Seminar zu aktuellen Themen der Numerik im Wintersemester 2010/2011

Transcription

1 Seminar zu aktuellen Themen der Numerik im Wintersemester 2010/2011 Modeling Two-Phase-Two-Component Processes in Porous Media Using an Upscaling-Multi-Scale Method Elin Solberg 27 January 2011

2 Contents 1 Introduction 1 2 Problem Statement Basic Definitions Governing Equations Two-Phase Model Two-Phase-Two-Component Model Flash Calculations Modeling with Multiple Scales Solution Strategy Overview Upscaling of the Saturation Equation Solution Strategy Solution Tools Time and Space Discretization Compression and Reconstruction Operators The Full Algorithm Results and Conclusion 16

3 Figure 1: Set-up to be modeled. 1 Introduction In environmental engineering modeling flow and transport of contaminants in the subsurface is a topic of interest. The subsurface is a heterogeneous porous medium and in the unsaturated zone, i.e. above the water table, its pore volume is occupied by water, air and possibly contaminants. In this report we shall restrict ourselves to the case of only water and air. In a large domain, with a side length of up to hundreds or thousands of meters, fine-scale effects, such as fine-scale heterogeneities of the solid matrix (i.e. the solid part of the subsurface) influencing the fluid flow and mass transfer processes at the interface between the liquid and gas phases influencing the phase saturations, need to be resolved by the model. Thus on the one hand, using a coarse resolution would lead to loss of fine-scale information. On the other hand, using a fine resolution in the entire domain leads to high computational costs. In this report, a multi-scale-upscaling strategy to reduce the computational cost while still capturing the fine-scale effects, proposed by Niessner and Helmig in [7] and by Niessner in [6], is presented. The set-up which will be considered is shown in Figure 1. The domain is a twodimensional slab of the unsaturated zone, in which heterogeneities are present everywhere. In a small part of the domain the amount of gas (air) is large and at the interface between this zone and its surroundings complex two-phase-two-component mass transfer processes will take place. Elsewhere in the domain a simple two-phase model will be adequate. The model will be simplified according to the following assumptions: fluids (including air) and solid matrix are assumed incompressible capillary pressure is neglected gravity is set to zero 1

4 2 Problem Statement no external sinks/sources are present. It is stressed by Niessner and Helmig, that their aim is to develop a first framework for this kind of complex processes in heterogeneous porous media, and that to model real-life systems, the above assumptions must be revoked by further development of the basic framework presented here. The remainder of this report is organized as follows: In section 2 first some basic concepts, central in the following discussion, and thereafter the basic mathematical model, are introduced. In section 3 attention is paid to the multiple scales present in the model. A first overview of the multi-scale-upscaling strategy is given, and an upscaled version of one of the governing equations, the saturation equation, is derived. Having the upscaled saturation equation, the problem to be attacked by the multi-scale-upscaling strategy can be formulated in its final form. Before the full solution strategy is presented step by step in section 4.2 the choice of time and space discretizations as well as compression and reconstruction operators, i.e. operators to move from fine- to coarse-scale quantities and vice versa, is given in section 4.1. Finally, in section 5, some results are presented and conclusions are drawn. 2 Problem Statement 2.1 Basic Definitions In the previous section, an intuitive understanding of concepts such as phase, component and saturation was assumed, however, to formulate a mathematical model we need a more precise understanding of these and some further concepts. In this section the three mentioned concepts, as well as absolute and relative permeability, are briefly introduced. For a more detailed discussion of these and related issues the reader is referred to [6]. Phases, Components The definition of a fluid phase, as given in [6], is a continuum of fluid which has a sharp interface to other such continua, where over this interface physical fluid properties are discontinuous. In the model presented in this report, two phases will be considered: a wetting phase and a non-wetting (gas) phase. Further two components, water and air, will be considered. In most of the domain the wetting phase will be assumed to consist only of water, and the non-wetting phase only of air. This situation is referred to as a two-phase system. However, near the interface between the two phases they will be considered to some extent soluble in one another, i.e. there will be air dissolved in the wetting phase and water molecules evaporated in the non-wetting phase. Hence, both components are present in each phase, and we refer to this as a two-phase-two-component system. Saturation 2

