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 the post-fracturing pressure-and rate-transient behavior, and hence, estimate the key parameters that affect well performance. However, these analytical models mainly consider the 2D flow, neglecting the fluids' flow from the upper/lower reservoir when the vertical fractures partially penetrate the reservoir. Although for linear flow models, Olarewaju and Lee (1989) and Azari et al. (1990, 1991) have studied the effects of fracture height, they merely used a skin factor. Moreover, the reservoir heterogeneity is seldom included. This paper presents an analytical model for multi-fractured horizontal wells (MFHWs) in tight sand reservoirs, accounting for the upper/lower reservoir contributions, reservoir heterogeneity and threshold-pressure gradient (TPG). The model is extended from the "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 a hydraulic fracture flow region. Reservoir heterogeneity along the horizontal wellbore is considered, thus, the effects of the fracture pattern in a heterogeneous reservoir are documented. Fracture interference is simulated by locating a virtual no-flow boundary between two adjacent fractures. The exact location of a no-flow boundary is determined based on the boundary's pressure. Thus, the no-flow boundary has a minimum pressure difference between its two sides during the well production, making the no-flow assumption reasonable. The experimentally observed TPG and the pressure drop within the horizontal wellbore are included. Modeling results are compared with those from the well-testing software KAPPA Ecrin, obtaining a good match in most flow regimes. Both the constant-rate and constant-pressure conditions are studied. Results suggest that the fracture penetration ratio dominates the early-mid time pressure responses and the start time of the boundary-dominated flow. For production responses, it determines the initial productivity and the production decline rate. The existence of the TPG results in a higher pressure drop and accelerates the production decline during the middle-late times. But this influence is marginal when the TPG is small (TPG<0.4psi/ft). Effects of other relative parameters, such as the formation permeability, fracture length, conductivity, fracture pattern, wellbore storage, flow-capacity ratio and storativity ratio for dual-porosity reservoir blocks are systematically investigated. Besides, some field data are analyzed and compared graphically, using type curve matching, and reliable results are obtained. Low CPU demands and the minimal data requirement of this model enable the operators to predict well-testing results under different fracture patterns in heterogeneous reservoirs with TPGs in a simple but effective way.
