ON THE QUASISTATIC APPROXIMATION IN THE STOKES-DARCY MODEL OF GROUNDWATER-SURFACE WATER FLOWS

Transcription

1 ON THE QUASSTATC APPROXMATON N THE STOKES-DARCY MODEL OF GROUNDWATER-SURFACE WATER FLOWS Marina Moraiti Department of Mathematics University of Pittsburgh Pittsburgh, PA 156, U.S.A. Tel: +1 (41) mam38@pitt.edu Abstract We study the validity of the quasistatic approximation in the fully evolutionary Stokes-Darcy problem for the coupling of groundwater and surface water flows, as well as the dependence of the problem on the specific storage parameter. n the coupled equations that describe the groundwater and surface water flows for an incompressible fluid, the specific storage, S, represents the volume of water that a fully saturated porous medium will expel (or absorb) per unit volume per unit change in hydraulic head. n confined aquifers, S takes values ranging from 1 6 or smaller to 1. n this work we analyze the validity of the previously studied quasistatic approximation (setting S = in the Stokes- Darcy equations) by proving that the weak solution of the fully evolutionary Stokes-Darcy problem approaches the weak solution of the quasistatic problem as S. We also estimate the rate of convergence. Keywords: Storage. Stokes-Darcy, Groundwater flow, Surface water flow, Specific 1. ntroduction n the time-dependent Stokes-Darcy problem that models the coupling of φ groundwater with surface water flows, given in (1.1), the term S t, where φ is the hydraulic head and S is the specific storage, arises because aquifers are poroelastic media. This means that the space between the pores responds to changes in the pressure of the water occupying the pores. See e.g., Bear [1] for a clear derivation of (1.1) from basic conservation laws. There has been considerable study of the poroelastic effect and its many consequences, see e.g., Biot [] and Wang [3]. n confined aquifers the values of S range from 1 6 or smaller for rock to 1 for plastic clay, see Domenico [4], while in unconfined Preprint submitted to Elsevier April 3, 1

2 aquifers, S can be larger 1. n Table 1 we give a few representative values for S in confined aquifers, see Anderson [5], Batu [6], Domenico and Mifflin [7], and Johnson [8]. The quasistatic approximation is obtained by setting S = in the Stokes-Darcy problem. t is thus equivalent to an inelastic assumption on the aquifer and is used in e.g., Cesmelioglu and Riviére [9], and Badea, Discacciati, and Quarteroni [1]. n this report we justify the validity of this quasistatic approximation. We prove in Theorems 4.1 and 4. that the solution of the fully evolutionary Stokes-Darcy problem, (u, φ), where u is the velocity in the fluid region and φ is the hydraulic head in the porous media region, converges to the solution of the quasistatic Stokes-Darcy model, denoted (u QS, φ QS ), as S converges to zero, under mild assumptions on the initial data and body forces, and the shape of the fluid and porous media domains. We obtain one half or first order convergence depending on the assumptions. Table 1: Specific Storage (S ) values for different materials Material Specific Storage S (m 1 ) plastic clay stiff clay medium hard clay loose sand dense sand dense sandy gravel rock, fissured jointed rock, sound less than Let the fluid and porous media domains be denoted by Ω f and Ω p respectively, Ω f/p R d, d = or 3, and assume they lie across an interface,, from each other as shown in Figure 1. Both domains are assumed to be bounded and regular. Before introducing the Stokes-Darcy problem and its quasistatic approximation, we list below the variables and parameters of the problem: u = u(x, t) = (u 1 (x, t),..., u d (x, t)) = fluid velocity in Ω f, φ = φ(x, t) = hydraulic head in Ω p, x = (x 1,..., x d ) Ω f/p, f f, f p = body forces in fluid region and sources or sinks in porous region, K = hydraulic conductivity tensor (symmetric positive definite), g = gravitational acceleration constant, ν = kinematic viscosity of fluid, ˆn f/p = unit outward pointing normal on Ω f/p, 1 A confined aquifer is one bounded above and below by impervious formations. n a well penetrating such an aquifer, the water level will rise above the base of the confining formation. An unconfined aquifer is one with a water table serving as its upper boundary, see Bear [1].

