On the Homogenization of Geological Fissured Systems With Curved non-periodic Cracks
OON THE HOMOGENIZATION OF GEOLOGICAL FISSUREDSYSTEMS WITH CURVED NON-PERIODIC CRACKS
FERNANDO A. MORALES
Abstract.
We analyze the steady fluid flow in a porous medium containinga network of thin fissures i.e. width O ( (cid:15) ), where all the cracks are generatedby the rigid translation of a continuous piecewise C functions in a fixed di-rection. The phenomenon is modeled in mixed variational formulation, usingthe stationary Darcy’s law and setting coefficients of low resistance O ( (cid:15) ) onthe network. The singularities are removed performing asymptotic analysis as (cid:15) → Introduction
Groundwater and oil reservoirs are frequently fissured or layered i.e. the bedrock contains fissures of characteristic dimensions considerably higher than thoseof the average pore size of the rock. The modeling of saturated flow through geo-logical structures such as these, gives rise to singular problems of partial differentialequations [19]. On one hand the singularities are due to the drastic change of per-meability from the rock matrix to the fissures. On the other hand a geometricsingularity is introduced due to the thinness of the fractures. The presence of sin-gularities in the model has non-desirable effects in their numerical implementation;some of these are ill-condition matrices, high computational costs, numerical stabil-ity, etc. This subject is a very active research field, see [2, 5, 9, 11, 12] for numericalanalysis aspects, [8, 10] for modeling discussion and [1, 3, 4, 13, 14] for rigorousmathematical treatment of the phenomenon. Homogenization and asymptotic anal-ysis techniques are a common approach for the analytical point of view. However,the remarkable achievements in the field require very restrictive hypotheses for thedescription of the geometry such as uniformly distributed, regular geometric shapesor periodic arrayed structures [7, 16]. In general the variational methods for partialdifferential equations can formulate successfully a wide class of geometric domains,the limited treatment of the geometry comes from the notorious difficulties it in-troduces in the asymptotic analysis of the problem.In the present work, the geometric possibilities of the medium are broaden to anunprecedented setting: free from the aforementioned hypotheses. We use the mixedmixed formulation and the scaling for the flow resistance coefficients presented in
Mathematics Subject Classification.
Key words and phrases. fissured media, tangential flow, interface geometry, coupled Darcyflow system, upscaling, mixed formulations.The author was supported by projects HERMES 14917, HERMES 17194 from UniversidadNacional de Colombia, Sede Medell´ın, and by the Department of Energy, Office of Science, USAthrough grant 98089. a r X i v : . [ m a t h . A P ] D ec FERNANDO A. MORALES [15], then a careful choice of directions or “stream lines”, consistent with the nat-ural scaling of the problem permits a successful asymptotic analysis of the model.This leads to a system coupled though multiple two dimensional manifolds repre-senting the fissures in the upscaled model. Additionally, the formulation allowsremarkable generality in the fluid exchange balance conditions between the rockmatrix and the channels, substantial efficiency for handling the system of equationsas well as the information (coefficients, matrices, etc) describing the geometry ofthe fractures, mostly due to the fact that it does not demand coupling constraintson the underlying spaces of functions. The main goal of the paper is to emphasizeon the geometry, consequently the study is limited to the steady case. We describeflow with Darcy’s law a ( · ) u + ∇ p + g = 0 , (1.1a)together with the conservation law ∇ · u = F. (1.1b)Drained and non-flux boundary conditions on different parts of the domain bound-ary will be specified to set a boundary value problem. The fluid exchange acrossthe interface separating the regions are given by p − p = α u and (1.1c) u · (cid:98) n − u · (cid:98) n = f Γ on Γ . (1.1d)Here, the coefficient a ( · ) is the flow resistance i.e. the fluid viscosity times theinverse of the permeability of the medium, to be scaled consistently with the fastand slow flow regions of the medium. Finally, the coefficient α indicates the fluidentry resistance of the rock matrix.In the following section we define the geometric setting, formulate the problemin mixed mixed variational formulation and establish its well-posedness. In sectionthree the problem is referred to a common geometric setting in order make possiblethe asymptotic analysis, the existence of a-priori estimates and the structure of thelimiting solution are also shown. Section four studies the formulation and well-posedness of the limiting problem and finds its strong form, particularly importantfor boundary and interface conditions and proves the strong convergence of thesolutions. Section five sets the limiting problem as a coupled system with twodimensional interfaces and section six discusses the possibilities and limitations ofthe technique as well as related future work.2. Formulation and Geometric Setting
Vectors are denoted by boldface letters as are vector-valued functions and corre-sponding function spaces. We use (cid:101) x to indicate a vector in R ; if x ∈ R then the R × { } projection is identified with (cid:101) x def = ( x , x ) so that x = ( (cid:101) x , x ). The symbol (cid:101) ∇ represents the gradient in the first first two directions: (cid:98) i , (cid:98) j . Given a function f : R → R then (cid:82) M f dS is the notation for its surface integral on the R manifold M ⊆ R . (cid:82) A f d x stands for the volume integral in the set A ⊆ R ; whenever thecontext is clear we simply write (cid:82) A f . In the same fashion, whenever there is noconfusion (cid:80) i , (cid:81) i indicate (cid:80) Ii =1 and (cid:81) Ii =1 respectively. A + t def = (cid:110) x + t (cid:98) k : x ∈ A (cid:111) (2.1) EOLOGICAL FISSURED SYSTEMS 3
Figure 1.
Unidirectional Translation Generated Fissures Γ Γ Γ h ̂ k h ̂ k Ω Ω Ω Ω( h , Γ )Ω( h , Γ )Ω h ̂ k Ω( h , Γ ) The symbol (cid:98) ν denotes the outwards normal vector on the boundary of a givendomain O and (cid:98) n denotes the normal upwards vector to a given surface i.e. (cid:98) n · (cid:98) k ≥ A ⊆ R and t ∈ R we define its t -vertical shift by2.1. General Geometric Setting.
The present work will be limited to the studyof fractured media where each fissures can be described in a specific way.
Definition 2.1.
Let G ⊆ R be open a bounded open simply connected set and ζ ∈ C ( G ) be a piecewise C . Define the surface Γ def = { [ (cid:101) x , ζ ( (cid:101) x )] : (cid:101) x ∈ G } . (2.2) We say Γ is a surface eligible for vertical translation fissure generation if ess inf { (cid:98) n ( s ) · (cid:98) k : s ∈ Γ } > . Given vertical height h > define the fissure of height h generatedby a rigid vertical translation of Γ by the domain Ω ( h, Γ) def = { ( (cid:101) x , y ) : ζ ( (cid:101) x ) < y < ζ ( (cid:101) x ) + h } . (2.3) Remark 2.1.
Notice that in the definition of Ω( h, Γ) we mention h as the heightand not as the width of the crack. Figure (4) shows that, depending on the gradientof the surface the height h can become significantly different from the actual width. The analysis will be limited to the type of geological system shown in figure (1).It depicts a region Ω ⊆ R containing a network of fissures generated by verticalrigid translation continuous piecewise C surfaces. Such a region is completelycharacterized in the following definition Definition 2.2.
We say a totally fractured medium of vertical translation generatedfissures is a finite collection ofSurface functions { ζ i ∈ C ( G i ) : G i ⊆ R open bounded simply connected region ; ζ i piecewise C functions such that ess inf (cid:98) n ( i ) · (cid:98) k > , ≤ i ≤ I } . (2.4a) FERNANDO A. MORALES vertical heights { h i > ≤ i ≤ I } , (2.4b) and rock-matrix regions (cid:8) Ω i ⊆ R : Ω i (cid:54) = ∅ open bounded simply connected region , ≤ i ≤ I (cid:9) . (2.4c) Verifying the following properties:Non-overlapping condition and indexed ordered sup { ζ i ( (cid:101) x ) + h i : (cid:101) x ∈ G i } < inf { ζ i +1 ( (cid:101) x ) : (cid:101) x ∈ G i +1 } ∀ ≤ i ≤ I − . (2.5a) The interface-domain condition ∂ Ω i ∩ ∂ Ω( h i +1 , Γ i +1 ) = Γ i +1 ∀ ≤ i ≤ I − ,∂ Ω i ∩ ∂ Ω( h i , Γ i ) = Γ i + h i ∀ ≤ i ≤ I. (2.5b) And the condition of connectivity only through fissures cl (Ω (cid:96) ) ∩ cl (Ω k ) = ∅ , whenever (cid:96) (cid:54) = k. (2.5c) For convenience of notation define Γ def = ∂ Ω − Γ and h def = 0 . The fissuredsystem described above will be denoted { (Γ i , h i , Ω i ) : 0 ≤ i ≤ I } . The sets Ω , Ω are the rock matrix and the fissures regions respectively i.e.Ω def = I (cid:91) i =0 Ω i , Ω def = I (cid:91) i =1 Ω ( h i , Γ i )Ω def = Ω ∪ Ω . (2.6) The global bottom and top interfaces are defined by Γ t def = I (cid:91) i =1 Γ i , Γ b def = I (cid:91) i =1 Γ i + h i Γ def = Γ b ∪ Γ t (2.7) Finally, (cid:98) n ( i ) indicates the upwards normal vector to the surface Γ i i.e. (cid:98) n ( i ) def = ( − (cid:101) ∇ ζ i , | ( − (cid:101) ∇ ζ i , | . (2.8) When there is no confusion (cid:98) n denotes the normal vector with respect to the surfaceof the crack. Remark 2.2.
The condition (2.5c) of connectivity only through fissures is notrequired for modeling the problem in mixed formulation as it is presented in section (2.4) ; however it is necessary for the asymptotic analysis of the system. The sameholds for the requirement of simply connected domains.
A Local System of Coordinates.