5 2.2 Governing Equations In the following discussion, saturation will be the quantity of main interest to us. Considering a representative elementary volume (REV) with side length of order 1 dm the saturation S α of phase α is defined as S α := V α V tot where V α is the volume of phase α in the REV and V tot is the total pore volume (i.e. the volume not occupied by the solid matrix) in the REV. From this definition it is clear that in our system, consisting of only two phases, we have S w + S n = 1 (1) where subscripts w and n refer to wetting and non-wetting phase, respectively. Absolute and Relative Permeability The structure of the solid matrix is described by the absolute permeability K, which is in the general case a tensor, however in the results section it will be restricted to a scalar, corresponding to an isotropic medium. We will be interested in modeling a heterogeneous porous medium. Heterogeneity is achieved by letting K vary in space, i.e. K = K(x). Local (fine-scale) variations of K(x) will impact the overall velocity field, and hence need to be resolved on a fine scale. Whereas the absolute permeability is a property of the solid matrix, the relative permeability of phase α, denoted by k rα, is a property of the phase depending on the saturation of that phase, i.e. k rα = k rα (S α ). Using (1) we may however express the relative permeability of either phase as a function of S w. We will consider two alternative parameterizations of the relative permeabilities, first a simple linear approach: k rw (S w ) = S w k rn (S w ) = 1 S w and second a non-linear approach suggested by Brooks and Corey [4]: where λ BC is a form parameter. k rw (S w ) = S (2+3λ BC)/λ BC w k rn (S w ) = (1 S w ) 2 (1 S (2+λ BC)/λ BC w ), 2.2 Governing Equations We are now ready to formulate the equations governing our model of the system described in section 1. First the case of a two-phase system is considered, i.e. the part of the domain where only a negligible amount of mass transfer between the phases takes place. Once 3

6 2 Problem Statement the two-phase model has been established, the governing equation of the two-phase-twocomponent model, valid in all parts of the domain, but specifically near the gas plume (see Figure 1) where a considerable amount mass transfer takes place between the two phases, is formulated Two-Phase Model Using the fully coupled formulation, the two-phase model is governed by the mass conservation and momentum balance equations, which under the simplifying assumptions stated in section 1 are given by S α t + v α = 0, α = w, n and (2) v α = k rα µ α K p, α = w, n, (3) respectively, where v α = v α (x, t) is the Darcy velocity of phase α, p = p(x, t) is the pressure of either phase (p w = p n = p since capillary pressure is neglected) and µ α is the viscosity of phase α (a constant scalar). The phase saturations S α, the relative permeabilities k rα and the absolute permeability K were described in section 2.1. In summary, we have the four equations (2) and (3) which may be solved for the five unknowns S w, S n, v w, v n and p, using additionally the relation (1). We will now rewrite the fully coupled formulation of the model, deriving instead the fractional flow formulation, in which one differential equation for pressure and one for wetting phase saturation will need to be solved. For the derivation we first define the total velocity: v := v w + v n, the phase mobility of phase α: λ α := k rα /µ α, the total mobility: λ := λ w + λ n, the fractional flow of phase α: f α := λ α /λ. Note that from the dependency of k rα on S w it follows that λ α, λ and f α are also all functions of S w. Summing the momentum balance equations (3) over both phases and expressing the result in terms of v and λ gives us a relation between pressure and total velocity: v = λk p. (4) Summing the mass conservation equations over both phases and noting that (Sw+Sn) t = 1 t = 0 yields v = 0, (5) in which (4) may be inserted, rendering the elliptic pressure equation: [λk p] = 0. (6) 4

