Numerical fractional flow modelling of inhomogeneous

Transcription

1 Numerical fractional flow modelling of inhomogeneous air sparging E.F. Kaasschieter\ G.J. Mulder^ & J.D. van der Werff ten Bosch^ Department of Mathematics and Computing Science, hoven University of Technology, P.O. Box 513, 5600 MB hoven, The Netherlands, lands, Resource Analysis, Zuiderstraat 110, 2611 SJ Netherlands, jasper, v. nl Abstract Spillage of organic compounds into the subsurface environment can result in costly remediation. As a possibly effective remediation technique, the injection of air into the groundwater (air sparging) has gained attention. The injected air migrates towards the unsaturated zone, volatising contaminants from the groundwater and delivering oxygen to biota for biodegradation (see The two-phase flow of air and water turns out to be predominantly driven by convection. It is modelled using a fractional flow approach, which yields a hyperbolic governing equation for the air saturation. The Godunov method is evaluated for the numerical solution of the one-dimensional problem in an inhomogeneous porous medium. 1 Governing equations Air sparging is a three-dimensional two-phase flow process of air and water, starting in the saturated zone of a porous medium and passing to the unsaturated zone.

2 52 Computer Methods in Water Resources XII It is assumed that the flow of both air and water can be adequately described by Darcy's law. The porous medium is assumed to be isotropic. With respect to the flow, both phases are assumed to have constant and uniform viscosity, and to be incompressible. If the phases are immiscible, then the following equations are representative: OQ <^(x)-^- + V.vi = 0, l = a,w, (1) v; = -- -V(p, + m#4, f = o, w. (2) A Equation (1) expresses conservation of volume. Herein, </> [ ] denotes the porosity while Si [ ] and Vf [m s~*] respectively denote effective saturation and specific discharge. The specific discharge is given by Darcy's law, i.e. (2), in which p\ [Pa], p\ [kg m~^], /// [Pa s], k\ [ } are the fluid pressure, fluid density, viscosity and relative permeability. Clearly, the index / can either be a (air phase) or w (water phase). Also in (2), we have y [m s~^] denoting the acceleration due to gravity and K [m^] the absolute permeability. The two phases are linked by the identity & +,^ = l. (3) The following empirical relations are currently widely used to describe how capillary pressure PC [Pa] and relative permeability in a two-phase system depend on saturation (see [2]): a(x) (4) where in [-] and a [m *] are empirical soil parameters satisfying 0 < m < I and a > 0. Not included in (4) is the phenomenon of hysteresis. The system (l)-(4) is referred to as the mixed form of Richards equation (see [1]). 2 The fractional flow model A formulation that is often used for multi-phase flow problems is the so-called fractional flow model. It is equivalent to (l)-(4) and yields

3 Computer Methods in Water Resources XII 53 two separate equations for saturation and specific discharge. One of the major advantages of this formulation is that the model becomes more accessible to analysis. Adding the equations (2) for both phases and using (3) we obtain V-v = 0, in which v = v% + v%, is the total specific discharge. To obtain the equation for air saturation, (2) is rewritten as v/ / = &,%;, (5) where e~ is the unity vector in the positive ^-direction and the mobility of phase /, A/ [m^ Pa~* s~*], is defined as, Kki. A/ = -, / = a, w. Vl We subtract (5), / = w, from (5), / = a, to obtain v% = A[v + A^((/)^ - /)%)#ez - Vpc)], where the air fractional flow function fa [ ] is defined by A, fa = a + The systems (1) and (2) can now be rewritten in terms of air saturation and total specific discharge: <Xx) - (% + V [F(x, 5% v) - v A(x, / 2)Vpc(x, ^ \ yj ^)]? = 0, ^g^ V-v = 0, _, where The vector-valued function F [m s *] and the scalar function A [m^ Pa~^ s~^] are respectively referred to as the flux and mobility function. The flux function represents convective transport of the air phase and includes gravitational effects. Its non-linear behaviour is illustrated in Figure 1 in which a graph of F% is depicted

4 54 Computer Methods in Water Resources XII 4.5,x Figure 1: Typical flux function S for K = ^ nf and v, = mm s"*. The values used for the other relevant parameters can be found listed at the end of this article. Figure 1 shows that F? is not necessarily monotone in S. This is due to the presence of gravity in the system. FZ is an increasing function if, and only if, (see [5]). From numerical experiments it appears that our system is dominated by convection. Note that the saturation equation changes from a parabolic into a hyperbolic partial differential equation with the disappearance of the capillary pressure. This observation will enable us to identify appropriate numerical techniques. 3 One-dimensional modelling In this article the simple one-dimensional instationary case is studied in order to gain insight in the various physical and numerical problems