Some aspects of the flow through thefissures are handled more conveniently when the velocities are expressed in a coor-dinate system consistent with the geometry of the surface that generates the crack.Let Γ be a surface as defined in (2.1) and (cid:98) n the upwards normal to the surface Γi.e. (cid:98) n = (cid:98) n ( s ) = (cid:98) n ( (cid:101) x ). Now, for each point (cid:101) x we choose a local orthonormal basisin the following way B ( (cid:101) x ) def = { (cid:98) e ( (cid:101) x ) , (cid:98) e ( (cid:101) x ) , (cid:98) n ( (cid:101) x ) } (2.9) EOLOGICAL FISSURED SYSTEMS 5
Let M = M ( (cid:101) x ) be the orthogonal matrix relating the global canonical basis withthe local one i.e. M ( (cid:101) x ) (cid:98) i = (cid:98) e ( (cid:101) x ) (2.10a) M ( (cid:101) x ) (cid:98) j = (cid:98) e ( (cid:101) x ) (2.10b) M ( (cid:101) x ) (cid:98) k = (cid:98) n ( (cid:101) x ) (2.10c)The block matrix notation for this local matrix will be M ( (cid:101) x ) def = M T, τ M T, (cid:98) n M (cid:98) k , τ M (cid:98) k , (cid:98) n ( (cid:101) x ) (2.11)Here the index T stands for the first two components in the directions (cid:98) i , (cid:98) j whilethe index τ stands for the expression of the velocity orthogonal to the componentin the direction (cid:98) n . Then w = [ w τ , w (cid:98) n ]( (cid:101) x ) with the following relations w (cid:98) n def = w · (cid:98) n ( (cid:101) x ) (2.12a) w τ def = ( w · (cid:98) e ( (cid:101) x ) , w · (cid:98) e ( (cid:101) x ) ) (2.12b)Clearly, the relationship between velocities is given by w ( (cid:101) x , x ) = (cid:40) (cid:101) ww · (cid:98) k (cid:41) ( (cid:101) x , x ) = M ( (cid:101) x ) (cid:40) w τ w · (cid:98) n (cid:41) ( (cid:101) x , x )= M T, τ ( (cid:101) x ) M T, (cid:98) n ( (cid:101) x ) M (cid:98) k , τ ( (cid:101) x ) M (cid:98) k , (cid:98) n ( (cid:101) x ) (cid:40) w τ w · (cid:98) n (cid:41) ( (cid:101) x , x ) (2.13) Proposition 2.3.
Let h > , Γ , Ω( h, Γ) be as in definition (2.1) ; (cid:98) n be the upwardsnormal to the surface Γ and M be the matrix defined by (2.10) . Then(i) The map w (cid:55)→ M ( (cid:101) x ) w is an isometry in L (Ω( h, Γ)) . In particular if w τ , w · (cid:98) n are defined as in (2.12) then w ∈ L (Ω( h, Γ)) if and only if w τ ∈ L (Ω( h, Γ)) × L (Ω( h, Γ)) and w · (cid:98) n ∈ L (Ω( h, Γ)) .(ii) If w ∈ L (Ω( h, Γ)) is such that ∂ z w ∈ L (Ω( h, Γ)) then ∂ z w ( (cid:101) x , z ) = M ( (cid:101) x ) (cid:40) ∂ z w τ ( ∂ z w ) · (cid:98) n (cid:41) ( (cid:101) x , z ) (2.14) Proof. (i) For (cid:101) x fixed the matrix M ( (cid:101) x ) is orthogonal i.e. for arbitrary functions v , w ∈ L (Ω( h, Γ)) and x ∈ Ω( h, Γ) holds v ( x ) · w ( x ) = M ( (cid:101) x ) v ( x ) · M ( (cid:101) x ) w ( x ).Hence (cid:90) Ω( h, Γ) v ( x ) · w ( x ) d x = (cid:90) Ω( h, Γ) M ( (cid:101) x ) v ( x ) · M ( (cid:101) x ) w ( x ) d x = (cid:90) Ω( h, Γ) v τ ( x ) · w τ ( x ) d x + (cid:90) Ω( h, Γ) ( v · (cid:98) n )( x ) · ( w · (cid:98) n )( x ) d x The equality of the second line shows the necessity and sufficiency of the tan-gential and normal components been square integrable in the domain Ω( h, Γ).(ii) It follows from a direct calculation of distributions with ϕ ∈ [ C ∞ (Ω( h, Γ))] arbitrary and the fact that ∂ z M = 0. (cid:3) FERNANDO A. MORALES
The Problem and its Formulation.
In this section we define the problemin a rigorous way and give a variational formulation in which it is well-posed.Let { (Γ i , h i , Ω i ) : 1 ≤ i ≤ I } be a totally fractured domain of vertical translationgenerated fissures. We denote v , p the velocity and pressure in the rock matrixregion Ω . In the same fashion v , p denote the velocity and pressure in the fissuresregion Ω . Consider the problem a u + ∇ p + g = 0 and (2.15a) ∇ · u = F in Ω . (2.15b) p = 0 on ∂ Ω − Γ . (2.15c) p − p = α u · (cid:98) n Γ b − α u · (cid:98) n Γ t and (2.15d) (cid:0) u − u (cid:1) · (cid:98) n Γ t − (cid:0) u − u (cid:1) · (cid:98) n Γ b = f Γ on Γ . (2.15e) a u + ∇ p + g = 0 and (2.15f) ∇ · u = F in Ω . (2.15g) u · (cid:98) n = 0 on ∂ Ω − Γ . (2.15h)The flow resistance coefficients a , a and the fluid entry resistance coefficient α are assumed to be positively bounded from below and above, see [15]. In equations(2.15d), (2.15e) the split of cases is made in order to be consistent with the sign ofthe upwards normal vector (cid:98) n .2.4. Mixed Formulation of the Problem.
We start defining the spaces of ve-locities and pressures V def = { v ∈ L (Ω) : ∇ · v ∈ L (Ω ) , v · (cid:98) n | Γ ∈ L (Γ) } . (2.16a) Q def = { q ∈ L (Ω) : ∇ q ∈ L (Ω ) } (2.16b)Endowed with their natural norms (cid:107) v (cid:107) V def = { (cid:107) v (cid:107) L (Ω) + (cid:107) ∇ · v (cid:107) L (Ω ) + (cid:107) v · (cid:98) n (cid:107) L (Γ) } / (2.16c) (cid:107) q (cid:107) Q def = { (cid:107) q (cid:107) L (Ω) + (cid:107) ∇ q (cid:107) L (Ω ) } / (2.16d) Remark 2.3.
In the spaces above it is understood that (cid:107) v · (cid:98) n (cid:107) L = (cid:107) v · (cid:98) n (cid:107) L b ) + (cid:107) v · (cid:98) n (cid:107) L t ) = I (cid:88) i =1 (cid:107) v · (cid:98) n ( i ) (cid:107) L i ) + I (cid:88) i =1 (cid:107) v · (cid:98) n ( i ) (cid:107) L i + hi ) (2.17)Consider the problem Find p ∈ Q, u ∈ V (cid:90) Ω a u · v + (cid:90) Ω a u · v − (cid:90) Ω p ∇ · v + (cid:90) Ω ∇ p · v + α (cid:90) Γ (cid:0) u · (cid:98) n (cid:1) (cid:0) v · (cid:98) n (cid:1) dS − (cid:90) Γ t p (cid:0) v · (cid:98) n (cid:1) dS + (cid:90) Γ b p (cid:0) v · (cid:98) n (cid:1) dS = − (cid:90) Ω g · v (2.18a) (cid:90) Ω ∇ · u q − (cid:90) Ω u · ∇ q EOLOGICAL FISSURED SYSTEMS 7 + (cid:90) Γ t (cid:0) u · (cid:98) n (cid:1) q dS − (cid:90) Γ b (cid:0) u · (cid:98) n (cid:1) q dS = (cid:90) Ω F q + (cid:90) Γ f Γ q dS (2.18b)for all q ∈ Q, v ∈ VRemark 2.4.
In the formulation above the non-symmetric interface terms are splitin two pieces in order to express everything in terms of the upwards normal vector (cid:98) n . In the case of the symmetric term (cid:82) Γ ( u · (cid:98) n )( v · (cid:98) n ) dS in (2.18a) such splitbecomes unnecessary since the sign of the normal vector changes in both factorscanceling each other. Define the bilinear forms A : V → V (cid:48) , B : V → Q (cid:48) , C : Q → Q (cid:48) by A v ( w ) def = (cid:90) Ω a v · w + (cid:90) Ω a v · w + α (cid:90) Γ (cid:0) v · (cid:98) n (cid:1) (cid:0) w · (cid:98) n (cid:1) dS (2.19a) B v ( q ) def = − (cid:90) Ω ∇ · v q + (cid:90) Ω v · ∇ q − (cid:90) Γ t ( v · (cid:98) n ) q dS + (cid:90) Γ b ( v · (cid:98) n ) q dS (2.19b)Then, the system (2.18) is a mixed formulation for the problem (2.15) with theabstract form u ∈ V , p ∈ Q : A u + B (cid:48) p = − g in V (cid:48) , −B u = f in Q (cid:48) . (2.20)For the sake of completeness recall some well known results Theorem 2.4.
Let V , Q be Hilbert spaces and (cid:107)·(cid:107) V , (cid:107)·(cid:107) Q be their respective norms.Let A : V → V (cid:48) , B : V → Q (cid:48) be continuous linear operators such that(i) A is non-negative and V -coercive on ker B .(ii) The operator B satisfies the inf-sup condition inf q ∈ Q sup v ∈ V |B v ( q ) |(cid:107) v (cid:107) V (cid:107) q (cid:107) Q > . (2.21) Then, for each g ∈ V (cid:48) and f ∈ Q (cid:48) there exists a unique solution [ u , p ] ∈ V × Q to the problem (2.20) . Moreover, it satisfies the estimate (cid:107) u (cid:107) V + (cid:107) p (cid:107) Q ≤ K ( (cid:107) g (cid:107) V (cid:48) + (cid:107) f (cid:107) Q (cid:48) ) (2.22) Proof.