7 2.2 Governing Equations Our second differential equation in the fractional flow formulation will be similar to the mass conservation equation for the wetting phase, but will depend on total velocity rather than on wetting phase velocity. We derive it by first, for both phases, dividing both sides of (3) by λ α = k rα /µ α to see that v w λ w = K p = v n λ n = (v v w) λ n where the last equality follows from the definition of v. Resolving this expression for v w we see that v w = λw λ v = f wv = f w (S w )v, which may be inserted into the wetting phase mass conservation equation (2) to finally yield the hyperbolic saturation equation: S w t + v f w (S w ) = 0, (7) where use has been made of (5) to perform the simplification (f w v) = v f w +f w v = v f w. We thus have in the fractional flow formulation two differential equations, the pressure equation (6) and the saturation equation (7), which may be solved for the unknowns S w, v and p using additionally the relation (4). Once S w and p are known, S n and the phase velocities can be computed, if needed, using the relations (1) and (3) respectively Two-Phase-Two-Component Model In the previous section a two-phase system was considered, in which each phase was assumed to consist of only one component. Near the interface between two phases this is not the case, and to model the mass transfer processes between phases not only saturations, but also concentrations need to be taken into account. The concentration of component κ in phase α is computed as C κ α = ρ α X κ α, (8) where ρ α is the density of phase α and X κ α is the mass fraction of component κ in phase α. Note that due to the incompressibility assumption ρ α is constant for each α. The total concentration of component κ is then defined as C κ = C κ ws w + C κ ns n. (9) In our case the components are water, denoted by κ = 1, and air, denoted by κ = 2, and the governing equations of our two-phase-two-component system are the concentration equations: C κ + v f α (S w )Cα κ = 0, κ = 1, 2. (10) t α 5

8 2 Problem Statement Note that in the case of a two-phase system C 1 n = 0 and the concentration equation for component water reduces to the wetting phase saturation equation (7). Before solving the concentration equations, the pressure equation is solved and the velocity field is computed from the pressure field using (4). Hence the velocity in the concentration equations can be considered known. Using so called flash calculations, explained in section 2.2.3, the mass fractions X κ α can be computed as functions of the pressure p. The mass fractions are thus independent of total concentrations and saturations. From the mass fractions all C κ α can be computed, using (8). Consequently all C κ α depend on pressure but are independent of total concentrations and saturations, and enter the concentration equations as known quantities. The only remaining unknowns of the two equations (10) are thus C 1, C 2 and S w. Finally, using (9) and (1), we may express S w as an affine mapping of either total concentration S w = Cκ Cn κ Cw κ Cn κ, (11) which corresponds to the final step of the flash calculations described in the next section. One interpretation of this is that we may substitute (11) into the concentration equations (10), leaving us with two (uncoupled) equations for C 1 and C 2. By Niessner and Helmig [7] the relation (11) is rather described as a computational step being performed once each time step when solving the concentration equation, to update the wetting phase saturation. In practice, since we are in the end merely interested in saturations and not concentrations, only one of the concentration equations (in our case the one for C 1 ) is solved Flash Calculations As mentioned above, flash calculations are needed to compute mass fractions X κ α, knowing the pressure p. The flash calculations presented here, a so called K κ flash, are due to Young and Stephenson [8], and build upon the assumption that the ratio K κ in which each component is distributed among the phases is a known function of pressure (and, under less strict assumptions than those stated in section 1, of density, temperature and total concentration), i.e. x κ n = K κ x κ w, κ = 1, 2 (12) with x κ α being the mole ratio of component κ in phase α and K κ = K κ (p). For the components water and air we have, respectively K 1 = p1 vap p K 2 = 1 H 2 p, 6

9 where p 1 vap is the vapor pressure of water and H 2 is the Henry coefficient of air. By definition of mole fractions we have x 1 α + x 2 α = 1 for α = w, n, which together with (12) gives us a linear system of four equations, which may be solved for the four unknowns x κ α (α = w, n; κ = 1, 2). Once the mole fractions are known, mass fractions are easily computed using the molar masses M 1 and M 2 of the components: X κ α = x κ αm κ x 1 αm 1 + x 2 αm 2. In the final step of the flash calculations the computed mass fractions are used together with the solution C 1 of the concentration equation for water, to compute S w using (11) as described in the previous section 1. 3 Modeling with Multiple Scales Having stated the basic mathematical model in the previous section, we will now turn to the issue of different scales being present in the model. First, the main sources of fine-scale effects are discussed and a method to deal with different scales is outlined. The method is based on treating an upscaled version of the saturation equation, and the main part of this section is devoted to deriving it from the saturation equation formulated in section Solution Strategy Overview In this section the main idea of the solution strategy is outlined. A detailed step-by-step presentation of the full algorithm will be given in section 4.2. Before exploiting the different scales inherent in the problem, we will consider how to deal with the pressure, saturation and concentration equations as they are given in section 2.2. Letting T denote the entire domain of interest and U the subdomain in which two-phase-two-component processes are not negligible, the pressure equation is valid in all of T, whereas the saturation equation is valid only in T \U. To render the saturation equation valid (i.e. mass conservative) also in U a source/sink term, resulting from the solution of the concentration equation will have to be added. Hence at the beginning of a new time step we first solve the pressure equation for p in T and then the concentration equation for C 1 only in U, from which S C w, with the superscript C indicating that the saturation value comes from solving the concentration equation, can be computed as 1 In [7] a different expression than (11) is used to compute S w from C 1 : S w = C1 K 1 C 1 K 2 K 1 ρ n+k 1 K 1 ρ n (K 2 1)(K 1 ρ n ρ w). This expression is derived in [6] using - in addition to (1) - the fact that X 1 α +X 2 α = 1 for both phases. It can be shown that the two expressions are equivalent, assuming that in the above expression the last term of the numerator should be K 1 K 2 ρ n. The reason for using the more complicated expression might be related to a more natural extension to multiple phases and components. 7

