Abstract
On the basis of the laws of conservation and the principles of thermodynamics, a mathematical model of the flow of a two-phase granular fluid is proposed. One of the phases is the viscoplastic granular Bingham fluid; the other phase is a viscous Newtonian fluid. The equations for flows in the Hele–Shaw cell are analyzed asymptotically, i.e., when the flat-channel width is much less than its length. The correlations between the phase flow rates and the pressure gradient leading to equations of filtration for a two-phase granular viscoplastic fluid are constructed. The criterion is found for the initiation of motion of a granular phase in a porous medium. It is established that, depending on the shear-yield stress, such a phase does not flow if either the pressure gradient or the channel thickness is small. The phase flow rates are analyzed numerically at various input parameters such as the phase viscosities, phase resistivities, ultimate shear stress, etc. The factors slowing down the penetrating motion of the solid phase into the porous medium are revealed.
Similar content being viewed by others
REFERENCES
X. Chateau, G. Ovarlez, and K. L. Trung, J. Rheol. 52, 489 (2008).
A. C. Eringen, Microcontinuum Field Theories (Springer-Verlag, New York, 1999).
J. Mewis and N. Wagner, Colloidal Suspension Rheology (Cambridge University, Cambridge, 2012).
V. V. Shelukhin, J. Non-Newtonian Fluid Mechanics, in Press. https://doi.org/10.1016/j.jnnfm.2018.02.004
V. V. Shelukhin, J. Math. Fluid Mech. 4, 109 (2002).
V. V. Shelukhin and M. Ruzicka, Z. Angew. Math. Mech. 93 (1), 57 (2013).
Funding
This work was carried out as part of Project 72 of the Complex Program for Fundamental Research of the Siberian Branch of the Russian Academy of Sciences II.1 (development of the mathematical model) and supported by the Government of the Russian Federation, grant no. 14 W03.31.0002 (development of the computational algorithm and calculation).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by V. Bukhanov
Rights and permissions
About this article
Cite this article
Shelukhin, V.V., Kontorovich, A.E. Behavior of Viscoplastic Rocks near Fractures: Mathematical Modeling. Dokl. Phys. 64, 461–465 (2019). https://doi.org/10.1134/S1028335819120036
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1028335819120036