See [6] (cid:3)
Lemma 2.5.
Let O be an open connected bounded set in R N and G ⊆ ∂ O withnon-null R N − -Lebesgue measure, then there exists κ = κ ( O ) > such that (cid:107) ∇ η (cid:107) L ( O ) + (cid:107) η (cid:107) G ≥ κ (cid:107) η (cid:107) H ( O ) (2.23) for all η ∈ H ( O ) .Proof. See proposition 5.2 of [18] or lemma 1.2 in [15]. (cid:3)
Corollary 2.6.
There exists a constant κ > such that (cid:107) ∇ q (cid:107) L (Ω ) + (cid:107) q (cid:107) L (Γ) ≥ κ (cid:107) q (cid:107) L (Ω ) . (2.24) For all q ∈ H (Ω ) Proof.
Apply lemma (2.5) on each connected component Ω( h i , Γ i ) and choose κ asthe minimum constant associated to each domain. (cid:3) Lemma 2.7.
The operator B satisfies the inf-sup condition (2.21) . FERNANDO A. MORALES
Proof.
We use the same strategy presented lemma 1.3 in [15] with a slight modifi-cation in the construction of the particular test function. Fix q ∈ Q and denote ξ j the unique solution of the problem − ∇ · ∇ ξ j = q in Ω j , ∇ ξ j · (cid:98) n = q on Γ j , ∇ ξ j · (cid:98) n = − q on Γ j + h j ,ξ j = 0 on ∂ Ω j − Γ j − (Γ j + h j ) . (2.25)Define v def = (cid:80) Ij =0 ∇ ξ j Ω j . Thus, − ∇ · v = (cid:80) Ij =0 q Ω j and v · (cid:98) n = I (cid:88) i =1 q Γ i − I (cid:88) i =1 q Γ i + h i . Due to the Poincar´e inequality c (cid:107) v (cid:107) H div (Ω ) ≤ (cid:107) q (cid:107) L (Ω ) + (cid:107) q (cid:107) L (Γ) . Hence,setting v def = ∇ q we have B v ( q ) = − (cid:90) Ω ∇ · v q + (cid:90) Ω v · ∇ q − (cid:90) Γ t ( v · (cid:98) n ) q dS + (cid:90) Γ b ( v · (cid:98) n ) q dS = (cid:90) Ω | q | + (cid:90) Ω | ∇ q | + (cid:90) Γ t | q | dS + (cid:90) Γ b | q | dS ≥ (cid:90) Ω | q | + κ (cid:90) Ω | q | + 12 (cid:18)(cid:90) Ω | q | + (cid:90) Γ | q | dS (cid:19) ≥ c (cid:107) v (cid:107) V (cid:107) q (cid:107) Q (2.26)For c def = min { c , , κ } , which gives the inf-sup condition of the operator B . (cid:3) Theorem 2.5.
Suppose that ≤ α , a i ( · ) ∈ L ∞ (Ω) and a ∗ def = min i = 1 , ess inf { a i ( x ) : x ∈ Ω i } (2.27) If a ∗ is positive then, the mixed variational formulation (2.20) (or equivalently, thesystem (2.18) ) is well-posed.Proof. Clearly A is non-negative and V -coercive on ker B . The operator B satisfiesthe inf-sup condition as seen in the preceding lemma. Due to theorem (2.4) theresult follows. (cid:3) Scaling the Problem and Convergence Statements
In order to perform the asymptotic analysis for a the problem (2.18) in a mediumof thin fractures, the heights and resistance coefficients have to be scaled. We havethe following definition (see figure (2)).
Definition 3.1.
Let { ( ζ i , h i , Ω i ) : 1 ≤ i ≤ I } be a fractured medium of verticaltranslation generated fissures. For (cid:15) ∈ (0 , we define its associated (cid:15) -scaled fissuredsystem { ( ζ (cid:15)i , (cid:15) h i , Ω (cid:15)i ) : 1 ≤ i ≤ I } by ζ (cid:15)i = ζ i − (1 − (cid:15) ) i − (cid:88) (cid:96) = 0 h (cid:96) , ≤ i ≤ I. (3.1a) { (cid:15) h i > ≤ i ≤ I } . (3.1b)Ω (cid:15)j def = Ω j − (1 − (cid:15) ) j (cid:88) (cid:96) = 0 h (cid:96) , ≤ j ≤ I. (3.1c) EOLOGICAL FISSURED SYSTEMS 9
Figure 2.
Domains Mapping Γ ϵ Γ ϵ Γ ϵ ϵ h ϵ h Ω ϵ Ω ϵ Ω ϵ Ω(ϵ h , Γ ϵ )Ω(ϵ h , Γ ϵ )Ω ϵ ϵ h Ω(ϵ h , Γ ϵ ) Γ Γ Γ h h Ω Ω Ω Ω( h , Γ ) Ω( h , Γ )Ω h Ω( h , Γ ) φ The domains Ω (cid:15) , Ω (cid:15) , Ω (cid:15) and the surfaces Γ (cid:15) t , Γ (cid:15) b , Γ (cid:15) are defined as in (2.4c) , (2.6) respectively. Remark 3.1.
Clearly the systems { (Γ (cid:15)i , (cid:15) h i , Ω (cid:15)i ) : 1 ≤ i ≤ I } satisfies the condi-tions of definition (2.2) . Isomorphisms of Spaces and Formulation.
Let Ω (cid:15) , Ω (cid:15) , Ω (cid:15) and Γ (cid:15) t , Γ (cid:15) b , Γ (cid:15) be the domains and surfaces associated to the family { ( ζ (cid:15)i , (cid:15) h i , Ω (cid:15)i ) : 1 ≤ i ≤ I } as in definition (3.1). Define the spaces V (cid:15) def = { v ∈ L (Ω (cid:15) ) : ∇ · v ∈ L (Ω (cid:15) ) , v · (cid:98) n | Γ (cid:15) ∈ L (Γ (cid:15) ) } , (3.2a) Q (cid:15) def = { q ∈ L (Ω (cid:15) ) : ∇ q ∈ L (Ω (cid:15) ) } . (3.2b)We endow the spaces with the norms coming from the natural inner product (cid:107) v (cid:107) V (cid:15) def = { (cid:107) v (cid:107) L2 (Ω (cid:15) ) + (cid:107) ∇ · v (cid:107) L (cid:15) + (cid:107) v · (cid:98) n (cid:107) L (cid:15) ) } / (3.2c) (cid:107) q (cid:107) Q(cid:15) def = { (cid:107) q (cid:107) L (cid:15) ) + (cid:107) ∇ q (cid:107) L (cid:15) } / (3.2d)Consider the scaled problem Find p (cid:15) ∈ Q (cid:15) , u (cid:15) ∈ V (cid:15) : (cid:90) Ω (cid:15) a u (cid:15) · v d y + (cid:15) (cid:90) Ω (cid:15) a u (cid:15) · v d y − (cid:90) Ω (cid:15) p (cid:15) ∇ · v d y + (cid:90) Ω (cid:15) ∇ p (cid:15) · v d y + α (cid:90) Γ (cid:15) ( u (cid:15), · (cid:98) n ) ( v · (cid:98) n ) dS − (cid:90) Γ (cid:15) t p (cid:15), ( v · (cid:98) n ) dS + (cid:90) Γ (cid:15) b p (cid:15), ( v · (cid:98) n ) dS = − (cid:90) Ω (cid:15) g (cid:15) · v d y (3.3a) (cid:90) Ω (cid:15) ∇ · u (cid:15) q d y − (cid:90) Ω (cid:15) u (cid:15) · ∇ q d y + (cid:90) Γ (cid:15) t (cid:0) u (cid:15), · (cid:98) n (cid:1) q dS − (cid:90) Γ (cid:15) b (cid:0) u (cid:15), · (cid:98) n (cid:1) q dS = (cid:90) Ω (cid:15) F (cid:15) q d y + (cid:90) Γ (cid:15) f (cid:15) Γ (cid:15) q dS (3.3b)for all q ∈ Q (cid:15) , v ∈ V (cid:15) Clearly, the problem (3.3) is well-posed since it verifies all the hypothesis of theorem(2.5). In order to analyze the asymptotic behavior of the solution ( u (cid:15) , p (cid:15) ) as (cid:15) ↓ (cid:15) -domains must be mapped to a common domain of reference.3.2. The (cid:15) -Problems in a Reference Domain.
We introduce the change ofvariable (see figure (2)) ϕ : Ω (cid:15) → Ω defined by ϕ ( y ) def = I (cid:88) j =0 (cid:32)(cid:101) y , y + (1 − (cid:15) ) j (cid:88) (cid:96) = 0 h (cid:96) (cid:33) Ω (cid:15)j ( y )+ I (cid:88) i =0 (cid:32)(cid:101) y , (cid:15) ( y − ζ (cid:15)i ( (cid:101) y )) + ζ (cid:15)i ( (cid:101) y ) + (1 − (cid:15) ) i − (cid:88) (cid:96) = 0 h (cid:96) (cid:33) Ω( (cid:15)h i , Γ (cid:15)i ) ( y ) (3.4)Defining ( (cid:101) x , z ) def = ϕ ( y ) the gradients are related as follows ∇ y = (cid:40) (cid:101) ∇ x ∂ z (cid:41) Ω (cid:15) + (cid:88) i I (1 − (cid:15) ) (cid:101) ∇ x ζ i ( (cid:101) x )0 1 (cid:15) (cid:40) (cid:101) ∇ x ∂ z (cid:41) Ω( (cid:15)h i , Γ (cid:15)i ) (3.5)Here, it is understood that I is the identity matrix in ∈ R × . We write ζ i insteadof ζ (cid:15)i for the sake of simplicity recalling that both surfaces differ only by a constantof vertical translation. Theorem 3.2.