10 3 Modeling with Multiple Scales described in section The source/sink term is computed as q w = (S S w S C w )/ t, with S S w coming from the saturation equation solution in the last time step. Finally q w is added to the saturation equation to make it mass conservative, and it can then be solved in all of T for S w at the present time step. Solving the above described system using a fine discretization in the entire domain is costly in terms of computing power and data to be collected. On the other hand, solving it using a coarse grid, fine-scale effects of two types will be lost: 1. Effects on the flow field due to fine-scale heterogeneities of the porous medium, i.e. local variations in the absolute permeability K, present in the pressure equation. 2. Effects on the concentrations due to two-phase-two-component mass transfer processes, local in nature since they only occur near the surface between two phases. The method suggested by Niessner and Helmig in [7] to reduce the computational cost, still accounting for both types of fine-scale effects is to solve an upscaled version of the saturation equation on a coarser grid. The upscaled saturation equation has a macrodispersion term, accounting for fine-scale heterogeneities and the concentration equation is still solved on a fine grid (note however, that it is only solved in the subdomain U, which is assumed much smaller than T ) to account for two-phase-two-component effects. The overall idea is thus to 1. solve the pressure equation on a fine grid in T, 2. solve the concentration equation on a fine grid in U, 3. solve the upscaled saturation on a coarse grid in T. It is recognized by Niessner and Helmig [7] that also the pressure equation could be upscaled, however, this is left for future work. Before coming to details of the above solution steps, the upscaled saturation equation is derived in the next section. 3.2 Upscaling of the Saturation Equation The upscaling of the saturation equation (7) presented in this section follows the derivation by Efendiev, Durlofsky and Lee in [5], and considers only the case of linear relative permeabilities and µ w = µ n = 1. Hence we have f w (S w ) = S w, and the saturation equation is reduced to S w + v S w = 0. (13) t The upscaled equation is semi-discrete in the sense that its derivation depends on a specific (coarse) grid. Now, let D be a coarse-grid block. We may express the unknowns 8

11 3.2 Upscaling of the Saturation Equation as: with v = v + v (14) S w = S w + S w (15) = 1 (x, t)dx, (16) D D i.e. we split the velocity and wetting phase saturation into two components, an averaged coarse grid component and a fluctuation component, respectively. Our aim is to approximate (13) expressed in terms of averaged quantities, still keeping a macrodispersion term including fluctuating velocity, which lets the effect of fine-scale heterogeneities on the pressure field influence the coarse-scale wetting phase saturation. More precisely, we will at the end of the derivation arrive at the following upscaled equation S w t + 1 v j n j Sw dl = 1 D D [ t v i(x)v j(x(τ))dτ 0 ] n i j S w (x, t)dl, (17) where summation over indices i and j is implicit and where the right hand side is the mentioned macrodispersion term 2 Inserting (14) and (15) in the saturation equation (13) and averaging over D gives the averaged saturation equation: S w t + v S w + v ( S w ) = 0, (18) where S w is the volume average (16) of S w over D and ( S w ) = S w S w. Since v is constant over D it follows that v = 0. We may use this (and the divergence theorem) to rewrite the second term of (18) as v S w = 1 v S w dx = 1 ( vs w )dx D D D D = 1 ( v n)s w dl = 1 ( v n)( D D S w + S w)dl = 1 ( v n) D S w dl, (19) 2 This is not the final right hand side from [5]. The last technical steps of the derivation are needed for the discussion in [5] on how to in practice compute the right hand side but are omitted here, since the steps taken in the following derivation are considered sufficient to capture the idea of the upscaling method. The derivation presented here further deviates from the derivation in [5] in that the last term of the upscaled equation here includes a factor js w(x, t) whereas the corresponding factor in the original derivation has the form j Sw(x, t). This difference will be present throughout the derivation. 9

