By substituting net pressure values for the fluid mechanics of typical fracture growth calculations, numerical models for calculating fracture growth from these pressures, using any of three popular fracture width equations, have been developed. One model uses expected net pressure behavior, ideally obtained from a similar treatment in an offset well, to predict fracture growth for design calculations. A second model uses collected pressures and injection or flow-back rates to calculate fracture dimensions over time, allowing fracture behavior to be monitored during a treatment or analyzed afterward.
The governing equations for these models were examined under specific conditions, leading to guidelines for interpreting fracturing pressures under a variety of conditions.
And, by considering fluid mechanics, as well as fracture mechanics and a volume balance, expected pressure behaviors for power-law fluids in all three fracture geometries have been determined.
Normal hydraulic fracturing treatment design calculations combine fracture mechanics, fluid mechanics, and a volume balance to predict fracture growth with time. Fracture mechanics relate fracture width to pressure and fracture length, height, or radius; fluid mechanics relate pressure to injection rate, width, and length or radius and the volume balance relates fracture volume to injection and fluid-loss rates.
As pointed out by Shlypaborsky et. al.,1,2 pressures obtained during fracturing treatments do not always agree with pressures predicted by fracture design models. They listed five factors that had the potential for causing this disagreement: (1) high perforation friction pressure, (2) high friction pressure in the fracture, (3) the generation of multiple parallel fractures, (4) higher actual fracture toughness values than measured in the lab, and (5) a non-penetrating region near the fracture tip. To isolate the cause of the disagreement, they measured overpressure, the difference between down-hole instantaneous shut-in pressure and the least principal stress, and thus eliminated the three friction-related effects. The overpressure, the result of one or both of the remaining two factors, was then used to determine an apparent fracture toughness. The apparent fracture toughness was subsequently used in a geometry model that considered fracture toughness in its solution to the fracture mechanics portion of the problem.
To allow all four of the factors that may occur within the fracture to be compensated for and to provide more flexibility, models were developed that substitute net pressure – – i.e., the difference between bottom-hole treating pressure and least principal stress – –for fluid mechanics calculations. By substituting either a given net (excess) pressure value, a correlation between net pressure and time, or a set of net pressure values for the net pressures determined through fluid mechanics relationships, fracture geometry models that can be used in fracturing treatment design, monitoring and analysis have been developed. The models allow calculations to be made for radial (penny-shaped) fracture geometry and for geometries based on Khristianovic-Zheltov3 and Sneddon4 width equations for constant height fractures. By considering the variation of injection rate and pressure with time, the version of the model developed for treatment monitoring and analysis can be used to calculate fracture behavior during shut-in and flow-back as well as during injection.