HOMOGENIZATION OF THERMAL-HYDRO-MASS TRANSFER PROCESSES

. In the repository, multi-physics processes are induced due to the long-time heat-emitting from the nuclear waste, which is modeled as a nonlinear system with oscillating coeﬃcients. In this paper we ﬁrst derive the homogenized system for the thermal-hydro-mass transfer processes by the technique of two-scale convergence, then present some error estimates for the ﬁrst order expansions.

1. Introduction. Accompanied by the developing of the nuclear power, more and more nuclear waste is produced. The half-life period of radioactivity of nuclear waste is usual very long, especially for the high radioactive waste, which may be several tens of thousands years long. The safe disposal of nuclear waste is an important problem.
Due to the long-term heat emitted by radioactivity, the rock and the underwater in the repository will be heated up. Since the water and rock have different thermal expansivity, thermal input may cause significant pore pressure change which will induce convective flow in porous media. The temperature builds up to a certain level and then decreases. It will take 15-100 years to attain the peak of temperature near the waste repository and 200-1000 years for the far field. That means the thermalhydro-mechanical processes will last a very long time. A lot of researches have been done to the thermal-hydro-mechanical processes (see [6,7,14,21,23,24,25]). There also exits the possibility of the leakage of nuclear waste. If so, the nuclear waste will be dissolved in water and transferred to the far field by the underwater flow. Then the diffusion-convection processes of radioactive nuclear waste must be considered and it leads to the thermal-hydro-mass transfer processes in porous media. In order to describe these processes mathematically, We first show some notations that will be used.
Suppose a cubic domain Ω ⊂ R 3 is occupied by the porous media which is adjacent to the waste at the left boundary Γ 1 , through which the heat and waste comes into the porous media. heat conductivity coefficient c s specific heat of solid K ε permeability coefficient ρ f density of fluid D ε c diffusion coefficient of waste in porous media c f specific heat of fluid β 1 heat exchange coefficient φ ε porosity β 2 mass exchange coefficient v ε Darcy velocity of fluid T out temperature outside left boundary T ε temperature C out concentration of waste outside left boundary p ε pressure T r reference temperature (constant) C ε concentration of waste ρ 0 initial density of fluid Table 1. Notations. Here 0 < ε 1 is the ratio of the characteristic length of micro scale to the whole field By the conservation of energy, the thermal process is governed by By the conservation law of mass of fluid, we have By the conservation law of mass of nuclear waste, the mass transfer process is controlled by As a constitution relation, the Darcy's law is needed: To close the above model, we still need the following state equation by Boussinesq approximation to the density of fluid ( [11]): where ρ 0 is constant standing for the density of fluid at initial temperature T r and α is the thermal expansion coefficient of water. In reality, α will be very small, i.e. 0 < α 1 and the thermal expansion coefficient of rock matrix is even much smaller than α. And note that the small change of the density of fluid only has significant effect on the pressure of fluid, not directly on the temperature and concentration. So we take some assumptions on the density: 1. ρ s is a constant in this study; 2. ρ f = ρ 0 in equations (1) and (3). Then the thermal-hydro-mass transport processes can be described as: with