12 3 Modeling with Multiple Scales where in the last step it has been assumed that we may interpret averaged quantities as averages over coarse block edges, rather than over entire coarse blocks, and analogously for fluctuating quantities. Similarly, using the incompressibility to see that 0 = v = v + v = v on D, we may rewrite the third term of (18) as v ( S w ) = 1 D = 1 D = 1 D = 1 D D D v ( S w ) dx = 1 v ( S w S w )dx D D v S w dx = 1 (S w v )dx D D S w (v n)dl = 1 ( D S w + S w)(v n)dl S w(v n)dl, (20) where in the last step again averaging over block edges is assumed. Substituting (19) and (20) into (18) we arrive at S w + 1 ( v n) t D S w dl + 1 S D w(v n)dl = 0, (21) whose first two terms are recognized as the left hand side of (17), the upscaled equation we want to derive. Our next goal is to approximate the term 1 D S w(v n)dl. We start by subtracting (18) from the saturation equation (13) we obtain the fluctuating equation: S w t + v ( S w ) + v S w + v ( S w ) = v ( S w ) (22) Projecting the fluctuating equation onto coarse-grid streamlines dx/dt = v, assuming ds w(x(t), t) dt = S w t + v ( S w ) we get ds w(x(t), t) + v dt j j S w + v j( j S w ) = v j ( js w ), (23) where again summation over index j is implied. Integrating over (0, t), with x = x(t), yields 3 S w(x, t) = + t 0 t 0 [ v j(x(τ)) j S w (x(τ), τ) + v j(x(τ))( j S w (x(τ), τ)) ] dτ v j (x(τ))( js w (x(τ), τ)) dτ. (24) 3 The assumption S (0,x(0))=0 is made but not explicitly stated in [5]. 10

13 Multiplying (24) by v i (x) and integrating over the boundaries of a coarse-grid block D further yields S w(x, t)v i(x)n i dl = t v i(x)v j(x(τ))n i j S w (x(τ), τ)dτdl 0 t v i(x)v j(x(τ))n i ( j S w (x(τ), τ)) dτdl 0 t + v i(x)v j (x(τ))n i( j S w (x(τ), τ)) dτdl. 0 Here the last term equals zero, since v = 0 on each edge (interpreting averages and fluctuations as being defined over edges rather than entire blocks). In general we assume fluctuations to be small, and hence the second term of the right hand side, being a product of three fluctuating quantities, may be neglected. We are thus left with the approximation t S w(x, t)v i(x)n i dl v i(x)v j(x(τ))n i j S w (x(τ), τ)dτdl. 0 The final approximation is reached by noting that S w depends only weakly on t along streamlines, so that we have [ t ] S w(x, t)v i(x)n i dl v i(x)v j(x(τ))dτ n i j S w (x, t)dl which is substituted into (21) to finally render the upscaled saturation equation: S w + 1 v j n j Sw dl = 1 [ t ] v t D D i(x)v j(x(τ))dτ n i j Sw (x, t)dl 4 Solution Strategy 0 In this section, the solution strategy suggested by Niessner and Helmig in [7] is outlined. In the first subsection the choices of some tools, needed in the algorithm, are presented. With these tools at hand, we are then ready to formulate the full algorithm, step by step. 4.1 Solution Tools Time and Space Discretization The three equations to be solved during the steps of the full algorithm are the pressure equation, the upscaled saturation equation and the concentration equation. For all 0 11