Let ϕ : Ω (cid:15) → Ω be the change of variable defined in equation (3.4) . Then, the maps defined Φ : V → V (cid:15) , Φ : Q → Q (cid:15) defined respectively by (Φ v ) ( y ) def = v ( ϕ ( y )) and (Φ q ) ( y ) def = q ( ϕ ( y )) are isomorphisms.Proof. First notice for v ∈ V and q ∈ Q the functions Φ v and Φ q are defined onΩ (cid:15) . Moreover, for (cid:96) = 1 , ϕ : Ω (cid:15)(cid:96) → Ω (cid:96) is a bijection. Therefore v ( · ) ∈ L (Ω (cid:96) ) if and only if v ( ϕ ( · )) ∈ L (Ω (cid:15)(cid:96) )and q ( · ) ∈ L (Ω (cid:96) ) if and only if q ( ϕ ( · )) ∈ L (Ω (cid:15)(cid:96) ). Even more, ϕ : Γ (cid:15)i → Γ i and ϕ : Γ (cid:15)i + (cid:15)h i → Γ i + h i are bijective rigid translations. Therefore, the isomorphisms L (Γ (cid:15)i ) (cid:39) L (Γ i ), L (Γ (cid:15)i + (cid:15) h i ) (cid:39) L (Γ i + h i ) follow for all 1 ≤ i ≤ I .For the isomorphism Φ take v ∈ V which is equivalent to v ( y ) ∈ L (Ω) and ∇ y · v ( y ) ∈ L (Ω ). Due to the previous discussion these two conditions areequivalent to v ( ϕ ( y )) ∈ L (Ω (cid:15) ) and ∇ y · v ( ϕ ( y )) = ∇ y · v ( x ) ∈ L (Ω (cid:15) ). However,equation (3.4) yields ∇ y · v ( ϕ ( y )) = ∇ y · v ( x ) = ∇ x · v ( x ) whenever x ∈ Ω ; i.e. ∇ y · v ( y ) ∈ L (Ω (cid:15) ) if and only if ∇ x · v ( x ) = ∇ x · v ( ϕ ( y )) ∈ L (Ω ) as desired.For the map Φ , the L -integrability condition between spaces Q and Q (cid:15) isshown using the same arguments of the first paragraph. It remains to show the L -integrability condition on the gradient. First observe that the last row in the matrixequation (3.5) implies that ∂∂y q ( y ) ∈ L (Ω (cid:15) ) if and only if ∂∂z q ( x ) ∈ L (Ω ).Second, for the derivatives in the first two directions equation (3.5) yields ∂∂ y (cid:96) q ( y ) = ∂∂ x (cid:96) q ( x ) + (cid:18) − (cid:15) (cid:19) ∂∂ x (cid:96) ζ i ( x ) ∂∂z q ( x ) , (cid:96) = 1 , . EOLOGICAL FISSURED SYSTEMS 11
Recalling the gradient of ζ i is bounded we conclude ∂∂y (cid:96) q ( y ) ∈ L (Ω (cid:15) ) if an onlyif ∂∂x (cid:96) q ( x ) ∈ L (Ω ) for (cid:96) = 1 ,
2. Since ∂∂z q ( x ) ∈ L (Ω ) is immediate the proof iscomplete. (cid:3) We are to apply the change of variable ϕ : Ω (cid:15) → Ω in the problem (3.3), tothis end, it is more convenient to write the system in terms of the quantities anddirections which yield estimates agreeable with the asymptotic analysis. Hence, re-calling the definition of the upwards normal vector (2.8) the following relationshipshold | ( − (cid:101) ∇ ζ i , | v · (cid:98) n ( i ) = − (cid:101) v · (cid:101) ∇ ζ i + v , (3.6a)( (cid:101) v , (cid:101) v · (cid:101) ∇ ζ i ) · (cid:98) n ( i ) = 0 in Ω( h i , Γ i ) . (3.6b)Applying the change of variable (3.4) to the problem (3.3) and combining with therelation (3.6a) we get the following variational statement:Find p (cid:15) ∈ Q, u (cid:15) ∈ V : (cid:90) Ω a u (cid:15) · v + (cid:15) (cid:90) Ω a u (cid:15) · v − (cid:90) Ω p (cid:15) ∇ · v + (cid:88) i (cid:90) Ω( h i , Γ i ) (cid:15) (cid:16) (cid:101) ∇ p (cid:15) + ∂ z p (cid:15) (cid:101) ∇ ζ i (cid:17) · (cid:101) v + | ( − (cid:101) ∇ ζ i , | ∂ z p (cid:15) ( v · (cid:98) n ( i ) ) − (cid:90) Γ t p (cid:15), (cid:0) v · (cid:98) n (cid:1) dS + (cid:90) Γ b p (cid:15), (cid:0) v · (cid:98) n (cid:1) dS + α (cid:90) Γ (cid:0) u (cid:15), · (cid:98) n (cid:1) (cid:0) v · (cid:98) n (cid:1) dS = − (cid:90) Ω g (cid:15) · v − (cid:15) (cid:90) Ω g (cid:15) · v (3.7a) (cid:90) Ω ∇ · u (cid:15) q − (cid:88) i (cid:90) Ω( h i , Γ i ) (cid:15) (cid:101) u (cid:15), · (cid:16) (cid:101) ∇ q + ∂ z q (cid:101) ∇ ζ i (cid:17) + | ( − (cid:101) ∇ ζ i , | ( u (cid:15), · (cid:98) n ( i ) ) ∂ z q + (cid:90) Γ t (cid:0) u (cid:15), · (cid:98) n (cid:1) q dS − (cid:90) Γ b ( u (cid:15), · (cid:98) n ) q dS = (cid:90) Ω F (cid:15), q + (cid:15) (cid:90) Ω F (cid:15), q + (cid:90) Γ f (cid:15) Γ q dS (3.7b)for all q ∈ Q, v ∈ V Finally, due to the theorem (3.2) on isomorphisms of function spaces we concludethat the problems (3.7) and (3.3) are equivalent.3.2.1.
The Strong Rescaled Problem.
The solution of the problem (3.7) is the weaksolution of the following system of equations a u (cid:15), + ∇ p (cid:15), + g = 0 and (3.8a) ∇ · u (cid:15), = f (cid:15), in Ω . (3.8b) p (cid:15), = 0 on ∂ Ω − Γ . (3.8c) p (cid:15), − p (cid:15), = α u · (cid:98) n Γ b − α u · (cid:98) n Γ t and (3.8d) (cid:0) u (cid:15), − u (cid:15), (cid:1) · (cid:98) n Γ t − (cid:0) u (cid:15), − u (cid:15), (cid:1) · (cid:98) n Γ b = f (cid:15) Γ on Γ . (3.8e) (cid:88) i (cid:20) (cid:15) a (cid:101) u (cid:15), + (cid:101) ∇ p (cid:15), + (1 − (cid:15) ) ∂ z p (cid:15), (cid:101) ∇ ζ i + (cid:101) g (cid:15) (cid:21) Ω( h i , Γ i ) = 0 , (3.8f) (cid:15) a u (cid:15), + ∂ z p (cid:15), + (cid:15) g (cid:15) = 0 and (3.8g) (cid:88) i ∇ · (cid:16) (cid:15) (cid:101) u (cid:15), , (cid:15) (cid:101) u (cid:15), · (cid:101) ∇ ζ i + | ( − (cid:101) ∇ ζ i , | ( u (cid:15), · (cid:98) n ( i ) ) (cid:17) Ω( h i , Γ i ) = (cid:15) F (cid:15), in Ω . (3.8h) (cid:101) u (cid:15), · (cid:101) ν ( i ) = 0 on ∂ Ω( h i , Γ i ) − Γ for all 1 ≤ i ≤ I. (3.8i)As before equations (3.8d), (3.8e) have the separation of cases Γ b , Γ t in orderto be consistent with the upwards normal vector (cid:98) n . However, the equations (3.8e)and (3.8i) need further clarification. We start fixing an index i ∈ { , . . . , I } of thesum in the equation (3.7b); reordering and integrating by parts yield − (cid:90) Ω( h i , Γ i ) (cid:15) (cid:101) u (cid:15), · ( (cid:101) ∇ q + ∂ z q (cid:101) ∇ ζ i ) + | ( − (cid:101) ∇ ζ i , | ( u (cid:15), · (cid:98) n ( i ) ) ∂ z q = − (cid:90) Ω( h i , Γ i ) (cid:16) (cid:15) (cid:101) u (cid:15), , (cid:15) (cid:101) u (cid:15), · (cid:101) ∇ ζ i + | ( − (cid:101) ∇ ζ i , | ( u (cid:15), · (cid:98) n ( i ) ) (cid:17) · ∇ q = (cid:90) Ω( h i , Γ i ) ∇ · (cid:16) (cid:15) (cid:101) u (cid:15), , (cid:15) (cid:101) u (cid:15), · (cid:101) ∇ ζ i + | ( − (cid:101) ∇ ζ i , | ( u (cid:15), · (cid:98) n ( i ) ) (cid:17) q − (cid:90) ∂ Ω( h i , Γ i ) q (cid:16) (cid:15) (cid:101) u (cid:15), , (cid:15) (cid:101) u (cid:15), · (cid:101) ∇ ζ i + | ( − (cid:101) ∇ ζ i , | ( u (cid:15), · (cid:98) n ( i ) ) (cid:17) · (cid:98) ν ( i ) dS. Where (cid:98) ν ( i ) is the outwards pointing unit normal field of the boundary ∂ Ω( h i , Γ i ).We focus on the boundary term (cid:90) ∂ Ω( h i , Γ i ) q (cid:16) (cid:15) (cid:101) u (cid:15), , (cid:15) (cid:101) u (cid:15), · (cid:101) ∇ ζ i + | ( − (cid:101) ∇ ζ i , | ( u (cid:15), · (cid:98) n ( i ) ) (cid:17) · (cid:98) ν ( i ) dS = (cid:90) ∂ Ω( h i , Γ i ) − (Γ i ∪ h i +Γ i ) q (cid:16) (cid:15) (cid:101) u (cid:15), , (cid:15) (cid:101) u (cid:15), · (cid:101) ∇ ζ i + | ( − (cid:101) ∇ ζ i , | ( u (cid:15), · (cid:98) n ( i ) ) (cid:17) · (cid:98) ν ( i ) dS + (cid:88) (cid:96) = 0 , (cid:90) (cid:96) h i +Γ i q (cid:16) (cid:15) (cid:101) u (cid:15), , (cid:15) (cid:101) u (cid:15), · (cid:101) ∇ ζ i + | ( − (cid:101) ∇ ζ i , | ( u (cid:15), · (cid:98) n ( i ) ) (cid:17) · (cid:98) ν ( i ) dS. The equality (cid:98) ν ( i ) · (cid:98) k = 0 holds on the portion of the vertical wall ∂ Ω( h i , Γ i ) − (Γ i ∪ h i + Γ i ) i.e. the equation (3.8i) follows. For the remaining pieces of the boundaryrecall (cid:98) n ( i ) = (cid:98) ν ( i ) on h i + Γ i and (cid:98) n ( i ) = − (cid:98) ν ( i ) on Γ i ; together with the equation(2.8), we get − (cid:90) (cid:96) h i +Γ i q (cid:16) (cid:15) (cid:101) u (cid:15), , (cid:15) (cid:101) u (cid:15), · (cid:101) ∇ ζ i + | ( − (cid:101) ∇ ζ i , | ( u (cid:15), · (cid:98) n ( i ) ) (cid:17) · (cid:98) ν ( i ) dS = ( − (cid:96) (cid:90) (cid:96) h i +Γ i q (cid:16) (cid:15) (cid:101) u (cid:15), , (cid:15) (cid:101) u (cid:15), · (cid:101) ∇ ζ i + | ( − (cid:101) ∇ ζ i , | ( u (cid:15), · (cid:98) n ( i ) ) (cid:17) · ( − (cid:101) ∇ ζ i , | ( − (cid:101) ∇ ζ i , | dS = ( − (cid:96) (cid:90) (cid:96) h i +Γ i q ( u (cid:15), · (cid:98) n ( i ) ) dS for (cid:96) = 0 , . Combining this last identity with the interface terms in equation (3.7b), the strongnormal flux balance condition (3.8e) follows.3.3.
