Temperature dependent multiphase flow and transport

Transcription

1 Temperature dependent multiphase flow and transport J.F. Sykes, A.G. Merry and J. Zhu Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Abstract Many sites that are contaminated with non-aqueous phase liquids include the distribution of these materials in the unsaturated zone with the greatest saturation often occurring within meters of the ground surface. Reductions in the NAPL saturation occur as a result of drainage, dissolution and volatilization. These processes can be highly temperature sensitive. Seasonal changes in the temperature at the ground surface can also result in the development of a frozen layer. During the winter season, this layer reduces the dissipation of soil gas to the atmosphere. In the spring, the thaw of this layer and downward drainage of melt water will contribute to the NAPL dissolution and redistribution. The prediction of NAPL distributions in soil must therefore consider the effect of temperature. A temperature component was added to a two-dimensional finite element air-water-organic model. Solute transport in the water phase is modeled using the method of characteristics while the organic phase is assumed to be immobile. Transport in the active air phase was modeled using a traditional advection-diffusion formulation while heat transport included the heat sink/source term associated with a water-ice phase change. Rate-limited dissolution of the organic phase is modeled using Gilland-Sherwood correlations. Important factors in the analysis include the temperature at which the water-ice phase change occurs for soils with various saturations of air, water and immiscible organic, and the boundary condition changes that occur at freezing and thawing.

2 20 Computer Methods in Water Resources XII 1 Introduction Groundwater contamination by organic chemicals in the form of non-aqueous phase liquids (NAPLs) is a widespread problem that poses a serious threat to groundwater resources. Temperature can have a significant impact on the distribution of the organic in the unsaturated zone. It affects fluid viscosity, aqueous solubility, vapor pressure, Henry's constant //and, most importantly, the effective mobility of the various phases at the ground surface as a result of the water-ice phase change that can occur in winter. The release of pore water as a result of the thaw of a frozen layer can account for much of the annual infiltration that occurs in some regions. The understanding of temperature variations in the unsaturated zone and their influence on the mass transfer processes in the subsurface environment is very limited. In this study, the temperature in the unsaturated zone will be simulated using a two-dimensional Galerkin finite element model that incorporates the heat sink/source term associated with water-ice phase changes. The transport equations for the two-phase flow of water and gas with an immobile NAPL are solved using a backward method of characteristics formulation. The volatilization, water/gas partitioning and dissolution processes are assumed to be rate-limited. 2 Mathematical Model Development 2.1 Fluid Flow Equations Equations of fluid flow in porous media are obtained by combining Darcy's law with the mass continuity equation. For gas flow in the unsaturated zone, the compressibility of the porous medium can be neglected and the equation governing the gas flow is as follows [3] A, For the aqueous phase, the flow equation can be written as j' where the subscript a represents aqueous phase, and g represents gas

3 Computer Methods in Water Resources XII 21 phase; p is the fluid density; & are the components of the intrinsic permeability tensor; k^ is the relative permeability; ju is the absolute viscosity; p is the fluid pressure; g is the gravitational constant; q represents sources or sinks; (j) is the porosity of the porous medium; S^ is the volumetric saturation of the aqueous phase; Sg is the saturation of the gas phase; y^ is the gas compressibility; C^ is the gas phase concentration;/? is the compressibility of water; a is the compressibility of the porous medium; and can be calculated by fg = dpg/dcg ~ 1 ~Prg/Pv where p^ and Py is the density of the pure organic vapor is the reference air density, The equations for the gas and water phases are linked by the capillary pressures with p^, = p^ - p^. The volumetric saturations are related by S^ +Sg+S^ + S^ =1.0 where S^ pore volume that is frozen. S^ is related to p^ by r ~\ is the fraction of the ^,=^Y^= i+wy ^-^ ^ where a and ware empirical parameters determined experimentally; m = 1-1 / n, /^ is the water equivalent capillary head defined by he ~ PC I PaS The relative permeabilities of the liquid water and gas, k^ and k^, are developed from the functions of S^ with *,=(1-SJ(1-. (5) The fraction of the liquid water that will freeze at a given temperature below 0 C is given by ^=[l-^-(l-^x^% where r. is the fraction of un-freezable water assumed to be 5% in this paper, and co is a fitting parameter [4]. 2.2 Transport Equations Ignoring the fluid sources or sinks due to pumping or injection, the transport equations for the water and gas phases can be represented by

