A self-similar analytical solution of spontaneous and forced imbibition in porous media

Abstract

Both viscous and capillary forces control the two-phase flow in porous media. The Buckley
Leverett solution for viscous flow in porous media has been proposed for over a half
century. While the corresponding studies of capillary dominated solutions are mainly based
on the capillary tube based models. The continuum solutions are just prevail in recently
years. The analytical solution of the combination of both effects is rarely investigated.
A self-similar analytical solution of spontaneous and forced imbibition in porous media
is proposed in this work and the corresponding concise algorithms are presented. The
proposed solution successfully solves this typical non-linear partial differential equation
by introducing a transformation variable and the capillary fractional flow function analog
to the fractional flow function of Buckley Leverett solution. Finally, the case study is
performed, which demonstrates the feasibility and accuracy of this proposed solution to a
general two-phase flow condition.

References

Bjørnara, T.I., Mathias, S.A. A pseudospectral approach to the McWhorter and Sunada equation for two-phase flow in porous media with capillary pressure. Comput. Geosci. 2013, 17(6): 889-897.

Buckley, S.E., Leverett, M.C. Mechanism of fluid displace-ment in sands. Trans. AIME 1942, 146(1): 107-116.

Cai, J., Perfect, E., Cheng, C., et al. Generalized modeling of spontaneous imbibition based on Hagen-Poiseuille flow in tortuous capillaries with variably shaped apertures. Langmuir 2014, 30(18): 5142-5151.

Cai, J., Yu, B. A discussion of the effect of tortuosity on the capillary imbibition in porous media. Transp. Porous Media 2011, 89(2): 251-263.

Chapra, S., Canale, R.P. Numerical Methods for Engineers (Vol. 2). New York, USA, McGraw-Hill, 1988.

Chen, Z., Bodvarsson, G.S., Witherspoon, P.A. An Integral equation formulation for two-phase flow and other non-linear flow problems through porous media. Paper SPE 20517 Presented at SPE Annual Technical Conference and Exhibition, New Orleans, LA, 23-16 September, 1990.

Dake, L.P. Developments in Petroleum Science. Amsterdam, Netherlands, Elsevier Science, 1978.

Fayers, F.J., Sheldon, J.W. The effect of capillary pressure and gravity on two-phase fluid flow in a porous medium. J. Pet. Technol. 1959, 216: 147-155.

Handy, L.L. Determination of effective capillary pressures for porous media from imbibition data. Trans. AIME 1960, 219: 75-80.

Li, C., Shen, Y., Ge, H., et al. Analysis of spontaneous imbibition in fractal tree-like network system. Fractals 2016, 24(3): 1650035.

Li, K., Horne, R.N. Characterization of spontaneous water imbibition into gas-saturated rocks. Paper SPE 62552 Presented at SPE/AAPG Western Regional Meeting, Long Beach, California, 19-22 June, 2000.

Lucas, R. Rate of capillary ascension of liquids. Kolloid Z. 1981, 23(15): 15-22.

McWhorter, D.B., Sunada, D.K. Exact integral solutions for two-phase flow. Water Resour. Res. 1990, 26(3): 399-413.

Meng, Q., Liu, H., Wang, J. A critical review on fundamental mechanisms of spontaneous imbibition and the impact of boundary condition, fluid viscosity and wettability. Adv. Geo-Energy Res. 2017, 1(1): 1-17.

Peters, E.J. Advanced petrophysics: Dispersion, interfacial phenomena (Vol. 2). Greenleaf Book Group, 2012.

Schmid, K.S., Alyafei, N., Geiger, S., et al. Analytical solutions for spontaneous imbibition: Fractional-flow theory and experimental analysis. SPE J. 2016, 21(6): 2308-2316.

Schmid, K.S., Geiger, S., Sorbie, K.S. Semianalytical solutions for cocurrent and countercurrent imbibition and dispersion of solutes in immiscible twophase flow. Water Resour. Res. 2011, 47(2): 2144-2150.

Schmid, K.S., Geiger, S. Universal scaling of spontaneous imbibition for arbitrary petrophysical properties: Water-wet and mixed-wet states and Handy’s conjecture. J. Pet. Sci. Eng. 2013, 101: 44-61.

Shampine, L.F., Reichelt, M.W. The matlab ode suite. SIAM J. Sci. Comput. 1997, 18(1): 1-22.

Shen, Y., Ge, H., Zhang, X., et al. Impact of fracturing liquid absorption on the production and water-block unlocking for shale gas reservoir. Adv. Geo-Energy Res. 2018, 2(2): 163-172.

Washburn, E.W. The yynamics of capillary flow. Phys. Rev. Ser. 1921, 17(3): 273-283.

Xiao, J., Cai J., Xu, J. Saturated imbibition under the influence of gravity and geometry. J. Colloid Interface Sci. 2018, 521: 226-231.

Yokoyama, Y., Lake, L. The effects of capillary pressure on immiscible displacements in stratified porous media. Paper SPE 10109 Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, USA, 4-7 October, 1981.