A three-dimensional mesh-free model for analyzing multi-phase flow in deforming porous media

Abstract

Fully coupled flow-deformation analysis of deformable multiphase porous media saturated by several immiscible fluids has attracted the attention of researchers in widely different fields of engineering. This paper presents a new numerical tool to simulate the complicated process of two-phase fluid flow through deforming porous materials using a mesh-free technique, called element-free Galerkin (EFG) method. The numerical treatment of the governing partial differential equations involving the equilibrium and continuity equations of pore fluids is based on Galerkin’s weighted residual approach and employing the penalty method to introduce the essential boundary conditions into the weak forms. The resulting constrained Galerkin formulation is discretized in space using the same EFG shape functions for the displacements and pore fluid pressures which are taken as the primary unknowns. Temporal discretization is achieved by utilizing a fully implicit scheme to guarantee no spurious oscillatory response. The validity of the developed EFG code is assessed via conducting a series of simulations. According to the obtained numerical results, adopting the appropriate values for the EFG numerical factors can warrant the satisfactory application of the proposed mesh-free model for coupled hydro-mechanical analysis of applied engineering problems such as unsaturated soil consolidation and infiltration of contaminant into subsurface soil layers.

Appendix

The nodal matrices and vectors in Eq. (27) are defined as:

$$ C_{11\,IJ} = \int\limits_{\Omega } {B_{I}^{T} \,D_{T} \,B_{J} \,d\Omega } $$

(32)

$$ C_{u\,IJ}^{\alpha } = \int\limits_{{\Gamma_{u} }} {\Phi_{I}^{T} \,\alpha_{pu} \,\Phi_{J} \,d\Gamma } $$

(33)

$$ C_{12\,IJ} = \int\limits_{\Omega } {B_{I}^{T} \,\alpha \,m\,\left[ {s_{w} + p_{c} \,{{\partial s_{w} } \mathord{\left/ {\vphantom {{\partial s_{w} } {\partial p_{c} }}} \right. \kern-0pt} {\partial p_{c} }}} \right]\,\varphi_{J} \,d\Omega } $$

(34)

$$ C_{13\,IJ} = \int\limits_{\Omega } {B_{I}^{T} \,\alpha \,m\,\left[ {\left( {1 - s_{w} } \right) - p_{c} \,{{\partial s_{w} } \mathord{\left/ {\vphantom {{\partial s_{w} } {\partial p_{c} }}} \right. \kern-0pt} {\partial p_{c} }}} \right]\,\varphi_{J} \,d\Omega } $$

(35)

$$ F_{u\,I} = \int\limits_{\Omega } {\Phi_{I}^{T} \,\rho g\,d\Omega + } \int\limits_{{\Gamma_{\sigma } }} {\Phi_{I}^{T} \,\bar{t}\,d\Gamma } $$

(36)

$$ F_{u\,I}^{\alpha } = \int\limits_{{\Gamma_{u} }} {\Phi_{I}^{T} \,\alpha_{pu} \,\bar{u}\,d\Gamma } $$

(37)

$$ C_{21\,IJ} = \int\limits_{\Omega } {\phi_{I} \,\alpha \,s_{w} \,m^{T} \,B_{J} \,d\Omega } $$

(38)

$$ C_{22\,IJ} = \int\limits_{\Omega } {\phi_{I} \,\left[ {s_{w} \frac{\alpha - n}{{K_{s} }}\left( {s_{w} + \frac{{\partial s_{w} }}{{\partial p_{c} }}p_{c} } \right) + \frac{{n\,s_{w} \,}}{{K_{w} }} - n\frac{{\partial s_{w} }}{{\partial p_{c} }}} \right]\,\varphi_{J} \,d\Omega } $$

