Analytical solutions for water flow and solute transport in the unsaturated zone

Transcription

1 Models for Assessing and Monitoring Groundwater Quality (Procsedines of a Boulder Symposium July 1995). IAHS Publ. no. 227, Analytical solutions for water flow and solute transport in the unsaturated zone M. H. NACHABE, A. L. ISLAS & T. H. ILLANGASEKARE University of Colorado at Boulder, Campus Box 428, Boulder, Colorado, 80309, USA Abstract Analytical solutions for the water flow and solute transport equations in the unsaturated zone are presented. We use the Broadbridge & White non-linear model to solve the Richards' equation for vertical flow under a constant infiltration rate. Then we extend the water flow solution and develop an exact parametric solution for the advectiondispersion equation. The location of a solute front is determined with the method of characteristics. The dispersion component is incorporated into the final solution using a singular perturbation method. The analytical solutions are simple and can be generated without resorting to computationally demanding numerical schemes. Indeed, the analytical solutions provide tools to verify the accuracy of numerical models. Comparison with a finite element numerical solution indicates that a good match for the predicted water content is achieved when the mesh grid is one fourth the capillary length scale of the porous medium. However, numerical dispersion and spatial oscillations were significant in solving the solute transport equation at this level of discretization. INTRODUCTION The movement of water and solutes through the unsaturated zone has been of importance in traditional applications of ground water hydrology, soil physics, and agronomy. In recent years, the need to understand the behavior of hazardous waste and toxic chemicals in soils has resulted in a renewed interest in this subject. One of the primary concerns is that dissolved contaminants may migrate through the unsaturated zone, reach the saturated zone, and contaminate the ground water. The search for analytical solutions to model flow and solute transport continues to be of scientific interest. Typically, water flow and solute transport in unsaturated soils result in transient phenomena, making it a challenging problem. The nature of soil hydraulic properties like diffusivity and hydraulic conductivity renders the governing flow equation non-linear. In this paper, the Broadbridge & White (1988) (B&W, hereafter) non-linear flow model for constant infiltration rate is adopted. The ability of this model to conform to a variety of soil types, as demonstrated by White & Broadbridge (1988) and White & Perroux (1989), is very encouraging in field applications The objective of this paper is to use B&W analytical flow model to simulate solute movement in soils so that a closed form analytical solution for solute

2 126 M. H. Nachabe et al. transport in the unsaturated zone is available. The analytical solutions are compared with a numerical model to assess the accuracy of the numerical approximations. THEORY Two fundamental equations govern water flow and solute transport in unsaturated soils. Richards' equation is obtained by combining Darcy's law for unsaturated flow with the continuity equation. For one dimensional vertical flow, this equation is dt dz Z>(0), 30" -#-Jf(6) (1) dzj dz The transport of a conservative solute (i.e. non-volatile, non-sorbing, non-degrading) in the unsaturated zone is described by the advection-dispersion equation written in the form -Be -(0C) = ^{qc)+^\ EÏf (2) d dt 3z dzl dz In both equations, t is time, z is depth taken zero at the soil surface and positive downward, Q(z,t) is volumetric soil water content, Z>(0) is soil water diffusivity, K(Q) is unsaturated soil hydraulic conductivity, c(z,t) is solute concentration (mass of solute over volume of solution), q(z,t) is Darcy's water flux q(z,t) =K(B)-D(Q) jr (3) dz and is a hydrodynamic dispersion coefficient, a parameter that lumps the impacts of molecular diffusion and mechanical mixing. Equations (1), (2) and (3) aie coupled and they should be solved sequentially. First, the flow equation is solved for 0(z,r) and the subsequent substitution of Q(z,t) into (3) provides the unsaturated Darcy's flux q(z,t). The solutions of Q(z,t) and q(z,t) become input for (2). Equation (2) is reduced to a more practical mathematical form using the continuity equation for water flow 38 dq n Substitution of (4) into (2) yields 9r q dz + dz^t'dz ) {5) Equation (5) applies for transient transport of conservative solutes in unsaturated soils. ANALYTICAL SOLUTIONS The water content Q(z,t) solution of (1) was determined by B&W (1988) for a constant infiltration rate. This solution is presented here for the sake of completeness. The dimensionless form of Richard's equation is

