A Semi-Analytical Mathematical Model of Off-Center Multi-Stage Fractured Horizontal Well in Circle Bi-Zonal Gas Reservoir

Reservoir permeability and stress heterogeneous distribution lead to different hydraulic fracture lengths and angles. Gas reservoirs' heterogeneous characteristics lead that well is not the center of the circle high-permeability regions after large-scale fracturing. Traditional pressure transient model cannot be used in this case. When a horizontal well length is relatively small, radial composite model can be used to simulate wellbore pressure. Therefore, the aim of this paper is to present a semi-analytical mathematical model of the off-center fractured horizontal well with a circle high-permeability region. The coupling approach of hydraulic fracture and matrix model is employed to solve mathematical model. The wellbore pressure transient solution can be obtained by the Laplace transform, Gauss elimination and the Stehfest numerical inversion. The results show that the wellbore pressure response curve of this model includes eleven flow regimes. When the well is not the center of the reservoir, an obvious derivative curve “upwards” will emerge after the inner radial flow regime, which is different from the previous fractured horizontal well model. Hydraulic fracture properties and geometries distribution have obvious influence on derivative curves characteristic of the early regime. Off-center distance and reservoir properties have distinct influences on derivative curves characteristic of the middle regime. This semi-analytical mathematical model can provide a guide on high-permeability region depiction, hydraulic fracture, and reservoir parameter inversion.