14 4 Solution Strategy three equations a space discretization is needed, and for the two latter additionally a time discretization is needed. The choice of discretization scheme will of course influence the overall algorithm. Niessner and Helmig [7] choose a Discontinuous Galerkin scheme as space discretization, pointing out that it introduces not too much numerical diffusion and that it can be used for differential equations of different types, from elliptic to hyperbolic, and hence can be used for all three equations. For time discretization, a Runge-Kutta scheme is chosen, as it can easily be combined with DG schemes. Both the saturation equation and the concentration equation are solved time-explicitly, implying that the CFL condition has to be satisfied. This in turn means that the time steps taken when solving the concentration equation (on the finescale grid) should be smaller than those for the upscaled saturation equation, which is solved on the coarse-scale grid. These time steps will be referred to as micro and macro time steps, respectively Compression and Reconstruction Operators For the fine-scale pressure equation, the value of S w, coming from solving the upscaled saturation equation on the coarse-scale grid, is needed as a fine-scale parameter. Likewise, Sw is needed on the fine grid to compute boundary conditions for the fine-scale concentration equation. We thus need a reconstruction operator R S, mapping the coarsescale saturation S w onto a fine-scale saturation S w. It is mentioned by Niessner and Helmig that due to the hyperbolic character of the saturation equation, an upstreamweighted value of the saturation should be used in the latter case, i.e. when computing boundary conditions for the concentration equation. However, in the version presented in [7] the simplest possible reconstruction operator is used in both cases: for each point in space and time, the fine-scale value is taken to be equal to the coarse-scale value at that point, i.e. at time t we have S w (x, t) = R S Sw (x, t) := S w (x, t), x T. Conversely, for the upscaled saturation equation we need coarse-scale values of the fine-scale velocity v computed from the solution of the pressure equation and of the finescale saturation S w coming from solving the concentration equation. The coarse-scale values are computed from the fine-scale values using one compression operator Q v for velocity and another one, Q S, for saturation. The velocity compression Q v is a volume averaging over coarse element edges E: v(x, t) = Q v v(x, t) = 1 h E E v(x, t)dl, x E where h E is the length of E. In case needed, the velocity field in the element interior is computed by linear interpolation in x- and y-direction between the edges. 12

15 4.2 The Full Algorithm The saturation compression operator Q S is a volume averaging over entire coarse elements D in U: S w (x, t) = Q S S w (x, t) = 1 A S w (ξ, t)dξ, x D, where A is the area of D. 4.2 The Full Algorithm We now nearly have all the components needed to present the full upscaled-multi-scale algorithm. The problem is closed by prescribing initial conditions for S w and boundary conditions for S w and p. Note that due to the elliptic character of the pressure equation, no initial conditions for p are needed. The full algorithm suggested by Niessner and Helmig in [7] is presented below. Figure 2 schematically illustrates the algorithm, and will be repeatedly referred to in the following. At the beginning of a macro time step, say at time t, the (coarse-scale) wetting phase saturation S w (x, t) is known from the last time step (and is denoted by S S w(x, t)) for all x in T, and the aim is to compute S w (x, t + t), where t denotes one macro time step. D 1. Solve the pressure equation [λ(r S SS w )K p] = 0 on the fine grid in all of T, using the (reconstructed) coarse-scale wetting phase saturation to approximate the total mobility λ (see the green grid and step c in Figure 2). Known after this step: in T : S S w(t), p(t). 2. Compute fine-scale velocity field in T as v discont = λ(r S SS w )K p, and use a BDM 0 1 projection (details in [6, 7, 2, 3, 1]) to make the velocity field continuous across element edges. Known after this step: in T : S S w(t), p(t), v(t). 3. Compute averaged (coarse-scale) and fluctuating (fine-scale) velocity in T at time t: v = Q v v v = v v, where Q v is the velocity compression operator defined in section Known after this step: in T : S S w(t), p(t), v(t), v(t), v (t). 13

