Innovative Application of the Ritz Method to Oil-Gas Seepage Problems: A Novel Variational Approach for Solving Underground Flow Equations

State-of-the-art commercial simulators (e.g., Eclipse, CMG) predominantly employ finite difference schemes, which face persistent challenges in modeling strongly nonlinear seepage dynamics. This study explores the application of the Ritz method, grounded in variational theory, to solve underground oil seepage problems in reservoir engineering. The research focuses on deriving the variational form of steady-state seepage equations and presents a systematic procedure for solving these equations in finite domains. Using a one-dimensional steady-state seepage problem as a case study (which can effectively represent a wide range of typical flow regimes), the study compares the approximate solutions obtained by the Ritz method (both monomial and binomial forms) with exact solutions. The results demonstrate that the binomial approximate solution achieves high accuracy, with an average deviation of only 0.30% from the exact solution, significantly outperforming the monomial solution. The findings validate the Ritz method as an effective tool for addressing seepage problems and highlight its potential for broader applications in oil and gas reservoir modeling.