3 Figure 1: Fluid and porous media domains The specific storage, S, represents the volume of water that a portion of a fully saturated porous medium releases from storage, per unit volume, per unit change in hydraulic head, see Freeze and Cherry [1], and Hantush [13]. All material and fluid parameters above are positive. Moreover, < λ k max, where λ λ(k), and λ(k) is the spectrum of the hydraulic conductivity tensor, K. The fluid velocity u = u(x, t), defined on Ω f, and porous media hydraulic head φ = φ(x, t), defined on Ω p, satisfy u t ν u + p = f f (x, t), u =, in Ω f, S φ t (K φ) = f p (x, t), in Ω p, u(x, ) = u, in Ω f, φ(x, ) = φ, in Ω p, (1.1) u(x, t) =, in Ω f \, and φ(x, t) =, in Ω p \, + coupling conditions across, where the pressure in the fluid domain, p, as well as f p are rescaled by the fluid density. We assume Dirichlet boundary conditions on the exterior boundaries (not including the interface ); our analysis extends to other boundary conditions as well. The coupling conditions are conservation of mass across the interface u ˆn f K φ ˆn p =, on, and balance of forces across the interface p νˆn f u ˆn f = gφ, on. The last condition is a condition on the tangential velocity on. Let ˆτ i, i = The specific storage is defined as S = S, where S the storativity coefficient (dimensionless) and b the height of the aquifer. For more information see Watson and Burnett b [11]. 3

4 1,..., d 1, denote an orthonormal basis of tangent vectors on, d = or 3. We use the Beavers-Joseph-Saffman-Jones condition, see Joseph [14] and Saffman [15]: ν ˆτ i u ˆn f = u ˆτ i, for i = 1,..., d 1, on, (1.) ˆτi K ˆτ i which is a simplification of the original and more physically realistic Beavers- Joseph condition, see Beavers and Joseph [16]. The latter states that the tangential component of the normal stress of the flow in the conduit at the interface is proportional to the tangential velocity in the conduit at the interface. n (1.), > is a dimensionless, experimentally determined constant. For more information on this condition see e.g., Mikeli`c and Jäger [17], and Payne and Straughan [18]. One common model used in e.g., Cesmelioglu and Riviére [9], and Badea, Discacciati, and Quarteroni [1], is based on the assumption that the porous media pressure adjusts instantly to changes in the fluid velocity, in other words, the term S φ t is dropped in the Stokes-Darcy equations. This leads to replacing (1.1) by the quasistatic approximation: u QS t ν u QS + p QS = f f (x, t), u QS =, in Ω f, (1.3) (K φ QS) = f p (x, t), in Ω p, with the same interface coupling and boundary conditions for u QS, φ QS, and initial condition for u QS. We consider herein the mathematical foundation for this simplification. Problems of the type ɛu t + Au =, ɛ small, are treated in Lions [19]. However, problem (1.1), with S small, does not fit within the general theory in [19]. The rest of this paper is organized as follows. Section presents the variational formulation for both the Stokes-Darcy and the quasistatic problems. n Section 3 we obtain à priori bounds on the velocity and hydraulic head for both problems. n Section 4 we justify that u u QS and φ φ QS, as S, through Theorems 4.1 and 4.. We prove one half and first order convergence in S of the Stokes-Darcy solution to the quasistatic solution, under assumptions on the initial data and body forces. Convergence of the pressure is then standard. This analysis justifies the inelastic or quasistatic approximation provided S <<. Finally, we present conclusions in Section 5.. Variational formulation We denote the L norms on Ω f/p by f/p respectively, and the L norm on the interface,, by ; the corresponding inner products on Ω f/p are denoted 4

5 by (, ) f/p. Moreover, the H 1 norm on Ω f/p is denoted by 1,f/p. Define the spaces X f := v (H 1 (Ω f )) d : v = on Ω f \, X p := ψ H 1 (Ω p ) : ψ = on Ω p \, Q := L (Ω f ), V f := v X f : ( v, q) f = for all q Q. Define the norms on the dual spaces X f and X p by f 1,f/p = (f, v) f/p sup. v X f/p v f/p n what follows, we will use the basic estimates u L ( Ω f ) C(Ω f ) u f u f (.1) φ L ( Ω p) C(Ω p ) φ p φ p, (.) where by a scaling argument, C(Ω f/p ) = O( L f/p ), L f/p = diameter(ω f/p ). Define the bilinear forms d 1 a f (u, v) = (ν u, v) f + a p (φ, ψ) = (K φ, ψ) p, c (u, φ) = g φu ˆn f ds. Lemma.1. The bilinear forms a f (, ), a p (, ) satisfy ˆτi K ˆτ i (u ˆτ i )(v ˆτ i ) ds, a f (u, v) max ν + 1, C(Ω f ) u 1,f v 1,f, (.3) a f (u, u) ν u f + d 1 (u ˆτ i ) dσ =: ν u f + u ˆτ kmax kmax, (.4) a p (φ, ψ) k max φ p ψ p, (.5) a p (φ, φ) φ p, (.6) for all u, v X f and all φ, ψ X p. Proof. Let φ, ψ X p. Since K is positive definite, and < λ(k) k max, (.5) and (.6) are straightforward. For u, v X f, and using ˆτ i K ˆτ i, 5

