# Numerical implementation of the one-dimensional microscopic model of in-situ leaching

### Abstract

This publication is devoted to the numerical implementation of the mathematical model on a microscopic level of in-situ leaching process in the case of one space variable. The mathematical model is based on the common system of differential equations, when the fluid dynamics is described by the equation of motion of an incompressible fluid filling the pores of absolute solid ground skeleton, and dynamics of the active solution is described by the equation of diffusion-convection with point boundary conditions on the unknown free boundary between the fluid and the solid skeleton, expressing the conservation law of reagents. Numerical simulation by finite difference method is applied for the numerical solution of the problem. The nonlinear boundary conditions defined on the unknown free boundary is numerically solved by the iterative Newton’s method. For a more precise description of the movement of the free boundary, interpolation method is detailed. The significance of computer modelling of in-situ leaching process on a micro scales is an ability to study the basic mechanisms of the flow of physical and chemical process comprises reacting an active solutions with a solid skeleton and its movement through the capillary. The article presents the results of the problem in the case of one space variable in the form of graphs, obtained in mathematical environment Matlab.### References

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[2] Cohen C.E., Ding D., Quintard M., Bazin B. From pore scale to wellbore scale: Impact of geometry on wormhole growth in carbonate acidization // Chemical Engineering Science. - 2008. - P. 3088-3099.

[3] Panga M.K.R., Ziauddin M., Balakotaiah V. Two-scale continuum model for simulation of wormholes incarbonate acidization // A.I.Ch.E.Journal. - 2005. - P. 3231-3248.

[4] Rogov E.I. Sistemnyi analiz v gornom dele. - Alma-Ata: Nauka, 1976.

[5] Rogov E.I., Rogov S.E., Rogov A.E. Nachalo osnov teorii tekhnologii dobychi poleznyh iskopaemyh. - Almaty, 2001.

[6] Rogov E.I., Yazikov V.G., Rogov A.E. Matematicheskoe modelirovanie v gornom dele. - Almaty, 2002.

[7] Burridge R., Keller G.B. Poroelasticity equations derived from microstructure // Journal of Acoustic Society of America. - 1981. - No. 4. - P. 1140-1146.

[8] Sanchez-Palencia E. Non-Homogeneous Media and Vibration Theory - New York: Springer-Verlag, 1980.

[9] Nguetseng G. Asymptotic analysis for a stiff variational problem arising in mechanics // SIAM J. Math. Anal. - 1990. - P. 1394-1414.

[10] Buchanan J.L., Gilbert R.P. Transition loss in the farfield for an ocean with a Biot sediment over an elastic substrate // ZAMM. - 1997. - P. 121-135.

[11] Buckingham M.J. Seismic wave propagation in rocks and marine sediments: a new theoretical approach. // 4th European Conference on Underwater Acoustics. - Rome: CNR-IDAC, 1998. - P. 301-306.

[12] Gilbert R.P., Mikelic A. Homogenizing the acoustic properties of the seabed: Part I // Nonlinear Analysis. - 2000. - V. 40. - P. 185-212.

[13] Clopeau T.H., Ferrin J.L., Gilbert R.P., Mikelic A. Homogenizing the acoustic properties of the seabed: Part II // Mathematical and Computer Modelling. - 2001. - V. 33. - P. 821-841.

[14] Ferrin J.L., Mikelic A. Homogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluids. // Math. Meth. Appl. Sci. - 2003. - V. 26. - P. 831-859.

[15] Lukkassen D., Nguetseng G., Wall P. Two-scale convergence // Int. J. Pure and Appl. Math. - 2002. - No. 1. - P. 35-86.

[16] Meirmanov A. Mathematical models for poroelastic flows. - Paris: Atlantis Press, 2013.

[17] Meirmanov A. Nguetseng’s two-scale convergence method for filtration and seismic acoustic problems in elastic porous media // Siberian Mathematical Journal. - 2007. - V. 48. - P. 519- 538.

[18] Meirmanov A. Double porosity models in incompressible poroelastic media // Mathematical Models and Methods in Applied Sciences. - 2010. - V. 20, No. 4. - P. 635- 659.

[19] Ovsyannikov L.V. Vvedenie v mekhaniku sploshnyh sred. - Novosibirsk: Novosibirski Gosudarstvennyi universitet, 1977.

Published

2017-11-24

How to Cite

ЖУМАЛИ, А. С..
Numerical implementation of the one-dimensional microscopic model of in-situ leaching.

**KazNU Bulletin. Mathematics, Mechanics, Computer Science Series**, [S.l.], v. 89, n. 2, p. 27-34, nov. 2017. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/349>. Date accessed: 17 aug. 2018.
Section

Mathematics

**Keywords:**leaching, free boundary, microscopic model, numerical implementation