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Journal of Geophysical Research: Solid Earth
Experiments and Simulations of Fully Hydro-Mechanically
Coupled Response of Rough Fractures Exposed
to High-Pressure Fluid Injection
D. Vogler1,2 , R. R. Settgast3, C. Annavarapu3, C. Madonna4,P. Bayer
5, and F. Amann4,6
1Geothermal Energy and Geofluids, Institute of Geophysics, ETH Zurich, Zurich, Switzerland, 2Transport Processes and
Reactions Laboratory, Institute of Process Engineering, ETH Zurich, Zurich, Switzerland, 3Atmospheric, Earth, and Energy
Division, Lawrence Livermore National Laboratory, Livermore, CA, USA, 4Department of Earth Sciences, ETH Zurich, Zurich,
Switzerland, 5Institute of new Energy Systems, Ingolstadt University of Applied Sciences (THI), Ingolstadt, Germany,
6Chair of Engineering Geology, RWTH Aachen, Aachen, Germany
Abstract In this work, we present the application of a fully coupled hydro-mechanical method to
investigate the effect of fracture heterogeneity on fluid flow through fractures at the laboratory scale.
Experimental and numerical studies of fracture closure behavior in the presence of heterogeneous
mechanical and hydraulic properties are presented. We compare the results of two sets of laboratory
experiments on granodiorite specimens against numerical simulations in order to investigate the
mechanical fracture closure and the hydro-mechanical effects, respectively. The model captures fracture
closure behavior and predicts a nonlinear increase in fluid injection pressure with loading. Results from
this study indicate that the heterogeneous aperture distributions measured for experiment specimens can
be used as model input for a local cubic law model in a heterogeneous fracture to capture fracture closure
behavior and corresponding fluid pressure response.
1. Introduction
Hydro-mechanically (HM) coupled fluid flow is an actively researched topic in reservoir engineering,enhanced
geothermal systems (EGSs) and CO2sequestration (Figueiredo, Tsang, Rutqvist, Bensabat, & Niemi, 2015;
Figueiredo, Tsang, Rutqvist, & Niemi, 2015; Gu et al., 2014; Guo et al., 2016; Rutqvist & Stephansson, 2003).
Specifically, hydro-mechanically coupled processes in EGS often have critical effects on the efficiency of heat
extraction from the reservoir (Fu et al., 2016; Guo et al., 2016). Induced seismic hazards associated with EGS
and CO2sequestration can also be mitigated through a greater understanding of hydro-mechanical processes
that take place in the subsurface (McClure & Horne, 2014a, 2014b).
Hydro-mechanically coupled processes span several length scales (e.g., laboratory to field scale). Here the
fracture aperture and asperity length scales at smaller scales often significantly affect the fluid flow pathways
through the fractures at the macroscopic scale. Several studies have therefore focused on studying this effect
at the laboratory scale (e.g., Esaki et al., 1999; Raven & Gale, 1985; Renshaw, 1995; Watanabe et al., 2008, 2009;
Zimmerman & Bodvarsson, 1996; Zimmerman et al., 1991). The mechanical and hydraulic effects in fractures
are commonly represented with the mechanical and hydraulic aperture (amech and ahyd), respectively. Here the
mechanical aperture signifies the distance between the two fracture surfaces. The hydraulic aperture relates
flow rate and pressure gradient across a fracture to an averaged aperture which can be computed under the
assumption that the fracture consists of two parallel plates (cubic law) (e.g., Louis, 1969; Witherspoon et al.,
1980). Laboratory investigations on fluid flow in fractures, both with and without mechanical deformation,
have concluded that averages of the mechanical and hydraulic aperture across the fracture can normally
not be equated (e. g., Brown, 1987; Chen et al., 2000; Cook, 1992; Esaki et al., 1999; Hakami & Larsson, 1996;
Lee & Cho, 2002; Oron & Berkowitz, 1998; Renshaw, 1995; Raven & Gale, 1985; Vogler, Amann, et al., 2016;
Zimmerman & Bodvarsson, 1996; Zou et al., 2015). While the specific relationship between hydraulic and
mechanical aperture is not fully understood, a number of studies found the hydraulic aperture to be generally
smaller than the mechanical aperture, both with and without mechanical loading of the fracture (Esaki et al.,
1999; Hakami & Larsson, 1996; Raven & Gale, 1985; Vogler, Amann, et al., 2016).
