and a signif-icant error can only be seen in an area where no fracturesare present, i.e., the fracture solution is very accurate andcouples significantly back to the matrix. A finer grid is onlynecessary in areas where the (non-linear) solution is not cap-tured well enough with the bi-linear standard finite elements(Fig. 13).

4 Boundary conditions

This section deals with a way of prescribing Dirichletboundary conditions to a fractured porous medium. Bound-ary conditions for fractured porous media systems withexplicitly modeled fractures are not easy to define. Thesimplest choice is to prescribe a constant pressure along adomain boundary or a linear change for example for the caseof a hydrostatic pressure distribution.

Applying constant or linear pressure over a larger partof a domain boundary often does not reflect the highlyheterogeneous structure in the case of fractured porousmedia systems. That again leads to a strong influence of

Fig. 14 Partitioning of the boundary according to intersectingfractures

the boundary conditions on the solution if the domain is notchosen large enough. For field scale simulations, one usu-ally obtains pointwise pressure information from which thebest boundary conditions are to be picked. We present a pos-sibility to interpolate pointwise pressure data along a givenboundary including the information of the geometrical posi-tion and geological parameters (aperture, permeability) ofthe fractures intersecting with this boundary.

In this section, we look only at the boundary of theproblem domain, to be more precise a segment of arbitrary(non-zero) length of that boundary, Fig. 14. We define thissegment as ω having local coordinates from zero to oneχ ∈ [0, 1]. There are nf fractures intersecting with thisboundary segment so that we get nf + 1 parts of the domainΩm,i adjacent to the boundary ω = ∪

iωi . In our nomen-

clature, this fact is accounted for by counting the boundarysegments from zero to nf. The position of the fractures,χi, i = 1, . . . , nf, intersecting with the boundary segmentare given in boundary local coordinates. Additionally, wedefine χ0 = 0, χnf+1 = 1 as the ends of the boundarysegment we are looking at,

∂Ωm,i = [χi, χi+1] =.. ωi, |ωi | = χi+1 − χi.

We want to find an interpolation method to interpolatebetween χ0 and χn+1, i.e., we set the pressure at χ = 0 andχ = 1. In our implementation for the complete 2d domain,we use (bi-)linear finite elements, therefore we assume it tobe sufficient if the interpolation on the boundary is piece-wise linear with respect to the intervals ωi . The pressure canthen be written for every boundary section as

pb(χ) = miχ + bi so that∇ pb = mi on ωi, (8)

with the two unknowns mi, bi . We admit a jump in thepressure across the fracture-boundary intersections. In par-ticular, the jump at intersection j is given by the left andright pressure limit, namely,

[[pb ]]j = limχ↘χj

pb(χj ) − limχ↗χj

pb(χj )

= mjχj + bj − mj−1χj − bj−1, (9)

and it is positive in the normal direction from ωi−1 towardsωi .

For the new problem we want to solve on ω, we introducethe test functions which are inH1(ωi) and zero on the globalboundary.

qb,i ={

nfχ − i on ωi

0 else(10)

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Fig. 15 Boundary interpolationfor variations of aperture andmatrix-fracture permeabilityratio. pleft = 1.0, pright = 2.0,left: k/kf,n = 1.0, right: d = 1.0

This particular choice of testfunctions allows us to use

∇ qb,i = nf on ωi, i = 0, . . . , nf,

[[ qb ]]i = −1, i = 1, . . . , nf,

[[ qb,i ]]i = nfχi − i, i = 0, . . . , nf,

[[ qb,i ]]i+1 = −nfχi+1 + i, i = 0, . . . , nf.

The arbitrary linear combination of all qb,i is abbreviatedas qb - without index.

To close the system there are two obvious choices. First,the pressure in the fracture can be set equal to the aver-age of the adjacent matrix pressures. This has the advantagethat the modeler does not need any extra information, i.e.,does not have to know the pressure in the fractures at theboundary. However, this is a strong assumption which is notnecessarily physically correct within the domain. The otherpossible choice, which can be required by the applicationsetup, is to set the pressure in the fractures additionally tothe two matrix pressures at χ = 0 and χ = 1.

