**1. Introduction**

The propagation of hydraulic fractures is a highly non-linear fluid-solid interaction problem involving a moving boundary (i.e., the propagating fracture front in the neighboroughood of which the governing equations degenerate). Simulating this class of problem numerically is challenging, especially properly tracking the evolving fracture front.

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©2012 Lecampion et al., licensee InTech. This is an open access chapter distributed under the terms of the


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where *Qo* is the volumetric injection rate per unit length in the out-of plane direction, *<sup>E</sup>*′ denotes the plane-strain elastic modulus, *<sup>µ</sup>*′ <sup>=</sup> <sup>12</sup>*µ<sup>f</sup>* is an equivalent viscosity (with *<sup>µ</sup><sup>f</sup>* the fluid viscosity), and *<sup>K</sup>*′ <sup>=</sup> <sup>√</sup>32/*<sup>π</sup> KIc* with *KIc* the fracture toughness (see [14] for more details). Equivalently a dimensionless viscosity M = K−<sup>4</sup> can be used. The complete solutions for the fracture length evolution, fracture width and net pressure have been obtained for the limiting cases of zero dimensionless toughness (equivalently infinite M) and zero dimensionless viscosity (infinite K) First order solutions for either small toughness or viscosity are also available [3, 4]. Semi-analytical solutions for any finite values of

We will restrict our comparisons to the case of relatively small dimensionless toughness (e.g. K < 1), which is known to be the more difficult condition to reproduce numerically. Therefore we express the solution in a so-called viscosity scaling. Because the solution is self-similar the time dependence can be obtained using dimensional analysis. We aim to compare the solutions provided by different numerical codes, which are typically developed in space-time. We thus introduce a dimensionless time *τ* = *t*/*tc* and a scaled coordinate *ξ* = *x*/ℓ, where *tc* is a characteristic time scale and ℓ is the fracture length. The fracture

dimensionless toughness or viscosity are also available [5].

length, opening, and net pressure can be written as follows:

1/6

1/3*µ*′1/6

*<sup>γ</sup>m*(K) =

<sup>Π</sup>*m*(*ξ*, <sup>K</sup>) = *<sup>E</sup>*′2/3*µ*′2/3

where we have highlighted the correspondence between results obtained using a time-based algorithm (say *γ*, Ω, Π) to the self-similar solution dimensionless solution F*m*(K) =

The dimensionless solution F*m*(K) = {*γm*, Ω*m*, Π*m*} for small toughness developed in [3] will be compared with the numerical solutions from different simulators. More precisely, for three small values of dimensionless toughness K = 0.01, 0.1, 0.5, we will focus on the comparisons of the dimensionless fracture length *γm*, opening profile Ω*m*(*ξ*) close to the fracture tip and error in the fracture volume etc. We are especially interested in the evolution of the error of the numerical solutions with respect to mesh sizes in the near-tip region of the fracture, where gradients are the largest. The solution, for small toughness (K < 1), is in fact governed by the hydraulic fracture viscosity tip asymptote: the opening behaves as *<sup>w</sup>* ∼ (ℓ − *<sup>x</sup>*)2/3 and the net pressure as *<sup>p</sup>* ∼ (ℓ − *<sup>x</sup>*)−1/3 close to the fracture tip (see [10] for

*<sup>E</sup>*′1/6 <sup>Ω</sup>*m*(*ξ*, <sup>K</sup>) = *<sup>Q</sup>*1/2

*E*′ *Q*3 *o t* 4 *c µ*′

> *<sup>o</sup> t* 1/3 *<sup>c</sup> <sup>µ</sup>*′1/6

1/6

*<sup>t</sup>*1/3 *<sup>τ</sup>*−1/3Π*m*(*ξ*, <sup>K</sup>)

*τ*2/3*γm*(K) *γ*(*τ*,K)

*<sup>E</sup>*′1/6 *<sup>τ</sup>*1/3Ω*m*(*ξ*, <sup>K</sup>)

Π(*ξ*,*τ*,K)

Ω(*ξ*,*τ*,K)

Accuracy and Convergence of Hydraulic Fracture Simulators

http://dx.doi.org/10.5772/56212

(1)

ℓ = *E*′ *Q*3 *o t* 4 *µ*′

{*γm*, Ω*m*, Π*m*}.

*<sup>w</sup>* <sup>=</sup> *<sup>Q</sup>*1/2 *<sup>o</sup> t*

*<sup>p</sup>* <sup>=</sup> *<sup>E</sup>*′2/3*µ*′2/3 *t* 1/3 *c*

**Table 1.** List of available solutions for the propagation of hydraulic fractures driven by a Newtonian fluid under constant rate of injection (zero leak-off).

In geoscience applications, hydraulic fractures propagate in a complex, often poorly characterized medium. Nevertheless, the description of the medium must be simplified in order to apply theoretical models. It is thus crucial that numerical implementations of such models for fracture growth be accurate such that differences from field observations can be attributed to model assumptions rather than poor numerical solution.

In the last ten years, a number of reference solutions (analytical and semi-analytical) have been obtained for propagating plane-strain [1–5] and radial hydraulic fractures [6, 7] (see Table 1). These solutions provide invaluable benchmarks for numerical simulators. We compare a number of simulators (2D and 3D) that use different propagation algorithms against these reference solutions for hydraulic fractures driven by a Newtonian fluid under a constant injection rate. For the sake of clarity, we do not address fluid leak-off in our discussion. Of particular interest is the accuracy of the different simulators in tracking the moving fracture front, particularly in the so-called viscosity-dominated regime of propagation.

An outstanding question relates to the convergence and robustness of numerical simulators with respect to the multiscale near-tip behavior of hydraulic fractures. The coupled lubrication (fluid flow) and elasticity equations are known to degenerate near the fracture tip, such that the solution of a semi-infinite fracture propagating at a constant velocity is characterized by a multiscale singular behavior near the tip [8–10]. The nature of the dominant singularity depends on the relative importance of two dissipative processes (viscous forces and fracture energy), as well as the reference length scale. Such a multiscale behavior near the fracture tip in turn governs the evolution of the velocity of a finite hydraulic fracture during injection. We discuss the degree to which a numerical simulator needs to include and resolve the near-tip behavior in order to accurately match reference solutions in the light of different benchmarks.
