Introduction to Aspects of Multiscale Modeling as Applied to Porous Media

Transcription

1 Introduction to Aspects of Multiscale Modeling as Applied to Porous Media Part III Todd Arbogast Department of Mathematics and Center for Subsurface Modeling, Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin

2 Mathematical Homogenization

3 Periodicity The solution u has high frequency wiggles due to the heterogeneity of k. u(x) ū(x) x We want the average behavior! Can we find ū(x) without knowing u(x)? The wiggles are irregular, so they are hard to deal with. Idea : Assume that the heterogeneity is periodic, so that the wiggles are regular, and thus easily identified. This is basically our closure assumption. Idea : Let the period of oscillation be ǫ, and let ǫ. This should remove the wiggles.

4 Convergence of Wiggles Example: Let u ǫ (x) = sin(x/ǫ). As ǫ, sin(x/ǫ) does not converge, it just oscillates more and more between - and. But it becomes a blur, so is it not? (At least in some sense?)

5 Test Functions Idea 3: Consider a weaker form of convergence. Theorem: If then f = g. f(x) φ(x) dx = g(x) φ(x) dx φ C φ(x) We test f by multiplying by φ and integrating. If we know all such tests, we know f. f(x)

6 Convergence of Wiggles Let φ(x) C and consider uǫ (x) φ(x) dx = sin(x/ǫ) φ(x) dx = ǫcos(x/ǫ) φ (x) dx We call this weak convergence, and write u ǫ

7 Definition: If lim ǫ u ǫ (x) φ(x) dx = Weak Convergence u(x) φ(x) dx φ C then u ǫ converges weakly to u. We write u ǫ u. Definition: The L -norm of a function u is u { u(x) dx } / Theorem: If there is C > independent of ǫ such that u ǫ C < then there exists u such that u ǫ u.

8 Obtaining Periodic Wiggles Suppose that the domain Ω has a periodic structure with period ǫy. As ǫ, we obtain our macro-scale model for the average flow. Y Question: How do we proceed? Homogenization is very mathematical, and involves deep analysis and partial differential equations. Fortunately there is a simpler, more physical way to view homogenization.

9 Formal Homogenization Scaling: We assume that the space variable has both a slow (x) and fast (y) component. x x + ǫy At any point x, y allows us to see the local details, which disappear as ǫ. x x + ǫy y Y Ω Formal assumption: We assume without proof that we can expand the true solution p(x) into a power series involving ǫ: p(x) p (x, y) + ǫ p (x, y) + ǫ p (x, y) + wherein y = x/ǫ and each p k is periodic in y. Gradient scaling: Then x + ǫ y

10 Homogenization of Darcy Flow Recall the model of Darcy flow (ignore outer BCs): (k ǫ p ǫ ) = f in Ω Make the closure assumption that k ǫ (x) = k(x, x/ǫ), where k(x, y) is periodic of period in y, so k ǫ (x) has periodic oscillations of period ǫ. Note that k ǫ (x) can vary slowly over the domain with only the local variability being periodic. Formal expansion: Substitute p ǫ (x) = p (x, y) + ǫ p (x, y) + ǫ p (x, y) + and = x + ǫ y to find (ǫ y + x ) [k(x, y)(ǫ y + x ) ( p (x, y) + ǫ p (x, y) + ǫ p (x, y) + )] = f in Ω Y

11 We rewrite our expansion Homogenization of Darcy Flow (ǫ y + x ) [k(x, y)(ǫ y + x ) ( p (x, y) as a power series in ǫ ǫ { y [ k(x, y) y p (x, y) ]} + ǫ p (x, y) + ǫ p (x, y) + )] = f in Ω Y + ǫ { y [ k(x, y) ( y p (x, y) + x p (x, y) )] x [ k(x, y) y p (x, y) ]} + ǫ { y [ k(x, y) ( y p (x, y) + x p (x, y) )] + = f in Ω Y x [ k(x, y) ( y p (x, y) + x p (x, y) )]} This should hold for all ǫ as ǫ, so it must hold for each term.

12 Step, ǫ -terms: Homogenization of Darcy Flow 3 y [k(x, y) y p (x, y) ] = in Ω Y p (x, y) is periodic in y Note that there are no derivatives in x, so x is just a parameter. We have a partial differential equation (PDE) in y only. It is not particularly difficult to see that p (x, y) must be constant in y. Conclusion: p = p (x) only. Question: Why is this result important? That is, why should the leading order of the solution not depend on y?