RESEARCH ARTICLE
10.1002/2017JB015057
Special Section:
Rock Physics of the Upper Crust
Key Points:
• Experimental measurements of
mechanical fracture closure and fluid
flow in rough fractures are presented
• A method to predict fully
hydro-mechanically coupled effects in
rough fractures is demonstrated
• Mechanical fracture closure behavior
and corresponding fluid pressure
response are captured by numerical
simulations
Correspondence to:
D. Vogle r,
davogler@ethz.ch
Citation:
Vogler, D., Settgast, R. R.,
Annavarapu, C., Madonna, C.,
Bayer, P., & Amann, F. (2018).
Experiments and simulations of fully
hydro-mechanically coupled response
of rough fractures exposed to
high-pressure fluid injection. Journal of
Geophysical Research: Solid Earth,123.
https://doi.org/10.1002/2017JB015057
Received 3 OCT 2017
Accepted 9 JAN 2018
Accepted article online 11 JAN 2018
©2018. American Geophysical Union.
All Rights Reserved.
VOGLER ET AL. 1
Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
To further understanding of the relationship between mechanical and hydraulic aperture (Esaki et al., 1999;
Rutqvist & Stephansson, 2003; Xiong et al., 2011; Zou et al., 2015), a number of research efforts have been
dedicated to determining the specific relation between amech and ahyd (e. g., Cook, 1992; Park et al., 2013;
Wang et al., 2015; Zimmerman et al., 1991; Zimmerman & Bodvarsson, 1996). Specifically, a reduction of the
mechanical aperture has been linked to an increase in tortuosity, which in turn weakens assumptions made in
the cubic law (e. g., Brown, 1987; Tsang, 1984). The impact of fracture normal strength and topography on the
hydro-mechanical behavior in a fracture has therefore been thoroughly scrutinized (e. g., Li et al., 2008; Park
et al., 2013; Pyrak-Nolte & Morris, 2000; Walsh, 2003; Yeo et al., 1998), and a number of empirical relationships
relating effective normal stress and fracture normal stiffness (e. g., Barton et al., 1985; Gu et al., 2014) have
been proposed.
It is understood that the mechanical aperture, contact stresses, and the fluid flow field and pressures asso-
ciated with the hydraulic aperture can exhibit a tightly coupled relationship. A number of studies have
investigated permeability enhancement due to pressure injection on the reservoir scale, where asperity-scale
processes such as the influence of contact stresses and regions on stress and flow distribution in the fracture
are averaged with a Barton and Bandis relationship or poroelastic deformation (Barton et al., 1985; Figueiredo,
Tsang, Rutqvist, Bensabat, & Niemi, 2015; Figueiredo, Tsang, Rutqvist, & Niemi, 2015; Gu et al., 2014). However,
to our knowledge, a numerical study that computes the tightly coupled behavior between mechanical defor-
mation and fluid flow on the asperity scale for heterogeneous fractures, as well as the corresponding stress
field in the rock mass, does not yet exist. On the laboratory scale, most previous simulation approaches for
processes in rough fractures focus either on fluid flow or the mechanical behavior of the fracture and do not
incorporate full hydro-mechanical coupling (e. g., Nemoto et al., 2009; Tsang, 1984; Watanabe et al., 2008,
2009; Wang et al., 2015). On the field scale, fractures are commonly approximated as parallel plates (Derode
et al., 2013; Min et al., 2013), and mechanical and hydraulic properties are averaged over large regions (Cappa
& Rutqvist, 2011; Derode et al., 2013; Figueiredo, Tsang, Rutqvist, Bensabat, & Niemi, 2015; Figueiredo, Tsang,
Rutqvist, & Niemi, 2015). Predictive simulations need to be able to compute effective mechanical and hydraulic
properties and behavior from geometric information of the fracture and bulk rock properties, which are either
well known or can be measured in simple experiments.