A-priori Estimates and Convergence Statements.
In order to get a-priori estimates on the norm of the solutions the following hypothesis are assumed (cid:107) F (cid:15) (cid:107) L (Ω) is bounded and F ,(cid:15) w (cid:42) F in L (Ω ) , (3.9a) EOLOGICAL FISSURED SYSTEMS 13 g (cid:15) w (cid:42) g in L (Ω ) , g ,(cid:15) ( (cid:101) x , (cid:15) z ) w (cid:42) g ( (cid:101) x ) in L (Ω ) , (3.9b)and f (cid:15) Γ w (cid:42) f Γ in L (Γ) . (3.9c)Now test equation (3.7a) with u (cid:15) and equation (3.7b) with p (cid:15) add them togetherand get a ∗ (cid:0) (cid:107) u (cid:15), (cid:107) , Ω + (cid:107) (cid:15) u (cid:15), (cid:107) , Ω (cid:1) + α (cid:13)(cid:13) u (cid:15), · (cid:98) n (cid:13)(cid:13) L (Γ) = (cid:90) Ω F (cid:15), p (cid:15) + (cid:15) (cid:90) Ω F (cid:15), p (cid:15) + (cid:90) Γ f (cid:15) Γ p (cid:15), dS − (cid:90) Ω g · u (cid:15) − (cid:90) Ω g · (cid:15) u (cid:15) ≤ C ( (cid:107) F (cid:15) (cid:107) , Ω + (cid:107) f (cid:15) Γ (cid:107) , Γ ) (cid:107) p (cid:15) (cid:107) Q + (cid:107) g (cid:15) (cid:107) , Ω (cid:0) (cid:107) u (cid:15), (cid:107) , Ω + (cid:107) (cid:15) u (cid:15), (cid:107) , Ω (cid:1) . (3.10)Here, the constant C > (cid:15) >
0. Next the term (cid:107) p (cid:15) (cid:107) Q must bebounded in terms of the flux u (cid:15), Ω + (cid:15) u (cid:15), Ω and the forcing terms. Due to theequation (3.8g) we have (cid:107) (cid:15) ∂ z p (cid:15), (cid:107) , Ω ≤ (cid:15) (cid:107) a (cid:107) L ∞ (Ω2) (cid:107) u (cid:15), (cid:107) , Ω + (cid:107) g (cid:15) (cid:107) , Ω . (3.11a)Combined with equation (3.8f) yields (cid:107) (cid:101) ∇ p (cid:15), (cid:107) , Ω ≤ C (cid:0) (cid:107) a (cid:107) L ∞ (Ω2) (cid:107) (cid:15) u (cid:15), (cid:107) , Ω + (cid:107) g (cid:15) (cid:107) , Ω (cid:1) . (3.11b)For C > (cid:107) ∇ p (cid:15), (cid:107) , Ω ≤ C (cid:0) (cid:107) a (cid:107) L ∞ (Ω2) (cid:107) (cid:15) u (cid:15), (cid:107) , Ω + (cid:107) g (cid:107) , Ω (cid:1) . (3.12)With C > (cid:15) >
0. Additionally, the equation (3.8a)yields (cid:107) ∇ p (cid:15), (cid:107) , Ω ≤ (cid:107) a (cid:107) L ∞ (Ω2) (cid:107) u (cid:15), (cid:107) , Ω + (cid:107) g (cid:107) , Ω . (3.13)The boundary condition (3.8c) together with Poincar´e inequality give the con-trol (cid:107) p (cid:15), (cid:107) H (Ω ) ≤ C (cid:107) ∇ p (cid:15) (cid:107) , Ω . On the other hand, the inequality (2.24) implies (cid:107) p (cid:15) (cid:107) , Ω ≤ C ( (cid:107) p (cid:15) (cid:107) , Γ + (cid:107) p (cid:15) (cid:107) , Ω ); combined with the normal stress balance condi-tions (3.8d) we conclude: (cid:107) p (cid:15) (cid:107) Q ≤ (cid:107) p (cid:15) (cid:107) , Ω ≤ C (cid:107) ∇ p (cid:15) (cid:107) , Ω . (3.14)And C > (cid:15) >
0. Finally, a combination of inequalities (3.14),(3.13) and (3.12) imply that the left hand side of inequality (3.10) is bounded.
Remark 3.2.
The previous estimate on (cid:107) p (cid:15), (cid:107) H (Ω ) could have been attained with-out requiring the drained condition (3.8c) on the whole matrix rock region exter-nal boundary. It was enough to set the drained condition on a subset of posi-tive measure contained in ∂ Ω j − Γ for j fixed to have control on (cid:107) p (cid:15), (cid:107) , Ω j by (cid:107) ∇ p (cid:15), (cid:107) , Ω j . Combining this fact with the normal stress balance conditions (3.8d) ,an inequality of the type (3.14) can be deduced for the union of adjacent domains Ω( h j , Γ j ) ∪ Ω j ∪ Ω( h j +1 , Γ j +1 ) and continue the process until the whole domain Ω is covered and the global inequality (3.14) is obtained. Due to the observations above we conclude that the following sequences arebounded (cid:107) u (cid:15), (cid:107) , Ω , (cid:107) (cid:15) u (cid:15), (cid:107) , Ω , √ α (cid:107) u (cid:15), · (cid:98) n (cid:107) L (Γ) (3.15a) (cid:107) p (cid:15), (cid:107) H (Ω ) , (cid:107) p (cid:15), (cid:107) H (Ω ) , (cid:107) (cid:15) ∂ z p (cid:15) (cid:107) , Ω , (cid:107) ∇ · u (cid:15), (cid:107) L (Ω ) . (3.15b) Remark 3.3.
The change of variable ϕ modifies the structure of the divergenceon the domains Ω( h i , Γ i ) for all ≤ i ≤ I , therefore it can only be claimed thatthe linear combination (cid:15) (cid:101) ∇ · (cid:101) u (cid:15), + (cid:15) (1 − (cid:15) ) ∂ z ( (cid:101) ∇ ζ i · (cid:101) u (cid:15), ) + ∂ z u (cid:15), is bounded in L (Ω( h i , Γ i )) . Weak Limits.
