Fluid imbibition from hydraulic fractures into shale formations is mainly affected by a combination of capillary forces and viscous resistance, both of which are closely related to the pore geometry. This study established five self-imbibition models with idealized pore structures and conducted a comparative analysis of these models. These models include circular, square, and equilateral triangular capillaries; a triangular star-shaped cross-section formed by three tangent spherical particles; and a traditional porous medium representation method. All these models are derived based on Newton’s second law, where capillary pressure is described by the Young-Laplace equation and viscous resistance is characterized by the Hagen-Poiret equation and Darcy’s law. All derived models predict that the fluid imbibition distance is proportional to the square root of time, in accordance with the classical Lucas-Washburn law. However, different pore structures exhibit significantly different characteristic imbibition rates. Compared to the single pore model, the conventional Darcy’s law-based model for porous media predicts significantly lower imbibition rates, which is consistent with the relatively slower uptake rates in actual shale nanoscale pore networks. These findings emphasize the important role played by pore geometry in fluid imbibition dynamics and further point to the need for optimizing pore structure to extend fluid imbibition duration in shale reservoirs in practical operations.

Spontaneous imbibition has recently received a great deal of research attention for improving hydrocarbon recovery from shale gas and oil reservoirs. It is highly desirable to know the true significance and the role of fluid imbibition in the recovery process. Using a Krüss Drop Shape Analyzer 100S with Krüss’ Advance software, water imbibition depth was measured in this study on dry cores from four shale gas/oil reservoirs namely Tuscaloosa Marine Shale, Eagle Fort Shale, Marcellus Shale, and Green River Shale. The initial water-contact angles on the Tuscaloosa Marine Shale, Eagle Fort Shale, Marcellus Shale and Green River Shale core surfaces were measured to be 36.62°, 66.68°, 52.78° and 84.73°, respectively. The contact angle and thus volume of liquid droplet changed due to fluid imbibition into the core samples and evaporation. The change in droplet volume, together with the contact area and shale porosity, was used to calculate the imbibition depth. An analytical imbibition model was derived and tuned to upscale the tested fluid imbibition data to field level. The result of time-upscaling using the tuned imbibition models shows that the 1-month water imbibition depths for the Tuscaloosa Marine Shale, Eagle Fort Shale, Marcellus Shale and Green River Shale are 2.61, 1.59, 0.89 and 0.16 cm, respectively. These low values suggest that the direct effect of water imbibition into shale matrix on hydrocarbon recovery in shale reservoirs is insignificant in the practical scales of space and time. However, the imbibition-induced shale cracks can increase shale permeability significantly for mass transfer during the hydrocarbon recovery process. Water imbibition in the cracks should be investigated in future studies.

A mathematical model was derived in this study to reveal the mechanism of CO₂ imbibition in shale formations considering the combined effects of capillary force and viscous force in concave curved triangle pore channels surrounded by different solid materials with different wettability. The model reveals that CO₂ imbibition depth is proportional to the square root of CO₂ soaking time, square root of the pore size determined by grain size, square root of interfacial tension and cosine of contact angle, and inversely proportional to the square root of CO₂ viscosity. Up to three solid wall materials with different contact angles can be considered in the model. Using the average contact angle for the three materials over-estimates the imbibition distance. CO₂ imbibition is faster in concave curved triangle pores than in equivalent circular-shaped pores. The dimensionless geometry correction factor is less than unity $(\alpha =0.81)$. The newly developed imbibition model can be used for predicting the maximum time of imbibition between parallel fractures in multi-fractured shale formations.

Multi-stage hydraulic-fracturing horizontal wells has revolutionized today's oil and gas industry. Post-frac fluid soaking is essential for improving productivity of shale oil-gas wells. Optimization of soaking time is an open problem to solve in the petroleum industry. Understanding the post-frac soaking process is vitally important for solving the puzzle. Analytical solutions were developed in this study to describe the spontaneous imbibition processes in shale matrix and shale cracks during fluid soaking. Solutions show that the imbibition distance is directly proportional to the square root of imbibition time and the imbibition velocity is inversely proportional to the square root of imbibition time. The rate of spontaneous imbibition in shale cracks is much faster than that in shale matrix. Therefore, the optimum time for post-frac fluid soaking was further analyzed on the basis of the imbibition in shale cracks only. The solution was combined with pressure fall-off data to formulate a mathematical method for predicting the post-frac fluid soaking time required for the fluid to reach the mid-point between two adjacent hydraulic fractures. A case study with Tuscaloosa Marine Shale data suggests that the front of fluid imbibition should propagate 4 meters in 2 weeks and to 6 meters in 4 weeks. These numbers may be considered as the optimum times of post-frac fluid imbibition if the shale swell effect is negligible. Future research should quantify the effects of shale swelling on spontaneous imbibition so that the information can be incorporated in the soaking model to fully describe the imbibition process for better prediction of well productivity.

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.

Natural gas production from marine gas hydrate reservoirs has become attractive to the oil and gas industry in recent years. It is still a great challenge to recover natural gas from hydrate reservoirs efficiently mainly due to sand production and wellbore collapse problems associated with the production scheme of depressurization. The thermal recovery method has not been proven economical due to the high cost of energy consumption. This study focuses on using geothermal energy to assist the depressurization process so that well pressure drawdown can be reduced and thus sand production and wellbore collapse problems can be mitigated. The authors investigated the transfer of heat energy from a natural geothermal zone to a marine gas hydrate reservoir and its effect on gas well productivity using analytical models. The result of our investigation shows that the initial well productivity can be significantly improved using geothermal energy more than 10-fold. This work provides engineers with an analytical tool for the feasibility analysis of using geothermal energy to improve well performance in gas hydrate reservoirs.