13 Step, ǫ -terms: Homogenization of Darcy Flow 4 y [ k(x, y) ( y p (x, y) + x p (x, y) )] From Step, one term vanishes. Thus x [k(x, y) y p (x, y) ] in Ω Y y [ k(x, y) y p (x, y) ] = y [k(x, y) p (x) ] in Ω Y p (x, y) is periodic in y Again x is basically a parameter, and this is a PDE in y for p (x, y), if we are given x p (x). But x p (x) is a constant vector in y! Trick: Use the linearity! Since p = j e j j p, replace p by e j and solve.

14 Homogenization of Darcy Flow 5 We solve (for each fixed x of interest) y [ k(x, y) y ω j (x, y) ] = y [ k(x, y)e j ] ω j (x, y) is periodic in y Then, multiplying by j p and summing, j j p (x) y [ k(x, y) y ω j (x, y) ] = j in Ω Y j p (x) y [ k(x, y)e j ] Linearity and constancy in y of j p allow us to move things inside y [ k(x, y) y ω j (x, y) j p (x) ] = y [ k(x, y) p (x) ] Conclusion: j p (x, y) = j ω j (x, y) j p (x) solves our problem y [ k(x, y) y p (x, y) ] = y [k(x, y) p (x) ] in Ω Y p (x, y) is periodic in y

15 Homogenization of Darcy Flow 6 Step 3 (Final Step), ǫ -terms: y [ k(x, y) ( y p (x, y) + x p (x, y) )] x [k(x, y) ( y p (x, y) + p (x) )] = f in Ω Y Trick: Remove y by averaging (integrate over y and divide by the volume of Y, Y ). For the first piece, we get Y Y y [ k(x, y) ( y p (x, y) + x p (x, y) )] dy = Y due to periodicity! Y [ k(x, y) ( y p (x, y) + x p (x, y) )] ν ds(y) = Easily, the third piece is Y Y f(x) dy = f(x)

16 Note that Homogenization of Darcy Flow 8 p (x, y) = j ω j (x, y) j p (x) tells us that we know p if we know p. Thus the second piece is Y Y x [ k(x, y) ( y p (x, y) + p (x) )] dy = Y k(x, Y y)( y ω j (x, y) j p (x) + p (x) ) dy = i = i = i i Y j j i ( Y j k(x, Y y)( y i ω j(x, y) j p (x) + j j Y k(x, y)( y i ω j(x, y) + δ ij i (ˆk ij j p (x) ) = (ˆk p ). ) dy ) j p (x) δ ij j p (x) ) dy We have derived our homogenized coefficient ˆk from k. It is a tensor! Conclusion: Collecting pieces, we have our desired result: (ˆk p ) = f in Ω

17 Homogenization of Darcy Flow 9 Summary: Starting from (k ) ǫ p ǫ = f in Ω we found that, as ǫ, p ǫ p, where (ˆk p ) = f in Ω and ˆk(x) can be computed as the tensor ˆk ij (x) = Y k(x, Y y)( y i ω ) j(x, y) + δ ij dy and ω j (x, y) can be computed from the local cell problems: y [ k(x, y) y ω j (x, y) ] = y [ k(x, y)e j ] ω j (x, y) is periodic in y in Ω Y

18 The Homogenized Permeability Lemma: ˆk is symmetric and positive definite: ξ ˆkξ = i,j ξ iˆk ij ξ j > for all vectors ξ. Thus, ˆk has three principle eigenvectors and only positive eigenvalues. Question: Why is this important? Lemma (Voigt-Reiss Inequality): ˆk lies between the harmonic and arithmetic averages. More precisely, if ( ) k h = Y (k(x, Y y)) dy and k a = k(x, y) dy Y Y then ξ k h ξ ξ ˆkξ ξ k a ξ

19 Convergence Theorem: As ǫ, we have weak convergence p ǫ p In fact, p ǫ p and p ǫ p Cǫ Moreover, if p ǫ = p + ǫ j ω j (x, x/ǫ) j p (x), then (p ǫ p ǫ) C ǫ

20 Computational Upscaling via Homogenization In our small -D problem, we obtain the following. Log-permeability and xx and yy local averages (xy = yx set to ): Computed pressure: homogenized avg 8 8 computed average Relative errors: Harmonic.4, Homog..36, Computational.8.