16 4 Solution Strategy Figure 2: Upscaling-multi-scale algorithm overview. 14

17 4.2 The Full Algorithm 4. Compute C 1 α(x, t) (α = w, n) for x U, using flash calculations and equation (8). Known after this step: in T : S S w(t), p(t), v(t), v(t), v (t); in U : C 1 α(t). 5. Solve one concentration equation C 1 t + v α f α (S w )C 1 α = 0 on the fine grid in U (see the blue grid and step a in Figure 2). This is done in a sub-time loop, taking n micro time steps of size δt = t/n, where n is the ratio of the length of a coarse-grid edge to a fine-grid edge. After each micro time step S w is updated using (11), and the value of S w at the end of the sub-time loop is denoted by S C w to indicate that it comes from solving the concentration equation. At the boundary of U, Dirichlet boundary conditions are given by C 1 D = C 1 w(r S SS w ) + C 1 n(1 (R S SS w )), where R S is the reconstruction operator defined in section Known after this step: in T : S S w(t), p(t), v(t), v(t), v (t); in U : C 1 α(t), C 1 (t + t), S C w (t + t). 6. Compute averaged (coarse-scale) and fluctuating (fine-scale) saturation in U at time t: S C w = Q S S C w (25) S w C = S C w S C w, (26) where Q S is the saturation compression operator defined in section Known after this step: in T : S S w(t), p(t), v(t), v(t), v (t); in U : C 1 α(t), C 1 (t + t), S C w (t + t), SC w (t + t), S w C (t + t). 7. Compute sink/source term 4 q w (x) = { SC w (x,t+ t) S S w(x,t) t, x U 0, x T \U to be added to the upscaled saturation equation, in order to make it mass conservative. Known after this step: in T : S S w(t), p(t), v(t), v(t), v (t), q w ; in U : C 1 α(t), C 1 (t + t), S C w (t + t), SC w (t + t), S w C (t + t). 4 The sink/source term presented here has reversed sign compared to the term defined in [7]. 15

18 5 Results and Conclusion 8. Solve the upscaled saturation equation 5 S w t + 1 v j n j Sw dl = 1 [ t ] v D D i(x)v j(x(τ))dτ n i j Sw (x, t)dl + q w 0 on the coarse grid in T (see the purple grid and steps a and b in Figure 2). This final step gives us the (coarse-scale) wetting phase saturation at time t + t, which is used as input for the next macro time step. Known after this step: in T : S S w(t), p(t), v(t), v(t), v (t), q w, SS w (t + t); in U : C 1 α(t), C 1 (t + t), S C w (t + t), SC w (t + t), S w C (t + t). 5 Results and Conclusion A complex algorithm for the modeling of two-phase-two-component processes in heterogeneous porous media, using an upscaling of the saturation equation and a multi-scale method to account for local mass transfer processes in order to decrease the computational time while retaining a solution with high accuracy, has been presented throughout the previous sections. In a comparison between the upscaling-multi-scale algorithm (in the following referred to as UMSA) and a reference scheme in which the fine-scale pressure and concentration equations are both solved in all of T, we would thus hope for the solution of the UMSA to be as close as possible to the reference solution, but for the CPU time to be shorter. Due to the complexity of the algorithm, however, comparisons of several kinds are needed to assess the performance of the UMSA. One main step of the development of the UMSA was the upscaling of the saturation equation. Hence comparing solutions of the upscaled to the fine-scale saturation equation would show whether the upscaling was useful or not, i.e whether accuracy was retained while CPU time was decreased, or not. This comparison should further be done for different levels of coarsening, where we expect a high level of coarsening (many fine elements per coarse element) to give shorter computing times but worse accuracy. Additionally, these results may be compared to a coarse-scale saturation equation (again with different levels of coarsening) with the macrodispersion term set to zero, in order to assess the need for that term. All of the above comparisons can be made for the case of linear and non-linear relative permeabilities, respectively. Further, different input data may be considered, such as different initial and boundary data as well as different absolute permeability fields. Especially, different heterogeneous setups may be compared to a homogeneous solid matrix, i.e. an absolute permeability with no variations in space. This way the contribution to dispersion by heterogeneity may be compared to the contribution by mass transfer 5 Here the upscaled equation derived in section 3.2 assuming linear relative permeabilities is given. See [7] for the general upscaled equation. 16

19 Figure 3: Test case domain, initial conditions and boundary conditions. The absolute permeability field is indicated by colors. processes, since in the homogeneous case all dispersion in the direction perpendicular to the flow is due to mass transfer processes. Covering all these test cases is beyond the scope of [7] as well as of this report. In [6] the above mentioned, and several further tests are performed and evaluated, and I will confine myself to presenting only one set of results from [7] and then summarizing the most important findings from [6]. The set-up of the test case which will be studied here is displayed in Figure 5. For the top and bottom boundary of domain T Neumann no-flow boundary conditions are taken. For left and right boundaries Dirichlet boundary conditions are used, letting a difference in the pressure boundary conditions lead to flow from left to right. In a small subdomain V U the initial value of wetting phase saturation is taken as S w = 0.1, corresponding to a gas plume. In all other parts of T the initial value is S w = 0.9. For the total concentration C 1 Dirichlet boundary conditions are assumed at the boundary of U. These boundary conditions are updated in each macro time step, as stated in section 4.2, so that the gas plume can leave the subdomain U. The distribution of the absolute permeability (which is taken to be scalar) is geostatistically generated with a mean value of K = m 2. Densities of the fluid phases are taken to be ρ w = 1000 kg/m 3 and ρ n = 0.9 kg/m 3. Using this set-up, the results of the UMSA for the case of linear permeabilities and µ w = µ n = 1.0 kg/ms for different time steps is displayed in Figure 5. The (space) discretization length used is x = m for the fine grid and x = m for the coarse grid (i.e. two fine grid edges per coarse grid edge). Corresponding time steps are chosen in order to fulfill the CFL conditions. Details on the DG and Runge-Kutta schemes used can be taken from [7]. We see that the main features of the solution is that the gas plume is advected in the 17

