dualporosity model

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Comput Geosci (2008) 12:91104

DOI 10.1007/s10596-007-9062-x

ORIGINAL PAPER

Modeling fractures as interfaces: a modelfor Forchheimer fractures

Najla Frih Jean E. Roberts Ali Saada

Received: 18 May 2007 / Accepted: 6 November 2007 / Published online: 17 January 2008 Springer Science + Business Media B.V. 2007

Abstract In this paper we are concerned with modeling

single phase flow in a medium with known fractures.In particular we are interested in the case in which theflow rate in the fractures is large enough to make it

appropriate to use Forchheimers law for modeling the

flow in the fractures even though the flow in the sur-rounding domain is such that Darcys law is adequate.

We describe a model in which the fractures are treated

as interfaces. We also consider the case of intersecting

fractures and the case of nonconforming meshes.

Keywords Porous media Fractures

Forchheimers law Mixed finite elements

Domain decomposition Nonconforming meshes

1 Introduction

Modeling flow in porous media is made difficult by

the presence of heterogeneities in the characteristics

of the medium. In particular, it is difficult to take intoaccount fractures in a numerical model, because of their

width which is very small in comparison to the size of

the domain. Yet because their permeability may also

N. Frih (B) A. SaadaENIT-LAMSIN, BP 37, 1002 Tunis-le Belvdre, Tunisiae-mail: [email protected], [email protected]

A. Saadae-mail: [email protected]

J. E. RobertsINRIA-Rocquencourt, BP 105,78153 Le Chesnay Cedex, Francee-mail: [email protected]

differ greatly from that of the surrounding medium,

they may have a very important influence on the flowin the medium. They may act as privileged channels forflow or they may act as barriers.

Fractures may occur as networks of very fine frac-

tures, impossible to localize individually, [9]. These aretaken into account using homogenization techniques

or double porosity models, [7, 8]. Here however, we

are concerned with larger known fractures. In [1], a

model was introduced in which such larger fractures aremodeled as (n 1) dimensional interfaces in an n di-

mensional medium. The fractures were supposed to be

of high permeability so the pressure was assumed to be

continuous across the fractures. However the flux wasnot supposed to be continuous as the fluid could flow

into and out of as well as along the fractures. A model

introduced in [14] generalizes the earlier model so thatit can handle both large and small permeability in the

fracture. For this model it is no longer assumed that the

pressure is continuous across the fracture. In [2], a 3-Dmodel with intersecting fractures was introduced. For

all of these models, flow in the fractures as well as in

the surrounding medium is governed by Darcys law.In an earlier article, [12], a model was introduced in

which the flow in the fracture is sufficiently rapid that it

is governed by a nonlinear law, Forchheimers law, see

[6,11,17] and[20]. In the present article we developthe model of [12] in more detail, extending it to the

case of intersecting fractures and giving more numerical

results. We also show that nonmatching grids may beused for the subdomains and for the fractures.

We mention that others have also treated fractures

as interfaces: for Darcy flow, a model introduced byAngot et al. in [3] is based on Robin boundary condi-

tions at the interface and assumes the continuity of the


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92 Comput Geosci (2008) 12:91104

flux across the fracture. A model similar to that of[14]

was studied by Flauraud et al. in [10]. For a problem

of multiphase flow in a fractured medium, Helmig et al.

in[15], represented the fracture by lower dimensionalfinite elements.

In Section 2, we describe the different equations

governing single phase, incompressible flow in a porous

media. In Section3, we describe a simple model prob-lem with a domain containing a single fracture. Then,

in Section 4, the interface model of [12] is derived

for this simple problem and in Section 5we give thediscrete formulation along with some numerical results.

In Section6,we extend the simple model problem to a

problem with intersecting fractures. In Section 7,weusenonconforming meshes and we show some numerical

results with several intersecting fractures.

2 Equations governing flow in a porous medium

Flow of a single-phase, incompressible fluid in a porousmediumis governed by the law of mass conservation:

div(u) = q in , (1)

where q is a source term and u is a volumetric flow

rate or velocity. The velocity u is related to the gra-

dient of the pressure p(in the absence of gravity) byDarcys law:

u + K p = 0, in , (2)

whereKis the permeability of the medium.