6 i, the Cauchy-Schwarz inequality, and the trace inequality (.1) we have a f (u, v) ν u f v f + C(Ω f ) u f u f v f v f. kmin Applying the arithmetic-geometric mean inequality we obtain (.3). using ˆτ i K ˆτ i k max, i, we get a f (u, u) ν u f + d 1 kmax Finally, (u ˆτ i ) dσ =: ν u f + kmax u ˆτ. The key quantity in the analysis is the interface coupling term, c (, ). Lemma.. The bilinear form c (, ) satisfies for all u, φ X f, X p. c (u, φ) gc(ω f, Ω p ) u 1,f φ 1,p (.7) Proof. (.7) follows by applying the Cauchy-Schwarz inequality and the trace inequalities (.1)-(.). The variational formulation of the Stokes-Darcy problem then is to find u : [, ) V f, φ : [, ) X p satisfying (u t, v) f + a f (u, v) + c (v, φ) = (f f, v) f, (.8) gs (φ t, ψ) p + ga p (φ, ψ) c (u, ψ) = g(f p, ψ) p, (.9) v V f, ψ X p, where u(x, ) = u (x), φ(x, ) = φ (x) are given. Existence and uniqueness of a solution (u, φ) to problem (.8)-(.9) follow by the Hille - Yosida theorem, see Brézis []. The variational formulation of the quasistatic approximation is obtained by setting S = in (.8)-(.9): Find u QS : [, ) V f, φ QS : [, ) X p satisfying (u QS t, v) f + a f (u QS, v) + c (v, φ QS ) = (f f, v) f, (.1) ga p (φ QS, ψ) c (u QS, ψ) = g(f p, ψ) p, (.11) v V f, ψ X p, where u QS (x, ) = u (x) is given. through (.11), by solving φ QS (x, ) is defined ga p (φ QS (x, ), ψ(x)) = c (u (x), ψ(x)) + g(f p (x, ), ψ(x)), ψ X p, for the unknown φ QS (x, ). 6

7 3. À priori estimates The difference between variational formulations (.8)-(.9) and (.1)-(.11) is the term gs (φ t, ψ) p. Thus, convergence will hinge on à priori bounds on the hydraulic head φ. C = C (u, φ, f f/p, g, Ω p ) and C = C (u, φ, f f/p, ν, g,, k max ) will denote finite constants, independent of S, and L (, T ; X) = v : [, T ] X : T v(t) X dt <, L (, T ; X) = v : [, T ] X : sup v(t) X <. [,T ] Theorem 3.1 Part 1, gives à priori bounds for the velocity and hydraulic head for both problems. The second part gives bounds for the time derivatives of the same quantities for both problems. Theorem n the variational formulations (.8)-(.9) and (.1)-(.11) assume the initial data and body forces satisfy u L (Ω f ), f f L (, T ; H 1 (Ω f )), f p L (, T ; H 1 (Ω p )). Then for u QS given by (.1)-(.11) u QS L (, T ; L (Ω f )), u QS L (, T ; L (Ω f )), u QS ˆτ L (, T ; L ()), φ QS L (, T ; L (Ω p )). (3.1) f in addition φ L (Ω p ), then for u, φ given by (.8)-(.9) u L (, T ; L (Ω f )), S φ L (, T ; L (Ω p )), u L (, T ; L (Ω f )), u ˆτ L (, T ; L ()), φ L (, T ; L (Ω p )). (3.). Assume that the body forces satisfy f f,t L (, T ; H 1 (Ω f )), f p,t L (, T ; H 1 (Ω p )), where f f/p,t denotes differentiation with respect to time. f the initial data for (.8)-(.9) satisfy u t () L (Ω f ), φ t () L (Ω p ), then u t L (, T ; L (Ω f )), S φ t L (, T ; L (Ω p )), u t L (, T ; L (Ω f )), u t ˆτ L (, T ; L ()), φ t L (, T ; L (Ω p )). (3.3) f the initial data for (.1)-(.11) satisfy u QS t () L (Ω f ), then u QS t u QS t L (, T ; L (Ω f )), u QS t L (, T ; L (Ω f )), ˆτ L (, T ; L ()), φ QS t L (, T ; L (Ω p )). (3.4) 7

