Exact Solutions for Gravity-Segregated Flows in Porous Media

Abstract

The review is devoted to exact analytical solutions for quasi-2D gravity segregated flows or gravity currents in subterranean porous formations. The problems under consideration are quasi-linear. The driving forces are two components of the buoyancy—one exerting the bulk of the light fluid and one due to the curvilinearity of the interface between the fluids. In the case of homogeneous formation or where the seal slope is negligible, the transport equation is parabolic and allows for a wide set of self-similar solutions. In a large-scale approximation of the buoyancy domination, the governing equation is hyperbolic; the method of characteristics allows for a detailed analytical description of gravity current propagation with final accumulation in the geological trap. Analytical models for leakage via the caprock seal are also discussed. The work was completed by formulating some unsolved problems in segregated flows in porous media.

Separation of Variables and Exact Solutions to Nonlinear PDEs

“…Exact solutions of nonlinear PDEs are most often constructed using the classical method of symmetry reductions [3][4][5][6], the direct method of symmetry reductions [1,[7][8][9], the nonclassical symmetries methods [10][11][12][13], methods of generalized separation of variables [1,9,[14][15][16], methods of functional separation of variables [1,9,17,18], the method of differential constraints [1,9,19,20], and some other exact analytical methods (see, for example, [21][22][23][24][25][26][27]). On methods for constructing exact solutions of nonlinear delay PDEs and functional PDEs, see, for example, [2,[28][29][30][31][32][33][34].…”

“…Statement 6. Let the nonlinear PDE with constant delay (27) have a functional separable solution of the form u(x, t) = U (z), where z = m k=1 ϕ k (x)ψ k (t), (36) in which the linearly independent coordinate functions ϕ k (x) and the external function U (z) are independent on τ , and the functions ψ k = ψ k (t) are described by the ODE system with constant delay (29). Then the more complex PDE with variable delay (30), which is obtained from equation ( 27) by formally replacing the constant τ with an arbitrary function τ (t), admits an exact solution of the form (36), where the coordinate functions ϕ k (x) and the function U (z) are unchanged, and the functions ψ k = ψ k (t) are described by the ODE system with variable delay (31).…”