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# Fracture propagation models

The first fracture treatments were pumped just to see if a fracture could be created and if sand could be pumped into the fracture. In 1955, Howard and Fast published the first mathematical model that an engineer could use to design a fracture treatment. The Howard and Fast model assumed the fracture width was constant everywhere, allowing the engineer to compute fracture area on the basis of fracture fluid leakoff characteristics of the formation and the fracturing fluid. Modeling of fracture propagation has improved significantly with computing technology and a greater understanding of subsurface data.

## Two-dimensional fracture propagation models

The Howard and Fast model was a 2D model. In the following years, other 2D models were published. With a 2D model, the engineer fixes one of the dimensions, normally the fracture height, then calculates the width and length of the fracture. With experience and accurate data sets, 2D models can be used in certain formations with confidence, assuming the design engineer can estimate the created fracture height accurately.

Figs. 1 and 2 illustrate two of the most common 2D models used in fracture treatment design. The Perkins-Kern-Nordgren (PKN) geometry (Fig. 1) is normally used when the fracture length is much greater than the fracture height, while the Khristianovic-Geertsma-de Klerk (KGD) geometry (Fig. 2) is used if fracture height is more than the fracture length. In certain formations, either of these two models can be used successfully to design hydraulic fractures. The key is to use models (any model) to make decisions, rather than trying to calculate precise values for fracture dimensions. The design must always compare actual results with the predictions from model calculations. By “calibrating” the 2D model with field results, the 2D models can be used to make design changes and improve the success of stimulation treatments. If the correct fracture height value is used in a 2D model, the model will give reasonable estimates of created fracture length and width if other parameters, such as in-situ stress, Young’s modulus, formation permeability, and total leakoff coefficient, are also reasonably known and used.

To illustrate how certain variables affect fracture propagation, Eqs. 1 through 3 conform to the PKN fracture geometry assumptions. For fluid flow down an elliptical tube, ....................(1)

The PKN fracture mechanics equation is ....................(2)

and the PKN width equation is ....................(3)

Eq. 1 is the relationship used to compute the pressure distribution down the fracture for any given combination of injection rate, fracture fluid viscosity, fracture height, and fracture width. This equation, given certain physical dimensions and constraints, provides the pressure distribution in the fracture.

Eq. 2 provides the relationship between a given pressure distribution and what the dimensions of the fracture will be on the basis of rock mechanics theory. This equation, given a certain pressure distribution, provides the fracture width distribution. Eq. 1 and Eq. 2 are solved simultaneously to generate Eq. 3 . By reviewing Eq. 3 , one can observe that the fracture width will increase when the injection rate increases, the fracture fluid viscosity increases, the fracture length increases, or the formation modulus decreases. Similar equations have been derived by a number of authors. A complete discussion concerning the equations that describe the various 2D fracture models can be found in Refs. 6 and 7.

## Three-dimensional fracture propagation models

2D models have been used for decades with reasonable success. Today, with high-powered computers available to most engineers, pseudo-three-dimensional (P3D) models are used by most fracture design engineers. P3D models are better than 2D models for most situations because the P3D model computes the fracture height, width, and length distribution with the data for the pay zone and all the rock layers above and below the perforated interval.

Clifton provides a detailed explanation of how 3D fracture propagation theory is used to derive equations for programming 3D models, including P3D models. Figs. 3 and 4 illustrate typical results from a P3D model. P3D models give more realistic estimates of fracture geometry and dimensions, which can lead to better designs and better wells. P3D models are used to compute the shape of the hydraulic fracture as well as the dimensions. The key to any model, including 3D or P3D models, is to have a complete and accurate data set that describes the layers of the formation to be fracture treated, plus the layers of rock above and below the zone of interest. In most cases, the data set should contain information on 5 to 25 layers of rock that will or possibly could affect fracture growth. It is best to enter data on as many layers as feasible and let the model determine the fracture height growth as a function of where the fracture is started in the model. If the user only enters data on three to five layers, it is likely that the user is deciding the fracture shape rather than the model.

## Nomenclature

 G = Shear modulus, m/L3 H = fracture height, L L = fracture half-length, L, ft Q = injection rate, L3/t = average gas viscosity, cp ∂Δp = change in net pressure in the fracture, m/Lt2 ∂x = incremental distance down the fracture, L ν = Poisson’ s ratio w = fracture width, L μ = fluid viscosity, m/Lt