8 Proof. The result follows from Propositions below. Proposition 3.1 is the first energy estimate for the Stokes-Darcy weak formulation (.8)-(.9): Proposition 3.1. Consider the fully evolutionary Stokes-Darcy problem (.8)- (.9). Assume the initial data and body forces satisfy u L (Ω f ), φ L (Ω p ), f f L (, T ; H 1 (Ω f )), f p L (, T ; H 1 (Ω p )). We have T sup u(t) f + gs φ(t) p + [,T ] u f + gs φ p + ν u(t) f + T u ˆτ + g φ p kmax 1 ν f f 1,f + g f p 1,p dt dt C. (3.5) Proof. Fix t >. Set v = u(t), ψ = φ(t) in (.8)-(.9) and add. Note that the two coupling terms exactly cancel. The remainder follows by standard manipulations. Using coercivity of the bilinear forms and Young s inequality we obtain 1 d dt u(t) f + gs φ(t) p + ν u(t) f + kmax u ˆτ + g φ(t) p f f 1,f u f + g f p 1,p φ p ν u f + 1 ν f f 1,f + g φ p + g f p k 1,p. min Rearranging and integrating over [, t] for any t in (, T ] and T <, yields t u(t) f + gs φ(t) p + ν u(s) f + u ˆτ + g φ(s) p ds kmax t u() f + gs φ() 1 p + ν f f 1,f + g f p 1,p ds. Finally, the result follows by taking supremum over [, T ]. The next proposition gives the corresponding energy estimate for the quasistatic weak formulation (.1)-(.11). Proposition 3.. Consider the quasistatic weak formulation (.1)-(.11). Assume the initial data and body forces satisfy u L (Ω f ), f f L (, T ; H 1 (Ω f )), f p L (, T ; H 1 (Ω p )). 8

9 We have sup u QS f + [,T ] T ν u QS (t) f + u QS ˆτ + g φ QS (t) p dt k max g f p 1,p dt C. T 1 u f + ν f f 1,f + Proof. Fix t >. n (.1)-(.11) pick v = u QS (t), ψ = φ QS (t) and add. The coupling terms cancel and the result follows by manipulations similar to the ones in the proof of Proposition 3.1. Propositions 3.3 and 3.4 provide à priori bounds for the time derivatives of u and φ in the Stokes-Darcy and the quasistatic Stokes-Darcy problems, respectively. Proposition 3.3. Consider the fully evolutionary Stokes-Darcy problem (.8)- (.9). Assume the initial data and body forces satisfy Then u t () L (Ω f ), φ t () L (Ω p ), f f,t L (, T ; H 1 (Ω f )), f p,t L (, T ; H 1 (Ω p )). T sup ut f + gs φ t p + [,T ] u t() f + gs φ t() p + ν u t f + T u t ˆτ + g φ t p kmax 1 ν f f,t 1,f + g f p,t 1,p dt dt C. Proof. Start with the weak formulation (.8)-(.9), and take the derivative with respect to time (u tt, v) f + a f (u t, v) + c (v, φ t ) = (f f,t, v) f, (3.6) gs (φ tt, ψ) p + ga p (φ t, ψ) c (u t, ψ) = g(f p,t, ψ) p. (3.7) Fix t >. n (3.6)-(3.7) pick v = u t (t), ψ = φ t (t), and add. The coupling terms will cancel and the rest of the proof is similar to the proof of Proposition 3.1. Proposition 3.4. Consider the quasistatic weak formulation (.1)-(.11). f u QS t () L (Ω f ), f f,t L (, T ; H 1 (Ω f )), f p,t L (, T ; H 1 (Ω p )) 9