5 Computer Methods in Water Resources XII 55 encountered in modelling the air sparging process. We will consider vertical upward flow resulting from a constant vertical injection of air at a certain depth. Thus, consider the one-dimensional fractional flow model for air saturation resulting from (6): dv + F(Z, S) - X(z, S)-pc(z, S) = 0, (7) where Ffz,^) = A(^)t; + A(z,^)(^-/)J^. From (8) it is clear that v is uniform. Suppose that (7) holds for -A& < z < ht. The lower boundary z = hb is located at the position of the vertical air injection, i.e. the boundary condition at z = hf> is r\ f (z, f) - A(z, ^)g;pc(^ 61 = ^. (9) From this it follows that v = v^. We locate the upper boundary sufficiently high above the natural water table at z - 0. We assume the natural no-flow equilibrium as initial condition. Thus, initially the water table is located at z - 0, i.e. S(z) = 0 if z < 0. The noflow saturation distribution above the water table now follows as the solution of the one-dimensional analogon of (2). Thus, p^ = -p^gz and pa = -pagz for z > 0, and from thefirstequation of (4) it follows that 4 The Godunov scheme Since (7) is assumed to be dominated by convection, we will consider the hyperbolic conservation law ^W + A^) = o do) with the boundary condition F(z, S] - ^ at z = -h\>. To discretise (10) the domain [-/^,/^] is divided into M cells of equal width Az. This gives M nodes z, = -^ + (; - l/2)az, ; = 1,...,M, where Az - (hb + ht)/m. The nodes Zj are the centres of the grid cells. Using this block-centred approach, the air saturation S is approximated

6 56 Computer Methods in Water Resources XII at the cell centres, giving unique values per cell. After collecting those values, a piecewise constant approximation for the saturation profile is constructed. Time-sampling takes place at t^ =raatf,n =0,1,..., where A/ is the time step. We write S for a numerical approximation of the exact solution S(zj,tn)- The discretisation scheme considered is the Godunov scheme (see [6] and [3]). The scheme is explicit, which is a very attractive feature. However, the time step is limited by the Courant-Friedrichs-Lewy condition CFLj < 1, where max is the Courant number. The Godunov scheme is based on solving Riemann problems at the interfaces ~j+i/2 between grid cells. At each time level the numerical solution of the previous time level serves as initial data. The solutions of the Riemann problems are assembled to form a piecewise constant numerical solution at the new time level. Provided that CFLj < 1, j = 1,..., Af, the Godunov scheme is ; S7_i,S7))], (11) where <j>j = </>(zj) and Fj(S) = F(z^S). The function STZ(»; S,S+i) is denned by where W is the solution of the Riemann problem for the interface O TT7 ^ +1 Equation (11) results from averaging the juxtaposition 5^T*~* of the Riemann solutions ^( j 5^"*""^, S^^} over the corresponding grid cells at time level /nii i-e.

7 Computer Methods in Water Resources XII 57 / = 240 = Figure 2: Solution for the Godunov scheme It is possible to compute S'^ without solving Riemann problems explicitly. The construction of S +* has to be chosen differently in order to incorporate the inflow boundary condition (9):?n+l _ A/ 5 A numerical experiment Let A%z) = 5.3'10-^m^ifz> Oorz < - if 0 < z < 1, and the other parameters be chosen as listed at the end of this article. The CFL-condition results into the requirement Af < Az. We choose Az = m and Af = 7.5 s. The results are displayed in Figure 2.

8 58 Computer Methods in Water Resources XII Parameter default values symbol g ht m a Pa meaning acceleration due to gravity depth of computational domain height of computational domain Van Genuchten soil constant Darcy injection flux in the air phase Van Genuchten soil parameter viscosity of the air phase viscosity of the water phase density of the air phase density of the water phase porosity value 9.81 m s"2 6 m 4 m 2/ m! 2m-: s Pa s Pa s 1.24 kg m-3 1 # kg m References [1] Van Dijke, M.I.J., Van der Zee, S.E.A.T.M. & Van Duijn, C.J., Multi-phase flow modelling of air sparging, Advances in Water 18, pp , [2] Van Genuchten, M.Th., A closed form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Science So- 4, pp , [3] Godlewski, E. & Raviart, P. -A., Hyperbolic Systems of Conservation Laws, Ellipses, Paris, [4] Hinchee, R.E., Miller, R.N. & Johnson, P.C., fn 5% Air Sparging, Bioventing, and Related Remediation Processes, Battelle Press, Columbus, [5] Kaasschieter, E.F., Van der Werff ten Bosch, J.D., & Mulder, G.J., A numerical fractional flow model for air sparging, Report RAN A of the Department of Mathematics and Computing Science, Eindhoven University of Technology, [6] LeVeque, R.J., Numerical Methods for Conservation Laws, Birkhauser Verlag, Basel, 1990.