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Research Papers

# Propagation of a Plane Strain Hydraulic Fracture With a Fluid Lag in Permeable Rock

[+] Author and Article Information
B. Chen

Energy Safety Research Institute,
College of Engineering,
Swansea University Bay Campus,
Swansea SA1 8EN, UK;
Zienkiewicz Centre for Computational
Engineering, College of Engineering,
Swansea University Bay Campus,
Swansea SA1 8EN, UK

Andrew R. Barron

Energy Safety Research Institute,
College of Engineering,
Swansea University Bay Campus,
Swansea SA1 8EN, UK;
Department of Chemistry,
Rice University,
Houston, TX 77005;
Department of Materials Science and
Nanoengineering,
Rice University,
Houston, TX 77005

D. R. J. Owen

Zienkiewicz Centre for Computational
Engineering,
College of Engineering,
Swansea University Bay Campus,
Swansea SA1 8EN, UK

Chen-Feng Li

Energy Safety Research Institute,
College of Engineering,
Swansea University Bay Campus,
Swansea SA1 8EN, UK;
Zienkiewicz Centre for Computational
Engineering, College of Engineering,
Swansea University Bay Campus,
Swansea SA1 8EN, UK;
Department of Chemistry,
Rice University,
Houston, TX 77005;
Department of Materials Science
and Nanoengineering,
Rice University,
Houston, TX 77005

Manuscript received February 22, 2018; final manuscript received May 13, 2018; published online June 14, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(9), 091003 (Jun 14, 2018) (10 pages) Paper No: JAM-18-1113; doi: 10.1115/1.4040331 History: Received February 22, 2018; Revised May 13, 2018

## Abstract

Based on the KGD scheme, this paper investigates, with both analytical and numerical approaches, the propagation of a hydraulic fracture with a fluid lag in permeable rock. On the analytical aspect, the general form of normalized governing equations is first formulated to take into account both fluid lag and leak-off during the process of hydraulic fracturing. Then a new self-similar solution corresponding to the limiting case of zero dimensionless confining stress ($T=0$) and infinite dimensionless leak-off coefficient ($L=∞$) is obtained. A dimensionless parameter $R$ is proposed to indicate the propagation regimes of hydraulic fracture in more general cases, where $R$ is defined as the ratio of the two time-scales related to the dimensionless confining stress $T$ and the dimensionless leak-off coefficient $L$. In addition, a robust finite element-based KGD model has been developed to simulate the transient process from $L=0$ to $L=∞$ under $T=0$, and the numerical solutions converge and agree well with the self-similar solution at $T=0$ and $L=∞$. More general processes from $T=0$ and $L=0$ to $T=∞$ and $L=∞$ for three different values of $R$ are also simulated, which proves the effectiveness of the proposed dimensionless parameter $R$ for indicating fracture regimes.

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## Figures

Fig. 1

Parametric space of plane strain hydraulic fracturing and limiting propagation regimes

Fig. 2

Sketch of the KGD model

Fig. 3

Computational model for quarter of the KGD scheme. The injection point locates in the origin. (l, 0) and (lf, 0) are the location of crack tip and fluid front respectively. a = 50l at the beginning of the simulation and doubles when l > a/25.

Fig. 4

Finite element mesh. Left: the entire simulation domain. Right: the mesh in (0, 2.5L0) × (0, 2.5L0).

Fig. 5

Self-similar solutions of normalized fracture width on ÕK̃ edge for various values of ξf from 0.001 to 0.999 (corresponding values are shown in Table 1)

Fig. 6

Self-similar solutions of normalized fluid pressure on ÕK̃ edge for various values of ξf (corresponding values are shown in Table 1). Top: 0.001 to 0.03 and Bottom: 0.1 to 0.999.

Fig. 11

Hydraulic fracturing paths in the case of K =0.5 and various value of R

Fig. 7

Evolution of the normalized fracture half-length with respect to dimensionless time under zero dimensionless confining stress

Fig. 8

Evolution of the normalized fracture width (top) and normalized fluid pressure (bottom) under zero dimensionless confining stress and K=0.498. Solid lines represent the numerical solutions at different dimensionless time t/tl and the dashed line represents the self-similar solution at ÕK̃ edge.

Fig. 9

Evolution of the fracture half-length with respect to dimensionless time under nonzero dimensionless confining stress

Fig. 10

Hydraulic fracture path in the case of K =0.5 and R=0.0177

## Errata

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