4.1 Fracture pressure as output

Assuming the average of the pressure left and right of thefracture to be equal to the pressure in the fracture leads to thefollowing boundary functions for single and multiple frac-tures. We use the weak formulation for the matrix (4a), butnow only on the boundary with zero dimensional fractures

and the assumption pf = {{pm}}, i.e., we do not need (4b),and obtain

2∑

i=1

((K is · s)∇ pb, ∇ qb)ωi+ kf,n

d[[pb ]][[ qb ]]

∣∣∣∣ω1∩ω2

= 0.

(11)

Here, s is a vector which is oriented along the boundaryof ∂Ωm,i , so that we can define the effective permeabilityof the boundary section ωi to be Kis · s =.. kb,i . ExtendingEq. 11 to multiple fractures gives

∑j

i=j−1

∫

ωi

kb,i ∇ pb · ∇ qb ds

= − kf,n

d

∣∣∣∣j

[[pb ]]j [[ qb ]]j for j = 1, . . . , nf

(12)

With the given boundary conditions that we want to inter-polate, namely pb(0) = pleft and pb(1) = pright, andthe additional constraint assumption, pf,i = {{pb}}i , weget a general solution for the linear pressure parts on theboundary,

mi = pright − pleft

ki

, (13)

Fig. 16 Pressure boundarycondition on the bottom (left)and on the top (right) of thedomain for three interpolationmethods

Comput Geosci

Fig. 17 Pressure in the fracture for three different boundary condi-tions on the rock matrix boundaries

with

ki = kb,iωi

q0 0

q−1 0

i∏

�=0

q(�−1)�

q��

·⎧⎨

⎩

nf∑

j=1

q(j−1)j

j∏

k=1

qkk

q(k−1)k

[d

kf,n

∣∣∣∣j

(1

q(j−1)j+ 1

qjj

)

+ χj

(1

kb,jωj

1

q(j−1)j− 1

kb,j−1ωj−1

1

qjj

)]

+ 1

kb,nωωnω

nf∏

�=1

q��

q(�−1)�

⎫⎬

⎭for i = 0, . . . , nω,

(14)

where qij ..= [[ qb,i ]]j , q0 0 and q−1 0 are only introducedfor ease of notation, are not assigned any value and alwayscancel out. Moreover, the coefficients bi are given as

bi =i∑

�=1

⎛

⎝χ�(m�−1 − m�)

+ d

kf,n

∣∣∣∣�

⎛

⎝�∑

k=�−1

kb,kmkωk

⎞

⎠

⎞

⎠+ b0 (15)

For the reformulations above, we require the fracturepositions to be different than the roots of the jump terms ofthe test functions to avoid division by zero.

For one single fracture, nf = 1, this yields

m0 = pright − pleftkb,0kf,n

d + |ω1| + |ω0| kb,0kb,1

+ (kb,0kb,1

− 1), (16)

m1 = pright − pleftkb,1kf,n

d + |ω0| + |ω1| kb,1kb,0

+ (1 − kb,1kb,0

). (17)

As an example, we look at the symmetric case of onefracture intersecting a homogeneous rock matrix boundary,k = kb,0 = kb,1, |ω0| = |ω1| = χ1 = 0.5. Then

m0 = m1 = pright − pleft

1 + kkf,n

d, (18)

[[p ]] = pright − pleft

1 + kf,nk

1d

. (19)

From Eqs. 18 and 19, we can see, cf. Fig. 15, for thedifferent limit cases that

– For a really small fracture width, limd→0

, the pressure gra-

dient is the same as without a fracture and the pressurejump is zero.

– For a highly conductive fracture, limk/kf,n →0

, there is

almost no effect on the pressure gradient and the jumpis zero (not shown in Fig. 15).

– For a totally impermeable fracture, limkf,n /k→0

, the pres-

sure gradient is zero and therefore [[p ]] = pright −pleft.

For visualization purposes, the fracture aperture is cho-sen as d = 1.0, which does not represent a physicallymeaningful case and only shows the behavior in the limit.

