Summary
The theory describing a pseudo-three-dimensional (pseudo-3D) hydraulic fracturing model that solves the coupled fluid-flow and elastic-rock-deformation problem associated with a fracture propagating into a zone composed of three or more layers is presented. The fracture is initiated in the center layer. Fracture growth is formulated from the critical-stress-intensity-factor criterion, and fracture width is obtained from plane-strain elasticity solutions. Fluid fronts and proppant settling during fracture closure are tracked during the treatment. Fracture parameters obtained by this model show excellent agreement (6% maximum difference) with the solution given by a 3D simulator. Also, designs of hydraulic fracturing treatments depicting ways to minimize fracture growth and to optimize proppant distribution are described. The explicit expressions developed for modeling the fracture growth and fracture opening have reduced the complexity of the formulation and the computational effort.
Introduction
The coupled problem of fluid flow and elastic rock deformation associated with hydraulically induced vertical fractures in hydrocarbon-bearing reservoirs has been considered by many authors. Earlier work was based on lateral extension of fixed-height fractures. Constant-height models, particularly of the Perkins and Kern formulation, were subsequently extended to include vertical growth prediction. These models are known as the pseudo-3D simulators. Most of these models assume one-dimensional (1D) fluid flow in the direction of the length (no pressure drop in the vertical direction). This assumption is consistent with fractures in which the length is significantly greater than the height. Other models assume a modified pressure drop in the vertical direction on the basis of the pressure drop in the lateral direction. Three-dimensional hydraulic fracturing simulators were introduced in the literature during the last 9 years. Computational time for 3D modeling is comparable to a two-dimensional finite-element problem of equivalent mesh density, but run times become substantially high for elongated fractures (fractures with height-to-length ratios much smaller than one). Pseudo-3D formulation on the other hand, provides a significant reduction in computational effort and results in good agreement with 3D formulation for elongated fractures. The theory describing a pseudo-3D model including applications is described below.
Hydraulic Fracturing Model
The pseudo-3D formulation presented here models the fracturing phenomenon under the following assumptions. 1. All rock formations are assumed to behave as isotropic linear elastic materials. 2. Dominant fluid flow is in the direction of the length. 3. Modulus and stress contrasts between the pay zone and barriers are allowed. 4. Proppant settling during closure is considered. The linear-elastic-response assumption permits application of the principle of superposition to determine the combined effects of the stress contrasts, fluid pressure gradients, and formation pressure gradients in the fracturing mechanism. This principle is valid when the rock deformation caused by the fracture does not affect the magnitude of the far-field in-situ stresses (limited pumping times). Dominant fluid flow in the direction of the length is consistent with fractures that have height-to-length ratios smaller than one. For these fractures, the pressure drop in the lateral direction is much greater than that in the vertical direction. A vertical pressure drop estimated from the lateral pressure drop is used to prevent unstable vertical growth in fractures with height-to-length ratios approximately equal to one. The elastic solution for a crack crossing an interface 16 is used to determine the effects of modulus and Poisson's ratio contrasts. This effect is significant only for soft formations bounded by stiffer formationsi.e., coal formations bounded by higher-modulus sand-stones or shales. High-viscosity fluids have shown small settling velocities during pumping. Depending on the permeability of the formation, however, settling during closure can be significant. Settling velocities are computed with Novotny's equations.
Description of the Fracturing Process.
The injection of high-pressure fluids into the reservoir leads to fracture initiation followed by fracture propagation into the reservoir and, in most cases, into the bounding formations. This formulation models the fracture propagation process caused by localized failure of the near-crack-tip region. The extent of the fracture migration depends on fluid-induced pressures, the volume of fluid pumped, and the in-situ stress contrasts provided by the bounding layers.
Fracture-Propagation Criterion.
One fracture-growth criterion that describes rock failure states that fracture propagation occurs as the stress-intensity factor, K, at the propagating front (Mode 1 considered here) equals the critical stress-intensity factor. Kc. Vertical growth is calculated on the basis of this criterion. Lateral fracture growth is obtained by advancing the fracture in steps of constant length increments at variable times. The time increment is obtained on the basis of the volume-balance equations.
Governing Equations.
The equations defining pressure drop, vertical fracture growth, fracture width, fluid loss, volume, and mass balance for a discretized fracture in sections of equal incremental lengths and variable heights (Fig. 1) are described below. Pressure-Drop Equation. The pressure drop along an elliptical cross-sectional fracture for ID flow of a power-law fluid is given by the relation
,..............................(1)
where, .............................(2)
,........................(3)
p=fluid pressure above reservoir minimum in-situ stress (pf - p) determined at pay-zone center, q1/2=flow rate along length, w max = maximum fracture opening in cross section, fw = width function at x, H = fracture half-height, n' = fluid-flow behavior index, K'= consistency index for fluid, and Fc = factor defining correspondence between parallel plates and elliptical conduits. To prevent unstable, and unrealistic, vertical fracture migration from being predicted, a vertical pressure gradient, gv. is superposed on the lateral pressure to define the drop in pressure for the vertical flow:
.........................................(4)
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