10 then sup u QS t (t) f + [,T ] T ν u QS t f + kmax u QS t u QS t () f + T ˆτ + g φ QS t p 1 ν f f,t 1,f + g f p,t 1,p dt dt C. Proof. Starting with the weak formulation (.1)-(.11), take the derivative with respect to time to get (u QS tt Fix t >. coupling terms cancel. Proposition 3.1. Pick v = u QS t, v) f + a f (u QS t, v) + c (v, φ QS t ) = (f f,t, v) f, (3.8) ga p (φ QS t, ψ) c (u QS t, ψ) = g(f p,t, ψ) p. (3.9) (t), ψ = φ QS t (t) in (3.8)-(3.9) and add so that the The remainder of the proof is similar to the one of n the subsection that follows we obtain á priori bounds assuming less regularity on the body forces, but introducing a constraint on the domains Ω f, Ω p À priori estimates using less regular body forces By assuming less regularity on the body forces we obtain á priori bounds on the velocity and hydraulic head. n this case, however, we restrict the domains Ω f and Ω p by assuming that there exists a C 1 diffeomorphism from Ω f to Ω p. We begin with a Lemma that gives a bound on the interface term, that is essential in the analysis of the less regular case. Lemma 3.1. Assume there exists a C 1 diffeomorphism F : Ω f Ω p. Then there exists a constant C such that where v H(div,f) = v f + v f. Proof. Define φ : Ω f Ω p by c (u, φ) gc u H(div,f) φ 1,p, (3.1) φ(x) = (φ F )(x), x Ωf φ(x), x. Since F is a C 1 diffeomorphism, there exist constants C 1, C such that 1 det(f ) C 1, in Ω f, (3.11) F Hilb C, in Ω f, (3.1) 1

11 where F is the Jacobian matrix of F, and Hilb denotes the Hilbert norm. We have ( ) φu ˆn f ds = φu ˆn f ds = φu ˆnf ds = φu ds Ω f Ω f = φ u dx + φ u dx, Ω f Ω f by the divergence theorem. Thus we obtain c (u, φ) g φ u dx + g φ u dx Ω f Ω f g u H(div;f) φ 1,f, (3.13) by the Cauchy-Schwarz inequality. The change of variables theorem yields φ 1,f = ( ( Ω f ( φ F + (φ F ) ) det(f ) det(f ) dx ) 1 Ω f ( φ F + (φ F ) ) det(f ) C 1 dx = C 1 ( Ω p ( φ + x φ ) dη ) 1 C 1 ( Ω p ( φ + F Hilb η φ ) dη C 1 max1, C ( ) 1 Ω p ( φ + η φ ) dη ) 1 ) 1 = C φ 1,p, (3.14) where C := C 1 max1, C, x = (x1,...,x d ), x Ω f, denotes the gradient operator in Ω f, and η = (η1,...,η d ), η Ω p, denotes the gradient operator in Ω p. The inequality now follows by combining (3.13) and (3.14). t is worth noting that in the special case when the interface is flat and Ω f, Ω p are any domains (Figure ), inequality (3.1) holds with constant C = 1. Lemma 3.. f the interface is flat (i.e., is the line x = in d or the plane x 3 = in 3d), then c (u, φ) g u H(div,f) φ 1,p. Proof. Embed the two domains in equal sized boxes as shown in Figure. Extend u by zero on B and φ by zero on B, and denote the extended functions 11

12 Figure : Fluid and porous media domains with flat interface by u and φ respectively. Let F (x 1, x,..., x d 1, x d ) = (x 1, x,..., x d 1, x d ), (x 1,..., x d ) B. Then det(f ) = 1. Define φ : B R by φ(x 1,..., x d ) = (φ F )(x1,..., x d ) = φ(x 1,..., x d ), on B φ(x 1,..., x d ), on. Using the same steps as in the proof of (3.1), with B replacing Ω f, we obtain φu ˆn f ds = φu ˆn f ds = φu ˆn f ds B = ( φu) dx = φ u dx + φ u dx B u H(div,f) ( = u H(div,f) ( = u H(div,f) ( B B B = u H(div,f) φ 1,p, B ( φ + x φ ) dx B ) 1 ) 1 ( φ + x φ ) dη ) 1 ( φ + η φ ) dη where, as in the general proof of the inequality, x/η denotes the gradient operator in Ω f/p respectively. Thus, the inequality follows. n our problem, u = in Ω f, in which case the proof of (3.1) yields the following inequality instead c (u, φ) gc u f φ p. (3.15) 1

