Accurate early-time and late-time modeling of countercurrent spontaneous imbibition

Spontaneous countercurrent 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 1-D, for incompressible fluids and homogeneous rocks, the onset of imbibition can be captured by self-similar solutions and the imbibed volume scales with t. 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 semianalytically. The onset is captured by a semianalytical solution. We also provide an a priori estimate of the time until which the imbibed volume scales with root t. 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 behavior and note that it can be readily included into dual-porosity models.