One of the principal criteria for development of the boundary element method (BEM) in porous media is derivation of the required fundamental solutions in the boundary integral equations (BIE). Furthermore, setting up the governing BIEs based on the governing partial differential equations (PDE) is another challenge in solving a physical phenomenon using BEM. In this regard, the governing BIEs for unsaturated porous media have been developed using the available derived fundamental solutions. In this research, a perturbation type approximation is exploited for developing a system of BIEs for the quasi-static unsaturated porous media with moderate variations in its properties. Nevertheless, the fundamental solutions of the medium with constant properties are applied. The method produces two sets of equations with constant parameters instead of the original equations. Besides, the required boundary conditions have been formulated. This type of BIEs is essential to be used in the BEM for unsaturated porous media as the fundamental solutions for a medium with coordinates dependent properties is not available so far. The resulted introduced BIEs may be used directly in a BEM numerical model for an unsaturated porous media in one, two or three dimensional conditions.
I. D. Moldovan and T. D. Cao, "Hybrid-Trefftz Displacement Finite Elements for Elastic Unsaturated Soils," International Journal of Computational Methods, vol. 11, no. 2, p. 1342005, 2014.
Schanz, "Poroelastodynamics: Linear Models, Analytical Solutions, and Numerical Methods," Applied mechanics reviews, vol. 62, no. 3, p. 030803, 2009.
M. P. Cleary, "Fundamental solutions for a fluid-saturated porous solid.," International Journal of Solids and Structures, vol. 13, no. 9, pp. 785-806, 1977.
B. Gatmiri and E. Jabbary, "Time-domain Green’s functions for unsaturated soils. Part I: Two-dimensional solution," International Journal of Solids and Structures, vol. 42, no. 23, pp. 5971-5990, 2005.
B. Gatmiri and E. Jabbari, "Time-domain Green’s functions for unsaturated soils. Part II: Three-dimensional solution," International Journal of Solids and Structures, vol. 42, no. 23, pp. 5991-6002, 2005.
P. Maghoul, B. Gatmiri and D. Duhamel, "Boundary integral formulation and two-dimensional fundamental solutions for dynamic behavior analysis of unsaturated soils," Soil Dynamics and Earthquake Engineering, vol. 31, no. 11, pp. 1480-1495, 2011.
M. Schanz and P. Li, "Wave propagation in a 1-D partially saturated poroelastic column," ZAMMâJournal of Applied Mathematics and Mechanics, vol. 81, no. S3, pp. 591-592, 2001.
Ghorbani, J., M. Nazem, and J. P. Carter. "Numerical modelling of multiphase flow in unsaturated deforming porous media." Computers and Geotechnics 71 (2016): 195-206.
Igumnov, L. A., S. Yu Litvinchuk, A. N. Petrov, and A. A. Ipatov. "Numerically Analytical Modeling the Dynamics of a Prismatic Body of Two-and Three-Component Materials." In Advanced Materials, pp. 505-516. Springer, Cham, 2016.
Igumnov, L. A., A. N. Petrov, and I. V. Vorobtsov. "One-dimensional wave propagation in a three phase poroelastic column." Key Engineering Materials 685 (2016).
H. Wang and Q.-H. Qin, "Boundary integral based graded element for elastic analysis of 2D functionally graded plates," European Journal of Mechanics-A/Solids, vol. 33, pp. 12-13, 2012.
I. Ashayeri, M. Kamalian and M. Jafari, "Transient Boundary Integral Equation of Dynamic Unsaturated Poroelastic Media," in Geotechnical Earthquake Engineering and Soil Dynamics IV, 2008.
O. C. Zienkiewicz, Y. Xie, B. A. Schrefler, A. Ledesma and N. Bicanic, "Static and dynamic behaviour of soils: a rational approach to quantitative solutions. II. Semi-saturated problems," in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1990.
D. G. Fredlund and H. Rahardjo, Soil Mechanics for Unsaturated Soils, John Wiley & Sons, 1993.
o. Lafe and H.-D. Cheng, "A perturbation boundary element code for steady state groundwater flow in heterogeneous aquifers," Water Resources Research, vol. 23, no. 6, pp. 1079-1084, 1987.
K. Sato and R. N. Horn, "Perturbation boundary element method for heterogeneous reservoirs: Part 1-Steady-state flow problems," SPE Formation Evaluation, vol. 8, no. 04, pp. 306-314, 1993.
I. Ashayeri, M. Kamalian, M. K. Jafari and B. Gatmiri, "Analytical 3D transient elastodynamic fundamental solution of unsaturated soils," International Journal for Numerical and Analytical Methods in Geomechanics, vol. 35, no. 17, pp. 1801-1829, 2011.
K.-V. Nguyen and B. Gatmiri, "Numerical implementation of fundamental solution for solving 2D transient poroelastodynamic problems," Wave motion, vol. 44, no. 3, pp. 137-152, 2007.
K. Wapenaar and J. Foklema, "Reciprocity theorems for diffusion, flow and waves," TRANSACTIONS-AMERICAN SOCIETY OF MECHANICAL ENGINEERS JOURNAL OF APPLIED MECHANICS, vol. 71, no. 1, pp. 145-150, 2004.
P. Li and M. Schanz, "Time domain boundary element formulation for partially saturated poroelasticity," Engineering Analysis with Boundary Elements, vol. 37, no. 11, pp. 1483-1498, 2013.
S. N. Atluri and T. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, vol. 22, Computational mechanics, 1998.
B. Gatmiri, P. Delage and M. Cerrolaza, "UDAM: A powerful finite element software for the analysis of unsaturated porous media," Advances in Engineering Software, vol. 29, no. 1, pp. 29-43, 1998.
Jabbari, E., & Behnia, M. (2018). Boundary Integral Equations for Quasi-Static Unsaturated Porous Media. Journal of Hydraulic Structures, 4(2), 42-59. doi: 10.22055/jhs.2018.27811.1091
MLA
Ehsan Jabbari; Mazda Behnia. "Boundary Integral Equations for Quasi-Static Unsaturated Porous Media". Journal of Hydraulic Structures, 4, 2, 2018, 42-59. doi: 10.22055/jhs.2018.27811.1091
HARVARD
Jabbari, E., Behnia, M. (2018). 'Boundary Integral Equations for Quasi-Static Unsaturated Porous Media', Journal of Hydraulic Structures, 4(2), pp. 42-59. doi: 10.22055/jhs.2018.27811.1091
VANCOUVER
Jabbari, E., Behnia, M. Boundary Integral Equations for Quasi-Static Unsaturated Porous Media. Journal of Hydraulic Structures, 2018; 4(2): 42-59. doi: 10.22055/jhs.2018.27811.1091