13 The constant C in (3.15) is given instead by C := C 1 C, where C 1, C are the constants in (3.11)-(3.1). Theorem 3.. Assume the initial data and body forces satisfy u, u() L (Ω f ), u ˆτ i t= L (), i = 1,..., d 1, φ() L (Ω p ), f f L (, T ; L (Ω f )), f p L (, T ; L (Ω p )), and that the domains Ω f, Ω p are such that (3.15) holds. We have + gkmin φ p Then, specifically 1 T T u t f dt + gs (t) T gc + sup u f [,T ] φ t p dt + sup ν u f + [,T ] kmax u ˆτ f f f dt + g T T f p p dt + 8(gC) φ p dt S (t) + d 1 ν u f + + g(k φ, φ) p c (u, φ) u t L (, T ; L (Ω f )), S φ t L (, T ; L (Ω p )), ˆτi K ˆτ i (u ˆτ i) ds () C. u L (, T ; L (Ω f )), u ˆτ L (, T ; L ()), φ L (, T ; L (Ω p )). Proof. Fix t > and set v = u t (t), ψ = φ t (t) in (.8)-(.9) and add: Thus, (u t, u t ) f + gs (φ t, φ t ) p + a f (u, u t ) + ga p (φ, φ t ) + c (u t, φ) c (u, φ t ) d 1 u t (t) f + gs φ t (t) p + ν( u, u t ) f + = (f f, u t ) f + g(f p, φ t ) p. ˆτi K ˆτ i (u ˆτ i )(u t ˆτ i ) ds + g(k φ, φ t ) p + c (u t, φ) c (u, φ t ) = (f f, u t ) f + g(f p, φ t ) p. Using the Cauchy-Schwarz and Young s inequalities we obtain u t f + gs φ t p + 1 d d 1 ν u f + (u ˆτ i ) dt ˆτi ds K ˆτ i + g(k φ, φ) p + c (u t, φ) c (u, φ t ) f f f u t f + g f p p φ t p 1 u t f + 1 f f f + gs φ t p + g S f p p. 13

14 Rearranging then gives u t f + gs φ t p + d d 1 ν u f + dt ˆτi K ˆτ i (u ˆτ i ) ds + g(k φ, φ) p + c (u t, φ) c (u, φ t ) f f f + g S f p p. Using c (u t, φ) c (u, φ t ) = d dt c (u, φ) + c (u t, φ) we get u t f + gs φ t p + d d 1 ν u f + dt + g(k φ, φ) p c (u, φ) ˆτi K ˆτ i (u ˆτ i ) ds f f f + g S f p p 4c (u t, φ). (3.16) By (3.15) and Young s inequality we obtain c (u, φ) gc u f φ p g C u f g φ p. (3.17) Moreover, since u = in Ω f implies that u t = in Ω f, we also get 4c (u t, φ) 4gC u t f φ p 1 u t f + 8(gC) φ p. (3.18) Using (3.18) in (3.16) and rearranging terms gives 1 u t f + gs φ t p + d ν u f + dt + g(k φ, φ) p c (u, φ) d 1 ˆτi K ˆτ i (u ˆτ i ) ds f f f + g S f p p + 8(gC) φ p. 14

15 ntegrating over (, t], t T, then gives 1 t t d 1 u t f dt + gs φ t p dt + ν u f + + g(k φ, φ) p c (u, φ) (t) t t d 1 + 8(gC) φ p dt + ν u f + Using (3.17), we estimate d 1 ν u f + ˆτi K ˆτ i (u ˆτ i ) ds f f f dt + g t f p p dt S ˆτi K ˆτ i (u ˆτ i ) ds + g(k φ, φ) p c (u, φ) (). (3.19) (u ˆτ i) ˆτi ds + g(k φ, φ) p c (u, φ) (t) K ˆτ i ν u f + u ˆτ i + g φ p gc u f gkmin φ p (t). kmax (3.) With (3.) and after rearranging, (3.19) becomes 1 t t u t f dt + gs φ t p dt + ν u f + u ˆτ + gkmin φ p (t) kmax t f f f dt + g t t f p p dt + 8(gC) φ p dt + gc u f S d 1 + ν u f + (u ˆτ i) ˆτi ds + g(k φ, φ) p c (u, φ) (), K ˆτ i t T. Taking supremum over [, T ] yields + 1 T T u t f dt + gs φ t p dt + sup [,T ] d 1 ν u f + + g φ p (t) T ν u f + kmax u ˆτ f f f dt + g T f p p dt S T + 8(gC) φ p dt + gc sup u f (t) [,T ] ˆτi K ˆτ i (u ˆτ i ) ds + g(k φ, φ) p c (u, φ) (). 15

