The classical Lucas-Washburn (L-W) capillary-driven model calculates capillary pressure using the Young-Laplace equation, which leads to a deviation from the real values for a pipe with small diameter. In this work we use Tolman length to improve the Young-Laplace equation, and propose an improved L-W model. Moreover, circular pipes with rough and variable sections are considered rather than the circular pipe with a constant diameter. Relation between time and wetting fluid injection length in the circular pipe with variable sections is established. The mathematical model is described as a second-order nonlinear ordinary differential equation, which cannot be solved analytically, and thus an efficient numerical method is developed. A concrete pipe with variable cross-section is selected, and the relation between water length and time is calulated with numerical simulation. Numerical results are analyzed, and the effectiveness of Tolman length is verified. It shows that the local shrinkage of circular pipe changes the states of water motion significantly, and there are three kinds of movement modes. The local expansion of circular pipe changes slowly the state of seepage and water absorption, and only one single movement mode takes place.

Mathematical models of imbibition under different boundary conditions were built with piston-like displacement assumption. Effect of gravity and capillary pressure on imbibition under OEO (single boundary) and TEO-OW (mixed boundary) boundary conditions are studied. It shows that oil is produced only by counter-current imbibition under OEO boundary conditions. Gravity inhibits imbibition as the open face is at the lower end face of core and gravity promotes imbibition as the open face is at the upper end face of core. Oil is produced by combination of co-and counter-current imbibition under TEO-OW boundary condition. Gravity inhibits counter-current imbibition and promotes co-current imbibition under TEO-OW boundary condition. Gravity is dominant driving force as the inverse Bond number is smaller and capillary pressure is dominant driving force as inverse Bond number is greater. In addition, effect of gravity and capillary pressure on imbibition under OEO and TEO-OW boundary conditions are studied. It shows that oil production by gravity under OEO boundary condition is greater than that under TEO-OW boundary condition with same inverse Bond number. The rate of oil production under OEO boundary condition is faster than that under TEO-OW boundary condition as inverse Bond number is smaller and opposite phenomena is observed as inverse Bond number is greater.

Spontaneous imbibition in porous media is ubiquitous in natural and industrial applications. It is of great value to probe into the microscopic and macroscopic multiphase flow mechanisms of spontaneous imbibition in science and engineering aspects. However, due to the multi-influencing factors such as petrophysical properties, fluid properties and boundary conditions in porous media, the microscopic and macroscopic flow mechanisms in spontaneous imbibition are difficult to be understood thoroughly. Currently, the spontaneous imbibition process under the influence of multiple factors is widely studied and mainly focused on laboratory experiments, theoretical analysis and numerical simulations. In this paper, the key issues and recent progress in spontaneous imbibition are analyzed, and some suggestions and an in-depth understanding of capillarity in porous media are provided for future work.