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 √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.