Accurate estimation of fracture width is necessary for successful hydraulic fracturing or FracPac treatments. This information becomes even more critical when most of the fluid leakoff is forced through the proppant pack and out through the tip of the fracture during hydraulic fracturing or FracPac treatments. With the development of new, low-leakoff fluid systems, the coupling of fluid flow through porous media and subsequent pressure responses must be incorporated in predicting fracture-width profile.

The generalized width equations, although well-known, are seldom solved for cases other than constant pressure throughout the fracture or for simple pressure distributions. This paper demonstrates how these complex fracture-width equations can be solved for an arbitrary pressure distribution in a simplified approach. The generalized width equations for fracturing calculate a fracture’s width along its length or radius, w(x) or w(r), given a pressure profile, p(x) or p(r).

Two main width techniques are discussed, one based on a modified Perkins-Krech approach and the other on a corrected Valko-Economides approach. A comparison to the Jaeger-Cook deflection equation is made. In the Perkins-Krech and the Jaeger-Cook cases, the method of superposition is used. In the Valko-Economides case, the pressure is fit with a polynomial equation, and the integration is carried out term by term. The results are comparable with a finite element analysis (FEA) model, and good agreement has been observed. Examples are presented on how these techniques can be set up in spreadsheets, allowing the engineer to investigate concepts like "dry tip" and stresses around a fracture perimeter.

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