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Parameter identification for a two-scale model of reactive transport inporous media — some first steps
Sebastian Meier1∗ and Michael Bohm1 ∗∗
1 Centre for Industrial Mathematics, FB 3, University of Bremen, Postfach 330 440, 28334 Bremen, Germany.
We formulate a mathematical setting for recovering microscopic parameters from macroscopic measurements by means of atwo-scale model. The model describes reaction and diffusion of a chemical substance in two phases of a porous medium.
1 Introduction
When modelling transport of reactive species in porous media, one often encounters highly different magnitudes of the timescales involved. We consider a special case where a gas A diffuses fast in the pore air and slowly as a solute in the pore water.In the pore water, A is subjected to one or more chemical reactions. The situation is prototypical for chemical degradationof concrete structures; see [1]. It is known by homogenisation theory that the mass balance for A is well approximated by atwo-scale model, in which the processes in the pore water are described by local cell problems [2].
The additional computational costs of a two-scale model are justifiable only when proper values of the pore scale parametersare known, such as the microscopic diffusivity or a pore radius. In practice, they usually have to be identified from macroscopicmeasurements. While recovering a diffusion coefficient in a purely macroscopic model is well-studied (see [3], e.g., for theelliptic 2d case, or [4] for a parabolic 1d equation), this issue seems to be new for a two-scale model. In this work, we presentsome first steps towards a rigourous mathematical setting of these identification problems. We rely on results from [5] for theforward model.
2 The two-scale problem
Fig. 1 Geometry of the distributed microstructure model (2.1)–(2.6).
We consider an unsaturated porous medium, in which the distribution of pore water is assumed as completely known andtransport of water is at rest. Let S := [0, T ], Y := (0, l)n and Ω ⊂ �n be a bounded domain. For each x ∈ Ω, assume we aregiven bounded domains Ys,x, Yx ⊂ Y representing the solid and the liquid phase near x such that Ys,x ∩ Yx = ∅. We assumethat the pore air is connected, whereas the liquid domains Yx are individually isolated, i.e., Y x ⊂ Y . This corresponds to alow humidity of the medium. An example of such a geometry is depicted in Figure 1. We denote Γx := ∂Yx \ ∂Ys,x �= ∅ andassume that the family {(Yx, Γx) : x ∈ Ω} is a regular distribution of cells in the sense of Def. 3.10 in [6].1 Let u = u(x, t)and U = (t, x, y), t ∈ S, x ∈ Ω, y ∈ Yx, be the mass concentrations of A in air and water, respectively. Then the two-scalemodel describing the mass balance of A is given as follows (cf. [6, 5]).
∂t(θ(x)u(t, x)) − div(d(x)∇u) = −1
|Y |
∫Γx
k(Hu − γxU) dσy, t ∈ S, x ∈ Ω, (2.1)
∂tU(t, x, y) − divy(D∇yU) = −g(t, x, y, U), t ∈ S, x ∈ Ω, y ∈ Yx, (2.2)
−D∇yU(t, x, y) · νx = k(γxU(t, x, y) − Hu(t, x)), t ∈ S, x ∈ Ω, y ∈ Γx, (2.3)
−D∇yU(t, x, y) · νx = 0, t ∈ S, x ∈ Ω, y ∈ ∂Yx \ Γx, (2.4)
−d∇u(t, x) · ν = b(u − ue(t, x)), t ∈ S, x ∈ ∂Ω, (2.5)
u(0, x) = u0(x), U(0, x, y) = U0(x, y), x ∈ Ω, y ∈ Yx. (2.6)
∗ Corresponding author E-mail: [email protected], Phone: +00 421 218 63842, Fax: +00 421 218 9406∗∗ E-mail: [email protected], Phone: +00 421 218 63841, Fax: +00 421 218 94061 See also [5]. Roughly, this means that we we can integrate functions over the (2n−1)-dimensional submanifold
Sx∈Ω
({x} × Γx).
PAMM · Proc. Appl. Math. Mech. 7, 1061003–1061004 (2007) / DOI 10.1002/pamm.200700675
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Here the symbols ∇y and divy indicate differentiation by y. The given function θ(x) := |Yg,x|/|Y | is the local volume fractionof the pore air, d(x) ∈ �n×n is the local effective diffusion tensor in air, D, b, k, H are positive constants and ue(t, x) is agiven external concentration. The reaction term g(t, x, y, U) can be nonlinear.