4 22 Computer Methods in Water Resources XII where D? is the hydrodynamic dispersion tensor; C# is the mass concentration of the NAPL; and Q represents sources or sinks due to mass transfer from other phases. Initial conditions for (6) include the specification of initial gas and water phase concentrations. 2.3 Interphase Mass Transfer Interphase mass transfer processes are represented by first-order expressions in which the mass transfer driving force is proportional to the difference between equilibrium and pore concentrations. This approach results in the source/sink terms for the water and gas phase being given by "a = ^ (C. - CJ + (WyUQ - HC,) (7) Q,=A,(C,-C,)-^(C,-tfCJ (8) where //is the Henry's law coefficient; A^ is the dissolution rate coefficient; /^ is the volatilization rate coefficient; and/l^ is the mass transfer rate coefficient for partitioning. These rate coefficients reflect the contribution of the geometry of the porous media and the distribution of the residual NAPL. In this paper, the dissolution rate coefficient correlation of Imhoff et al. [1] is used to represent the dissolution process Sh =340Re"'0 (djzr =WlM (9) where d^ is the mean grain diameter; % is the distance into the NAPL; Sh is the Sherwood number; Re is the Reynolds number; /)* is the molecular diffusivity in the aqueous phase; and 6^ is the NAPL volumetric fraction. In this study, it has been assumed that A^ and A,^ are constant. 2.4 Temperature Equations Assuming that the system is comprised of moving water, air, immobile NAPL and soil, the equation governing energy balance is + Q,=-[(cpLT\ (10) a

5 Computer Methods in Water Resources XII 23 where c^ is specific heat of the water; T is temperature; k ^ is the average thermal conductivity; (cp)^ is the average heat capacity per unit volume of the soil-ice-napl-air system; Q^ is heat sink/source term; and t is time. Equation (10) is simplified by assuming that all the thermal properties are constant and that (/)S^c^p^D«k ^ and is therefore negligible. The sink/source term representing the energy lost/gained at the phase change front is determined with conditions that must be met as the phase change front advances and retreats. Suppose that the surface of separation between the ice and water phases is at X(t}, two boundary conditions to be satisfied at this surface are Ta = T>ce =?m when x = X(t) and where 7^ is the melting-point; ^ and k^ are the thermal conductivities of water and ice respectively; 7^ and T^ are the temperatures on the water and ice sides respectively; dt^/dx^ dt-^/dx-^ are the temperature gradients on the water and ice sides respectively; L is the latent heat of fusion of water; and dx/dt is the speed at which the phase change front is moving. In developing the above relationship, the change of volume on freezing will be neglected, so that the density Pa will be the same in both ice and water phases. Equation (11) is valid for a water/ice phase system. An effective latent heat of fusion L^ must be used that is proportional to the volumetric water content of the porous media to recognize the fact that the unsaturated zone is a NAPL/water/ice/solid/air phase system, giving L«r = 4S.L. 2.5 Computational Scheme The set of partial differential equations and constitutive equations described in the preceding paragraphs are solved for a twodimensional, vertical cross section using a Lagrangian formulation for transport and the Galerkin finite element method with linear triangular elements [3]. The rate-limited dissolution and volatilization rate coefficients are accounted for and incorporated into the model. The Backwards Method of Characteristics (BMOC) is applied to the solution and