We address these issues through a fully coupled hydro-mechanical model with a provision for heterogeneous
mechanical and hydraulic properties in fractures. Our objective is to capture the effect of fracture roughness at
the asperity scales on the macroscopic stress perturbations in the rock mass, flow channeling in the fractures,
and fluid pressure response to mechanical deformation. We compare our numerical model with experiments
investigating (1) bulk mechanical behavior of the fracture and (2) hydro-mechanical effects during loading.
The proposed method is implemented in GEOS: a flexible multiscale, multiphysics simulation environment
developed at Lawrence Livermore National Laboratory (Settgast et al., 2017).
2. Methods
2.1. Experimental Setup
Granodiorite specimens from the Grimsel Test Site, Switzerland, were used for this set of experiments, with
uniformly distributed mineral grains with a size of 3 to 8 mm. Three specimens of cylindrical shape with a diam-
eter of 122 mm and a height of 243, 225, and 206 mm (Specimens A, B, and C) were prepared (Figures 1b and
1c). Saw cut notches were cut in the specimens at the targeted height of the fracture and subsequently loaded
with two prisms from the top and bottom. This generated artificial fractures in the middle of the specimens,
with a fracture plane approximately normal to the cylinder axis (Figure 1). A 2,000 kN servocontrolled uniaxial
press (Walter and Bai AG, Switzerland) was used to apply axial loading to the specimens. For all experiment
sets, preloads of 0.14 MPa were applied to the specimens. Two sets of experiments were performed:
1. Bulk mechanical behavior of the fracture. First, the dry fracture was incrementally loaded from 0.25 to 10 MPa,
after which the load was subsequently decreased again to 0.25 MPa. Step sizes were 0.25 MPa for loads
between 0.25 and 1 MPa, while step size between 1 and 10 MPa was 1 MPa.
2. Hydro-mechanical effects during loading. A centric hole of 8 mm diameter was drilled into the top half of the
specimen along the specimen axis. Metal tubing with an outer diameter of 8 mm and an inner diameter
of 4 mm was placed in the hole to prevent leakage of fluid into the specimen before the fluid reached the
fracture. Fluid injection was controlled with a syringe pump (Teledyne Isco, Model 260D). The ISCO pump
used to inject fluid into the rock specimen has an associated error of 0.5% of the set point at the full range
(up to 51.7 MPa), which results in a maximum error of 0.25 MPa. The pump was used to inject a constant
VOGLER ET AL. 2
Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
Figure 1. Experimental setup with cylindrical specimen. The fracture is situated half-way up the cylinder, with the
fracture plane normal to the cylinder axis. (c) A schematic of experiment Set II, with supplementary figures depicting
(a) a tested fracture plane, (b) the testing configuration with displacement transducers, and (d) the displacement
transducers. The displacement transducer measures if top and bottom part of the sensor are displaced relative to
each other.
flow rate of 5 mL/min into the fracture for all specimens. Axial loads were stepwise increased from 0.25 to
1 MPa (i.e., in steps of 0.25 MPa) and subsequently from 1 to 10 MPa (i.e., in steps of 1.0 MPa) during fluid
injection, and changes in injection pressure in response to axial load changes were recorded. Figure 1 shows
the experimental setup with the pump, which was used to inject fluid into the fracture and measure the
injection pressure, the axial stress 𝜎Zwhich was applied to the top of the specimen, the borehole from
the center of the top of the specimen to the center of the fracture, the displacement transducers, and the
outflow of the fracture at the fracture boundary to atmospheric fluid pressure conditions.
A displacement transducer (type DD1, HBM, Germany) was employed in both experiment sets to measure
specimen deformation during loading. Specimen deformation is measured with four contact points to the
specimen, with two contact points approximately 5 cm apart along the specimen axis on each side the speci-
men (positioned diagonally across the specimen, see Figure 1c). The maximum error associated with the DD1
sensor measurement is ±0.05% of the measured displacement. The error associated with the applied nor-
mal loads has a maximum value of 1 kN for the maximum normal load applied in the presented experiments
(120 kN).