Darcys law is valid for low flow rates for which

inertial effects are negligible. For higher flow rateshowever, the results obtained by Darcys law do not

coincide with experimental results; the flow rate pre-

dicted by Darcys law is too high. Forchheimers lawintroduces an inertial term, a quadratic term, which

slows down the flow. So, when the flow is sufficiently

rapid, Forchheimers law gives a more accurate relation

between the gradient of the pressure and the flow rate.

This law is a nonlinear law given by

(1 + b |u|)u +Kp= 0 in , (3)

whereb denotes the Forchheimer constant also called

the inertial constant. In some works b is taken to bea tensor, but here we have restricted our attention

to the case b is a scalar. For both experimental and

theoretical derivations of Forchheimers law see the list

of references in the introduction of [13].

3 Description of a simple model problem

Suppose that is a convex domain in Rn, n= 2 or 3,

and denote by = the boundary of . We sup-pose that the flow in is governed by a conservation

equation together with Forchheimers law relating the

gradient of the pressure pto the flow velocityu:

div u = q in

(1 + b |u|)u = Kp in

p= pD

on, (4)

where q is a source term, bthe Forchheimer coefficient,p

D the given pressure on the boundary and Ka di-

agonal permeability (or hydraulic conductivity) tensor

which is supposed to satisfy

0< Kmin x2

0 ( Kx,x) Kmax x

2

0


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Comput Geosci (2008) 12:91104 93

where d(s) denotes the thickness of the fracture at

s . We also suppose that f separates into two

disjoint, connected subdomains:

\f =1 2, 1 2 = .

We will use the notationifor the part of the boundary

ofiwhich lies on ,i = 1, 2, f

i = i , i= 1, 2, f

and we denote by i the part of the boundary of i,

i= 1, 2which lies on f

i = i f , i= 1, 2.

The case of interest for us is that the velocity in

the subdomains is small enough for Darcys law to be

sufficient while that in the fracture requires the use ofForchheimers law. In this case in order to avoid having

to solve a nonlinear problem in all of it would seem

appropriate to formulate the problem as a transmissionproblem using the nonlinear law only in the fracture. If

we assume that the flow in the subdomains is governed

by Darcys law and if we denote by pi, ui,Ki, and qithe restrictions ofp, u,K,andqrespectively toi, i=

1, 2, f, and by pDithe restriction ofpD to i, i= 1, 2, f

we can rewrite the above problem (4) as a transmission

problem:

div ui = qi ini, i= 1, 2, f,

ui = Kipi ini, i= 1, 2,

(1 + b |uf|)uf = Kfpf in f,

pi = pDi oni, i= 1, 2, f,

pi = pf oni, i= 1, 2,

ui n= uf n oni, i= 1, 2. (5)

The system (5) could then be solved by using a con-

ventional nonoverlapping domain decomposition tech-

nique in which f would be considered simply as athird subdomain. Following [1, 14] and [2], we pro-

pose here as in[12] the alternative technique of treat-

ing f not as a subdomain, but as an interface

between the subdomains 1 and 2 (see Fig. 1) on

which nonlocal transmission conditions are imposed.

The advantage of using domain decomposition formodeling the fractured medium is that one no longer

needs to have local refinement around the fracture.

However if the fracture is taken to be a subdomain

itself this introduces a subdomain much thinner thanother subdomains which is not usually desirable for

domain decomposition algorithms. Further with a stan-

dard domain decomposition approach the elements inthe fractures would either have one dimension much

smaller than the other dimensions or the grids would

be nonconforming requiring the use of mortar elements

or other special techniques. By treating the fracture

not as a thin subdomain but as an interface this prob-

lem is eliminated. An added advantage of the inter-face approach in the present case with Forchheimer

flow in the fracture is that one has only an (n 1)

dimensional nonlinear problem to solve instead of an n

dimensional one.

4 Derivation of the interface model

The model in which the subdomain f is replacedby the interface , is obtained by using the technique

of averaging across the fracture. For this however, a

simplifying hypothesis, that flow in the fracture in thedirection normal to the central hyperplane is much

smaller than that in the tangential direction, is used.

This hypothesis seems reasonable in light of the verysmall ratio, width to length of the fracture. Thus we sup-

pose that the flow in the direction normal to the fracture

is adequately described by Darcys law and that in the

tangential direction, it is described by Forchheimerslaw.

The first step toward deriving the model is to decom-

poseu fasu f =u f,n+ uf,withu f,n = (uf n) n(recallthat n= n1 = n2), and to introduce the notation and div for the tangential gradient and divergence

operators andn and divnfor the normal gradient anddivergence operators.