16 Since 1,f/p C f/p, by (3.) of Theorem 3.1, we have u L (, T ; L (Ω f )) and φ L (, T ; L (Ω p )). Thus, the claim of the theorem follows. 4. Convergence to the quasistatic solution n this section we prove that the Stokes-Darcy solution, (u, φ), given by (.8)-(.9), converges to the quasistatic solution, (u QS, φ QS ), given by (.1)- (.11), as S approaches zero. We use the à priori estimates from the previous section to obtain estimates for the errors in the velocity and hydraulic head between the two problems. For the less regular body forces case we obtain one half order convergence in S. For the more regular case, we obtain first order convergence. Let e u (x, t) := u(x, t) u QS (x, t), e φ (x, t) := φ(x, t) φ QS (x, t), denote the errors in u and φ respectively. Then we have e u (x, ) = and e φ = φ (x) φ QS (x, ). Subtracting (.1) from (.8) and (.11) from (.9) we find that the errors satisfy the quasistatic weak formulation (.1)-(.11): (e ut, v) f + a f (e u, v) + c (v, e φ ) =, (4.1) ga p (e φ, ψ) c (e u, ψ) = gs (φ t, ψ) p. (4.) This can also be written in the form of the Stokes-Darcy weak formulation (.8)-(.9): (e ut, v) f + a f (e u, v) + c (v, e φ ) =, (4.3) gs (e φt, ψ) p + ga p (e φ, ψ) c (e u, ψ) = gs (φ QS t, ψ) p. (4.4) Theorem 4.1 gives a result of first order convergence of the solution (u, φ) to the quasistatic solution (u QS, φ QS ), as S converges to zero. Theorem 4.1. Consider the weak formulation (4.3)-(4.4). Assume the initial data and body forces satisfy Then + g e φ (t) p u QS t () L (Ω f ), φ t () 1,p <, f f,t L (, T ; H 1 (Ω f )), f p,t L (, T ; H 1 (Ω p )). T sup eu(t) f + gs e φ (t) p + [,T ] ν e u(t) f + kmax e u ˆτ dt gs φ φ QS () p + S C C ( S + 1)S. 16

17 Proof. We apply the energy estimate obtained in Proposition 3.1 to the weak formulation (4.3)-(4.4) for the error, with f f, f p = S φ QS t, e u replacing u and e φ replacing φ. Using the Poincarè-Friedrichs inequality we have φ QS t 1,p C(Ω p ) φ QS t p. By (3.4) of Theorem 3.1 we have φ QS t L (, T ; L (Ω p )). The above inequality then implies that φ QS t L (, T ; H 1 (Ω p )). Therefore, T sup e u(t) f + gs e φ (t) p + [,T ] ( ν e u(t) f + eu ˆτ kmax + g e φ (t) p gs φ φ QS () p + gs T φ QS t 1,p dt gs φ φ QS () p + C S k, min (4.5) which proves the first part of the theorem. For the last inequality, set t = in (.9) and (.11) and subtract the second from the first equation to obtain gs (φ t (), ψ) p + ga p (φ φ QS (), ψ) =, ψ X p, (4.6) where we used that u QS (x, ) = u (x). Take ψ = φ φ QS () in (4.6) to get ga p (φ φ QS (), φ φ QS ()) = gs (φ t (), φ QS () φ ) p. Using coercivity of the bilinear form a p (, ) and the definition of the 1 norm we have so that (φ φ QS ()) p S φ t () 1,p (φ φ QS ()) p, (φ φ QS ()) p S φ t () 1,p. (4.7) Finally, using the Poincarè-Friedrichs inequality on the left hand side of (4.7) yields φ φ QS () p C(Ω p)s φ t () 1,p. (4.8) The last inequality of the theorem now follows by combining inequalities (4.5) and (4.8). n Theorem 4. we assume less regularity on the body forces and prove one half order convergence of the Stokes-Darcy solution to the quasistatic solution as S. Theorem 4.. Consider the weak formulation (4.1)-(4.). Assume the initial data and body forces satisfy u, u() L (Ω f ), u ˆτ i t= L (), i = 1,..., d 1, φ L (Ω p ), ) dt 17

