A Study On Gas And Water Two-Phase Seepage In Coal-bed Methane Wells With Vertical Fractures

Coal-bed methane(CBM)is a clean and unconventional gas resource that plays an important role in natural gas exploitation of China.The gas-water two-phase seepage flow appears in the process of water drainage and gas recovery,resulting in changes of pressure in CBM reservoirs.Therefore,the goal of this paper is to solve mathematical models of wells and vertically fractured wells in two-phase CBM reservoirs,based on the characteristics of gas desorption and the principle of drainage in CBM reservoirs.Also,the mathematical models are applied to calculation of cases and parameter analysis.And this is of theoretical significance for predicting the bottom-hole pressure and investigating production performances.It can also guide the gas and water production in CBM reservoir engineering.The desorbed gas volume,gas deviation coefficient,viscosity and comprehensive compression coefficient are all functions of pressure,thus pseudopressure and pseudotime functions different from those of conventional gas reservoirs can be introduced to linearize the control equations of two-phase CBM reservoirs.The resulting linearized equations can be non-dimensionalized to obtain equations similar to those of single-phase flow.Therefore,according to the methods of single-phase seepage flow,we can obtain the dimensionless pseudopressure solutions.Combined with the material balance equation(MBE),the differential equation of material balance and the comprehensive compression coefficient of two-phase seepage flow,the dimensionless pseudopressure can be adopted to solve the pressure and average gas saturation variations in CBM reservoirs.In this paper,different methods are used to analyze three kinds of mathematical models in two-phase CBM reservoirs.Firstly,a radial-flow well model is established in two-phase CBM reservoirs.The Laplace transform and Stehfest numerical inversion are used to solve the dimensionless pseudopressure for reservoirs under constant flow rate.Furthermore,the solutions of an application example are verified by numerical simulations.Secondly,a vertically fractured well model is built based on one-dimensional flow in two-phase CBM reservoirs.In the case of infinite CBM reservoirs,a similarity transformation method is used to obtain the dimensionless pseudopressure.In another case of CBM reservoirs with closed boundary,the Laplace transform and Stehfest numerical inversion are adopted to obtain dimensionless pseudopressure solutions under constant flow rate.Instead,the Duhamel convolution method is applied to solve the dimensionless rate solutions under constant bottom-hole pressure.Additionally,type curves of dimensionless pseudopressure are almost coincided with those of dimensionless rate by introducing a definition of material-balance pseudopressure.Finally,a vertically fractured well model is also proposed based on the two-dimensional flow in two-phase CBM reservoirs.Due to the existence of two-dimensional variables in space,the mathematical model is successively solved by Fourier and Laplace transform.And then the Stehfest numerical inversion is used to obtain the dimensionless pseudopressure.According to the three kinds of mathematical models above,applications of two-phase CBM reservoirs and the analysis of parameter sensitivity are provided.This work can form a reliable method to predict dynamics of pressure in two-phase CBM reservoirs.