Minimizing the Fluid Used to Induce Fracturing

Abstract:

The less fluid injected to induce fracturing means less fluid needing to be produced before gas is produced. One method is to inject as fast as possible until the desired fracture length is obtained. Presented is an alternative injection strategy derived by applying optimal system control theory to the macroscopic mass balance. The picture is that the fracture is constant in aperture, fluid is injected at a controlled rate at the near end, and the fracture unzips at the far end until the desired length is obtained. The velocity of the fluid is governed by Darcy’s law with larger permeability for flow along the fracture length. Fracture growth is monitored through micro-seismicity.

Since the fluid is assumed to be incompressible, the rate at which fluid is injected is balanced by rate of fracture growth and rate of loss to bounding rock. Minimizing injected fluid loss to the bounding rock is the same as minimizing total injected fluid

How to change the injection rate so as to minimize the total injected fluid is a problem in optimal control. For a given total length, the variation of the injected rate is determined by variations in overall time needed to obtain the desired fracture length, the length at any time, and the rate at which the fracture is growing at that time.

Optimal control theory leads to a boundary condition and an ordinary differential equation in time whose solution is an injection protocol that minimizes the fluid used under the stated assumptions. That method is to monitor the rate at which the square of the fracture length is growing and adjust the injection rate proportionately.