Multi-stage fracturing is currently key technique to develop shale reservoirs. Different analytical models have been proposed to fast investigate post-fracturing pressure- and rate-transient behaviors, and hence, estimate key parameters that affect well performance. However, previous models neglect reservoir heterogeneity and typical seepage characters of shale, such as adsorption/desorption, gas slippage, and diffusion effects. This paper presents an analytical model for pressure- and rate-transient analysis of multi-stage fractured shale reservoir, considering heterogeneity, typical seepage characters and, specifically, fluids flow from upper/lower reservoir when vertical fractures partially penetrate the formation.

This model is similar to five-region-flow model, but subdivides the reservoir into seven parts, namely, two upper/lower-reservoir regions, two outer-reservoir regions, two inner-reservoir regions, and hydraulic fracture region, which are all transient dual porosity media except the hydraulic fracture. As reservoir heterogeneity along the horizontal wellbore is included, the fracture distribution can be various. Fracture interference is simulated by locating a no-flow boundary between two adjacent fractures. The actual locations of these no-flow boundaries of a specific heterogeneous reservoir are determined based on the pressure value which varies with time and space. Thus, the two sides of this boundary has minimum pressure difference, satisfying the no-flow assumption. Adsorption/desorption, gas slippage and diffusion effects are included for rigorous modeling of flow in shale.

This model is validated by comparing with commercial well testing software, obtaining a good match in most flow regimes. And the simplified form of our formulas (no upper/lower reservoir contribution, no desorption, slippage and diffusion effects) have exactly the same solutions as Brown's (2011) model. Log-log dimensionless pressure, pressure derivative and production type curves are generated to conduct sensitivity analysis. Results suggest that larger desorption coefficient causes smaller pressure and its derivative value as a larger proportion of gas is desorbed in formation and contributes to productivity. We also compared our solutions with Azari's (1990, 1991) work, where the effect of fracture height is merely treated as a skin factor. Results show that our model is more accurate in partially penetrating cases, and errors of Azari's method become particularly noticeable in early-middle time response. The influence of other parameters, such as matrix permeability, matrix block size, secondary fracture permeability and hydraulic fracture conductivity, are also discussed. Optimal fracture pattern is selected based on cumulative production. Besides, field data are analyzed and compared graphically with modeling solutions, and reliable results are obtained.

As numerical and semianalytical methods require extensive computing processing, this model is a practical alternative to predict well-testing results and select optimal well pattern of shale reservoirs. Reservoir heterogeneities in vertical direction can be further added to our model by vertically subdividing the reservoir into more parts.

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