HOMOGENIZATION OF THERMAL-HYDRO-MASS TRANSFER PROCESSES 57
As to the boundary conditions, Robin boundary conditions for T ε and C ε are assumed on Γ 1 due to the heat dissipation and leakage of waste. For pressure p ε , we assume that the fluid is impermeable on Γ 1 . On the right boundary Γ 2 , which is far away from the waste, the Dirichlet boundary conditions are applied. On the other boundaries Γ 3 = ∂Ω \ (Γ 1 ∪ Γ 2 ), impermeable conditions for T ε , C ε and p ε are imposed. We also need the initial values of T ε and C ε , which we assume to be zero, the same as the far field data.
where ν is the unit outward normal vector. System (6) is a nonlinear partial differential system with high oscillating coefficients. From a numerical point of view, resolving the microscopic details of (6) using typical numerical methods would require at least a cost of O(ε −n ) (n=3) or more. This often becomes prohibitively expensive since ε 1. One way of avoiding this is to solve instead the homogenized equation of the problem (6). The general theory of homogenization can be found in [4,5,10,26] for simple model problems. This paper is devoted to establish the homogenization theory of thermal-hydro-mass transfer processes (6). Two-scale convergence method is employed to deal with the coupled nonlinear terms and the weak convergence for p ε . Two-scale convergence was first introduced by G. Allaire [1,2] and G. Nguetseng [18]. In [3,15], two-scale convergence for time dependent problem was considered. At the same time, the error estimate between the solutions of original problem and their first order expansions is also very import in homogeneous theory. The usual error estimate method for elliptic problems can be found in [10] and [26]. In order to reduce the regularity assumptions for the homogenized problems, the skew-symmetric matrix technique [26] and boundary correctors are always used. For parabolic problems, the initial value corrector is also needed [8]. For problem (6)-(7), it is more complicate since the mixed Robbin-Dirichlet boundary conditions are imposed. Following the idea of [9], we derive the error estimates between the solutions of problem (6)-(7) and their first order expansions. There is one thing also worth pointing out that the homogenized behavior of Equation (3) is different with a single diffusion-convection equation with a prescribed multiscale velocity field. For a single diffusion-convection equation with a prescribed multiscale velocity field, the homogenized velocity is determined not only by micro velocity itself, but also by the micro diffusion coefficient [26]. But here in a system, the homogenized velocity is only determined by the micro pressure equation (2), not related with the micro diffusion coefficient D ε c . Throughout the paper, we also use some mathematical notations such as W k,p (Ω), H k (Ω) = W k,2 (Ω) for general Sobolev spaces, V = {v ∈ H 1 (Ω), v = 0, on Γ 2 }, W = {v ∈ L 2 (0, T ; V ); ∂v ∂t ∈ L 2 (0, T ; H −1 (Ω))} and C ∞ Γ2 (Ω) = {v ∈ C ∞ (Ω); v = 0 on Γ 2 }. C > 0 is a general constant independent of ε, which may be different at different occurrences. Einstein summation convention is also applied. The outline of paper is as follows: in §2 we first get some a priori estimates of the system (6); in §3 we use two-scale method to derive the homogenized system; in §4 we present some error estimates between the solutions of (6) and their first order expansions. As to the multiscale numerical method for problem (6), we will show it in another paper.

