“Triple porosity” modeling of transient well test and rate decline analysis for multi-fractured horizontal well in shale gas reservoirs

•Source functions with the consideration of adsorption and desorption characteristics of shale gas reservoirs are derived.•The expressions of horizontal well transient pressure and rate decline in shale gas reservoir are acquired.•The effect of influence factors on discrete elements is also analyzed.

This paper presents a “tri-porosity” mathematical model to describe fluid flow from shale gas formation to a multi-fractured horizontal well. The shale gas is stored by either free or adsorption condition: free gas is stored in both the micro-pores in matrix and natural fractures and the adsorption gas is adsorbed on the surface of matrix particles, which will be desorbed into the micro-pores and then flow into the natural fractures. Based on the Langmuir adsorption isotherm theory, we obtained the point source solution in the Laplace space. Using the Stehfest algorithm, the log–log type curves of dimensionless pseudo-pressure and rate decline curves are obtained. We also analyzed the effects of horizontal well length, fracture number and Langmuir volume on type curves. The results obtained in this paper have important significance to understand the transient pressure and rate decline behaviors of multi-fractured horizontal well in this reservoir.