Fracturing-Pressure Analysis for Nonideal Behavior
Authors: Nolte Kenneth G. 
Published: December 2007

Fracturing-Pressure Analysis for . Non ideal Behavior

Summary.
The analyses of fracturing pressure, during injection and after shut-in, provide powerful tools for understanding and improving the fracturing process. The current framework of analyses is based on several idealized assumptions. This paper assesses these assumptions paper assesses these assumptions and quantifies the effect of the deviations. The deviations are investigated by numerical simulation, and the results are presented in terms of pressure/time plots, with dimensionless pressure/time plots, with dimensionless parameters used when appropriate. parameters used when appropriate. These plots, with the associated derivatives (slopes), provide diagnostics for interpreting the pressure responses and identifying deviations from assumptions. The focus is for vertical fractures with large horizontal penetrations relative to the vertical height penetrations relative to the vertical height -i.e., the Perkins-Kern assumption.

Introduction
The important of the analysis of fracturing pressure was recognized in 1958, and the pressure was recognized in 1958, and the current application was reviewed in Refs. 2 and 3. This section provides an introductory perspective, required for subsequent sections of this paper. Fig. 1 illustrates the evolution of the geometry and pressure during injection. The initial stage consists of expanding radial shapes (point source) or elliptical shapes (line source) with decreasing pressure. This stage ends when vertical growth is restricted, from above and below, by competent stress barriers. During the second stage, growth is primary horizontal with increasing pressure. As the net pressure approaches the stress difference, of either barrier formation, the vertical height increases significantly and the net pressure asymptotically approaches the stress differences. The diagnostic log-log plot of net pressure vs.time indicates essentially pressure vs.time indicates essentially straight lines for these three stages. The case of initial radial growth was investigated and characterized into periods dominated by either fluid viscosity or rock toughness. An analysis of elliptical growth showed that, for the limiting case of small aspect ratios (small penetration relative to height), the Khristianovich-Geertsma-de Klerk. (KGD) geometry assumption was valid. The second stage, with increasing horizontal propagation and very limited vertical propagation and very limited vertical growth, follows the Perkins-Kem-Nord-gmn (PKN) assumptions. The pressure, Perkins-Kem-Nord-gmn (PKN) assumptions. The pressure, response for this case and subsequent deviations are characterized by the slopes of the diagnostic log-log plot, as Fig. 2 illustrates. For this figure, Conditions I and II-a correspond to Stages 2 and 3 of Fig. 1. The zeroslope Condition B of Fig. 2 results from inefficient extension when the formation pressure capacity, defined by the in-situ stress pressure capacity, defined by the in-situ stress state (Fig. 14.14 of Ref. 2), is reached. The approximate unit slope (Condition III-a) results from proppant at the fracture extremities, restricting extension. The negative slope, Condition TV, results from penetrating a sum barrier with the subsequent penetrating a sum barrier with the subsequent unrestricted vertical growth into a lower-stress formation. The ideal assumptions for Stage 2 (Fig. 1) and deviations (Conditions II-a and III-a, Fig. 2) are addressed in the following section. After injection, the fluid remaining in the fracture is lost to the formation. From the material balance and negligible fluid compressibility, the fracture volume, Vf, at shut-in is related to the injected volume, Vi, by Vf=n Vi. The proportionality parameter, n, is called the fluid efficiency. After parameter, n, is called the fluid efficiency. After shut-in, the volume lost, VLs, equals the decrease in fracture volume, Vf. For the case without proppant, as, for a calibration treatment, the volume lost between shut-in and closure, VLs (tc), equals the fracture volume at shut-in. A convenient ratio is the volume to volume lost during pumping, VLp, or pumping, VLp, or (1) Also, (2) (3) (4) and (5)
where . The expression on the right side of Eq.1 is for no proppant and is the time to close. Eq. 4 expresses the change in fracture volume in terms of the change in average width and fracture area; Eq. 5 expresses the average width in terms of the net pressure at the well (is the fracture closure pressure). Eq 5 results from the elastic behavior of the fracture, to the net pressure, and the pressum gradient in the fracture from the fluid pressum gradient in the fracture from the fluid flow and rheology, characterized by . The fracture compliance, depends on the elastic moduli of the formation and a characteristic dimension. Both and can be defined for the idealized models appplicable to Stages I and 2 of Fig. 1. Combining Eqs. 3 through 5 provides
(6) Eq. 6 identifies ideal assumptions required for a direct inference of the fluid loss (volume per unit fracture area,) from the change in wellbore pressure, .
JPT
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