2.
A priori estimates. In this section, some a priori estimates of the problem (6) will be derived. We need some assumptions on the coefficients in order to ensure the regularity.
• (H0): The exchange coefficients β 1 and β 2 , the density ρ s , the specific heats c s and c f are positive constants. The conductive coefficients D ε T , D ε c and permeability coefficient K ε are in the forms of D ε x ε ) and D T (x, y), K(x, y), D c (x, y) satisfy that symmetric, continuous and periodic with respect to y ∈ Y = [0, 1] n , uniformly elliptic, i.e. there exist positive constants λ i and Λ For porosity φ ε , we give the following assumption, and φ(x, y) is continuous and periodic with respect to y ∈ Y = [0, 1] n .
So we have φ ε φ 0 (x) = Y φ(x, y)dy weakly * in L ∞ (Ω). The system (6) is not fully coupled in the sense that one computes first T ε , then p ε and finally C ε . By the weak maximum principle [16], there exists a constant C > 0 such that By the basic theory of elliptic and parabolic equations [12,13,16], we can get the existence and uniqueness of solutions, Theorem 2.1. Suppose hypotheses (H0), (H1) hold. If T out , C out ∈ L 2 ((0, T )×Γ 1 ) and α is sufficiently small, then there exist uniqueness solutions T ε ∈ W , p ε ∈ L 2 (0, T ; V ) and C ε ∈ W to system (6) and (7).
Next we show some a priori estimates for the solutions.
Consequently, we get that T ε is bounded in W . By Aubin-Lions lemma, there exists a compact injection W ⊂ L 2 (Ω T ) [10], so there exists a subsequence of T ε converges strongly in . By a variant of Aubin-Lions lemma [20,17], we also have a subsequence of C ε converging strongly in L 2 (Ω T ). In conclusion, we get the following lemma. Lemma 2.3. Let T ε , p ε and C ε be the solutions of problem (6). Under the same conditions of Theorem 2.2, there exist T 0 , C 0 and p 0 ∈ L 2 (0, T ; V ) such that up to a subsequence, Remark 1. Please note that the temperature T ε and concentration C ε converge strongly in L 2 (Ω T ) while the pressure p ε only converges weakly in L 2 (0, T ; V ) since there is no information on its derivative with respect to time. This leads to some difficulties to deal with the nonlinear coupled terms, which can be overcome by the method of two-scale convergence.
Before we prove Theorem 2.2, we first present the weak forms of system (6) and (7), which will be needed later. Find T ε ∈ W , p ε ∈ L 2 (0, T ; V ) and C ε ∈ W such that Proof of Theorem 2.2.
Taking v = T ε (·, t) in (11) and using (H0) and (H1), we have Here we have used Trace's Theorem, Hölder's inequality and Poincaré's inequality for T ε (·, t) ∈ V in the above estimate. Integrating over (0, t) with t ∈ [0, T ] on both 60 SHIXIN XU AND XINGYE YUE sides, we have that Multiplying the first equation of system (6) by ∂Tε ∂t and integrating by part, it yields Integrating over (0, t) with t ∈ [0, T ] and using the Trace Theorem, it follows .
Step 3. Estimate C ε . Multiplying C ε on both sides of the third equation of (6) and integrating by parts, it yields, where we have used the fact that C ε L ∞ is uniformly bounded from the maximum principle (9). Integrating over (0, t) with t ∈ [0, T ] on both sides and combining (18), we have that Combining the weak form of C ε (13) and estimate (18), we can get the The proof is completed.
3. Homogenization. In this section, we will derive the homogenized equations for the limit T 0 , C 0 and p 0 by the method of two-scale convergence for time dependent problem. Before stating the main results, we first review the concept on two-scale convergence.
Then there exists a subsequence, still denoted by u ε , and a function u 0 ∈ L 2 (Ω T × Y ) such that u ε 2 u 0 (t, x, y).
Moreover, u ε converges weakly in L 2 (Ω T ) to the average of the two-scale limit x, y) weakly in L 2 (Ω T ).

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and their two-scale limits p 0 , T 0 and C 0 satisfy the following homogenized system: where with N j (x, y) being the solution of the following cell problem with χ j (x, y) solving the cell problem −∇ y · (K(x, y)∇ y χ j (x, y)) = ∇ y · (K(x, y)e j ), in Y, χ j (x, y) is Y-periodic, and χ j (x, y) Y = 0, and with π j (x, y) solving the cell problem −∇ y · (D c (x, y)∇ y π j (x, y)) = ∇ y · (D c (x, y)e j ), in Y, π j (x, y) is Y-periodic, and π j (x, y) Y = 0.
Furthermore, there exist p 1 , T 1 and C 1 , which are in form as such that where χ k ε := χ(x, x ε ), N k ε := N k (x, x ε ), and π k ε := π k (x, x ε ). Remark 2. The third equation of (21) is a homogenized equation of a convectiondiffusion problem, wherein the homogenized velocity v 0 = −K 0 ∇p 0 is totally determined by the micro-scale velocity v ε = −K ε ∇p ε . This is interesting and different with the behavior of the homogenization for a single equation of convection-diffusion problem with prescribed multiscale velocity, where the homogenized velocity may be determined by both the micro-scale velocity v ε and the micro-scale diffusion coefficient D ε c [26], when the velocity field is not divergence-free. The weak forms of system (21) are as follows. Find T 0 ∈ W , p 0 ∈ L 2 (0, T ; V ) and C 0 ∈ W such that Theorem 3.3. Suppose (H0) and (H1) hold. If T out , C out ∈ L 2 ((0, T ) × Γ 1 ) and α is sufficiently small, then there exist uniqueness solutions T 0 ∈ W , p 0 ∈ L 2 (0, T ; V ) and C 0 ∈ W for system (21).
By Theorem 2.2, Lemma 2.3 and Proposition 1, we can get the following two-scale convergence results.
Lemma 3.4. If p ε , T ε and C ε solve problem (6), then up to a subsequence, still denoted by p ε , T ε , C ε , there exist p 1 (t, x, y), T 1 (t, x, y) and C 1 (t, x, y) ∈ L 2 (Ω T ; H per (Y )) such that Where p 0 , T 0 and C 0 are defined in (10). Let us check the convergence of the coupled term K ε ∇p ε C ε .

