Concluding remarks

MEDIA

Chapter 9 Chapter 9 Applications of MsFEM to upscaling of displacements in heterogeneous porous media

9.14 Concluding remarks

172

0 0 0.5 I 1.5 2 2.5

PVI

Figure 9.20: Case 1, = 0.3, ly = 0.01, a = 2. Dashed-dotted line is for fine fractional flow curve on 100 x 100 grid, solid line is for coarse model using three velocity values in each coarse block on 2 x 2 coarse grid, dotted lines is for the non-uniform coarsening approach on 2 x 2 and 5 x 5 coarse grid (2 x 2 is the worst one)

PVI

Figure 9.21: Case I, = 0.3, ly = 0.05, a = 2. Dashed-dotted line is for fine fractional flow curve on 100 x 100 grid, solid line is for coarse model using three velocity values in each coarse block on 2 x 2 coarse grid, dotted lines is for the non-uniform coarsening approach on 2 x 2 and 5 x 5 coarse grid (2 x 2 is the 'worst one)

PVI

Figure 9.22: Case 1, = 0.3, 1, = 0.2, a = 2. Dashed-dotted line is for fine fractional flow curve on 100 x 100 grid, solid line is for coarse model using three velocity values in each coarse block on 2 x 2 coarse grid, dotted lines is for the non-uniform coarsening approach on 2 x 2 and 5 x 5 coarse grid (2 x 2 is the worst one)

Appendix A Difference form of the cell term

To show that A; can be written in the difference form we need just to prove that

46,,Q,;,,, Q,$,24k,2

and

#8,14b,2 + 48,24h,l

are the same constants in each of two triangular elements with t h e common side kl. In this way the integrand is the same in the union of these two elements. Since the average of the integrand of A t is zero, it is the divergence of a periodic field. Consequently, A$ can be written as a difference of integrals over the boundaries.

To show that these constants axe the same for our configuration we need the following lemma.

Lemma A.O.l If Sl

U

Sz is a parallelogram then

45t,j46,p + &,j4(jp

is the same in Sl and

sz .

Proof. In this case we have

Note that the line segment kA is parallel to the line segment lB, which is also parallel to the line segment Ox, see Fig. A.1. Moreover, since IkAl = [I31 and &js, jk) ⁼ i, g5$js, (A) = 0 , 41s2(1) = 1, ~ b k I s ~ ( B ) = 0, we have

The same can be shown for j = 2.

t i

Remark A.0.1. It can be shown that this difference structure leads to the summation by parts in (7.48).

Now let us show that JK f okd3: can be written in the difference form

( j K

fX~$g,pdz is similar). Consider the configuration illustrated in Fig. A.2. By reordering the terms, we obtain

0 ^X

Figure A.l: Segments in a triangulation

(i- 1 j)

(i-lj-1)

Figure A.2: Element nodes in a triangulation

+(I

. f ~ + d x

-

^{fsi-lj-l dx)}

.

Kz

Here we have used tIk = 0 on K E Kh

.

This difference structure leads to the summation by parts in G ~ P ! .

Appendix B The estimate for linear functions

In this section we show that on any triangular element K

where 1 is a linear function and u satisfies the following equation on a triangular domain S > K

Let us consider the difference of u and I :

4

^{= u-1.} Clearly

45

satisfies the following equation:

Introducing the auxiliary function vi (i = 1,2) defined by v i a i j v j v k = -viai"n S,

vr, = 0 on 8s.

We can express the solution of (B.3) as a linear combination of vi. This gives

where ai = Vil are constants. Then we obtain

where A is the matrix with the following elements All = J K [ ( l

+

^{~ 1 ~}

+

¹( V 2 ~ 1 ) ~ ] d x , ^{) ~}

A12 = J K [ ( l

+

V 1 ~ 1 ) V 1 ~ 2

+

^{( 1}

+

T J ~ V ~ ) V ~ V I ] ~ X , A21 = J K [ ( l

+

V 1 ~ 1 ) 0 1 ~ 2

+

^{( 1}

+

V ~ V ~ ) V ~ V I ] ~ X ,

= J K [ ( l

+

^{~ 2 ~}

+

²( ~ 7 1 ~ 2 ) ~ ] d ~ . ^{) ~}

It can be checked that

and

(Aa, a )

>

0

are the sufficient conditions for (Aa, a ) 2 da2, for some d

>

0. For example under these conditions d can be chosen to be 2d ⁼ All

+

A22

-

J(All ^- A22)2

+

4(A12)2. Note that if d

>

0 then d

>

Ch2 from which it follows that da2

>

C1/VZ11L2(K). Assuming the opposite, i.e.? any of the inequalities does not hold we have that vl

+

^{xl or v2}

+

^{x2 or u} is constant in K. Let's note that vi ^f xi (i ⁼ 1,2) satisfy

Therefore vi

+

^{xi (i = 1 , 2 )} cannot be constant in K [39]. Consequently, All and A22 are strictly positive. Also J K ( O ~ ) 2 d x

>

0 if lVll

>

0 [39] which guarantees (Aa, a )

>

0. If

JVlj = 0 then (B.l) satisfies. This completes the proof of (B.l).

Appendix C Formulation based on dissipation energy

Indelman and Dagan [30] suggested the use of averaged dissipation energy for determining the grid block permeability, i.e.,

(Op

.

KOp)v = (VP'

.

KEVp')v, (C.1) where p-s the solution of (8.5) and p is the solution of

This formulation may be viewed as an approximation to the energy convergence in the homogenization theory (cf. [32]). Note that K cannot be uniquely determined from ((2.1) since adding any anti-symmetric tensor to K does not, change the equality. Thus, we enforce K to be symmetric.

Equation (C.l) is useful for calculating

K

only when Vp is known in advance. This can be achieved by specifying special boundary conditions. Let p = w

+

^e

.

x be the solution of (C.2), where e is a eonstmt vector. Then w d e r the condition w = 0 on aV or w being periodic in V, we have Vp = e on V from (C.2) since K is a constant tensor in V. Thus

(C.l) reduces to

This explicit formula is in fact equivalent to (8.13). We briefly outline the proof here.

First, because (C.3) holds for arbitrary e, choosing e = ei (i = 1,.

. . ,

d) and denoting the corresponding p' by pz, we obtain (8.13) for i = j . Now, choose e = ei

+

^{ej (i}

^#

^{3 ) .}

By using the symmetry of K (as enforced) and K c , as well as the previous result for i = j , it is easy to show that (8.13) holds for i

#

j. On the other hand, since any vector e can be written as a linear combination of ei, we obtain (C.3) from (8.13) by simple algebra.

For this, we use the facts that (8.5) is linear and homogeneous, (8.7) (or (8.8)) is invariant under linear superposition, and that the two sides of (8.13) are bilinear forms.

181

With the boundary condition (8.9), Vp = e only when K is diagonal; otherwise, Vp and K are coupled together and (C.l) could be difficult to use in actual computations.

We mention that reference [lo] showed the above equivalence under the periodic boundary condition with a different approach, but the conclusion for the linear pressure drop condition (corresponding to w = 0 on aV) was incorrect.