Approximation of Invariant Solutions to the Nonlinear Filtration Equation by Modified Pade Approximants

TL;DR

The paper addresses nonlinear oil filtration in porous media with pressure-dependent viscosity, examining self-similar solutions under classical Darcy flow and traveling-wave solutions under a relaxation Darcy law. A semi-analytic method based on modified Padé approximants is developed to approximate invariant solutions on semi-infinite domains by blending Taylor expansions with asymptotic erfc behavior for self-similar cases and an exponential-factor Padé form for traveling waves. The approach yields accurate, compact quasi-rational expressions that closely match numerical solutions even at low Padé orders, and it remains adaptable to variations in the hydraulic conductivity function $D(p)$. The results provide practical tools for modeling moving liquid fronts in porous media and can inform well-operation and front-propagation analyses in oil reservoirs and related heat/mass transfer problems. This method thus offers a robust, semi-analytic framework bridging nonlinear diffusion-type equations and their invariant solutions in geophysical and engineering contexts.

Abstract

This paper deals with a mathematical model for oil filtration in a porous medium and its self-similar and traveling wave regimes. The model consists of the equation for conservation mass and dependencies for porosity, permeability, and oil density on pressure. The oil viscosity is considered to be the experimentally expired parabolic relationship on pressure. To close the model, two types of Darcy law are used: the classic one and the dynamic one describing the relaxation processes during filtration. In the former case, self-similar solutions are studied, while in the latter case, traveling wave solutions are the focus. Using the invariant solutions, the initial model is reduced to the nonlinear ordinary differential equations possessing the trajectories vanishing at infinity and representing the moving liquid fronts in porous media. To approximate these solutions, we elaborate the semi-analytic procedure based on modified Pade approximants. In fact, we calculate sequentially Pade approximants up to 3d order for a two-point boundary value problem on the semi-infinite domain. A good agreement of evaluated Pade approximants and numerical solutions is observed. The approach provides relatively simple quasi-rational expressions of solutions and can be easily adapted for other types of model's nonlinearity.

Figures (3)

• Figure 1: The approximation of viscosity by the parabola $\mu=\mu_{0}\left(1+a\left(p-p_{0}\right)^{2}\right)$ with the vertex $(p_0;\mu_0)=(4.16855\cdot 10^{6},0.005)$ and $a=1.507 \cdot 10^{-14}$ Pa$^{-2}$. The experiment of Hocott et al. Hocott is marked with filled circles, while their parabolic approximation is drawn with the dashed line.
• Figure 2: a: The $P(\xi)$ profiles for the solutions of BVPs (\ref{['sk:red_eq']}) -- (\ref{['sk:boundary']}) evaluated numerically (regarded as "exact" solution and marked by the solid curve) and corresponding $PA_{[M/M]}$. b: The differences $\delta_M (\xi)=P-PA_{[M/M]}$ vs. $\xi$ for the profiles from the left panel.
• Figure 3: (a) The profiles of the solution of equation (\ref{['sk:dyns_relax']}) and Padé approximants $PA_{[1/1]}$ and $PA_{[2/2]}$, defined by (\ref{['sk:PadeFormRelax']}). The inset shows the bundle of trajectories starting from initial conditions $P(0)=1$ and $P'(0)$ from the range $[-2.218, -2.215]$. (b) The profile of $Y(\xi)$ describing the approach of conservative quantity to its limit value $\Delta$.