(39)

$$ C_{23\,IJ} = \int\limits_{\Omega } {\phi_{I} \,\left[ {s_{w} \frac{\alpha - n}{{K_{s} }}\left( {1 - s_{w} - \frac{{\partial s_{w} }}{{\partial p_{c} }}p_{c} } \right) +\, n\frac{{\partial s_{w} }}{{\partial p_{c} }}} \right]\,\varphi_{J} \,d\Omega } $$

(40)

$$ K_{22\,IJ} = \int\limits_{\Omega } {B_{pI}^{T} \,\frac{{k\,k_{rw} \,}}{{\mu_{w} }}\,B_{pJ} \,d\Omega } $$

(41)

$$ K_{{p_{w} \,IJ}}^{\alpha } = \int\limits_{{\Gamma_{{p_{w} }} }} {\varphi_{I} \,\alpha_{{pp_{w} }} \,\varphi_{J} \,d\Gamma } $$

(42)

$$ F_{{p_{w} \,I}} = \int\limits_{\Omega } {B_{pI}^{T} \,\frac{{k\,k_{rw} }}{{\mu_{w} }}\rho_{w} g\,d\Omega} \, - \int\limits_{{\Gamma_{{q_{w} }} }} {\varphi_{I} \,\bar{q}_{w} \,d\Gamma } $$

(43)

$$ F_{{p_{w} \,I}}^{\alpha } = \int\limits_{{\Gamma_{{p_{w} }} }} {\varphi_{I} \,\alpha_{{pp_{w} }} \,\bar{p}_{w} \,d\Gamma } $$

(44)

$$ C_{31\,IJ} = \int\limits_{\Omega } {\varphi_{I} \,\alpha \,\left( {1 - s_{w} } \right)\,m^{T} \,B_{J} \,d\Omega } $$

(45)

$$ C_{32\,IJ} = \int\limits_{\Omega } {\varphi_{I} \,\left[ {\left( {1 - s_{w} } \right)\frac{\alpha - n}{{K_{s} }}\left( {s_{w} + \frac{{\partial s_{w} }}{{\partial p_{c} }}p_{c} } \right) +\, n\frac{{\partial s_{w} }}{{\partial p_{c} }}} \right]\,\varphi_{J} \,d\Omega } $$

(46)

$$ C_{33\,IJ} = \int\limits_{\Omega } {\varphi_{I} \,\left[ {\left( {1 - s_{w} } \right)\frac{\alpha - n}{{K_{s} }}\left( {1 - s_{w} - \frac{{\partial s_{w} }}{{\partial p_{c} }}p_{c} } \right) -\, n\frac{{\partial s_{w} }}{{\partial p_{c} }} + \frac{{n\,\left( {1 - s_{w} } \right)}}{{K_{nw} }}} \right]\,\varphi_{J} \,d\Omega } $$

(47)

$$ K_{33\,IJ} = \int\limits_{\Omega } {B_{pI}^{T} \,\frac{{k\,k_{rnw} \,}}{{\mu_{nw} }}\,B_{pJ} \,d\Omega } $$

(48)

$$ K_{{p_{nw} \,IJ}}^{\alpha } = \int\limits_{{\Gamma_{{p_{nw} }} }} {\varphi_{I} \,\alpha_{{pp_{nw} }} \,\varphi_{J} \,d\Gamma } $$

(49)

$$ F_{{p_{nw} \,I}} = \int\limits_{\Omega } {B_{pI}^{T} \,\frac{{k\,k_{rnw} }}{{\mu_{nw} }}\rho_{nw} g\,d\Omega} \, - \int\limits_{{\Gamma_{{q_{nw} }} }} {\varphi_{I} \,\bar{q}_{nw} \,d\Gamma } $$

(50)

$$ F_{{p_{nw} \,I}}^{\alpha } = \int\limits_{{\Gamma_{{p_{nw} }} }} {\varphi_{I} \,\alpha_{{pp_{nw} }} \,\bar{p}_{nw} \,d\Gamma } $$

(51)

where m = {1, 1,1, 0, 0, 0}^T.