Compression tests on intact rock samples from the same core material used to create the experiment
specimens yielded elastic Young’s modulus between 10 and 12 GPa. This property allowed calculation of
deformation from the measurements of the displacement of the fractured core. Specifically, fracture closure
was calculated from the sensor data by subtracting the theoretical intact rock mass deformation from the
displacement transducer.
Before and after the experimental procedure, high-resolution photogrammetric scans of the surface were
recorded with an ATOS Core 3D scanner (GOM mbH, Germany) (GOM mbH, 2017). The ATOS core 3D scan-
ner projects fringe patterns on the object surface, which are recorded by two cameras. A phase shift based
on sinusoidal intensity distribution results in a pattern, which enables calculation of the three-dimensional
(3-D) surfaces. The photogrammetry scanner was calibrated with length deviation errors between 0.009 and
0.027 mm. The optimized calibration deviations were 0.014 ±0.001 Pixel. Specifically, this means that stan-
dardized objects (e.g., the diameter of a perfectly round sphere) can be measured with maximum deviations
between 9 and 27 μm. To avoid problems with different reflectivities of minerals, the fracture surfaces were
coated with a white spray. The applied spray can add a few micrometers to the height of the fracture surface.
VOGLER ET AL. 3
Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
After scanning, the fracture surfaces are represented as three-dimensional point clouds and have a spatial
density which is roughly between 30 and 60 unique vertices per square millimeter. The fractures are further
prepared by removing any extraneous details of the specimen scan, which is not part of the fracture surface.
Afterward, the fracture plane is reoriented such that the normal vector to the fracture surface is oriented along
the zaxis. The x-yplane is defined as the plane that provides the least squares fit of the given fracture surface.
Reorientation is obtained by aligning the eigenvectors of the fracture surface plane with the primary coor-
dinate system. This fit is only performed for the lower fracture surface, whereas the upper fracture surface is
fitted to the lower fracture surface. Fitting of the fracture surfaces is achieved with the commercial optimiza-
tion tool GOM Inspect (GOM mbH, Germany), which offers best fit alignment of objects (e.g., two fracture
surfaces) consisting of large number of degrees of freedom. The top fracture surface is aligned to the bottom
fracture surface such that contact between the two fracture surfaces is established in three points, while the
integral over the aperture field is minimized.
Once fracture top and bottom are aligned, the surfaces were mapped onto a regular grid in the x-yplane with
100 μm spacing between grid points in both directions. The aperture field can then be computed by subtract-
ing the zvalues of the lower fracture surface from the upper fracture surface. The resolution in the zdirection
is set to the precise distance (aperture) computed between the two gridded surfaces. Similar approaches to
derive fracture surfaces and corresponding aperture fields were also employed by Vogler et al. (Vogler, 2016;
Vogler, Amann, et al., 2016; Vogler, Walsh, Bayer, & Amann, 2017; Vogler, Walsh, Dombrovski, & Perras, 2017).
Between experiment Sets I and II, 30 loading cycles were performed on the specimens. This ensures that no
plastic deformation should be expected for experiment Set II. No gouge material production or fracture sur-
face damage was observed for experiment Set II. This yields one set of aperture fields for experiment Set I and
another set of aperture fields for experiment Set II.
2.2. Numerical Simulations
The numerical simulations presented in this work are performed using the GEOS simulation framework. GEOS
is a massively parallel multiphysics simulation framework developed at Lawrence Livermore National Labora-
tory, which offers coupling of the presented method with other capabilities of GEOS, including simulation of
hydraulic fracture stimulation in two and three dimensions (Fu et al., 2015; Settgast et al., 2012, 2014, 2017;
Vogler, Settgast, et al., 2017); modeling geochemical transport and reaction (Walsh et al., 2012, 2013) geother-
mic drawdown (Fu et al., 2016); fracture shearing (Annavarapu et al., 2015), matrix flow, and heat transport
(Guo et al., 2016); and simulations of immiscible fluid flow (Walsh & Carroll, 2013). Preliminary results of the
current study, using the GEOS framework, can also be found in Vogler, Settgast, et al. (2016) and Vogler (2016).
