Productivity equation of fractured well in CBM reservoirs Yu Lou a, b , Hongqing Song a, b, * , Jiaosheng Yang c , Xiaohe Huang a , He Dong a a Key Laboratory of Educational Ministry for High Efficient Mining and Safety in Metal Mine, University of Science and Technology Beijing, Beijing 100083, China b School of Civil and Environmental Engineering, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing 100083, China c Langfang Branch, Research Institute of Petroleum Exploration and Development, CNPC, Langfang, Hebei 065007, China article info Article history: Received 12 November 2012 Received in revised form 30 November 2012 Accepted 8 December 2012 Available online 26 January 2013 Keywords: Coalbed methane (CBM) Productivity equation Desorption Artificial fracture Mathematical model abstract Due to complex process of CBM transport associated with desorption and diffusion, there has not been an explicit and accurate prediction formula of CBM production for fractured wells. This article presented a productivity equation of fractured well in CBM reservoir regarding desorption and diffusion. Elliptical flow pattern exists around the hydraulically fractured well, and the flow field was divided into two regions. One is high-velocity non-linear flow in artificial fracture, and the other is Darcy flow in elliptical region controlled by artificial fracture. Mathematical models for the elliptical gas flow were established based on conservation of mass and momentum equations, in which Langmuir equation, Fick pseudo-steady state law, and function of pseudo pressure were combined to consider effect of desorption and diffusion. The productivity equation of hydraulically fractured well was presented by coupling the analytical solutions to elliptical gas flow. Effect on gas rate of reservoir properties and production parameters such as desorption rate, diffusion coefficient, drawdown pressure, half-length of artificial fracture, and flow conductivity were clarified based on the productivity equation. It is seen that there exist optimal drawdown pressure and optimal half-length of hydraulic fracture dependent on other parameters. In the excess of optimal values, gas rate will reach a plateau and respond with little increment along with the increase of drawdown pressure or half-length. Consequently, the research provides direct insight of the effect of various parameters on gas rate and theoretical foundation for optimization design of CBM development. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction Productivity equation provides an effective tool and direct insight for gas reservoir engineers to predict production capacity and optimize reservoir design (Rahman et al., 2007; Zhu et al., 2011). However, there has been lack of analytical models for productivity due to complex transport mechanism of CBM including desorption, diffusion and seepage. CBM reservoir is dual porous media (Singh, 2011), constituted by micro-fracture network and matrix. CBM desorbs from matrix and diffuses into micro- fractures, where seepage is the main transport mechanism. CBM reservoirs in China feature low-permeability reservoir properties, leading to wide application of hydraulic fracturing technique for enhancement of CBM production. There exist urgent needs of theoretical analysis of optimal design of hydraulic fracturing. Researchers have done a lot of work on evolution of coal permeability (Wu et al., 2010; Wang et al., 2011; Tao et al., 2012; Pan and Connell, 2012) and modeling for effect of various properties on CBM production (Nie et al., 2012; Adeboye and Bustin, 2011; Wang et al., 2012; Karacan and Okandan, 2000; Anna, 2003; Connell, 2009; Aminian and Ameri, 2009; Wang and Zhang, 2010). In general, numerical simulation, which is complex and time- consuming, is adopted to predict productivity of CBM well. The objective of the study is to present an analytical produc- tivity equation of a single vertical well with a hydraulic fracture, considering desorption and diffusion. It is assumed that elliptical flow regime dominates flow performance in low-permeability dual-porosity CBM reservoirs. The flow field around the hydraulic fracture was divided into two regions. One is high-velocity non- linear flow in artificial fracture, and the other is Darcy flow in elliptic region controlled by artificial fracture. Based on Langmuir equation and pseudo-steady diffusion in matrix and Darcy flow in the network of micro-fractures, mathematical models for methane flow in the above two regions were established. Analytical solu- tions were presented and productivity equation was derived. 2. Description of physical model Due to poor physical properties of CBM reservoirs, hydraulic fracturing is frequently adopted for enhancement of gas recovery. * Corresponding author. School of Civil and Environmental Engineering, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing 100083, China. Tel.: þ86 10 82376239. E-mail address: songhongqing@yahoo.com.cn (H. Song). Contents lists available at SciVerse ScienceDirect Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse 1875-5100/$ e see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jngse.2012.12.001 Journal of Natural Gas Science and Engineering 11 (2013) 39e45