Fracture Design in Liquid-Saturated Reservoirs
Authors: Wahl Harry A. 
Published: December 2007

Abstract
This paper presents methods of designing hydraulic fracture treatments in formations saturated with slightly compressible liquids. Howard and Fast describe the fluidloss control resulting from the viscosity and compressibility effects of the reservoir fluid in terms of a fracturing fluid coefficient. However, for horizontal fractures, this coefficient is applicable to infinitely thick reservoirs. These methods of evaluating fluid loss from horizontal fractures during treatment differ from previous methods in that:formation thickness is incorporated in the velocity function describing the rate of fluid loss to the formation; andfor a given thickness the relative position of the fracture to the horizontal boundaries of the reservoir or any intermediate permeability barrier is incorporated in the design procedure.
T 12 is theory is applied to several actual and hypothetical field cases. The results of these calculations demonstrate that the effect of formation thickness on fluid loss from horizontal fractures may result in fractures several times larger than predicted by other methods.

Introduction
Hydraulic fracturing has been the dominating well-stimulation technique during the last decade. From the time it was commercially available, fracturing has had a very favorable success ratio compared to other stimulation methods. However, numerous wells that were successfully stimulated remained economic failures. Only in the last few years has the oil industry recognized the importance of designing treatments for each well which will yield the maximum profit. This implies that for a given set of treatment conditions the size and general physical characteristics of the fracture can be determined. One of the outstanding contributions in the field of fracture design was presented by Howard and Fast.' In their paper, three flow mechanisms that control fluid loss from the fracture to the formation are defined:viscosity effects of the fracturing fluid,viscosity and compressibility effects of the reservoir fluid, andwall-building effects of a treated fracturing fluid.
The rate of fluid loss to the formation adjacent to a fracture is shown to be a function of time. For each flow mechanism the velocity function v(t) is expressed as a constant or fracturing fluid coefficient divided by the square root of time. This form of v(t) can be used in the Carter equation, as developed in the Howard and Fast paper, to estimate the areal extent of fractures. This equation may be written as
QW 22A(t) = ex . erfc X + X −1 .....(1)4C
where X = .W
In the following sections the effect of formation thickness on fluid loss from horizontal fractures is evaluated when the controlling flow mechanism is the viscosity and compressibility of the reservoir fluids. The major assumptions inherent in the Carter equation are brought out in subsequent mathematical developments.

THE EFFECT OF RESERVOIR FLUIDS ON FRACTURE EXTENSION
When a fracture treatment is performed using a fracturing fluid identical to the reservoir fluid, the only flow mechanism controlling fluid loss is the viscosity and compressibility effects of the reservoir fluids. In developing a fracturing fluid coefficient for this mechanism, Howard and Fast considered a segment of the formation bounded by a unit area on the fracture face and extending to infinity in the direction of leak-off as shown in Fig. 1(a). The direction of leak-off z is normal to the fracture face. The equation governing the linear isothermal flow of fluid into this segment is
..................................(2)
where =
Eq. 2 is a form of the diffusivity equation.
JPT
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