In this paper, we prove the existence of a solution of the strongly nonlinear degenerate p(x)-elliptic equation of type:where Ω is a bounded open subset of IR N , N ≥ 2, a is a Carathéodory function from Ω × IR × IR N into IR N , who satisfies assumptions of growth, ellipticity and strict monotonicity. The nonlinear term g: Ω × IR × IR N −→ IR checks assumptions of growth, sign condition and coercivity condition, while the right hand side f belongs to L 1 (Ω) .

In this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation- div\,\,a\left( {x,u,\nabla u} \right) = b\left( x \right){\left| u \right|^{p - 2}}u + \lambda H\left( {x,u,\nabla u} \right),where Ω is a bounded smooth domain of 𝕉N.

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A typical situation of oil reservoir simulation is considered in a porous medium where the resident oil is displaced by water injection. An explicit expression of the speed of the oil-water interface is given in a pseudo-2D case via the resolution of an auxiliary Riemann problem. The explicit 2D solution is then corroborated with numerical simulations by solving the transport equation with a generalized scheme of Harten type.