4.1 Averaging the conservation equation

With the above notation, the first equation of Eq.5fori= f, may be rewritten as

divn u f+ div uf =qf in f. (6)

Integrating in the direction normal to the fracture, oneobtains

uf n|2 uf n|1 + divU = Q on , (7)

whereU=

d2

d2

uf, dnis the flow rate through a nor-

mal cross section of the fracture and Q=

d2

d2

qf dn.

Then using the continuity of the fluxes across 1and2, the last equation of Eq. 5for i= 1 and 2, we may

write

div U = Q + (u1 n1|1 + u2 n2|2 ) on . (8)


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94 Comput Geosci (2008) 12:91104

This is the conservation equation on with the addi-

tional source termu1 n1|1 + u2 n2|2 representing the

difference between what flows into the fracture and

what flows out of the fracture at a given point of.

4.2 Averaging Forchheimers law

With the hypothesis that flow in the fracture in thedirection normal to the fracture is described by Darcys

law, the decomposition of the second equation of the

system (5) into its tangential direction and the normaldirection parts is given by

(1 + b |uf|)u f, = Kf, pf (a)

uf,n = Kf,n n pf. (b) (9)

Integrating the first equation of the system (Eq.9, a)

along the line segments

d

2 ,

d

2

, yields:

d2

d2

(1 + b |uf|)u f, dn= Kf,

d2

d2

pf dn, (10)

where we have assumed that Kf, is constant along

the segments

d

2,

d

2

. We approximate|uf| = |uf,+

uf,n|by|uf,|

1 +

1

2

u2f,n

u2f,

.

Then using the hypothesis that u f,n is very small incomparison touf,, we obtain the approximation

|uf| |uf,| |U|

d .

Replacing |uf| by |U|

d in Eq.10and integrating one

obtains1 +

b

d|U|

U = Kf, d P, (11)

where P =1

d

d

2

d2

pf dn is the average pressure along

a normal cross section of the fracture.

Equation 11is Forchheimers law in the (n 1) di-

mensional domain . Together Eqs. 8 and 11 give aflow equation in with a source term representing the

flow from the subdomains 1 and 2 into the fracture.

The second equation of Eq.9can now be used to giveboundary conditions along for the subdomains 1and2.

Integrating the second equation of the system

(Eq.9, b) along the line segments

d

2,

d

2

, yields:

d2

d2

uf,n = Kf,n

d2

d2

n pf, (12)

where we have assumed that Kf

,n

is constant along the

segments

d

2,

d

2

.

Equation12is equivalent to

p2| p1| = 1

Kf,n

d2

d2

uf,n dn (13)

With the hypotheses that the fracture width d is much

smaller than the fracture length, and the high perme-ability in the fracture, i.e Kf,n is large, then we can

suppose that the pressure p is continuous through

so that

p1| = P = p2|.

4.3 The strong formulation of the interface model

problem

The interface model for the fracture problem can now

be written

divui =q i ini, i = 1, 2

ui = Kipi ini, i = 1, 2

div U= Q +(u1 nu2 n)on 1+

b

d |U|

U= Kf,d P on

pi = pDi oni, i = 1, 2

pi = P on , i = 1, 2

P= PD on ,

(14)

where PDis the average value of PDf along the seg-

ment

d

2,

d

2

inf.

5 Numerical discretization for the interface model

5.1 The discrete formulation of the model problem

Let Th,i be a conforming finite element partition ofi, i = 1, 2, and suppose that the meshes Th,i, i= 1, 2

match at the interface between the subdomains i,

i.e. suppose that these partitions are such that the

two partitions induced on by these partitions Th,icoincide. In other words Th,1 Th,2 forms a standard

finite element partition of. Then denote by Th, the


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Comput Geosci (2008) 12:91104 95

induced mesh on consisting of all edges of elements

ofTh,ithat lie on . Then let Th = Th,i, i= 1, 2, . ThusTh consists of both n-dimensional elements in i and(n 1)-dimensional elements on.

To define an approximation space for the pressure,

let Mh,i, i= 1, 2 be the space of piecewise constant

functions constant on each element T of Th,i and let

Mh,be the approximation space of piecewise constantfunctions constant on each element E ofTh,. Then let

Mh = Mh,1 Mh,2 Mh,.

