Time-dependent shape factor for scaling of multi-dimensional counter-current imbibition with different boundary conditions

The scaling laws are critically important in researching the characteristics of spontaneous imbibition and predicting oil and gas recovery from fractured reservoirs. This paper focuses on developing a shape factor for scaling the multi-dimensional counter-current imbibition processes. We studied the multi-dimensional imbibition characteristics by establishing numerical simulation models considering nine boundary conditions. On the basis of the capillary diffusion equation of counter-current imbibition, the equations of normalized oil recovery versus imbibition time were obtained under different boundary conditions. Then, we derived a new shape factor by processing and comparing these equations. The results show that the relationship between front distance and imbibition time of 2D and 3D models is different from that of 1D models. The shape factor for the multi-dimensional counter-current imbibition processes is time-dependent and varies with normalized recovery, which is also the reason for the difficulty of using a constant shape factor to scale imbibition data. Numerical simulation results and experimental data validated that the scaling equation could correlate the effect of differences in boundary conditions by introducing the new shape factor.