This paper presents asymptotic analytical and numerical solutions for two-dimensional (2-D) and three-dimensional (3-D) type hydraulic fracture geometries. The fracture propagation models investigated include: a Geertsma-DeKlerk (GDK) a Perkins-Kern/Nordgren (PKN) and a 3-D type model. Perkins-Kern/Nordgren (PKN) and a 3-D type model. Comprehensive 2-D and 3-D design formulae for power-law fracturing fluids are given for various power-law fracturing fluids are given for various cases of fluid loss and containment. These formulae can be used to benchmark 2-D and 3-D hydraulic fracturing simulators. Characteristics of equilibrium 3-D height growth are also discussed. Parametric studies based on the design formulae are Parametric studies based on the design formulae are performed to show the effect of any one variable. performed to show the effect of any one variable. Typical fracture designs are simulated for various containment and leak-off examples using each of the fracture propagation models, along with a comparison of proppant scheduling and design. The simulation studies identify and illustrate the basic characteristics and design difference of each model.
A number of 2-D and 3-D hydraulic fracturing simulators have been developed that account for many of the complex phenomena affecting the fracturing process. Some of the 2-D models are based on the process. Some of the 2-D models are based on the GDK type fracture geometry while others are based on a PKN type fracture. A comparison of the various 2-D theories and underlying assumptions is given by Geertsma and Haafkens.
Similarly, a number of variable height (3-D) models have been developed. These models vary considerably in complexity, underlying assumptions and method of solution. Some of these simulators include a comprehensive treatment of fracture containment and proppant transport, while others only calculate fracture propagation characteristics. Thus, the user of hydraulic fracturing simulators should have a good understanding of the assumptions, limitations and applicability of the code in question. An overview of current hydraulic fracturing design and treatment technology is given by Veatch. Veatch's paper includes an extensive list of other authors who have contributed to 2-D and 3-D hydraulic fracturing simulation technology.
After code familiarization, one often speculates as to the accuracy of the numerical results. One way to help ensure numerical simulation accuracy is to benchmark the code against known analytical or theoretical solutions.
Simplified design formulae are presented for benchmarking GDK, PKN and 3-D type hydraulic fractures for asymptotic solutions of no leak-off and large leak-off. Limiting 3-D model formulae are also presented for well contained, equilibrium and penny shape type fractures. penny shape type fractures. After code benchmarking, the next phase is to determine which fracture model is most applicable properties and treatment design considerations. properties and treatment design considerations. This also includes determining the effect critical parameters have on fracture characteristics. parameters have on fracture characteristics. Numerically simulating parametric effects is usually very time consuming computer intense. The major parameters affecting fracture characteristics can be parameters affecting fracture characteristics can be identified by an analytical parametric study based on the design formulae given below.
A comparison of hydraulic fracturing simulation designs for GDK, PKN and 3-D models is presented illustrating the importance of using the proper fracture model based on treatment design and formation properties.