Similarly, to define an approximation space for the

velocity, let Wh,i, i= 1, 2 be the RaviartThomasNedelec space of lowest order defined on the n-

dimensional space i subordinate to the mesh Th,i,

and let Wh,be the RaviartThomas space of lowest

order defined on the (n 1)-dimensional interface

subordinate to the meshTh,. Then let

Wh = Wh,1 Wh,2 Wh,.

The discrete mixed finite element approximation

for the interface problem (14) can now be written as

follows:Find (uh,1, uh,2, Uh,) Whand (ph,1,ph,2,Ph,) Mh

such that for all (vh,1,vh,2,Vh,) Wh and (rh,1, rh,2,

rh,) Mh

i K1i uh,i vh,iiph,idivvh,i=

Ph,vh,i ni

i

pDivh,i ni, i= 1, 2

(Ph)

i

divuh,irh,i =

i

qirh,i, i= 1, 2

(d Kf,)1

1 +

b

d|Uh,|

Uh, Vh,

Ph,divVh, =

PDVh, n

divUh,rh,=

Qrh, +

(uh,1 n1 +uh,2 n2)rh,.

(15)

Following[1,2,14], we use a domain decomposition

type method [18] to solve the discrete problem (Ph).

In this way, the problem (15) is reduced to a problemonly on the interface . For i= 1, 2, uh,i and ph,i are

decomposed into two parts,

uh,i = u0

h,i + uh,i, ph,i = p

0

h,i+ ph,i .

For eachi, we use the SteklovPoincar operator

Si : Mh, Mh,rh, u

0

h,i ni

which associates to an element rh, Mh,the elementu0h,i ni Mh,where(u

0

h,i, i,p0

h,i) Wh,i Mh,iis the

solution of the problemFind(u0h,i,p

0

h,i) Wh,i Mh,i such that

P

0

i

i

K1i u0

h,i vh,i

i

p0h,idivvh,i

=

rh,vh,i ni vh,i Wh,i

i

divu0h,irh,i = 0 rh,i Mh,i . (16)

We also use the notation i = uh,i ni i = 1, 2, where

(uh,i,ph,i) Wh,i Mh,iis the solution of the problemFind(uh,i,p

h,i) Wh,i Mh,i such that

P

i

i

K1i uh,i vh,i

i

ph,idivvh,i

=

i

pD

vh,i ni vh,i Wh,i

i

divuh,irh,i =

i

qirh,i rh,i Mh,i . (17)

Then if (ph,1,ph,2,Ph,) with (uh,1, uh,2, Uh,) is the

solution of(Ph); (u0

h,i,p0

h,i)is the solution of(P0

i ) withrh, = Ph,and(u

h,i,p

h,i)is the solution of(P

i )one has

ph,i = p0

h,i+ ph,i and uh,i = u

0

h,i + uh,i. (18)

Thus uh,i ni = Si(Ph,) + i, and (Uh,,Ph,) is the

solution of the following problem:Find (Uh,,Ph,) Wh, Mh, such that for all

rh, Mh, andVh, Wh,

(P) (d Kf,)

1

1 +

b

d|Uh,|

Uh, Vh,

Ph,divVh, =

pDVh, n

divUh,rh, +

(S1(Ph,) + S2(Ph,))rh,

=

Qrh, +

(1+ 2)rh,. (19)

This problem is nonlinear but it is only an (n-1) di-

mensional nonlinear problem, and it can be solved by


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96 Comput Geosci (2008) 12:91104

Fig. 2 Test-case

using a fixed point iteration method or a quasi-Newton

method. In either case the iteration can be initializedwith the solution for Darcy flow in the fracture. The

solutions (uh,i,ph,i) in the subdomains can then be

calculated using Eq.18.

5.2 Some 2-D numerical results

A first numerical result for this type of problem is toobtain a good correspondence between the solution

obtained by the interface problem and a reference solu-

tion. We obtain a reference solution by using the model

problem where the fracture is considered as a thirdsubdomain and using a mixed finite element method.

5.2.1 A simple test case

In this section numerical results obtained with the inter-

face model will be compared with those obtained using

a standard model with 2-D fracture. The results areobtained for a simple model problem: the subdomains

1 and 2 are both squares of unit length and width,

and the fracture f separating the subdomains is of

unit length and of width d= 0.01. The permeability

in each of the subdomains is assumed to be constantand equal to 109, and the fracture is assumed to be of

much higher (also constant) permeability, Kf = 106.

