Analytical Solutions for a Quad-Linear Flow Model Derived for Multistage Fractured Horizontal Wells in Tight Oil Reservoirs

Analysis of microseismic field data shows that the stimulated reservoir volume (SRV) in unconventional reservoirs partially covers the area between hydraulic fracture stages. Consequently, we are often faced with an effective fracture network area (EFNA) rather than a full SRV in such reservoirs. In this paper, we develop a new semi-analytical solution for pressure of hydraulically fractured horizontal wells in tight oil reservoirs with various SRV sizes. Our model is based on four linear flow regions including the hydraulic fracture, the stimulated reservoir, the unstimulated reservoir, and the outer reservoir region. Flow in each region is represented by a set of governing equations and boundary conditions that are coupled to those of other regions. The dual-porosity formulation represents the SRV, while single-porosity formulation is used for other flow regions. We transform the coupled system of equations into Laplace domain, solve for wellbore pressure, and invert the solutions back to time domain numerically. We validate the semi-analytical solutions by comparing them to other semi-analytical solutions in the literature for the special case of trilinear flow. We further validate the quad-linear flow solutions using numerical simulation. Based on the semi-analytical solutions, we generate logarithmic plots of wellbore pressure and pressure derivative. Moreover, we perform sensitivity studies to present the degree to which the solutions vary as size and other properties of the SRV change.