The development of the basic theory and numerical implementation of a three-dimensional hydraulic fracturing simulator has been completed. The description of the model is being presented in two companion papers. The first paper (Part 1) discusses the analytical elements of the theory, while this paper considers the practical applications. The simulator can predict fracture geometry in a layered rock media under a wide range of in situ and fluid treatment conditions.

A total of 15 cases are discussed. The effects of fracture pressure gradient, stress contrast, reservoir layer stiffness, fluid viscosity and initial fracture geometry are evaluated. Resultant fracture shapes are presented and their implications on fracture design presented and their implications on fracture design are discussed. Fracture containment is shown to be strongly dependent on the elastic modulus of the formation, magnitude of stress barriers, and fluid viscosity.


Hydraulic fracture propagation is based on the close coupling of fluid mechanics with elasticity theory. A three-dimensional solution of this problem is quite complex, and in the past, has been almost intractable because of computational as well as analytical/developmental difficulties. Therefore, the problem had been simplified by using two-dimensional and quasi three-dimensional approaches. Each of the models resulting from these approaches has played an important role in the development of hydraulic fracturing simulators.

The two-dimensional models, Palmer and Covel (1983) have divided into two sub groups. The first group assumes a parallel vertical fracture with no slippage between the pay zone and bounding layers (no vertical stiffness). Investigators included in this group are Christianovich and Zheltov (1955), Geerstma and de Klerk (1969), Daneshy (1973), and Settari (1980). The second group (Perkins and Kern, 1979; Nordgren, 1972; Nolte, 1979, etc.) is based on the concept that the width along the pay zone is a function of the vertical fracture height (no horizontal stiffness). The assumption of constant fracture height and one-dimensional fluid flow is common to both groups.

Generally, quasi three-dimensional models are extensions of one or both of the above criteria. Variable fracture height is computed by means of a fracture toughness concept. Stress gradients and contrasts are considered in the vertical growth. Nolte and Smith (1981), Settari and Cleary (1982), and Palmer and Covel (1983) have made contributions in this area.

Simulators resulting from these investigations still have definite applications today, especially where there is leeway for variation in prediction. Yet, because increasingly more prediction. Yet, because increasingly more fracturing jobs require very exacting predictions, there are many situations where nothing short of a three-dimensional model will suffice for accurate problem solution. The current model has been problem solution. The current model has been developed to meet those needs.

The results presented herein are derived from an elastically three-dimensional model coupled with two-dimensional fluid flow (Abou-Sayed, et. al., 1984). There are no assumptions as to the relationship between fracture height, length, and width. Fracture propagation is based on a relationship derived from fracture toughness considerations. The fracture is allowed to develop as the material and fluid properties dictate through the governing equations. Each of the formation layers making up the problem are assumed to be homogeneous, isotropic and elastic. (The homogeneous restriction could be relaxed with relatively minor modifications.) In addition, wellbore symmetry is assumed. The model is also very economical due to its being formulated by the boundary element technique, which makes the computational effort comparable to that of the quasi three-dimensional simulators.

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