The Forchheimer coefficient b is taken to be 10. The

upper and lower boundaries of the two subdomains areassumed to be impermeable, and there is a pressure

drop from right to left of 106.The same pressure drop

from top to bottom of the fracture is imposed. SeeFig.2.

To compare the different results, one can see in Fig. 3

the reference pressure and also the reference velocity,both obtained by using the standard model (model with2-D fracture). In Fig. 4, those obtained by using the

interface model are given. In both cases a 15 by 15grid was used for each of the subdomains 1 and 2,

and in the case of the reference model a 15 by 15 grid

was also used for the fracture domain f. As we cansee, these results show good agreement between thesolutions obtained by the two models.

5.2.2 Comparison between Darcy and Forchheimer

velocity in the fracture

In the experiment above the data were chosen so thatthe Forchheimer term would have a non negligible ef-

fect. Here a comparison between the Darcy and Forch-

heimer velocities in the fracture is given. The same data

as in the above experiment are used only in one case theForchheimer term in the fracture is omitted.

In Fig. 5 the tangential Darcy velocity (which for

this simple 2-D model is just a scalar) is representedin blue while the Forchheimer velocity is given by the

00.5

11.5

22.5

0

0.2

0.4

0.6

0.8

10

2

4

6

8

10

x 105

x

Standard model

y

Pressur

e

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Velocityinthetotaldomain

Standard Model

Fig. 3 Reference pressure (left) and reference velocity (right) given by the standard model


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Comput Geosci (2008) 12:91104 97

00.5

11.5

22.5

0

0.2

0.4

0.6

0.8

10

2

4

6

8

10

x 105

x

Interface Model

y

Pressure

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Velocityinth

etotaldomain

Interface Model

Fig. 4 Numerical pressure (left) and numerical velocity (right) given by the interface model

red curve. On the left, the velocity calculated usingthe standard model along a central vertical section is

shown, and on the right is the velocity obtained using

the interface model.A good agreement between the results obtained by

the standard model and the interface model can be

observed and in addition, the difference between theForchheimer velocity and the Darcy velocity can be

seen. The Forchheimer velocity is of much smaller

magnitude than the Darcy velocity because of the large

inertial coefficient.Indeed, in this example the Forchheimer coefficient

b was taken to be extremely large to show that themodel is valid even when the nonlinear term is veryimportant. For more physically correct values ofb , the

percent decrease in the magnitude of the velocity whenthe Forchheimer law is used instead of simply the Darcy

law is much smaller. To illustrate this, we have shown

in Fig.6, the average percent decrease in the velocitydue to the Forchheimer term for different values of

the Forchheimer coefficient b. The percentage calcu-

lated using the interface model is represented in blueand that obtained using the standard model is given

in red.

5.2.3 Numerical study of the error

The object of this paragraph is to obtain a quantitativeappreciation of the error made when using the inter-face model. A relative L2 error in the pressure in the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

x

Velocityinthe

interface

Interface model

Forchheimer velocity

Darcy velocity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

x

Velocityinthe

fracture

Standard model

Forchheimer velocity

Darcy velocity

Fig. 5 Darcy and Forchheimer velocity in the fracture given both by the interface model (left) and the standard model (right)


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98 Comput Geosci (2008) 12:91104

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

b

average%v

elocitydecrease

Average percent velocity decreasedue to Forchheimer term for different b values

interface model

standard model

Fig. 6 Average percent velocity decrease due to Forchheimerterm for different values of the Forchheimer coefficientb

subdomains is calculated. A reference solution P isobtained using the standard model with 2-D fracture,

using mixed finite elements on a regular fine mesh Tof

square elements of side length , with = 1/192in the

subdomains 1 and2.Approximate solutions Ph fordifferent values ofhare calculated using the interface

model with a uniform mesh of square elements of side

length h in each subdomain and a 1-D mesh in thefracture with elements of length h.

The discretization parameter h for each of the cal-

culations below is chosen such that the fine mesh in thesubdomains is a refinement of the mesh of size h.Thena

projection Phof each approximate solution Phonto

the fine mesh is obtained in the obvious way. Then the

relativeL2 error in the pressure is given by

Ph P

2

L2rel()=

CT

( Ph P)

2|C|

CT

(P)2

|C|(20)

where |C| is the measure of the cellC.In these experiments there are two sources of error,

the usual numerical error dependent upon the discreti-

sation parameter h and an error due to the interfacemodel dependent upon the widthdof the fracture.