18 f f L (, T ; L (Ω f )), f p L (, T ; L (Ω p )), and that the domains Ω f and Ω p are such that inequality (3.15) holds. We have T sup e u f + ν e u f + e u ˆτ + g e φ p dt [,T ] k max gs T ( ) C S φ t p dt S. Proof. We will use the energy estimate obtained in Proposition 3. and apply it to the weak formulation (4.1)-(4.) for the error, with f f, f p = S φ t, e u in place of u QS and e φ in place of φ QS. By Theorem 3. we have in addition that S φ t L (, T ; L (Ω p )). Thus, we conclude T sup e u f + ν e u f + e u ˆτ + g e φ p dt [,T ] k max gs T ( ) C S φ t p dt S. Theorem 4. is important because it proves convergence of the Stokes-Darcy solution to the quasistatic solution as S converges to zero assuming less regular body forces. Notice that the assumption on the body forces in Theorem 4.1 is that the time derivatives of the body forces in Ω f,p belong to L (, T ; H 1 (Ω f,p )) respectively, while the requirement in Theorem 4. is that f f/p L (, T ; L (Ω f/p )). Less regular body forces occur, for instance, in settings involving wells. One half order convergence is obtained in Theorem 4. by making a few more assumptions on the initial data for the Stokes-Darcy solution (u, φ) given by (.8)-(.9) and assuming in addition that there exists a C 1 -diffeomorphism between the domains Ω f and Ω p. Remark 4.1. From the results in Theorems 4.1 and 4. it is clear that dropping the term S φ t from the fully evolutionary Stokes-Darcy equations, if S is small, is justified provided S <<. 5. Conclusions The solution of the fully evolutionary Stokes-Darcy problem converges to the quasistatic solution, as S approaches zero, under mild assumptions on the initial data and body forces. First order convergence is obtained. Provided that S is small and S <<, the term S φ t can be dropped from the Stokes-Darcy equations. 18

19 6. Acknowledgements would like to thank my advisor, Professor William J. Layton, for suggesting the problem and for his continuous support and guidance. This work was partially supported by NSF grant DMS References [1] J. Bear, Dynamics of fluids in porous media, Dover books on physics and chemistry, Dover, [] M. A. Biot, General theory of three-dimensional consolidation, Journal of Applied Physics 1 () (1941) [3] H. Wang, Theory of linear poroelasticity with applications to geomechanics and hydrogeology, Princeton series in geophysics, Princeton University Press,. [4] P. A. Domenico, Concepts and models in groundwater hydrology, nternational series in the earth and planetary sciences, McGraw-Hill, 197. [5] M. Anderson, W. Woessner, Applied groundwater modeling: simulation of flow and advective transport, no. v. 4 in Applied Groundwater Modeling: Simulation of Flow and Advective Transport, Academic Press, 199. [6] V. Batu, Aquifer hydraulics: a comprehensive guide to hydrogeologic data analysis, Wiley, [7] P. A. Domenico, M. D. Mifflin, Water from low-permeability sediments and land subsidence, Water Resources Research 1 (4) (1965) [8] A.. Johnson, Specific yield - compilation of specific yields for various materials: hydrologic properties of earth materials, Geological Survey water-supply paper, US Gov. Print. Off., [9] A. Cesmelioglu B. Riviére, Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow, Journal of Numerical Mathematics. [1] L. Badea, M. Discacciati, A. Quarteroni, Numerical analysis of the Navier-Stokes/Darcy coupling, Numerische Mathematik 115 (1) 195 7, 1.17/s [11]. Watson, A. Burnett, Hydrology: an environmental approach, CRC Lewis, [1] R. A. Freeze, J. A. Cherry, Groundwater, Prentice-Hall, [13] M. S. Hantush, Flow to an eccentric well in a leaky circular aquifer, American Geophysical Union, 196. [14]. Jones, Low Reynolds number flow past a porous spherical shell, Mathematical Proceedings of the Cambridge Philosophical Society 73 (1) (1973) [15] P. G. Saffman, On the boundary condition at the surface of a porous medium, Studies in Applied Mathematics 5 () (1971)

20 [16] G. S. Beavers, D. D. Joseph, Boundary conditions at a naturally permeable wall, Journal of Fluid Mechanics 3 (1) (1967) 197. [17] A. Mikeli`c, W. Jäger, On the interface boundary condition of Beavers, Joseph, and Saffman, SAM Journal on Applied Mathematics 6 (4) () [18] L. E. Payne, B. Straughan, Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modelling questions, J Math Pures Appl 77 (1998) [19] J. Lions, Perturbations Singulieres dans les Problemes aux Limites et en Controle Optimal, Springer, [] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext Series, Springer, 1.