The system (2.1)–(2.6) is a semilinear, weakly coupled system of parabolic PDEs. For a periodic pore geometry, it can bederived by homogenisation theory provided that a proper scaling is chosen [2]. Similar models have been introduced in [7] forflow in fissured media and in [8] for reactive transport. The wellposedness of (2.1)–(2.6) in the weak sense is proven in [6]. In[5], the setting is extended to a microstructure evolving in time.
3 Identification of process parameters
In practical applications, microscopic parameters are often difficult to measure. Assuming a linear reaction g(U) = aU ,a > 0, then the reaction–diffusion process on the microscale is determined by the diffusivity D, the interfacial exchangecoefficient k and the reaction constant a. The question arises: Can these three parameters be estimated by purely macroscopicmeasurements, i.e., having only information on parts of Ω?
For instance, let hT ∈ L2(Ω′) be a measurement of u within a subset Ω′ ⊂ Ω of positive measure at fixed time T . We call
Λ := M3 ⊂ [L2(Ω)]3 the set of admissible parameters, where M is assumed to satisfy
M ⊂ {v ∈ Cα(Ω) : cm ≤ v(x) ∀x ∈ Ω, ‖v‖Cα(Ω) ≤ cM}, α, cm, cM > 0,
and M is convex and closed in [L2(Ω)]3. Let u = u(t, x; λ) and U = U(t, x, y; λ) be the unique weak solution correspondingto data λ = (D, k, a) ∈ Λ. Then a straight-forward least-squares approach consists of finding
λ∗ := arg minλ∈Λ‖u(T, ·, λ) − hT (·)‖2L2(Ω′). (3.1)
Proposition 3.1 Under the “usual” regularity assumptions on the remaining data for problem (2.1)–(2.6), the minimum(3.1) exists.
Sketch of proof. The subset M ⊂ L2(Ω) is convex, closed and bounded. The solution operator F : λ �→ (u, U) isweak sequentially continuous in the setting F : L2(Ω) ⊃ M → C(S; L2(Ω)) × L2(S; L2(Ω; H1(Yx))), where the spaceL2(Ω; H1(Yx)) is constructed as an anisotropic Sobolev space on the domain
⋃x∈Ω({x} × Yx) ⊂ �2n (see [6]). It follows
that also the functional in (3.1) is weak sequentially continuous and therefore takes its minimum in Λ.
4 Identification of the pore geometry
The precise geometry of the microstructure is also unknown in many situations and needs to be determined. The above settingcan be extended to this case provided that the shape of the domain Yx can be described by a few parameters. For example, ifYx is spherical-symmetric, then it can be transformed to a fixed reference geometry by means of two radii R1(x) and R2(x).The identification problem for R1 and R2 can be formulated within the same setting as above.
Remark 4.1
1. Details on the transformation approach can be found in [5]. There it is used for proving wellposedness for a two-scalesystem with a microstructure evolving in time.
2. The numerical solution of the inverse problems can be performed, for instance, by a regularised Gauss-Newton method(cf. [4]). This remains for future work.
Acknowledgements The authors are grateful to the state of Bremen for funding this work through the PhD program Scientific Computingin Engineering (SCiE) and to the German Research Foundation for additional funding through the Collaborative Research Center SFB 570.
References
[1] S. A. Meier, M. A. Peter, and M. Bohm, Computational Materials Science 39, 29–34 (2007).[2] M. A. Peter, Coupled reaction-diffusion systems and evolving microstructure: mathematical modelling and homogenisation, PhD
thesis, University of Bremen, 2006, Logos Verlag, Berlin.[3] G. Chavent and K. Kunisch, ESAIM: Control, Optimization and Calculus of Variations 8, 423–440 (electronic) (2002).[4] V. Isakov and S. Kindermann, Inverse Problems 16(3), 665–680 (2000).[5] S. A. Meier, Two-scale models of reactive transport in porous media involving microstructural changes, PhD thesis, University of
Bremen, In preparation.[6] S. A. Meier and M. Bohm, A note on the construction of function spaces for distributed-microstructure models with spatially varying
cell geometry, Submitted to Electronic Journal of Differential Equations, 2007.[7] R. E. Showalter and N. J. Walkington, Journal of Mathematical Analysis and Applications 155, 1–20 (1991).[8] A. Friedman and A. T. Tzavaras, Journal of Differential Equations 70, 167–196 (1987).
ICIAM07 Minisymposia – 06 Optimization 1061004
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