The previous section state bounds independent from (cid:15) > u (cid:15), , (cid:15) u (cid:15), ] ∈ V and p (cid:15) = [ p (cid:15), , p (cid:15), ] ∈ H (Ω ) × H (Ω ), consequently in Q .Then, there must exist u ∈ V , p ∈ Q , η ∈ L (Ω ) and a subsequence, from now ondenoted the same, such that p (cid:15) w (cid:42) p in Q and strongly in L (Ω) , (3.16a) u (cid:15), w (cid:42) u in L (Ω ) and ∇ · u (cid:15), w (cid:42) ∇ · u in L (Ω ) , (3.16b) √ α u (cid:15), · (cid:98) n w (cid:42) √ α u · (cid:98) n in L (Γ) , (3.16c) (cid:15) u (cid:15), w (cid:42) u in L (Ω ) , (3.16d)1 (cid:15) ∂ z p (cid:15), w (cid:42) η in L (Ω ) and ∂ z p (cid:15), → L (Ω ) . (3.16e)Choose φ ∈ C ∞ (Ω( h i , Γ i )) arbitrary, test the equation (3.7b) with q def = (cid:15)φ and let (cid:15) ↓
0. Recalling (3.16d) this gives0 = lim (cid:15) ↓ (cid:90) Ω( h i , Γ i ) | ( − (cid:101) ∇ ζ i , | ( (cid:15) u (cid:15), · (cid:98) n ( i ) ) ∂ z φ = (cid:90) Ω | ( − (cid:101) ∇ ζ i , | ( u · (cid:98) n ( i ) ) ∂ z φ = − (cid:68) ∂ z | ( − (cid:101) ∇ ζ i , | ( u · (cid:98) n ( i ) ) , φ (cid:69) D (cid:48) (Ω( h i , Γ i )) ,D (Ω( h i , Γ i )) . Since ( − (cid:101) ∇ ζ i ,
1) does not depend on the vertical variable z and it is the non-zerovector almost everywhere we conclude ∂ z ( u · (cid:98) n ( i ) ) = 0 i.e. the component ofthe velocity normal to the surface Γ i is independent from z in Ω( h i , Γ i ) for all1 ≤ i ≤ I . Now choose q ∈ Q arbitrary, test (3.7b) with (cid:15) q and let (cid:15) ↓ (cid:88) i (cid:90) Ω( h i , Γ i ) | ( − (cid:101) ∇ ζ i , | ( u · (cid:98) n ( i ) ) ∂ z q d x = (cid:88) i (cid:90) G i (cid:90) ζ i ( (cid:101) x )+ h i ζ i ( (cid:101) x ) | ( − (cid:101) ∇ ζ i , | ( u · (cid:98) n ( i ) ) ∂ z q dz d (cid:101) x = (cid:88) i (cid:90) G i | ( − (cid:101) ∇ ζ i , | ( u · (cid:98) n ( i ) ) [ q ( (cid:101) x , ζ i ( (cid:101) x ) + h i ) − q ( (cid:101) x , ζ i ( (cid:101) x ))] d (cid:101) x . The above holds for all q ∈ Q , in particular choosing q ( (cid:101) x , ζ i ( (cid:101) x )) = φ ( (cid:101) x ) for φ ∈ C ∞ ( G i ) arbitrary and q ( (cid:101) x , ζ i ( (cid:101) x ) + h i ) = 0 the statement transforms in (cid:90) G i | ( − (cid:101) ∇ ζ i , | ( u · (cid:98) n ( i ) )( (cid:101) x , ζ i ( (cid:101) x )) φ ( (cid:101) x , ζ i ( (cid:101) x )) d (cid:101) x ∀ φ ∈ C ∞ ( G i ) . Therefore | ( − (cid:101) ∇ ζ i , | (cid:0) u · (cid:98) n (cid:1) must be null and since | ( − (cid:101) ∇ ζ i , | is non-zero almosteverywhere we conclude u · (cid:98) n ( i ) = 0 in Ω( h i , Γ i ) for each 1 ≤ i ≤ I. (3.17)The later implies that the Cartesian coordinates of u satisfy the following relation u = (cid:26) (cid:101) u u (cid:27) = (cid:26) (cid:101) u (cid:101) u · (cid:101) ∇ ζ i (cid:27) in Ω( h i , Γ i ) , ≤ i ≤ I. (3.18) EOLOGICAL FISSURED SYSTEMS 15
Now fix i ∈ { , . . . , I } and take a function v τ ∈ ( C ∞ (Ω( h i , Γ i )) . Recalling (2.13)define (cid:101) v def = M T, τ i v τ and v def = M (cid:98) k , τ v τ . Then, the function v def = 1 (cid:15) ( (cid:101) v , v ) hasthe structure (3.18) or equivalently v · (cid:98) n = 0 inside Ω( h i , Γ i ). Define v as thetrivial extension of v to the whole domain Ω, therefore v ∈ V . Test (3.7a) with v and let (cid:15) ↓
0, this gives (cid:90) Ω( h i , Γ i ) a ( x ) u · ( (cid:101) v , v ) + (cid:90) Ω( h i , Γ i ) (cid:101) ∇ p · (cid:101) v + (cid:90) Ω( h i , Γ i ) g · ( (cid:101) v , v ) = 0 . Consequently (cid:90) Ω( h i , Γ i ) a ( x ) u τ · v τ + (cid:90) Ω( h i , Γ i ) (cid:101) ∇ p · M T, τ i v τ + (cid:90) Ω( h i , Γ i ) g τ · v τ = 0 . The equation above holds for all v τ ∈ ( C ∞ (Ω( h i , Γ i )) and due to the isomorphismof proposition (2.3) we conclude a ( x ) u τ + (cid:16) M T, τ i (cid:17) (cid:48) (cid:101) ∇ p + g τ = 0 in Ω( h i , Γ i ) , ≤ i ≤ I. (3.19)The equation (3.16e) implies that p does not depend on the variable z on Ω i.e. p = p ( (cid:101) x ). Therefore assuming a = a ( (cid:101) x ) , (cid:101) g = (cid:101) g ( (cid:101) x ) in Ω (3.20)the equation (3.19) gives u τ = u τ ( (cid:101) x ) i.e. u τ is independent from z in Ω . Togetherwith the fact u · (cid:98) n = 0 in Ω we conclude that the whole vector velocity u isindependent from z in Ω . Remark 3.4.
Observe that due to the assumptions for the data (3.20) the equation (3.19) is independent from z becoming a lower-dimensional Darcy-type constitutivelaw on the stream lines parallel to ζ i . The Limit Problem
Define the subspaces V def = (cid:110) v ∈ V : ∂ z v = 0 , v · (cid:98) n ( i ) = 0 in Ω( h i , Γ i ) (cid:111) . (4.1a) Q def = { q ∈ Q : ∂ z q = 0 in Ω } . (4.1b) Remark 4.1.
Notice that if q ∈ Q the fact that ∂ z q = 0 in Ω implies q | Γ i = q | Γ i + h i for all ≤ i ≤ I . Therefore, the spaces (cid:110) v ∈ L (Ω ) : ∇ · v ∈ L (Ω ) , v · (cid:98) n ( i ) ∈ L (Γ i ) (cid:111) × (cid:89) i L (Γ i ) L (Ω ) × (cid:89) i H (Ω( h i , Γ i )) Are isomorphic to (4.1a) and (4.1b) respectively.
Due to the structure of the space if v = [ v , v ] ∈ V then the function [ v , (cid:15) v ]is also in V . Using the latter to test (3.7a) and q ∈ Q for testing (3.7b) we let (cid:15) ↓ u (cid:15), , (cid:15) u (cid:15), ] → u and p (cid:15) → p are a solution of the limit problem Find p ∈ Q , u ∈ V : (cid:90) Ω a u · v − (cid:90) Ω p ∇ · v + (cid:90) Ω a u τ · v τ + (cid:90) Ω (cid:101) ∇ p (cid:15) · (cid:101) v − (cid:90) Γ t p ( v · (cid:98) n ) dS + (cid:90) Γ b p ( v · (cid:98) n ) dS + α (cid:90) Γ ( u · (cid:98) n )( v · (cid:98) n ) dS = − (cid:90) Ω g · v − (cid:90) Ω g τ · v τ (4.2a) (cid:90) Ω ∇ · u q − (cid:90) Ω (cid:101) u · (cid:101) ∇ q + (cid:90) Γ t (cid:0) u · (cid:98) n (cid:1) q dS − (cid:90) Γ b (cid:0) u · (cid:98) n (cid:1) q dS = (cid:90) Ω F q + (cid:90) Γ f Γ q dS (4.2b)for all q ∈ Q , v ∈ V Well-Posedness of the Limit Problem.
The problem (4.2) is a mixedformulation of the type (2.20) with the operators A : V → V (cid:48) and B : V → Q (cid:48) defined by A v ( w ) def = (cid:90) Ω a v · w + (cid:90) Ω a v τ · w τ + α (cid:90) Γ (cid:0) v · (cid:98) n (cid:1) (cid:0) w · (cid:98) n (cid:1) dS (4.3a) B v ( q ) def = − (cid:90) Ω ∇ · v q + (cid:90) Ω (cid:101) v · (cid:101) ∇ q − (cid:90) Γ t ( v · (cid:98) n ) q + (cid:90) Γ b ( v · (cid:98) n ) q (4.3b) Theorem 4.1.
The operator B satisfies the inf-sup condition.Proof. The proof has the same structure as lemma (2.7), there is only one detailto be examined in the construction of the test functions. Fix q = [ q , q ] ∈ Q ,construct v in the same way it is built in problem (2.25). On the other hand since q ∈ H (Ω ), ∂ z q = 0, define v def = (cid:88) i ( (cid:101) ∇ q , (cid:101) v · (cid:101) ∇ ζ i ) Ω( h i , Γ i ) . Then v · (cid:98) n ( i ) = 0 and ∂ z v = 0 in Ω( h i , Γ i ) for all 1 ≤ i ≤ I , i.e. v ∈ V and (cid:107) v (cid:107) , Ω ≤ C (cid:107) q (cid:107) , Ω as desired. Repeating the inequalities presented in (2.26)the proof is complete. (cid:3) Since the inf-sup condition holds the theorem (2.5) applies to the operators(4.3) on the spaces V , Q and the limit problem (4.2) is well-posed. Due to theuniqueness of the solution of the limit problem it follows that the original sequenceconverges weakly to the limit u ∈ V , p ∈ Q .4.2. The Strong Form.
