Multi-stage fracturing is currently the most effective method to exploit tight sand reservoirs. Various analytical models have been proposed to fast and accurately investigate post-fracturing pressure- and rate-transient behaviors, and hence, estimate key parameters that affect well performance. However, these models mainly consider 2D flow, neglecting fluids flow from upper/lower reservoir when vertical fractures partially penetrate the reservoir. Although for linear flow model, Olarewaju et al. (1989) and Azari et al. (1990, 1991) have studied the effects of fracture height, they merely used a skin factor. Moreover, reservoir heterogeneity is seldom included. This paper presents an analytical model for multi-stage fractured horizontal wells (MFHWs) in tight sand reservoirs, accounting for upper/lower reservoir contributions, reservoir heterogeneity and threshold-pressure gradient (TPG).

The model is extended from "five flow region" model and subdivides the reservoir into seven parts including two upper/lower flow regions, two outer flow regions, two inner flow regions and hydraulic fracture flow region. Reservoir heterogeneity along the horizontal wellbore is considered, thus, the fracture distribution can be various, and fracture pattern optimization strategies are documented. Fracture interference is simulated by locating a no-flow boundary between two adjacent fractures. The exact locations of no-flow boundaries are determined based on boundary's pressure which is a function of time and space. Thus, the no-flow boundary has minimum pressure difference between its two sides during the well production, making the no-flow assumption reasonable. The experimentally observed TPG and pressure drop within the horizontal wellbore are included.

Modeling results are compared with those from well-testing software KAPPA Ecrin, obtaining a good match in most flow regimes. Specifically, the effects of upper/lower reservoir contributions and TPG are studied under constant-rate and constant-pressure conditions respectively. Log-log dimensionless pressure, pressure-derivative and production type curves are generated. Results suggest that fracture penetration ratio dominates the early-middle time pressure response. The start time of boundary-dominate flow are significantly influenced by penetration ratio. The larger the penetration ratio is, the earlier boundary-dominate flow regime will arrive. As for production response, with penetration ratio increases, the early dimensionless rate becomes larger, indicating higher flow rate, however, the late-time (tD>10) production becomes smaller, that is, the production declines quicker when penetration ratio is large. When TPG is considered, differences in both pressure and production response mainly appear in middle-late time. Greater TPG results in higher pressure drop and accelerates production decline. But this influence is marginal when TPG is small (TPG<0.4psi/ft). Effects of other relative parameters, such as formation permeability, heterogeneity, fracture length, conductivity, and wellbore storage are systematically investigated. Besides, field data are analyzed and compared graphically, using type curve matching, and reliable results are obtained.

Low CPU demands and minimal data requirement of this model enable the operators to predict well testing results under different fracture patterns in a simple but effective way.

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