In Fig.7,on the left, the log of the relative L2 pres-

sure error in the total domain is plotted as a function of

the log of the fracture widthd. Each curve correspondsto a different value of the mesh size h: h= 1/6, h=

1/12, h = 1/24,h = 1/48,h = 1/96, h = 1/192. On the

right, the log of the relative L2 pressure error in the

total domain is plotted as a function of the log of 1

/ h,the inverse of the mesh size, for a constant fracture

widthd = 0.01.In the first five curves on the left there is no de-

crease in the error when the fracture width diminishes

because the error due to the model is dominated by the

error due to the numerical discretization. Only in thesixth curve with h= 1/192 can we see that the error

diminishes with decreasing d just until the numerical

discretization error again becomes dominant. In thecurve on the right, as expected, the error decreases

as 1/ hincreases. The large drop in the error between

1/ h= 96 and 1/ h= 192 is due to the fact that thereference solution is not an analytic solution but a

solution obtained with a mesh for whichh = 1/192.

103

102

101

105

104

103

102

101

100

Log (width of the fracture)

Log(L2E

rrorpressure)

Mesh 192X192

Mesh 96X96

Mesh 48X48

Mesh 24X24

Mesh 12X12

Mesh 6X6

100

101

102

103

105

104

103

102

101

100

Log (1/ h)

Log(L2E

rrorpressure)

error= f(1/ h)

Fig. 7 Left the relative L2 error in the pressure in the totaldomain as a function of the fracture widthd for different valuesof the mesh size: h= 1/6, h= 1/12, h= 1/24, h= 1/48, h=

1/96,h = 1/192.Rightthe relative L2 error variation in the totaldomain as a function of1/ h for a constant fracture width d= 0.01


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Comput Geosci (2008) 12:91104 99

6 A model with intersecting fractures

6.1 Description of the problem

For this model problem, as earlier, is supposed tobe a convex domain in Rn, n= 2 or 3, with boundary = decomposed into a Dirichlet part D and a

Neumann partN. For simplicity, we suppose that thereare three fractures k,k = 1, 2, 3,dividing into three

subdomains i,i = 1, 2, 3,

=

3i=1

i, i j= and i j= k i, j, k

withi = j=k =i,

in such a way that the intersection of each pair ofdistinct fractures is the same set T:

1 2 = 1 3 = 2 3 = T.

The intersectionTis thus a point in the 2-D case and a

segment in the 3-D case. See Fig.8.

As before, in each subdomain i, i= 1, 2, 3, wehave the law of mass conservation together with

Darcys law and the boundary conditions on the exte-

rior boundary:

div ui =qi ini

ui = Kipi ini

pi = pDi on i D

ui ni = 0 on i N. (21)

Also as before, we assume that the pressure is con-tinuous across the fracture interfacesk:

pi = Pk = pj on k, i= j=k =i, (22)

Fig. 8 A domain with intersecting fractures

where Pk is the pressure on the fracture k. In each

fracture we have the law of mass conservation together

with Forchheimers law and the boundary conditions on

the exterior boundary:

divUk = Qk +

ui ni +uj nj

onk

1+

b k

dk |Uk |

Uk = dk Kk Pk onk

Pk = PDk on k ,

(23)

where Uk , Qk , Kk , bk and dk are respectively the

pressure, the flow velocity, the source term, the per-meability tensor, the Forchheimer coefficient and the

width of the fracture for k, and PDk is the average

value of pD

on k . On the intersection T of the

fractures, following [1] and[2], we impose the continu-ity of the pressure and also the continuity of the flux:

P1 = P2 = P3 onT

U1 1 + U2 2 + U3 3 = 0 onT, (24)

where k is the exterior normal to k on k, k=1, 2, 3.

6.2 Numerical discretization

As in the case of a single simple fracture, to obtain

a numerical solution of the system (Eqs. 21, ..., 24),

we use mixed finite elements of dimension n in the

subdomains and mixed finite elements of dimension(n 1) in the fractures. Thus as before the pressure

is approximated in each element (in dimension n and

in dimension n 1) by a constant, and the velocity isapproximated in the space of lowest order Raviart

ThomasNedelec elements. However, in addition to the

unknowns associated with these spaces there are scalar

unknowns associated with the intersection T. In thecase R2 so that T is simply a point this amounts

to one unknown which is a multiplier for enforcing

continuity of the pressure at the fracture intersection.In the case R3 so that T is a segment, there is a

naturally induced mesh onTfrom each fracture havingTas part of its boundary, and as we have assumed herethat the meshes are compatible, i.e. the union of the

meshes for the subdomains forms a conforming mesh

on all , the meshes induced on T from the different

fractures havingTas part of their boundaries are all thesame. Thus there is one additional unknown for each

element of this mesh.

