Challenges in mathematical modeling of dynamic mass transfer controlled by capillary and viscous forces in spontaneous fluid imbibition processes

Dynamic mass transfer due to spontaneous imbibition is of significant importance in various scientific and engineering applications, including environmental remediation, chemical reactors, microfluidic systems, and oil recovery processes. This article addresses the challenges in mathematical modeling of the dynamic mass transfer due to spontaneous imbibition controlled by capillary and viscous forces. A mathematical model was developed to seamlessly integrate the effects of capillary and viscous forces on mass transfer. The model was validated by comparison with numerical solution, which shows excellent consistency, indicating no error in the derivation of the analytical model. Case analysis suggested some limitations of the analytical model. The model does not work at the starting point of imbibition because of mathematical singularity. The current computing technology does not generate model results under all conditions due to the data-overflow issue associated with the exponential function involved in the analytical model. Although using numerical solution with finite difference method can eliminate the data-overflow problem, time step size must be small enough to achieve algorithm convergence and generate meaningful result.