Analysis of Hydraulic Fracture Behavior With a Three Dimensional Microcomputer Simulator
Authors: Chen Z.X. , Chen M. 
Published: December 2007

Abstract
The paper develops a three-dimensional (3D) hydraulic fracturing model that solves the coupled fluid flow and elastic rock deformation problems. In the model, fracture height growth is formulated from the critical stress intensity factor criterion and plane strain elasticity solutions, the fracturing slurry is modeled as a non Newtonian, power law fluid, the heat transfer mechanism between fracturing fluid and formations is analyzed, the sand transport simulation method for multiple stage injections is derived. A new numerical solution procedure is introduced to ensure the calculation efficiency. For the purpose of oil field use, a hydraulic fracturing simulator is developed, which runs under Microsoft Windows with friendly interface, high efficiency and in line help system. The simulator can be used to predict and monitor the fracture extension and its geometry during field operation. The parameter study is conducted with the simulator and the effect on fracture geometry and proppant settling profile of in-situ stress profile, rock fracture toughness, properties of fracturing fluid, pumping rate and Young's modulus of payzone are evaluated.

Introduction
Hydraulic fracturing has been used successfully for enhancing oil, gas, coal-bed methane production. The effectiveness of hydraulic fracturing in most applications is directly dependent upon the geometry of fractures created within the layered formations, especially in oil and gas production, where hydraulic fracturing is required to maximize fracture length and propped width while maintaining fracture height as close to pay zone height as possible. Therefore, a numerical model simulating hydraulic fracture propagation under reservoir and operation conditions must be established to optimize hydraulic fracturing treatment design.
Earlier fracturing treatment design procedures were based on two dimensional geometric fracture assumptions: either a penny shape or constant height was supposed. Unless the minimal horizontal stress is constant along the vertical direction or a high confining stress in the bounding layers these assumptions cannot be justified. In recent years, some three dimensional hydraulic fracturing models (3D) are introduced, which can simulate the fracture height variations during fracturing treatment. The theory describing a 3D model is described below.

Model Description
The model simulates the hydraulic fracturing process under the following assumptions.Rock formations are assumed to behave as isotropic linear elastic materials.Dominant fluid flow in the fracture is in the direction of the length.Stress contrasts between the pay zone and barriers are allowed.Fracture is assumed to be in elliptical shape both in vertical and in length directions.Fracture length is much longer than height.

Pressure Drop Equation.
The pressure drop along an elliptical cross-sectional fracture for 1 D flow of a power-low fluid is given by the relation: (1)

Fracture Width Equation.
The fracture is divided into a number of vertical sections, and each vertical section is regarded as a line crack in plane strain, which may penetrate into regions of upper or lower barriers with different in-situ stresses. Fig. 1 illustrates a vertical section, and the stress profile acting on the fracture section surface. The relationship between fracture width and net pressure (fluid pressure minus in-situ stress) in fracture can be obtained by using England and Green results and the superposition theorem:
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