A mathematical model for transient testing of naturally fractured shale gas reservoirs

Transient linear triple porosity diffusion models (TPM) have been developed to justify the unexpected high gas production observed in naturally fractured, nanoporous shale gas reservoirs (NNSGR's). However, a critical assessment of the predictive capabilities of these models reveal that if the presence of obstacles, disconnected pathways in the matrix, and/or poor fracture connectivity in the fracture-matrix system are to be considered, these models may be inappropriate to describe the gas transport mechanisms in these complex systems. To overcome these limitations, an anomalous triple porosity model (ATPM) is developed to describe the gas production from a horizontal well producing in a NNSGR, allowing for the possibility of sub-diffusion in the matrix and micro-fracture within the stimulated reservoir volume (SRV). The mathematical model entails the use of a modified constitutive flux relationship for the flow behavior in the matrix, micro and macro-fractures rather than the empirical Darcy's equation. The Laplace transform method is employed to handle the resulting mathematical model to obtain semi-analytical expressions for the dimensionless matrix pressure, micro fracture pressure, macro fracture pressure, and wellbore flowrate. Subsequently, qualitative and quantitative validation of the rate-transient at the wellbore against existing solutions in literature are demonstrated. The ATPM solution converges to both the transient linear double porosity diffusion mathematical model (DPM) and the sequential TPM solutions established in the literature under very limiting conditions. The sensitivity of the introduced parameters was analyzed and presented through numerically generated type curves. Parametric study suggests that the derived ATPM solution exhibits unique behaviors and trends not observed in the previous mathematical models. Furthermore, the parametric study reveals the magnitude of the diffusion exponent affects the slope of the straight line observed in the log-log plot of rate and time. The advantage of the ATPM is that detailed information of the matrix, micro fracture and macro fracture petrophysical properties can be modeled as compared to the TPM. Furthermore, this model may find wide spread application for predicting the performance of horizontal wells producing in NNSGR's where analog studies reveal low spacing aspect ratio in the reservoir.