6 24 Computer Methods in Water Resources XII of the transport equations for the aqueous phase. The latent heat term is implemented using the method described in Sykes et al. [2]. We assume that there is no mass transfer to or from the frozen aqueous phase. 3 Results And Discussion A 400 m long by 16 m deep two-dimensional cross-section with a water table coinciding with the straight line between (0 m, 11.5 m) and (400 m, 10 m) was investigated. The NAPL is assumed to occupy a rectangle between 84.8 m and 91.2 m in the x direction and between 12.0 m and 16.0 m in the y direction. The bottom boundary of the domain and left and right boundaries in the unsaturated zone are impermeable to aqueous flow. A specified water flux boundary condition (type II) is assumed on the ground surface to account for infiltration. The recharge rate at the ground surface is 292 mm/year. Specified pressure (type I) boundary conditions are assigned to the saturated zone of the left and right boundaries. The ground surface boundary for gas flow is type II (no gas flux) or type I (atmospheric pressure). For contaminant transport, all inflow boundaries for the aqueous phase, which include the ground surface and the left saturated zone boundary, are described with the type III boundary condition with no contaminant in the flux component. The free-exit boundary condition is used for the right saturated zone boundary. All other aqueous transport boundaries are type II (no flux). The ground surface gas phase transport boundary condition is type III with diffusion across a stagnant boundary. Parameters used in the simulations are listed in Table 1. The NAPL properties are representative of Trichloroethylene. Table 1. Parameters Used in Two-Dimensional Simulations Parameter c* Value 236(mg/L) Parameter N Value 2.0»\ Pvapor H Hs Ca k 4 & 0.6(m*/day) 5.64(kg/m*) xl(r*(Pa.s) 4200 (J/kg/ C) io-'v) 0.0 a <%L <XT ^xa D: </> L 1.0(l/ro) 4.0(m) 0.01O) 1000Og/L) \Q-\m^ /day) (J/kg)

7 Figure 1 (a): Water phase saturation at coldest date Computer Methods in Water Resources XII 25 Figure 1 (b): Water phase saturation at warmest date Figure 2(a): Temperature (degrees K) Figure 2(b): Temperature (degrees K) at coldest date at warmest date Figure 3(a): Water phase log cone. at day 365 (time = 0 on coldest date) Figure 3(b): Water phase log cone. at day 365 (time = 0 on warmest date) X Figure 4(a): Gas phase log cone, at day 365 (time = 0 on coldest date) 400 Figure 4(b): Gas phase log cone, at day 365 (time = 0 on warmest date) X "300" X

8 26 Computer Methods in Water Resources XII The first of two cases presented in this paper involves a simulation that begins at a surface temperature of 263 K. The second case involves a starting surface temperature of 303 K. The system parameters are identical for both cases and the results are presented in Figures 1 to 4. Evident in Figure l(a) is the increased water content in the surface layer due to the reduced permeability that occurs with freezing. Figure l(b) shows the corresponding water saturation in summer conditions. Figures 2(a) and (b) present the temperature distributions for the peak winter and summer periods respectively. In Figure 3(b) the aqueous phase plume is more advanced than that shown in 3(a) due to the higher Henry's coefficient and increased infiltration at the onset of the simulation (summer conditions). The warmer summer condition results in higher gas phase concentrations than occur in the winter conditions at one year (see Figure 4). 4 Concluding Remarks A model has been developed to simulate temperature and ratelimited mass transfer between the aqueous phase and residual NAPL trapped in a porous medium. The transport equations for the gaseous and aqueous phases were solved using a Langrangian formulation. The temperature regime included the latent heat term associated with water freeze/thaw. The presented analyses illustrate the importance of temperature in mass transfer and transport in the unsaturated zone. References [1] Imhoff, P. T., Jaffe, P. R., and Pinder, G. F., An experimental study of complete dissolution of a nonaqueous phase liquid in saturated porous media, Water Resour. Res., 30(2), , (Corrections, Water Resour. Res., 30(10), 2871, 1994.) [2] Sykes, J.F., Lennox, W.C. and Charlwood, R.G., Finite Element Permafrost Thaw Settlement Model, Journal of the Geotechnical Engineering Div., ASCE, Vol. 100, No. GT11, , Nov., 1974 [3] Thomson, N. R., Sykes, J. F., and Van Vliet, D., A numerical investigation into factors affecting gas and aqueous phase plumes in, the subsurface, J. Contam. Hydrol., 28, 39-70, [4] Williams, P.J., Unfrozen water content of frozen soils and soil moisture suction, Geotechnique, Vol. 14, , 1964.