Available online at www.sciencedirect.com ScienceDirect Mathematics and Computers in Simulation 137 (2017) 448–461 www.elsevier.com/locate/matcom Original articles Finite element solution for a coal-bed methane reservoir model R. Touzani a,∗ , L. Alessio b , D. Kuzmichev b , R. Buoy b a Universit´ e Clermont Auvergne, CNRS, Laboratoire de Math´ ematiques Blaise Pascal, 63000 Clermont-Ferrand, France b Leap Energy Partners Sdn. Bhd., Suite 14.08, Level 14, G Tower 199, Jalan Tun Razak, 50400 Kuala Lumpur, Malaysia Received 26 December 2015; received in revised form 22 December 2016; accepted 19 January 2017 Available online 28 January 2017 Abstract We present a mathematical model for fluid flow in porous media in the case of coal bed methane reservoirs. The model consists of a two-phase immiscible fluid flow with gas and water and takes into account desorption. We discretize the problem by a finite element method with adding a streamline diffusion term for the case of small capillary pressure. Time integration uses the fully implicit Euler scheme. Numerical experiments in radial and fully 2-D cases show the effectiveness of the model and numerical method. c ⃝ 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved. Keywords: Porous media; Two-phase flow; Finite elements 1. Introduction In Reservoir Engineering, CBM (Coal Bed Methane) refers to the natural gas contained within coal. In the coal matrix gas is stored by adsorption, i.e. gas molecules adhere to the surface. The pressure/volume relationship is described by the Langmuir isotherm. When the initial gas content of the coal is below the equilibrium curve in this isotherm, no free gas is available and the fractures in the coal are filled with water. When pressure is lowered, a dewatering process starts before gas extraction. This suggests that mathematical modelling of such a process invokes a two-phase flow in a porous medium. More details on the CBM process and its modelling can be found in [2,7,18]. The present work aims at deriving a mathematical model that predicts the gas extraction process from a CBM reservoir. For this we shall consider classical fluid flow in porous medium of an immiscible mixture of gas and water, driven by the desorption process. Study of two-phase flows is widely used in reservoir engineering and applied math- ematics literature (See [1,5,12,16] for instance, and the references therein). Our contribution here is the adjunction of desorption as a source for the gas mass conservation equation. There is also a specific issue to the presented modelling: Since in most applications we start by a dewatering process, we have to deal with a two-phase model that starts with only one phase. The resulting singularity at initial time imposes the use of a fully implicit time integration scheme. This results in a nonlinear system that is heavy and time-consuming to solve. Numerical experiments show indeed that any simpler technique (e.g. IMPES, Implicit Pressure Explicit Saturation) fails to converge. ∗ Corresponding author. E-mail addresses: rachid.touzani@uca.fr (R. Touzani), laurent.alessio@leap-energy.com (L. Alessio), dmitry.kuzmichev@leap-energy.com (D. Kuzmichev), buoy.rina@leap-energy.com (R. Buoy). http://dx.doi.org/10.1016/j.matcom.2017.01.005 0378-4754/ c ⃝ 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.