This paper presents analyses which permit some of the parameters that quantify a fracture and the fracturing process to be estimated from the pressure decline following fracturing. The primary assumptions are consistent with those of current practice—a vertical fracture of essentially constant height, propagating through a quasi-elastic formation, with continuous displacements (i.e., no slip) at the planes bounding the top and bottom of the fracture. The parameters which can be quantified from the pressure decline are the fluid loss coefficient, the fracture length and width, fluid efficiency, and time for the fracture to close.
At the present time there is no direct and simple procedure for evaluating the basic parameters achieved procedure for evaluating the basic parameters achieved during a fracture treatment, such as the length, width and fluid efficiency at the time the treatment ends. There exist fracture models which permit estimation of fracture length and width based on assumed heights, fluid loss coefficient, fluid viscosity, and formation modulus. However, at the end of the treatment there is no way of knowing if the assumed parameters were correct, with the possible exception of fracture height near the wellbore. Techniques for defining the fracture geometry from production performance do not satisfy this need completely because of the time delay required for the data, and the inferred lengths and widths (fracture conductivity) are average values over sections which significantly contribute to production and are at best lower bounds on the actual dimensions created by the fracturing fluid. As the volume and unit cost of fluids used for a typical fracture treatment continue to increase, so does the need for better definition of the fracture geometry created by a particular treatment in a particular zone. particular treatment in a particular zone.
Based on the concept of a vertical fracture as presented by Perkins and Kern and as extended by presented by Perkins and Kern and as extended by Nordgren, relationships based on the fracturing pressure decline are derived in this paper. pressure decline are derived in this paper. References and illustrations at end of paper. The basic assumptions for the applicability of the analyses are that the fracture:
has essentially constant height.
propagates through a quasi-elastic formation with negligible slip of bedding planes.
was created by a constant injection race of apower-law fluid into two symmetric wings.
propagates continuously during pumping and propagation stops when pumping stops.
closes freely without significant interference from proppant.
Since actual hydraulic fractures will deviate by varying degrees at various times from these idealized assumptions, the utility of the analyses presented for any application will depend on the degree of deviation and the sensitivity to the deviation. The ultimate utility will depend on the ability to provide realistic engineering correlations and predictions. As discussed in the "Discussion of Applications" section, the most likely deviations from assumptions 1, 4, and 5 above, would produce estimates erring on the conservative side for the fluid loss coefficient and length.
In the following, the derived relationships are summarized with equation numbers corresponding to those in later sections of the paper. The relationships are expressed in terms of the dimensionless shut-in time,= t/to where to is the time since pumping stopped (shut-in) and to is the pump time prior to shut-in.
The rate of pressure decline at time is
with f() defined by the bounds