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Proof. For any ϕ ε (t, x) = ϕ(t, x)+εϕ The first term at the right-hand side tends to 0 since we have from Lemma 2.3 that C ε (t, x) → C 0 strongly in L 2 (Ω T ) as ε → 0. So using Lemma 3.4, we obtain Remark 3. In the above proof, only the two scale convergence result (2.3) is used. So for temperature T ε , we also have where p 0 , T 0 , p 1 , and T 1 are defined in Theorem 3.2.
Now we give the proof of Theorem 3.2. Proof of Theorem 3.2. For the temperature equation, it is a direct result of homogenization theory for parabolic equation [10,20].
Hence we obtain the homogenized equation for the pressure with K 0 (x)∇ x p 0 · ν = 0 on ∂Ω\Γ 2 and T 0 | t=0 = 0. For the concentration equation in (6), choosing the same test functions as above, we also have By Lemmas 3.4 and 3.5, up to a subsequence, the above formula converges in twoscale to Setting ϕ(x) = 0 and noting that the cell problem (24) for the pressure, we get T 0 Ω Y D c (x, y)(∇ x C 0 + ∇ y C 1 (x, y))∇ y ϕ 1 (x, y)ψ(t)dydxdt = 0.

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Comparing with the cell problem (25) for concentration, we have C 1 = π j ε ∂C0 ∂xj . Choosing ϕ 1 = 0 in (38), we obtain So we get the homogenized equation for the concentration The whole sequence two-scale convergence comes from the uniqueness of the solution for the homogenized system (21). Hence we complete the proof of Theorem 3.2.

4.
Error estimates for first order expansions. In this section, we will present the error estimates between p ε , T ε , C ε and their first order expansions To this purpose, we need some regularity assumptions on homogenized problem (21) (H2) : In the pressure equation of system (6), since φ ε only weakly* converges to φ 0 in L ∞ (Ω), we can only get that the right hand side φ ε ∂Tε ∂t weakly converges in L 2 (0, T ; H −1 (Ω)). Here due to the special structure of right hand side, we have got the convergence results in Theorem 3.2. In order to get the error estimate for first order expansion, further assumption has to be imposed either on the porosity φ ε , wherein we may consider the situation of porosity with only small change in amplitude φ ε (x) = φ 0 + εφ 1 x ε or on the temperature field T ε , wherein we need more regularity on T ε . Here we choose the second case and assume that (H3) : T out satisfies some more compatibility conditions such that The following theorem on the error estimate for the first order expansions is also one of our main results.