20 5 Results and Conclusion Figure 4: Results for the linear case in a heterogeneous domain at time steps 0, 2, 5, 8, 10, 15, 20, 30 and 40. Figure 5: Results for the linear case in a homogeneous domain at time steps 0, 2, 5, 8, 10, 15, 20, 30 and

21 References flow direction due to the difference in pressure boundary conditions and dispersed due to mass transfer processes and heterogeneities. As mentioned above, since the boundary conditions for the total concentration at the boundary of U are updated according to the solution of the upscaled saturation equation, the gas plume can actually leave U. In [7], no CPU time measurements are given for the presented results. Further, since the coarsening is only a factor of two, it seems that the aim of the results presented is not to show the possible computational time gain of the UMSA, but rather to illustrate the capacity of the algorithm to produce reasonable results. To demonstrate the influence of the heterogeneity on the solution, the same test case was solved in a homogeneous domain, i.e. with constant absolute permeability, taking it to be equal to the average absolute permeability of the above case (K = m 2 ). The results are displayed in Figure 5, where we see that dispersion only takes place in the longitudinal direction. All differences between the solutions in Figure 5 and 5 is due to heterogeneities, and we can conclude that heterogeneity has a massive impact on the advection and dispersion of the gas plume, and must not be neglected. In [6], the results of a wide range of tests of the UMSA are presented and analyzed. It is concluded that with the levels of coarsening being tested (between 2 and 8 times coarsening) the UMSA achieves good accuracy as compared to reference solutions. It is also noticed that the results are still satisfactory when the gas plume leaves U, indicating that the unstationary Dirichlet conditions for total concentration at the boundary of U is a good choice. Finally CPU times are measured and compared to reference computations, in which two-phase-two-component processes are modeled globally. The results turn out to be highly dependent on the geostatistical properties of the absolute permeability field. Further, as expected, the results are highly dependent on the complexity of the model of two-phase-two-component processes; with a complex model, the CPU time of the UMSA is shorter compared to the reference time than when a simple model is used. In the latter case the CPU time of the UMSA is sometimes even longer than the reference time, whereas in the former case and with a high level of coarsening the UMSA is shown to be more efficient than globally modeling the two-phase-two-component processes. As a final remark, it is once more stressed that the algorithm presented in this report is by Niessner and Helmig ([7]) seen as a first step towards a general framework to simulate complex processes using an upscaling-multi-scale approach, and that much work still needs to be done in developing this framework further. References [1] P. Bastian and B. Rivière. Superconvergence and h(div)-projection for discontinuous galerkin methods. International Journal for Numerical Methods in Fluids, 42(10): ,

22 References [2] F. Brezzi, J. Douglas, and Marini L.D. Recent results on mixed finite element methods for second order elliptic problems. Optimization Software Publications, [3] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer, [4] A.N. Brooks and A.T. Corey. Hydraulic Properties of Porous Media. Colorado State University, [5] Y. Efendiev, L.J. Durlofsky, and S.H. Lee. Modeling of subgrid effects in coarse-scale simulations of transport in heterogeneous porous media. Water Resources Research, 36(8): , [6] J. Niessner. Multi-Scale Modeling of Multi-PhaseMulti-Component Processes in Heterogeneous Porous Media. PhD thesis, Universität Stuttgart, [7] J. Niessner and R. Helmig. Multi-scale modelling of two-phase-two-component processes in heterogeneous porous media. Numerical Linear Algebra with Applications, 13: , [8] L.C. Young and R.E. Stephenson. A general compositional approach for reservoir simulation. SPE Journal,