The Use of Wavelet Transforms in the Solution of Two-Phase Flow Problems
George Julius Moridis, Michael Nikolaou, Y. You · SPE Journal · 1996
Abstract In this paper we present the use of wavelets to solve the nonlinear Partial Differential Equation (PDE) of two-phase flow in one dimension. The wavelet transforms allow a drastically different approach in the discretization of space. In contrast to the traditional trigonometric basis functions, wavelets approximate a function not by cancellation but by placement of wavelets at appropriate locations. When an abrupt change, such as a shock wave or a spike, occurs in a function, only local coefficients in a wavelet approximation will be affected. The unique feature of wavelets is their Multi-Resolution Analysis (MRA) property, which allows seamless investigation at any spatial resolution. The use of wavelets is tested in the solution of the one-dimensional Buckley-Leverett problem against analytical solutions and solutions obtained from standard numerical models, Two classes of wavelet bases (Daubechies and Chui-Wang) and two methods (Galerkin and collocation) are investigated. We determine that the Chui-Wang wavelets and a collocation method provide the optimum wavelet solution for this type of problem. Increasing the resolution level improves the accuracy of the solution, but the order of the basis function seems to be less important. Our results indicate that wavelet transforms are an effective and accurate method which does not suffer from oscillations or numerical smearing in the presence of steep fronts.