On the nonlinear dynamics of a saline boundary layer formed by throughflow near the surface of a porous medium

We consider gravitational instability of saline boundary layers, observed at the subsurface of salt lakes. This boundary layer is the result of the convective transport induced by the evaporation at the horizontal surface of a confined porous medium. When this upward transport is balanced by salt dispersion, a steady state boundary layer is formed. However, this boundary layer can be unstable when perturbed. This results in complex groundwater motion and density fields. The aim of this paper is to investigate the existence of finite amplitude solutions describing these resulting patterns (both the number of solutions and their structure), their stability, and their dependency on the system Rayleigh and Péclet numbers. For this purpose we construct a low-dimensional dynamical system (a reduced model) by projecting the nonlinear model equations onto a relatively small set of eigenfunctions of the problem linearized at criticality. The Galerkin projection approach is complicated by the fact that the problem under consideration is non-self-adjoint due to the existing evaporation. This implies that the eigenfunctions do not form an orthogonal set and therefore the adjoint eigenfunctions are used for the projection. The reduced model is constructed in such a way that it is capable of providing solutions in the strongly nonlinear regime as well. Convergence of these solutions towards the fully nonlinear model results is shown by means of direct numerical simulations. Further, the reduced model seems to partly capture the complex nonlinear behavior as seen in Hele–Shaw experiments by Wooding et al. [R.A. Wooding, S.W. Tyler, I. White, P.A. Anderson, Convection in groundwater below an evaporating salt lake: 2. evolution of fingers or plumes, Water Resour. Res. 33 (6) (1997) 1219–1228]. The physical transition mechanism that explains the occurrence of some observed bifurcation types is presented as well

A numerical method to solve the Buckley-Leverett equation and some applications from petroleum engineering

Electrical Engineering, Mathematics and Computer Scienc

Stability criteria for the vertical boundary layer formed by throughflow near the surface of a porous medium

We consider gravitational instability of a saline boundary layer formed by evaporation induced upward throughflow at a horizontal surface of a porous medium. Two paths are followed to analyse stability: the energy method and the method of linearised stability. The energy method requires constraints on saturation and velocity perturbations. The usual constraint is based on the integrated Darcy equation. We give a fairly complete analytical treatment of this case and show that the corresponding stability bound equals the square of the first root of the Bessel function J0. This explains previous numerical investigations by Homsy & Sherwood [1975, 1976]. We also present an alternative energy method using the pointwise Darcy equation as constraint, and we consider the time dependent case of a growing boundary layer. This alternative energy method yields a substantially higher stability bound which is in excellent agreement with the experimental work of Wooding et al. [1997a, b]. The method of linearised stability is discussed for completeness because it exhibits a different stability bound. The theoretical bounds are verified by two-dimensional numerical computations. We also discuss some cases of growing instabilities. The presented results have applications to the theory of stability of salt lakes and the salinization of groundwater

Stability criteria for the vertical boundary layer formed by throughflow near the surface of a porous medium

We consider gravitational instability of a saline boundary layer formed by evaporation induced upward throughflow at a horizontal surface of a porous medium. Two paths are followed to analyse stability: the energy method and the method of linearised stability. The energy method requires constraints on saturation and velocity perturbations. The usual constraint is based on the integrated Darcy equation. We give a fairly complete analytical treatment of this case and show that the corresponding stability bound equals the square of the first root of the Bessel function J0. This explains previous numerical investigations by Homsy & Sherwood [1975, 1976]. We also present an alternative energy method using the pointwise Darcy equation as constraint, and we consider the time dependent case of a growing boundary layer. This alternative energy method yields a substantially higher stability bound which is in excellent agreement with the experimental work of Wooding et al. [1997a, b]. The method of linearised stability is discussed for completeness because it exhibits a different stability bound. The theoretical bounds are verified by two-dimensional numerical computations. We also discuss some cases of growing instabilities. The presented results have applications to the theory of stability of salt lakes and the salinization of groundwater