Fig. 18 Contour plot: linearinterpolation on the boundarywithout a jump (left) andallowing a jump (right)

Comput Geosci

Fig. 19 Matrix pressure vs. x-distance at constant y = 0.9 for theboundary with and without a jump

4.2 Fracture pressure as input

Following the same steps as in Section 4.1, but allowingpf = {{pm}}, the full equation on the boundary becomes

j∑

i=j−1

∫

ωi

kb,i ∇ pb · ∇ qb ds

+ kf,nd

∣∣∣j[[ pb ]]j [[ qb ]]j

+ 4kf,nd

∣∣∣j

({{pb}}j − pfj) {{qb}}j

= 0 forj = 1, . . . , nf. (20)

This equation includes the fracture pressure at every frac-ture intersecting the boundary and therefore needs this asinput information, i.e., as a first step the PDE (20) is solvedand used to define the boundary conditions for (4a).

Fig. 20 Matrix pressure jump plotted along the fracture for the bound-ary conditions allowing a jump and linear interpolation without ajump

4.3 Example 1: solving the complete system on theboundary

In this example, the assumption that the average matrix pres-sure equals the fracture pressure is not made but instead thefracture pressure is an additional parameter which has to beset by the modeler. This example shows a single fracturesystem with different boundary values on the top and onthe bottom of the domain. The fracture is vertically inclinedfrom [−0.7, −1.0] to [0.7,1.0], cf. Fig. 18. The left- andright-hand side of the domain get homogeneous Neumannboundary conditions. The pressure value on the top and bot-tom boundary on the left is set to pb,left = 1.0, and on theright it is set to pb,right = 2.0 and the pressure in the frac-ture is set to pf = 0.5 at the bottom and it is set to pf = 1.8at the top.

The values for the Dirichlet boundaries are plotted inFig. 16. There are three different cases, a constant value leftand right of the fracture respectively, a linear interpolationfrom the left and right matrix pressure value to the fracturepressure value without a jump, and the linear interpolationwhich includes a pressure jump across the fracture as a solu-tion of (20) on the boundary. Keep in mind that for thatcase, the pressure in the fracture not necessarily is equal tothe average pressure, this can clearly be seen at the bottomboundary.

The choice of the boundary values affects the solutionin the matrix as well as in the fracture significantly. Thepressure in the fracture for all three cases is plotted inFig. 17. The piecewise constant pressure leads to a signif-icantly different flow behavior in the fracture, very steepgradients close to the boundary, and a pressure gradient inthe domain which has the opposite direction than for theother cases. The linear interpolated pressure with and with-out a jump across the fracture show a similar behavior. Theconstant pressure on the boundary which does not accountfor the fracture pressure influence on the matrix, representsa physically totally different setup and is therefore neglectedin the following discussion.

The pressure contour in the matrix for the linear inter-polation without a jump and allowing a jump are shownin Fig. 18. The matrix pressure inside the domain lookssimilar, i.e., if the domain is large enough the influence ofthe different interpolation methods is negligible. However,there is a clearly visible jump across the fracture within thedomain. The jump inside the domain can easily be seen inFig. 20, where the jump is plotted along the fracture andin Fig. 19 where the matrix pressure is plotted for constanty = 0.9 (close to the boundary). The pressure continuityat the boundary forces the pressure jump inside the domainto be smaller. Unless there is a physical reason to prohibit

Comput Geosci

this jump at the boundary, allowing it certainly appears tobe more justified (Fig. 20).

5 Conclusion

We present a new method to handle fracture crossingsof arbitrary number and intersection angles for arbitrarypermeabilities, which is easy to implement and integratein existing (XFEM) codes and models and shows robustbehavior for the presented implementation. Different exam-ples for highly conductive as well as almost impermeablefractures show numerical convergence and physical flowbehavior. A comparison to a classical groundwater flowbenchmark is conducted and shows the efficiency as well asaccuracy of the presented model.

Furthermore, a new method to interpolate pressure dis-tributions on boundaries with fractures which then can beused as Dirichlet boundary condition values is presented.Depending on the problem to simulate and the given data,the interpolation can calculate the fracture pressures or takethem as additional input to give an even better approxi-mation of the pressure distribution on the boundaries. Thepresented example shows the influence of the boundaryconditions and the advantage of the novel approach.

Acknowledgments This work was funded by the German ResearchFoundation (DFG) within the International Research Training Group“Nonlinearities and upscaling in porous media” (NUPUS, IRTG 1398)at the University of Stuttgart.

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