A Fully Implicit Hydraulic Fracture Model
Authors: Nghiem L.X. , Forsyth P.A. , Behie A. 
Published: December 2007

Summary
This paper presents some basic features of a hydraulic fracture model and a method for solving the equations for reservoir flow, fracture flow, and fracture mechanics simultaneously. A method for handling the moving-boundary problem associated with fracture propagation is given, and the method's validity is tested by use of grid sensitivity analysis. The model's capabilities are illustrated with examples involving fracture growth, fracture cleanup, and propagation and closure of a waterflood-induced hydraulic fracture.

Introduction
Hydraulic fracturing has been widely used to increase injectivity and productivity of wells in tight formations. The objective of this paper is to present a numerical model of the dynamic fracture process that fully couples the reservoir flow with the flow and mechanics of the fracture. The design of a fracture stimulation can be divided into three stages: prediction of fracture geometry, prediction of fracture cleanup, and prediction of long-term prediction of fracture cleanup, and prediction of long-term fractured well performance. Two popular models for predicting fracture geometry are those of Perkins and predicting fracture geometry are those of Perkins and Kern and of Geertsma and de Klerk. These models predict only two-dimensional (2D) fracture geometry, predict only two-dimensional (2D) fracture geometry, and current research is directed toward prediction of three-dimensional fracture geometry. Hagoort et al. and Settari described a simulator that can predict all three stages of the hydraulic fracturing process. They use a sequential method of solution to solve the reservoir flow equations, the fracture flow equation, and the fracture geometry equation. It is possible that the sequential method may result in small possible that the sequential method may result in small timesteps, especially during the prediction of long-term well performance. The coupling of the moving fracture tip also is handled through a complex fluid loss distribution scheme among the blocks. A fully implicit hydraulic fracture model that can simulate fracture propagation, cleanup and long-term performance is proposed. The simultaneous solution of performance is proposed. The simultaneous solution of all equations enhances the stability of the model. A rigorous treatment of the moving-boundary condition is presented. The discussion is restricted to vertical presented. The discussion is restricted to vertical fractures described by Geertsma and de Klerk's model, but the method of solution can be extended to include any other models for fracture geometry.

The Physics of Hydraulic Fracturing
Fracture Initiation. The equation for fracture initiation pressure pfi has been derived previously as given by pressure pfi has been derived previously as given by(1) where pinit is the initial reservoir pressure. The other terms in Eq. 1 are defined in the Nomenclature.
Fracture Propagation and Closure. During fracture propagation, the pressure in the fracture is given propagation, the pressure in the fracture is given by
(2)(3)
where pfp = propagation pressure, pfp = propagation pressure, pfoc = fracture opening/closing pressure, pfoc = fracture opening/closing pressure, Kc = critical stress-intensity factor, and Lf = fracture half-length.
To keep the fracture propagating, fluid must be inejected to maintain the fracture pressure equal to pfp, and consequently dVf/dt greater than 0, where Vf is the fracture volume. The fracture will close if pf less than pfp or dVf/dt less than 0. For 2D vertical fractures, the height is constant, and during the closing of a fracture Lf remains fixed while the width decreases. If pf drops below pfoc, the fracture will close completely. For the reopening of a fracture, the criterion proposed by Hagoort et al. and Settari was used. During the reopening process, the fracture length will not increase until the fracture pressure reaches pfp (Lfm), where Lfm, is the maximum length attained in the previous propagation phase. phase. Fracture Geometry. Geertsma and de Klerk derived the following equations for the quarter volume Vf of a vertical fracture.
(4)
JPT
P. 1191