Accurate early and late time modelling of countercurrent spontaneous imbibition

Spontaneous counter-current imbibition into a finite porous medium is an important physical mechanism for many applications, included but not limited to irrigation, CO2 storage and oil recovery. Symmetry considerations that are often valid in fractured porous media allow us to study the process in a one-dimensional domain. In 1D, the onset of imbibition can be captured by self-similar solutions and the imbibed volume scales with the square-root of time. At later times, the imbibition rate decreases and the finite size of the medium has to be taken into account. This requires numerical solutions. Here, we present a new approach to approximate the whole imbibition process semi-analytically. While the onset is captured by a semi-analytical solution. We also provide an a priori estimate of the time until which the imbibed volume scales with image. This time is significantly longer than the time it takes until the imbibition front reaches the model boundary. The remainder of the imbibition process is obtained from a self-similarity solution. We test our approach against numerical solutions that employ parametrizations relevant for oil recovery and CO2 sequestration. We show that this concept improves common first order approaches that heavily underestimate early-time behaviour and note that it can be readily included into dual porosity models. This article is protected by copyright. All rights reserved.