PAMM · Proc. Appl. Math. Mech. 17, 95 – 98 (2017) / DOI 10.1002/pamm.201710028
Modelling of hydraulic fracturing and ﬂuid ﬂow change in saturated
and Bernd Markert
Institute of General Mechanics, RWTH Aachen University. Templergraben 64, 52062, Aachen, Germany
The underlying research work aims to develop a numerical model of pressure-driven fracturing of saturated porous media.
This is based on the combination of the phase-ﬁeld modelling (PFM) scheme together with a continuum-mechanical approach
of multi-phase materials. The proposed modelling framework accounts for the crack nucleation and propagation in the solid
matrix of the porous material, as well as the ﬂuid ﬂow change in the cracked region. The macroscopic description of the
saturated porous material is based on the theory of porous media (TPM), where the proposed scheme assumes a steady-state
behaviour (quasi-static) and neglects all thermal and chemical effects. Additionally, it assumes an open system with possible
ﬂuid mass production from external source. Special focus is laid on the description of the interface and change of the volume
fractions and the permeability parameter between the porous domain and the crack. Finally, a numerical example using the
ﬁnite element method is presented and compared with experimental data to show the ability of the proposed modelling strategy
in capturing the basic features of hydraulic fracturing.
2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Hydraulic fracturing is a very important subject in various engineering applications, especially in the energy sector, such as
in geothermal applications, mining and petroleum. In the ﬁeld of enhanced geothermal systems (EGS), which are applied
to generate geothermal electricity through hot water, high-pressure water is injected into deep rock layers with low perme-
ability in order to enhance the rock’s permeability. This leads to improving the system’s efﬁciency and helping to produce
electricity with lower prices. Hydraulic fracturing using pressurised liquids with chemical additives is also used in petroleum
engineering to extract shale gas. In a similar fashion, the developed numerical tools can be applied to simulate phenomena
like intervertebral disc herniation in biomechanics.
The development of the phase-ﬁeld modelling (PFM) of fracture can be traced back to Grifﬁth , who described in 1921
the fracturing of brittle solids using elastic-energy-based mathematical formulations, where a critical energy release rate has
been deﬁned to start crack propagation. This criterion for the initiation of cracks has been later extended by Irwin , by
introducing the strain-energy release rate and the fracture toughness concepts. The challenge in the numerical implementation
of those approaches using, e.g., the ﬁnite element method, is the consideration of the cracks as discontinuities inside the
continuous solid domain. Thus, the PFM has been introduced to tackle this problem, which approximates the sharp edges of
the crack by a diffusive interface using a scalar ﬁeld variable, called the phase-ﬁeld variable. Therefore, no discontinuities in
the geometry or the ﬁnite-element mesh takes place. Additionally, the PFM introduces an internal length scale parameter to
deﬁne the width of the diffusive edge of the crack.
Apart from fracture mechanics, the idea of diffuse interfaces is found in physics, where it has been used by, e. g., Cahn and
Hilliard  in 1958 to describe the interfaces in a heterogeneous system by a fourth-order partial differential equation. The
link between the PFM and fracture mechanics, also within a variational framework, is found in later scientiﬁc works, such as
in [4–8], allowing to model multi-dimensional, mixed-mode fracturing. Moreover, to estimate and study the physical meaning
of the PFM parameters, a recent comparison between molecular dynamics (MD) simulations of fracture on the nano-scale and
the continuum PFM scheme is found in .
The PFM has been later implemented to modelling of hydraulic fracturing in porous materials, induced by the increase of the
pore pressure, see [10–15]. This leads, however, to permanent changes of the local physics of the problem, such as of the
volume fractions and the permeability, which are discussed in more details in the following sections.
For the description of the behaviour of porous materials on a macroscopic scale, the theory of porous media (TPM) is applied
[16, 17]. This mathematical model is based on the assumptions of a materially incompressible solid but compressible ﬂuid, a
quasi-static behaviour, no thermal or chemical effects or any mass exchange between the phases. However, the formulation
allows for an external supply of the ﬂuid phase, as will be shown in the numerical examples.
Corresponding author: e-mail firstname.lastname@example.org, phone +49 241 80 98286, fax +49 241 80 92231
2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim