Estimating reservoir flow capacity is crucial for production estimation, hydraulic fracturing design and field development. Laboratory experiments can be used to measure the permeability of rock samples, but the results may not be representative at a field scale because of reservoir heterogeneity and pre-existing natural fracture systems. Diagnostic Fracture Injection Tests (DFIT) have now become standard practice to estimate formation pore pressure and formation permeability. However, in low permeability reservoirs, after-closure radial flow is often absent and this can cast significant uncertainties in interpreting DFIT data. In addition, the established methods for analyzing DFIT data make two oversimplified assumptions: (1) Carter's leak-off and, (2) Constant fracture compliance (or stiffness) during fracture closure. However, both assumptions are violated during fracture closure and this is why G-function based models and subsequent related works can lead to an incorrect interpretation and are not capable of consistently fitting both before and after closure data coherently (Wang and Sharma 2017). Moreover, current after-closure analysis relies on classic well-test solutions with constant injection rate. In reality, a "constant injection rate" does not equal "constant leak-off rate into the formation", because over 90% of the injected fluid stays inside the fracture at the end of pumping, instead of leaking into formation. The variable leak-off rate clearly violates the constant rate boundary condition used in existing well-test solutions.

In this study, we extend our previous work and derive time-convolution solutions to pressure transient behavior of a closing fracture with infinite and finite fracture conductivity. We show that G-function and the square root of time models are only special cases of our general solutions. In addition, we found that after-closure linear flow and bilinear flow analysis can only be used to infer pore pressure reliably, but fail to estimate other parameters correctly. Most importantly, we present a new approach to history match the entire duration of DFIT data to estimate formation flow capacity, even without knowing closure stress and the roughness properties of the fracture surface. Our approach adds tremendous value to DFIT interpretation and uncertainty analysis, especially in unconventional reservoirs where the absence of after-closure radial flow is the norm. Two representative field cases are also presented and discussed.