In order to describe the strong limit problem correspond-ing to (4.2) two features have to be exploited. First, the structure v · (cid:98) n ( i ) = 0 inΩ for all v ∈ V implying (cid:101) v = M T, τ v τ , for M T, τ the matrix defined in (2.13).Second, the independence of the velocities and pressures with respect to z in Ω .This last property allows to write the integrals over Ω( h i , Γ i ) as surface integralson Γ i . Hence, the system (4.2) transforms inFind p ∈ Q , u ∈ V : (cid:90) Ω a u · v − (cid:90) Ω p ∇ · v + (cid:88) i h i (cid:90) Γ i ( (cid:98) n ( i ) · (cid:98) k )( a u τ + ( M T, τ i ) (cid:48) (cid:101) ∇ p (cid:15) + g τ ) · v τ dS EOLOGICAL FISSURED SYSTEMS 17 − (cid:90) Γ t p (cid:0) v · (cid:98) n (cid:1) dS + (cid:90) Γ b p (cid:0) v · (cid:98) n (cid:1) dS + α (cid:90) Γ (cid:0) u · (cid:98) n (cid:1) (cid:0) v · (cid:98) n (cid:1) dS = − (cid:90) Ω g · v (4.4a) (cid:90) Ω ∇ · u q − (cid:88) i h i (cid:90) Γ i ( (cid:98) n ( i ) · (cid:98) k ) M T, τ i u τ · (cid:101) ∇ q dS + (cid:90) Γ t (cid:0) u · (cid:98) n (cid:1) q dS − (cid:90) Γ b (cid:0) u · (cid:98) n (cid:1) q dS = (cid:90) Ω F q + (cid:90) Γ f Γ q dS (4.4b)for all q ∈ Q , v ∈ V Integrating by parts the above statement we get the strong lower dimensional prob-lem a u + ∇ p + g = 0 , (4.5a) ∇ · u = F in Ω . (4.5b) p = 0 on ∂ Ω − Γ . (4.5c) u · (cid:98) n = 0 , ∂ z p = 0 , (4.5d) (cid:88) i (cid:104) a ( s ) u τ + ( M T, τ i ) (cid:48) (cid:101) ∇ p + g τ ( s ) (cid:105) Γ i = 0 , (4.5e) (cid:88) i (cid:104) ( u · (cid:98) n ( i ) | Γ i + h i − u · (cid:98) n ( i ) | Γ i ) (cid:105) Γ i + (cid:88) i h i ( (cid:98) n ( i ) · (cid:98) k ) (cid:101) ∇ · ( M T, τ i u τ ) Γ i = f Γ in Γ . (4.5f) p − p = α u · (cid:98) n Γ b − α u · (cid:98) n Γ t . (4.5g) u · (cid:98) ν ( i ) = 0 on ∂ G i for all 1 ≤ i ≤ I. (4.5h)The statement of equation (4.5e) was already shown in (3.19), however the state-ments (4.5f) and (4.5h) need further discussion.4.3. The Interface Integrals Setting.Definition 4.2.
Let G , ζ and Γ be as in definition (2.1) , define the spaces L (Γ) def = { h : Γ → R : (cid:90) Γ h ( s ) dS < + ∞} (4.6a) H (Γ) def = { h ∈ L (Γ) : (cid:101) ∇ h ∈ L (Γ) × L (Γ) } (4.6b) H (Γ) def = { h ∈ H (Γ) : h | ∂ G = 0 } (4.6c) Here (cid:101) ∇ indicates the gradient with respect to the variables ( x , x ) contained in G . The following isomorphism result is necessary
Theorem 4.3.
Let G , ζ and Γ be as in definition (2.1) . Consider the naturalembedding : Γ → G defined by ( (cid:101) x , ζ ( (cid:101) x )) def = (cid:101) x and the map ϕ (cid:55)→ ϕ ◦ (4.7) Then(i) The embedding (4.7) is an isomorphism between L ( G ) and L (Γ) .(ii) The embedding (4.7) is an isomorphism between H ( G ) and H (Γ) .(iii) The embedding (4.7) is an isomorphism between H ( G ) and H (Γ) . Proof.
By definition : G → Γ is linear and bijective, therefore the map (4.7) isbijective between spaces of functions.(i) Due to the hypothesis ζ satisfies C = ess inf { (cid:98) n ( s ) · (cid:98) k : s ∈ Γ } > φ ∈ L (Γ) (cid:90) Γ ( φ ◦ ) dS = (cid:90) G ( (cid:98) n ( (cid:101) x ) · (cid:98) k ) − φ ( (cid:101) x ) d (cid:101) x ≤ C (cid:90) G φ ( (cid:101) x ) d (cid:101) x . The inequality above gives the continuity of the application ϕ (cid:55)→ ϕ ◦ j . Dueto Banach’s inversion theorem the map is an isomorphism.(ii) By definition (cid:101) ∇ ( φ ◦ ) = (cid:101) ∇ φ ( (cid:101) x ) holds for any φ ∈ H ( G ).(iii) Is immediate from (ii). (cid:3) Fix i ∈ { , . . . , I } , choose q ∈ Q supported inside Ω( h i , Γ i ) and test equation(4.4b); hence − (cid:90) Ω( h i , Γ i ) (cid:101) u · (cid:101) ∇ q − (cid:90) Γ i ( u · (cid:98) n ( i ) ) q dS + (cid:90) Γ i + h i ( u · (cid:98) n ( i ) ) q dS = (cid:90) Γ i ∪ Γ i + h i f Γ i q dS. (4.8)We focus on the first term of the left hand side. First ∂ z q = 0 implies (cid:101) u · (cid:101) ∇ q = u · ∇ q , then − (cid:90) Ω( h i , Γ i ) u · ∇ q = (cid:90) Ω( h i , Γ i ) ∇ · u q − (cid:90) ∂ Ω( h i , Γ i ) q u · (cid:98) ν ( i ) dS. (4.9)The two summands of the left hand side are treated separately. For the first sum-mand the independence from the variable z implies ∇ · u = (cid:101) ∇ · (cid:101) u , the fact u · (cid:98) n ( i ) = 0 in Ω( h i , Γ i ) gives (cid:101) u = M T, τ u τ . Thus (cid:90) Ω( h i , Γ i ) (cid:101) ∇ · (cid:101) u q d x = h i (cid:90) G i (cid:101) ∇ · ( M T, τ u τ ) q d (cid:101) x = h i (cid:90) Γ i ( (cid:98) n ( i ) · (cid:98) k ) (cid:101) ∇ · ( M T, τ u τ ) q dS. (4.10)The boundary term in (4.9) can be written as − (cid:90) Γ i ∪ Γ i + h i q u · (cid:98) ν ( i ) dS − (cid:90) ∂ Ω( h i , Γ i ) −{ Γ i ∪ Γ i + h i } q u · (cid:98) ν ( i ) dS. The first summand vanishes since u · (cid:98) n ( i ) = 0 in Ω( h i , Γ i ). The boundary piecedescribed in the second summand is a vertical wall, then (cid:98) ν ( i ) · (cid:98) k = 0 and it canbe identified with the outwards normal vector to the set G i ⊆ R . Moreover, dueto the independence of the integrand with respect to the variable z , the surfaceintegral can be collapsed to a line integral over ∂ G i . Combining these observationswith (4.10) and (4.9) the equation (4.8) transforms in h i (cid:90) Γ i ( (cid:98) n ( i ) · (cid:98) k ) (cid:101) ∇ · ( M T, τ u τ ) q dS − h i (cid:90) ∂G i q u · (cid:98) ν ( i ) dC − (cid:90) Γ i ( u · (cid:98) n ( i ) ) q dS + (cid:90) Γ i + h i ( u · (cid:98) n ( i ) ) q dS = (cid:90) Γ i ∪ Γ i + h i f Γ q dS Where dC is the arc-length measure on ∂G i . The isomorphisms provided by theo-rem (4.3) imply that the quantifier q | Γ i can hit any function in the space H (Γ i ).Therefore, the equation (4.5f) follows. Finally, using again theorem (4.3) the trace EOLOGICAL FISSURED SYSTEMS 19
Figure 3.
System of 2-D Manifold Fissures Λ Λ Λ Θ Θ Θ Θ of test function q | Γ i can hit any function in the space H (Γ i ) and combined withequation (4.5f) give (4.5h).4.4. Strong Convergence of the Solutions.Theorem 4.4.
Under the hypothesis (cid:107) F (cid:15), − F (cid:107) , Ω → , (cid:107) f (cid:15) Γ − f Γ (cid:107) , Γ → and (cid:107) g (cid:15) − g (cid:107) , Ω → The solutions u (cid:15) , p (cid:15) satisfy the following strong convergence statements (cid:107) u (cid:15), − u (cid:107) , Ω → , (cid:107) (cid:15) u (cid:15), − u (cid:107) , Ω → , (4.12) (cid:107) p (cid:15), − p (cid:107) , Ω → , (cid:107) p (cid:15), − p (cid:107) , Ω → . (4.13) Proof.
The proof uses exactly the same arguments presented in [15], theorem 3.2. (cid:3)
Finally, assume that u τ (cid:54) = 0 and consider the quotients: (cid:13)(cid:13) u (cid:15), τ (cid:13)(cid:13) , Ω (cid:107) u (cid:15), · (cid:98) n (cid:107) , Ω = (cid:13)(cid:13) (cid:15) u (cid:15), τ (cid:13)(cid:13) , Ω (cid:107) (cid:15) u (cid:15), · (cid:98) n (cid:107) , Ω > (cid:13)(cid:13) u τ (cid:13)(cid:13) , Ω − δ (cid:107) (cid:15) u (cid:15), · (cid:98) n (cid:107) , Ω > (cid:15) > δ > u τ = 0, unlike the analysis for flat interfaces presented in [15],no conclusions can be obtained due to the complexity introduced by the geometryof the fissures.5. A Problem with two dimensional Manifolds
In this section, using the independence of the limit functions with respect to z in Ω it will be shown that the limiting problem (4.5) can be formulated as a system coupling Darcy flow in three dimensions with tangential flow hosted twodimensional manifolds as depicted in figure (3).5.1. Geometric Setting.Definition 5.1.
