Theoretical Analysis of Capillary Rise in Porous Media

In this work, we study the dynamics of capillary-driven fluid invasion in three different settings including: (1) a single capillary tube, (2) a homogeneous porous medium, and (3) a fractured porous medium. A Lambert W functional form is proposed to describe the invasion dynamics in a single capillary tube, that predicts both early-time Washburn-type behavior () and late-time behavior. We extend the formulation to describe homogenous porous media and to include viscosity, pressure, and gravity effects in both advancing and defending fluids. Solutions for closed systems, where the advancing fluid compresses the defending fluid, are then developed. Finally, we extend the theory to describe fractured systems and propose a convolution integral formulation and a new explicit solution for fluid invasion into a fractured porous medium.