2025/04/30 by Venkat Maduri, Maduri, V. S., K. B. Nakshatrala +1
Chemical Engineering · Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Heat and Mass Transfer in Porous Media #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Rheology and Fluid Dynamics Studies
paper · pdf · doi:10.48550/arxiv.2504.21603
openalex publication_date 2025/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Most organic liquids exhibit a pressure-dependent viscosity, making it crucial to consider this behavior in applications where pressures significantly exceed ambient conditions (e.g., geological carbon sequestration). Mathematical models describing flow through porous media while accounting for viscosity-pressure dependence are nonlinear (e.g., the Barus model). This nonlinearity complicates mathematical analysis and makes numerical solutions more time-intensive and prone to convergence issues. In this paper, we demonstrate that the Hopf-Cole transformation, originally developed for Burgers' equation, can recast the governing equations -- describing flow through porous media with pressure-dependent viscosity -- into a linear form. The transformed equations, resembling Darcy's equations in the transformed variables, enable (a) systematic mathematical analysis to establish uniqueness and maximum principles, (b) the derivation of a mechanics-based principle, and (c) the development of efficient numerical solutions using solvers optimized for Darcy equations. Notably, many properties of the linear Darcy equations naturally extend to nonlinear models that depend on pressure. For example, solutions to these nonlinear models adhere to a reciprocal relation analogous to that observed in Darcy's equations.