We say a totally fractured medium of two dimensional manifoldfissures is a finite collection ofSurface functions { λ i ∈ C ( G i ) : G i ⊆ R open bounded simply connected region ; λ i piecewise C function such that ess inf (cid:98) n ( i ) · (cid:98) k > , ≤ i ≤ I } . (5.1a) And rock-matrix regions (cid:8) Θ j ⊆ R : Θ j (cid:54) = ∅ open bounded simply connected region , ≤ j ≤ I (cid:9) . (5.1b) Verifying the following propertiesNon-overlapping condition and indexed ordered sup { λ i ( (cid:101) x ) : (cid:101) x ∈ G i } < inf { λ i +1 ( (cid:101) x ) : (cid:101) x ∈ G i +1 } , ∀ ≤ i ≤ I − The interface domain condition Λ i = ∂ Θ i ∩ ∂ Θ i − , ∀ ≤ i ≤ I (5.2b) for Λ i def = { [ (cid:101) x , λ i ( (cid:101) x ) : (cid:101) x ∈ G i ] } . And the connectivity through fissures condition cl (Θ (cid:96) ) ∩ cl (Θ k ) = ∅ whenever | (cid:96) − k | > . (5.2c) For convenience of notation define Λ def = ∂ Ω − Γ . We denote this fissuredsystem by { (Λ i , Θ i ) : 0 ≤ i ≤ I } . The rock matrix and fissures regions are the sets Θ def = I (cid:91) i =0 Θ i , Λ def = I (cid:91) i =0 Λ i , Θ FR def = Θ ∪ Λ . (5.3) (cid:98) n ( i ) indicated the upwards normal vector to the surface Λ i . Finally, we introducethe notations Λ + i and Λ − i for the upper and lower faces of the manifold Λ i . Spaces of Functions and Isomorphisms.Definition 5.2.
We define the following spaces for velocity and pressure V f def = { v ∈ L (Θ FR ) : ∇ · v ∈ L (Θ j ) , ≤ j ≤ I ; v · (cid:98) n ( i ) | Λ + i , v · (cid:98) n ( i ) | Λ − i ∈ L (Λ i ) , v | Λ i ∈ L (Λ i ) , ≤ i ≤ I } , (5.4a) Q f def = { q ∈ L (Θ FR ) : q | Λ i ∈ H (Λ i ) , ≤ i ≤ I } . (5.4b) Endowed with the norms coming from the natural inner products (cid:107) v (cid:107) V f def = {(cid:107) v (cid:107) L (Θ FR ) + (cid:107) ∇ · v (cid:107) L (Θ FR ) + (cid:88) i (cid:107) v · (cid:98) n ( i ) | Λ + i (cid:107) L (Λ i ) + (cid:107) v · (cid:98) n ( i ) | Λ − i (cid:107) L (Λ i ) + (cid:107) v (cid:107) L (Λ i ) } / , (5.4c) (cid:107) q (cid:107) Q f def = {(cid:107) q (cid:107) L (Θ FR ) + (cid:88) i (cid:107) q (cid:107) H (Λ i ) } / . (5.4d) EOLOGICAL FISSURED SYSTEMS 21
Remark 5.1.
Notice that definition (5.4a) demands only v ∈ H div (Θ i ) i.e. thedivergence is square integrable only on these subdomains. Therefore, both normaltraces v · (cid:98) n ( i ) | Λ + i and v · (cid:98) n ( i ) | Λ − i make sense in H − / (Γ i ) but we require the extracondition of been in L (Λ i ) . We do not demand the global condition v ∈ H div (Θ FR ) because this would imply the continuity of the normal traces across a surface i.e. u · (cid:98) n ( i ) | Λ + i = u · (cid:98) n ( i ) | Λ − i . Such condition can not model jumps across the fissuresas the normal stress balance interface (4.5g) and the limit equation (4.5f) . Next define a change of variable based on piecewise translations
Definition 5.3.
Let x = ( (cid:101) x , x ) and define the map T : Ω → R T x def = I (cid:88) j = 0 (cid:32)(cid:101) x , x − j (cid:88) (cid:96) = 0 h (cid:96) (cid:33) Ω j ( (cid:101) x , x ) − I (cid:88) i = 1 (cid:32)(cid:101) x , ζ i ( (cid:101) x ) − i − (cid:88) (cid:96) = 0 h (cid:96) (cid:33) Ω( h i , Γ i ) ( (cid:101) x , x )(5.5) Define Θ j def = T (Ω j ) and λ i : G i → R by λ i def = ζ i ( (cid:101) x ) − (cid:80) i − (cid:96) =0 h (cid:96) . Clearly the system { (Λ i , Θ i ) : 1 ≤ i ≤ I } satisfies the conditions of definition(5.1). With the previous definitions we have the following result Theorem 5.4. (i) The application v (cid:55)→ v ◦ T is an isometric isomorphism from V to V f .(ii) The application q (cid:55)→ q ◦ T is an isometric isomorphism from Q to Q f .Proof. (i) The proof is a direct application of part (i) in theorem (4.3). The onlydetail that needs further clarification is to observe that v · (cid:98) n ( i ) | Γ i (cid:55)→ ( v ◦ T ) · (cid:98) n ( i ) | Λ − i v · (cid:98) n ( i ) | Γ i + h i (cid:55)→ ( v ◦ T ) · (cid:98) n ( i ) | Λ + i (ii) It is a direct application of parts (i) and (ii) in theorem (4.3). (cid:3) The Lower Dimensional Mixed Problem.
Due to the previous theoremthe problem (4.2) is equivalent to the following mixed problem with two dimensionalmanifolds Find p ∈ Q f , u ∈ V f : (cid:90) Θ a u · v − (cid:90) Θ p ∇ · v + (cid:88) i h i (cid:90) Λ i ( (cid:98) n ( i ) · (cid:98) k )( a u τ + ( M T, τ i ) (cid:48) (cid:101) ∇ p + g τ ) · v τ dS + (cid:88) i α (cid:90) Λ i (cid:104) ( u · (cid:98) n ( i ) | Λ + i )( v · (cid:98) n ( i ) | Λ + i ) + ( u · (cid:98) n ( i ) | Λ − i )( v · (cid:98) n ( i ) | Λ − i ) (cid:105) dS − (cid:88) i (cid:90) Λ i p (cid:104) ( v · (cid:98) n ( i ) | Λ + i ) − ( v · (cid:98) n ( i ) | Λ − i ) (cid:105) dS = − (cid:90) Θ g · v − (cid:90) Λ g τ · v τ dS (5.6a) (cid:90) Θ ∇ · u q − (cid:88) i h i (cid:90) Λ i ( (cid:98) n ( i ) · (cid:98) k ) M T, τ i u τ · (cid:101) ∇ q dS + (cid:88) i (cid:90) Λ i (cid:104) ( u · (cid:98) n | Λ + i ) − ( u · (cid:98) n | Λ − i ) (cid:105) q dS = (cid:90) Θ F q + (cid:90) Λ f Γ q dS (5.6b) for all q ∈ Q f , v ∈ V f . Finally the equivalence of problems (4.2) and (4.4) gives the well-posedness of thesystem above. 6.
Final Discussion and Future Work (i) The formulation presented in this work can manage large amounts of infor-mation in a remarkably efficient way. One of the main reasons is the notationintroduced by Showalter in [15] for the description of function spaces.(ii) The results can be generalized immediately to the R N -setting using the samearguments presented here. The structure of the problems is analogous.(iii) The approach based on analytic semigroups theory presented in section [15]can be directly applied here to model the time dependent problem for totallyfissured systems with singularities.(iv) Although the mathematical analysis is solid, the approach used throughoutthe paper stops been suitable for surfaces with high gradients such as the onedepicted in the right hand side of figure (4) where (cid:98) n · (cid:98) k (cid:28) (cid:98) n · (cid:98) k . In thiscase the translation in the direction (cid:98) k generates a fissure whose cross sectionareas can be very different from one piece to another i.e. A (cid:28) A . Such afissure is not realistic. On the other hand consider a fissure such as the onedepicted in the left hand side of figure (4). Here the translation is made inthe bisector vector direction (cid:98) e = 1 | (cid:98) n + (cid:98) n | (cid:98) n + (cid:98) n Figure 4.
Translation Generated Fissures
ΓΓ+ h ̂ e ̂ e ΓΓ+ h ̂ k A ̂ n ̂ n =̂ k ̂ n ̂ n V V V V A A A EOLOGICAL FISSURED SYSTEMS 23
Such question will be addressed in future work by the introduction of cor-rection factors obtained comparing the flow energy dissipation in a real fissureand an artificial one e.g. replacing the presence of the fissure in the left handside of (4) with the one on the right side affected by a correction factor. Inthe same way, fissures defined by walls which are not rigid translations of theother will be compared to a fissure generated by vertical translation of its“average surface” and having the same “average width”.7.
Acknowledgements
The author thanks to Universidad Nacional de Colombia, Sede Medell´ın for par-tially supporting this work under the projects HERMES 17194 and HERMES 14917as well as the Department of Energy, Office of Science, USA for partially supportingthis work under grant 98089. Finally, the author wishes to thank Professor RalphShowalter from the Mathematics Department at Oregon State University for hishelpful insight, observations and suggestions.
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Fernando A. MoralesEscuela de Matem´aticas, Universidad Nacional de Colombia, Sede Medell´ın. Colombia
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