Again as the nonlinearity of the problem is asso-ciated only with the fractures, domain decomposition

techniques can be used to reduced the problem to a


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100 Comput Geosci (2008) 12:91104

problem posed only on the fractures. The resulting

nonlinear system can now be written as follows:

A1(U1 ) B1

0 0 0 0 M1

B1 S1,1 0 S1,2 0 S1,3 0

0 0 A2(U2 ) B2

0 0 M2

0 S2,1 B2 S2,2 0 S2,3 00 0 0 0 A3(U3 ) B

3

M3

0 S3,1 0 S3,2 B3 S3,3 0

M1 0 M2 0 M3 0 MT

U1P1U2

P2U3P3PT

=

0

(25)

whereUk and Pk fork = 1, 2, 3, are the velocity and

the pressure respectively on the interface k and PTthe trace of the pressure on the intersection Tof thefractures. The matricies Ak(Uk )and Bk, represent the

mixed finite element matricies for the flow equations

in the fracture k, k= 1, 2, 3. The matrix Si,j, i, j=

1, 2, 3, represents the domain decomposition matrixassociated with the SteklovPoincar operator taking

into account the values of the pressure on the fractureiand returning the values of the flux on the fracturej.

The matriciesMi,i = 1, 2, 3, T,are the matrices associ-

ated with the equations of continuity of the pressure

and continuity of the flux on the intersection of thefractures.

6.3 A numerical result for the case of intersecting

fractures

In this experiment the domain R2

is divided into4 subdomains i, i= 1, . . . , 4 which are separated by

4 fracturesk,k = 1, . . . , 4. These fractures intersect at

the point T; see Fig. 9. The width dk is the same inall fractures and is equal to 0.01. The permeability in

each of the subdomains is assumed to be constant and

equal to 109, and the permeability in each fracture

is assumed to be much higher (also constant) Kk =

106. The Forchheimer coefficient b k is the same in all

Fig. 9 A test-case

fractures and is taken to be 10. The upper and lowerexterior boundaries of the subdomains are assumed to

be impermeable, and there is a pressure drop from right

to left of106. A pressure of106 is imposed at the ends ofthe fractures lying on the upper and right-hand sides of

the domain and a pressure of 0is imposed on the ends

of the fractures on the left-hand and lower sides of thedomain.

In Fig.10the conforming mesh used for this test case

is shown. In Fig.11the calculated pressure is shown on

the left and the calculated velocity on the right.

7 Nonconforming meshes

In many applications one may want to use nonconform-ing meshes. For simplicity, in this section, we suppose

that the domain is divided into only two subdomains

separated by a single fracture. In the numerical discre-tization, the domain will be decomposed into nonover-lapping subdomains with the fracture as the interface

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 10 The mesh


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Comput Geosci (2008) 12:91104 101

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Fig. 11 The numerical pressure (left) and the numerical velocity (right)

between the two subdomains. The grids will be defined

independently in each subdomain and in the fracture,see Fig.12.

Let Th,i be a conforming finite element partition ofi, i = 1, 2, and let Th,be a conforming finite element

partition of the interface in (n 1)-dimensions. The

meshesTh,1and Th,2need not form a conforming meshon and Th, is independent of Th,1 and ofTh,2. The

notation Th,,i, i= 1, 2, will be used for the partition

induced on by the partition Th,i on i; i.e. Th,,i,i= 1, 2 consists of all of the (n 1) dimensional faces

of elements ofTh,iwhich lie on. See Fig.13.

As above, we use the lowest order RaviartThomas

Nedelec mixed finite element spaces associated withTh,i, i= 1, 2, and Th, so that the vector variable is

approximated in a space Wh = Wh,1 Wh,2 Wh,and the spaces of piecewise constants associated withTh,i, i= 1, 2, and Th,so that the scalar variable is ap-

proximated in a space Mh = Mh,1 Mh,2 Mh,.

