This paper presents a novel analytical solution to predict the expansion of drainage volume for compound transient linear flow system consisting of stimulated-reservoir volume (SRV) and unstimulated shale matrix. The distance of investigation (DOI) has been demonstrated to be a useful notion for production data analysis; e.g., optimization of spacing of hydraulic fractures, and evaluation of stimulated-reservoir volume (SRV). However, the DOI concept has been mainly used to evaluate dynamic drainage volume (DDV) for the linear flow system within SRV. Thus, in view of production from both SRV and unstimulated shale/tight sand matrix; the DOI (or DDV) of transient linear flow in compound multi-stage fractured reservoirs has not been determined yet.
In this work, we derive the DDV equation analytically for compound transient linear flow system under constant-flowing-pressure and constant-production-rate condition for the compound linear flow system. To that end, coupled with the analytical equation calculating the pressure and DDV of linear flow within only SRV, the compound linear flow solution within both SRV and unstimulated matrix is derived. Laplace transform and numerical inversion are implemented to obtain the semi-analytical solution. The pressure front is calculated by implementing the impulse respond concept, which is the maximum rate of pressure response. In addition, a relation was established between DOI and square root of time using multivariable regression, based on results of 2000 cases calculated from our semi-analytical solution. The cases capture the impact of several parameters, including different diffusivities within SRV and unstimulated matrix, hydraulic fracture length, and fracture spacing.
Our solution suggested that the DOI demonstrates a linear relation to the square-root-of-time for both linear flow within SRV and the compound linear flow system. The advancement of DDV within stimulated-reservoir volume is significantly faster than that within unstimulated matrix. The majority of production is attributed to size of stimulated-reservoir volume. To verify the accuracy of the new DDV equations, we analyze the synthetic production data from a series of fine-grid numerical simulations. Finally, real-time average reservoir pressure and ultimate oil recovery were evaluated for a few selected wells from one of the US shale plays.
This work has the following contributions: (1) provide a practical equation of DDV for multi-stage fractured horizontal well; (2) capability to ascertain DDV and associated average reservoir pressure throughout different transient-flow regimes for fractured horizontal well; (3) calculation of oil recover factor. The latter is important in project economics.