Lemma 4.2. [9]
Let θ ε ∈ V be the boundary corrector satisfying the following problem where A ε (x) = A(x, x ε ) is positive definite and uniformly bounded, u ∈ H 2 (Ω) This boundary corrector problem is slightly different with that defined in [9], where the Dirichlet boundary condition is missing and Q j ik is a problem specific matrix. However, the idea used in [9] still works. One can prove the above lemma similarly. Lemma 4.3. If η ε ∈ V is the boundary corrector satisfying the following problem in Ω, and (H0) -(H2) hold, then

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We omit the proof of this lemma since it is similar to Theorem 3.1 in [8].
Lemma 4.4. Let C ε 1 be defined as in (40) and C 0 be the solution of problems (21), respectively. If (H0) and (H2) hold, then there exists a positive constant C independent of ε such that, for any ϕ ∈ V , Proof. For φ ∈ V , according to the definition of C ε 1 , after some simple calculation, we have with Since g j i Y = 0 and ∇ y · g j = 0, there exists a skew-symmetric matrix ( [26]) With this notation, we can rewrite Then we have Let θ ε c be the boundary corrector as a solution of the following problem: From the skew-symmetry of matrix G j ik , the weak form of problem (54) reads Then we have By Lemma 4.2 and the definition of R ε 1 , R ε 2 , we get the estimate: So we complete the proof.
Remark 4. Suppose that (H0) and (H2) be valid. We can obtain in a same way as in Lemma 4.4 that for any ϕ ∈ V , Lemma 4.5. Let C ε , p ε be the solutions of problem (6) and C 0 , p 0 be the solutions of problem (21). If (H0) and (H2) hold, then there exists a positive constant C independent of ε such that for ∀ϕ ∈ V , For the first term, we have The term J 3 can be bounded by a similar way used to treat term J 1 in Lemma 4.4. But we need a slightly different corrector problem to treat the coupling with C 0 . By simple calculations, we have for i = 1, . . . , n, Since h j i Y = 0 and ∇ y · h j = 0, there exists a skew-symmetric

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With this notation, we can rewrite In summary, we obtain for i = 1, . . . , n, Let the boundary corrector θ ε p satisfying the following problem, which is slightly different with the boundary corrected problem (54), By the skew-symmetry of matrix H j ik , the weak form of the above problem is as follows: By (60), J 3 can be rewritten as From Lemma 4.2 and the definitions of r ε 1 , r ε 2 , we obtain the estimate of J 3 Combining (57) with (64), we complete the proof.
Remark 5. If (H0) and (H2) hold, then we can get, in a same way as the above Lemma, that for any ϕ ∈ V , In the estimates of Lemma 4.5 and Remark 5, T ε and C ε are uniformly bounded by (2.2) and ∇p ε 1 is also uniformly bounded since under the assumption (H2), there exists a constant C > 0 such that We also need the following lemma to deal with the terms containing φ ε .
Lemma 4.6. ( [19]) Let g(x, y) ∈ L ∞ (Ω × R N ) be periodic in Y with respect to y and satisfy Y g(x, y)dy = 0 for any x ∈ Ω. Then there exists a constant C > 0 independent of ε such that for any u, v ∈ H 1 (Ω), Now we give the proof of Theorem 4.1. Proof of Theorem 4.1.
Step 1. For T ε − T ε 1 , by the weak forms (11) for T ε and (27) for T 0 , we have Introduce the boundary corrector η ε defined in problem (48). A weak form for problem (48) reads, for every ϕ ∈ V , Then e T = T ε − T ε 1 − εη ε ∈ V . Thanks to (67) and setting ϕ = e T in (66), we obtain Using the uniformly elliptic condition, Poincare's inequality, Remark 4 and Lemma 4.6 , we have Integrating over (0, t) with t ∈ (0, T ] on both sides of above formula and using Gronwall's inequality, we obtain From the estimates for η ε in Lemma 4.3, the error estimate for temperature field (43) is obtained.
Step 2. Estimate p ε − p ε 1 . By the weak forms (12) for p ε and (28) for p 0 , for every ϕ ∈ V , term on the right hand side can be handled by Trace Theorem. So we can get the estimate of C ε − C ε Now we complete the proof of Theorem 4.1.

5.
Conclusion. In summary, a mathematical model is established for the thermalhydro-mass transfer processes in porous media. Then the corresponding homogenized system is derived with the help of two-scale convergence. Some error estimates are also presented for the first order expansions under some higher regular assumptions.