3 Analytical solutions for water flow and solute transport in the unsaturated zone 127 =À- 9r* dz D*(ej^oz* 3 < 6 > oz* where z* = zfk s and f* = t/t s. X s and f* are the capillary length and time scales of the porous medium (Broadbridge & White, 1988; Nachabe et al., 1994; Nachabe & Illangasekare, 1994). Similarly the initial and boundary conditions are expressed in dimensionless form as 0* = 0 r* = 0 z* > 0 (7) K* (0 ) - >*(e*);r- = <7*n z*=0,r*>0 (8) u oz* where q* 0 = (q 0 - K 0 )/AK. The solution for (6) subject to the conditions in (7) and (8) is (B&W, 1988) 6. =CI1 = I (9) 2p+ 1 - u «K z* = C _1 p 2 ( 1 + P _1 )T:+p(2 + p"' 1 )Ç-ln(w) (10) The analytical expressions in (9) and (10) are documented in B&W (1988). Equations (9) and (10) constitute an exact parametric solution with parameter Ç. PARAMETRIC SOLUTION FOR THE ADVECTION-DISPERSION EQUATION We capitalize on the analytical form for the water flow solution and pursue an extension of this result to the movement of a conservative solute in the unsaturated zone. This is achieved by coupling the advection-dispersion equation with the water content solution in (9) and (10). Wilson & Gelhar (1981) developed analytical solutions for the advection-dispersion equation, but as opposed to the result presented here, their solution for water flow in the soil was an adaptation of the Philip & Knight (1974) iterative procedure, which does not provide exact parametric solution. We assume that initially there is zero concentration of solute in the soil profile. A step of solute concentration c 0 is applied at the surface. By introducing the dimensionless variables c* = clc 0 and E* = E[t s /k s 2 ] into (5), the advection dispersion equation is expressed in the mathematical form 9 n"i 9c * 7) (( 0 n"l dc *\ K n dc e * - +^k = ^ll e - + Â5r5^J- (^+^)3i: (11) The dimensionless initial and boundary conditions are written as c* = l z = 0 f>0 (12) c*=0 r* = 0 z*>l (13)

4 128 M. H. Nachabe et al. The solution procedure to derive a parametric solution for (1) is similar to the one adopted by Wilson & Gelhar (1981). This solution procedure solves sequentially for the two mechanisms of solute transport, i.e. convection with the mean water velocity and dispersive mixing. The convection component is treated by the method of characteristics to determine the location of the solute front. The dispersive component is then superimposed on the resulting front using the method of singular perturbation. The convective component is obtained by setting the dispersive in (11) equal to zero with the result 0 \3c* K dc 9 \ n, ^Âëfe+^ + Âf^0 (14) Thus we find that the characteristic r\ is given by the expression (see Nachabe et al., 199 A) ^(à + 1 ) Cz *" c-( â +^o)^ (is) which using the parametric solution in (10) for z* can be written in terms of Ç and the time x as r) = A 1 t: + S 1 Ç-C 1 lnm(ç,x) (16) where A l5 B x and C\ are constants defined symbolically as ^1 = C I(P 2 + P ) - ( ^ 1 2 ) (17) B x = Cj(2p+1) -1 (18) c i = 1 + cfe (19) This solution for T is a parametric expression in Ç. The mathematical procedures to locate the position zj(u) of the sharp front include two steps: (1) set TI equal to zero and determine tf, and (2) set (15) equal to zero and solve for zy in terms of dimensionless time t*. PERTURBATION APPROXIMATION OF DISPERSION Zj(t*) is the depth of the solute sharp front. The solute concentration c*(z*,f*) is one beyond this front and zero behind it. This contrast at the front boundary produces a singular problem that is tackled mathematically using the method of asymptotic expansion. For E* «1, an inner expansion valid in the vicinity of the front is introduced by stretching the coordinate system and rewriting the dispersion equation in the new coordinate system. The solution is (Nachabe et al., 1994)

