# On one thermodynamically consistent model of clay shale swelling

### Abstract

A modified version of the linear poroelasticity theory described by three elastic parameters is applied to the mathematical modeling of shale swelling with an aqueous electrolyte. It is assumed that the shale behaves as an isotropic ideal ionic membrane, and in this case, swelling depends only on the total stress and on the chemical water potential in pores of the rock. A formula is obtained for the Poisson coefficient in terms of three elastic parameters, the physical densities of the saturated fluid of the porous medium, and the porosity coefficient. It is shown that the diffusion coefficient is a function of the coefficient of interfacial friction (permeability) and is inversely proportional to the coefficient of porosity. The formulas for the flat strain analysis around the wellbore were obtained.

### References

[1]

Mroz Z., Boukpeti N., and Drescher A., “Constitutive model for static liquefaction,” Int. J. Geomech. ASCE, 3, 133–144 (2003).

[2]

Rice J. R., “On the stability of dilatant hardening for saturated rock masses,” J. Geophys. Res., 80, No. 11, 1531–1536 (1975).

[3]

Boukpeti N., Mroz Z., and Drescher A., “A model for static liquefaction in triaxial compression and extension,” Can. Geotech. J., 39, 1243–1253 (2002).

[4]

Mody F. K. and Hale A. H., “A borehole stability model to couple the mechanics and chemistry of drilling fluid shale interaction,” J. Petr. Tech., 45, No. 11, 1093–1101 (1993).

[5]

Van Oort E., Hale A. H., Mody F. K., and Roy S., “Transport in shales and the design of improved water-based shale drilling fluids,” SPE Drilling & Completion, 8, No. 3, 137–146 (1996).

[6]

Tan C. P., Richards B. G., and Rahman S. S., “Managing physico-chemical wellbore instability in shales with the chemical potential mechanism,” in: SPE Asia Pacific Oil and Gas Conf. (Adelaide, Australia, Oct. 28–31, 1996), pp. 107–116, 36971, SPE (1996).

[7]

Ghassemi A., Diek A., and Dos Santos H., “Effects of ion diffusion and thermal osmosis on shale deterioration and borehole instability,” in: AADE Nat. Drilling Conf. (Houston, TX, March 27–29, 2001), AADE-01-NC-HO-7440.

[8]

Fjaer E., Holt R. M., Nes O. M., and Sonstebo E. F., “Mud chemistry effects on time-delayed borehole stability problems in shales,” in: Proc. SPE/ISRM Rock Mech. Conf. (Irving, TX), SPE/ISRM78163, SPE (2002).

[9]

Nguyen V., Abousleiman Y., and Hoang S., “Analysis of wellbore instability in drilling through chemically active fractured rock formations,” SPE J., 14, No. 2, 283–301 (2009).

[10]

Ghassemi A., Tao Q., and Diek A., “Influence of coupled chemo-poro-thermoelastic processes on pore pressure and stress distributions around a wellbore in swelling shale,” J. Petrol. Sci. Eng., 67, No. 1–2, 57–64 (2009).

[11]

Zhou X. and Ghassemi A., “Finite element analysis of coupled chemo-poro-thermo-mechanical effects around a wellbore in swelling shale,” Int. J. Rock Mech. Mining Sci., 46, No. 4, 769–778 (2009).

[12]

Hale A. H., Mody F. K., and Salisbury D. P., “The influence of chemical potential on wellbore stability,” SPE Drilling & Completion, 8, No. 3, 207–216 (1993).

[13]

Sherwood J. D., “Biot poroelasticity of a chemically active shale,” Proc. Roy. Soc. London, 440, 365–377 (1993).

[14]

Sherwood J. D., “A model of hindered solute transport in a poroelastic shale,” Proc. Roy. Soc. London, 445, 679–692 (1994).

[15]

Biot M. A., “General theory of three-dimensional consolidation,” J. Appl. Phys., 12, 155–164 (1941).

[16]

Detournay E. and Cheng A. H-D., Fundamentals of Poroelasticity, ch. 5 in: Comprehensive Rock Engineering, vol. 2, pp. 113–171, Pergamon Press, New York (1993).

[17]

Coussy O., Poromechanics, Wiley, New York (2004).

[18]

Terzaghi K., “Die Berechnung der Durchassigkeitsziffer des Tones aus dem Verlaufder hydrodynamischen Spannungsercheinungen,” Sitzungsber. Akad. Wiss. Wien Math. Naturwiss. Kl., Abt. 2A, 132, 105–124 (1923).

[19]

Terzaghi K., “The shearing resistance of saturated soils,” in: Proc. Int. Conf. Soil Mech. Found. Eng., pp. 54–55 (1936).

[20]

Biot M. A., “Theory of elasticity and consolidation for a porous anisotropic solid,” J. Appl. Phys., 26, 182–185 (1955).

[21]

Imomnazarov Kh. Kh., “Concentrated force in a porous half-space,” Bull. Novosibirsk Comput. Center, Ser. Math. Model. Geophys. No. 4, 75–77, Novosibirsk (1998).

[22]

Grachev E. V., Zhabborov N. M., and Imomnazarov Kh. Kh., “A concentrated force in an elastic porous half-space,” Dokl. Phys., 48, No. 7, 376–378 (2003).

[23]

Grachev E., Imomnazarov Kh., and Zhabborov N., “One nonclassical problem for the statics equations of elastic-deformed porous media in a half-plane,” Appl. Math. Lett., 17, No. 1, 31–34 (2004).

[24]

Zhabborov N. M. and Imomnazarov Kh. Kh., Some Initial Boundary Value Problems of Mechanics of Two-Velocity Media [in Russian], Publishing house NUUz after named Mirzo Ulugbek, Tashkent (2012).

[25]

Blokhin A. M. and Dorovsky V. N., Mathematical Modelling in the Theory of Multivelocity Continuum, Nova Sci., New York (1995).

[26]

Imomnazarov Kh. Kh., “Some remarks on the Biot system [in Russian],” Dokl. RAN, 373, No. 4, 536–537 (2000).

[27]

Imomnazarov Kh. Kh., “Some remarks on the Biot system of equations describing wave propagation in a porous medium,” Appl. Math. Lett., 13, No. 3, 33–35 (2000).

[28]

Rice J. R. and Clearly M. P., “Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents,” Rev. Geophys. Space Phys., 14, 227–241 (1976).

[29]

Ageev P. G., Koldoba A. V., Gasilova I. V., Poveshchenko N. Yu., Yakobovsky M. V., and Tkachenko S. I., “Complex model of reservoir response to a plasma-pulse effect [in Russian],” Math. Montisnigri, 28, 75–98 (2013).

[30]

Skempton A. W., “The pore-pressure coefficients A and B,” Geotechnique, 4, 143–147 (1954).

*Mathematical notes of NEFU*, 27(2), pp. 93-104. doi: https://doi.org/10.25587/SVFU.2020.43.24.006.

This work is licensed under a Creative Commons Attribution 4.0 International License.