Also as above, domain decomposition techniques

are used to reduce the discrete problem to a problemposed only on the interface , but here we can notuse directly the operator Si : Mh, Mh,as defined

in Section 7 because of the incompatibility between

the grids. Letting Mh,,i denote the space of piecewise

constants associated with Th,,i, i= 1, 2, we will, withsome slight abuse of notation, now denote by Si the

mapping Si : Mh,,i Mh,,i defined in the obvious

manner. Then, let Ri : Mh, Mh,,i be the projec-tion operator defined by

Ri : Mh, Mh,,i

rh, rh,,i

where

rh,,i|E =1

|E|

E

rh, E Th,,i.

Fig. 12 An example ofnonconforming grids for thefracture and the subdomains


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102 Comput Geosci (2008) 12:91104

Fig. 13 An example of anumerical discretization withnonconforming grids for thefracture and the subdomains

So, for the case of nonmatching grids the SteklovPoincar operator Si : Mh, Mh,of Section is re-

placed by the operator Si : Mh, Mh,with

Si = Rti Si Ri,

where Rti is just the transpose of the projection oper-

ator Ri.In the numerical experiments below, no mortar ele-

ments were introduced as there are already (n 1)-Delements on the fracture interfaces, nor were the usual

size restrictions associated with mortar elements, c.f.

[4,5] and[19], respected. We have shown (Frih et al.,in preparation) that this is sufficient in the linear case,

i.e. in the case of Darcy flow in the fractures, for the

model of [14].

Remark In the case of intersecting fractures in a

3-D domain, one would need either to have compatible

meshes at the intersection Tor to use mortar elementsonT.

7.1 A numerical experiment for nonconforming

meshes

In this experiment the domain R2 is divided into4 subdomains i, i= 1, ..., 4, by 5 fracture interfacesk, k= 1, ...5. The fractures 1, 2, and 3 intersect at

Fig. 14 Test-case

a pointT1and the fractures 3, 4,and 5intersect at apointT2, see Fig.14.

The widthdk is the same for all the fractures and is

equal to 0.01. The permeability in each of the subdo-

mains is assumed to be constant and equal to 109, andthe permeability in each fracture is assumed to be much

higher and also constant: Kk = 106 for each k. TheForchheimer coefficient bk is taken to be 10 in all ofthe fractures. The upper and lower exterior boundaries

of the subdomains are assumed to be impermeable, and

there is a pressure drop from right to left of 106. Apressure of106 is imposed at the exterior ends of4and5 and a pressure of 0 is imposed on the exterior ends

of1and 2.

The nonconforming mesh used for this test case isshown in Fig.15.It is made up of independently chosen

grids for the subdomains and grids for the fracture

interfaces. Here for simplicity each of the one dimen-

sional grids is a uniform grid. The mesh parametersfor the two dimensional subdomain grids and for the

one dimensional fracture interface grids are given in

the tables below, where by mesh parameter of a two or

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 15 The nonconforming meshes


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Comput Geosci (2008) 12:91104 103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 16 The pressure (left) and the velocity (right) computed solutions

three dimensional grid is meant the maximum length

of an edge of an element of the mesh and of a one

dimensional grid the maximum length of a segment ofthe mesh.

Mesh parameters for the subdomain girds

Subdomain 1 2 3 4

Mesh parameter 0.20 0.10 0.08 0.05

Mesh parameters for the fracture interface grids

Interface 1 2 3 4 5

Mesh 0.035 0.029 0.07 0.048 0.11parameter

The numerical solution obtained with the meshshown in Fig.15is given in Fig.16, where the approxi-

mate pressure is shown on the left and the approximate

velocity field on the right. A good solution was obtainedin spite of the nonconforming gird, even though no

additional mortar elements were introduced.

8 Conclusion

This article is concerned with a numerical model, in-troduced in the short article[12], for flow in a porous

medium with fractures. For this model it was assumed

that the flow rate in the fractures is large enough to

make it appropriate to use Forchheimers law for mod-eling the flow in the fractures even though the flow

in the surrounding domain is such that Darcys law

is adequate. The fractures were treated as interfacesbetween subdomains and nonlocal, nonlinear transmis-

sion conditions were imposed on the interfaces. Then

domain decomposition techniques were used to reduce

the problem to a problem posed on the interfaces.

In this article this model is described in more detailand numerical studies are given. The model is extended

to the case of intersecting fractures and to the case of

nonconforming meshes.

Acknowledgements The authors would like to thank RachidaBouhlila and Jrome Jaffr for helpful conversations concerningthis article.

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