5 Analytical solutions for water flow and solute transport in the unsaturated zone 129 c* C z *> t*) 2 erfc 2 * 1/2 (z* 9 (20) ILLUSTRATION The solutions for water flow and solute transport in (9), (10), (16) and (20) area analytic. So, the water content profile 8*(z*,r*) and the concentration of solute in the soil c*(z*,t*) are determined without iterative steps commonly used in numerical schemes. The input requirement for analytical simulation includes: (1) soil hydraulic properties: the saturated conductivity K s, the saturated water content Q s, the parameter C, and the capillary length scale X s ; (2) the soil initial moisture content 0 and the water flux boundary condition q* 0 and (3)the dispersion coefficient E*. For the sake of illustration, we choose C=1.2, X S =3M cm, K s =4.5 cm h" 1, q* 0 =0.2, 6^=0.4, 6 =0.08, *=10-3. The values of these parameters reflected the conditions at a site in Golden, Colorado. t s is calculated to be 0.27 h. Numerical simulations with the PRINCETON finite element model (Celia, 1991; Celia et al, 1990) were conducted for the same input conditions. The time step in the PRINCETON code is adjusted internally depending on the number of nonlinear iterations (Celia, 1991). For the finite element numerical scheme, we chose an element size of one fourth the capillary length scale (i.e. V 4 ) or 0.96 cm for this particular case. Figure 1 suggests a good match between the analytical and numerical solutions for the normalized water content 0* at t* equal 2, 10 and 24 at this level of discretization. However, spatial oscillations (overshot and undershoot) and numerical dispersion were significantly large for the advection dispersion equation. Figure 2 u - t* =2 ^ ~ - JS If -2 - * -4 - /7-6 - ' t* = I0 ^- t' = 3^ A/ AS If / / / / h i e. Fig. 1 Analytical (solid line) and numerical (dashed line) solutions of 9*.

6 130 M. H. Nachabe et al. -Z» f = 10-3 t* = 2i Fig. 2 Analytical (solid line) and numerical (dashed line) solutions of c*. C. compares the analytical and numerical solutions when we reduced the mesh grid in the finite element code to one eighth the capillary length scale. While the match between the concentrations is not very satisfactory, especially at t* = 24, the trade-off between numerical dispersion and spatial oscillations are typical for advection dominated solute transport like the case considered here. Indeed, Pinder & Huyakorn (1983, p. 207) cautioned "that because advection-dominated transport problems are difficult to solve numerically, the user of computer models needs to test the adequacy of selected mesh and time step discretizations to avoid obtaining misleading or meaningless solutions. For this reason we recommend the use of available analytical solutions for calibration..." CONCLUSION We extended the Broadbridge and White infiltration model to derive a closed form analytical solution for the advection-dispersion of a conservative solute in the unsaturated zone. The presented analytical solutions are exact and parametric, and the soil water content and the solute concentration solutions are easily generated and programmed. The computation time required to achieve a complete solution is insignificant when compared with iterative numerical schemes. The numerical problems involved in solving advection dominated transport are well recognized among practitioners. The developed analytical solutions should provide a mean for evaluating the accuracy of existing numerical codes.

7 REFERENCES Analytical solutions for water flow and solute transport in the unsaturated zone 131 Broadbridge, P. & White, I. (1988) Constant rate rainfall infiltration: a versatile nonlinear model, 1. Analytical Solution. Wat. Resour. Res. 24, Celia, M. A. (1991) The one-dimensional Princeton unsaturated code. Distributed by Water Resources Program, Princeton, University, Princeton, New Jersey. Celia, M. A., Bouloutas, E. T. & Zabra, R. L. (1990) A general mass-conservative numerical solution for the unsaturated flow equation. Wat. Resour. Res. 26, Huyakom, P. S. & Pinder, G. F. (1983) Computational Methods in Subsurface Flow. Academic Press, Inc. Nachabe, M. H. & Illangasekare, T.H. (1994) Use of tension infiltrometer data with unsaturated hydraulic conductivity models. Ground Water 32, Nachabe, M. H., Islas, A. L. & Illangasekare, T. H. (1995) Analytical solutions for water flow and solute transport in the unsaturated zone. Ground Water 33 (in press). Philip, J. R. & Knight, J. G. (1974) On solving the unsaturated flow equation, 3. New quasi-analytical technique. Soil Sci. 117,1-13. Wilson, J. L. & Gelhar, L. W. (1981) Analysis of longitudinal dispersion in unsaturated flow, 1. The analytical method. Wat. Resour. Res. 17, White, I. & Broadbridge, P. (1988) Constant rate rainfall infiltration: a versatile nonlinear model, 2. Applications of solutions. Wat. Resour. Res. 24, White, I.. & Perroux, K. M. (1989) Estimation of unsaturated hydraulic conductivity from field sorptivity measurements. Soil Sci. Soc. Am. J. 53.