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Introduction

Over thc past 25 years, numerous hydraulic fracture models have been developed to predict the behavior of underground fracture propagation. Typical fracture propagation models predict fracture height. length, and width based on formation mechanical and fracture fluid properties. Fracturing fluids and effects of leakoff also have great influence on ultimate fracture dimensions. A recent paper by Warpinski et al. illustrates the diversity that still exists in the fracture propagation modeling problem.

There are basically two types of three-dimensional fracture models. Pseudo-three-dimensional models, that run efficiently on microcomputers, are mostly applicable in reservoirs bounded by high in-situ stress layers that can suppress rapid fracture height growth. Full three-dimensional models, on the other hand, require more computer memory. run time, and user expertise to design and evaluate a fracture treatment. Reducing CPU time is important, particularly during a fracture design optimization process, since numerous fracture predictions and sensitivity analyses are needed to determine the optimal fracture design.

In this paper, we will illustrate how to couple the equations describing formation elasticity, fracture fluid flow, and fracture mechanics and develop an efficient semi-analytical technique to determine fracture geometry. We have assumed a quadratic pressure and stress profile fit to approximate the fluid pressure inside the fracture and the in- situ stress profile in a multilayered system. We have tested our new model using different reservoir conditions. We provide a detailed sensitivity analysis on reservoir and fluid parameters in order to help us understand how these parameters influence the design of a hydraulic fracture treatment.

DEVELOPMENT OF THE MODEL

Our semi-analytical hydraulic fracturing model has been developed by combining the mass balance equation, the pressure drop equation inside the fracture, the fracture equilibrium equation, and the propagation criteria equation. We will discuss the pertinent equations in this section.

Mass Balance Equation

For a hydraulically created fracture, the following is the mass balance equation.

(1)

where

and

Q(x,z) = volumetric injection rate,

h(x) = fracture height,

P. 221^

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