Productivity enhancement in hydrofractured oil reservoirs

Low permeability formations are often hydrofractured to increase the production rate of oil and gas. This process creates a thin, but highly permeable, fracture which provides an easy path for oil and gas to flow through the reservoir to the borehole. Here we examine the payoff of hydrofracturing by determining the increased production rate of a hydrofractured well. We find explicit formulas for the steady production rate in the three regimes of small, intermediate, and large (dimensionless) fracture conductivities. Previously, only the formula for the large fracture conductivity case was known.We assume that Darcy flow pertains throughout the reservoir. Then, the steady fluid flow through the reservoir is governed by Laplace's equation with a second-order boundary condition along the fracture. We first analyze this boundary value problem for the case of small fracture conductivities. An explicit formula for the production rate is obtained for this case, essentially by combining singular perturbation methods with spectral methods in a function space which places the second-order boundary condition on the same footing as Laplace's equation. Next, we re-cast Laplace's equation as a variational principle which has the second-order boundary condition as its natural boundary condition. This allows us to use simple trial functions to derive accurate estimates of the production rate in the intermediate conductivity case. Then, an asymptotic analysis is used to find the production rate for the large fracture conductivity case. Finally, the asymptotic and variationally-derived production rate formulasare compared to exact values of the production rate, which have been obtained numerically.It may be feasible to create more than a single fracture about a borehole. So we also develop similar asymptotic and variational formulas for the production rate of a well with multiple fractures.