Partial fractional differential model for gas-liquid spontaneous imbibition with special imbibition index: imbibition behavior and recovery analysis

Hydraulic fracturing technology is widely used for developing hydrocarbons in tight unconventional formations, such as tight sandstone and shale. A considerable amount of liquid is imbibed spontaneously into the matrix during and after hydraulic fracturing. Unlike the strongly water-wet matrix, the gas-liquid spontaneous imbibition behavior in tight matrix is special, that the volume/mass of imbibed liquid is not always proportional to the imbibition time's power of 0.5. Thus, providing a new model is important to meet the special imbibition behavior in tight matrix. This work extends the previous gas-liquid spontaneous imbibition model on strongly water-wet matrix (such as Handy's model) and provides a new model considering special liquid imbibition and diffusion behavior when the imbibition index is not equal to 0.5. Typical solutions for imbibed liquid volume and saturation distribution are provided. The solutions can match well with experimental data. The results indicate that typical influencing factors, such as viscosity, imbibition index, and the style of imbibition front have special influence on imbibition process. Some possible mechanisms about the specialty on imbibition index in tight matrix are discussed. This work provides a more general imbibition model for describing spontaneous imbibition with special imbibition index, which may be helpful for imbibition analysis within shut-in period after hydraulic fracturing in tight formation in field.