The fully coupled HM solver in GEOS utilizes a finite element method to model solid body deformation and
a finite volume method to model the fluid flow in fractures and the matrix. For this study, matrix flow is
neglected due to the low permeability of Grimsel granodiorite and the relatively short time scales involved.
Fluid flow in the fracture applies lubrication theory (i.e., parallel plate assumptions) on volumes between
fractured element faces (Settgast et al., 2017):
𝜕
𝜕t(𝜌wa)− 1
12𝜇∇⋅𝜌wa3∇(p+𝜌gz)=0,(1)
where 𝜇is the dynamic viscosity, pis the fluid pressure, ais the fracture aperture, and 𝜌wis the fluid density.
The aperture ais computed as a=u⋅nwith the displacement across the fracture surface uand the face
normal vector n. Solid deformation of the body Ωwith external boundaries Γis discretized using a standard
Galerkin finite element method on the equations of static equilibrium. Integration over the elements in the
solid domain gives the solid body residual as
(Re
solid)hi =∫Γe
t
ΦhtidA+∫Γe
Φh(fc
i+ci−pni)dA
−∫Ωe
Φh,jTijdV+∫Ωe
Φh𝜌mbidV,
(2)
with nodal index h, spatial direction i, finite element shape function Φ, solid body density 𝜌m, the Cauchy stress
tensor Tij, body force (gravity) bi, and externally applied traction ti. The penalty method is used to enforce
VOGLER ET AL. 4
Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
frictional contact conditions under small-deformation assumptions (Annavarapu et al., 2015). The residual for
the rate of fluid mass in a finite volume (r)isgivenby
(Rfluid)r=dmr
dt−∑
E∈r
̄𝜌w,er𝜅er (pe−pr)−qin
=d(𝜌rVr)
dt−∑
e∈r
̄𝜌er𝜅er (pe−pr)−qin
r,where
𝜅er =a3
rle
12𝜇ler
(3)
with fluid pressure pron the face r, fluid pressure peon the edge e, the density of fluxed volume ̄𝜌er into
the cell denoted with rfrom edge eand a source or sink term qin
r. Fluid pressure on the fracture surface and
solid deformation are coupled to obtain fracture aperture and corresponding fracture storativity. An in-depth
description of the framework can be found in Settgast et al. (2017).
Within this work, GEOS was extended to incorporate a locally variable mechanical and hydraulic aperture. Here
the fracture aperture changes across the fracture plane, where each element used for flow along the fracture
plane is assumed to consist of two parallel plates. This represents a local cubic law approach (Zimmerman
& Yeo, 2000), which aims to account for hydraulic and mechanical effects introduced by rough fracture
geometries, by assuming that smaller patches of the fracture can be approximated by a parallel plate model.
Zimmerman and Yeo (2000) showed that the local cubic law provides a suitable approximation for fluid flow,
but that it can lead to an overestimation of the local transmissivity, and that using Stokes equations will pro-
vide more accurate results. Factors which can weaken an approximation of fluid flow by the local cubic law
can be roughness and flow tortuosity in the direction normal to the fracture and the neglecting of inertial
forces (Wang et al., 2015). While mechanical and hydraulic apertures spatially vary, they are assumed to be
equal for respective points in the fracture.
The advantage of this approach lies in the possibility to compute hydro-mechanical effects in the fracture
plane, for rough fracture sets. While previous studies only simulate fluid flow in fractures (Wang et al., 2015;
Zimmerman & Yeo, 2000) or do not model mechanical deformation during loading (Watanabe et al., 2008,
2009), the presented approach allows to estimate mechanical fracture deformation and the resulting fluid
flow patterns and fluid pressure response in a fully coupled manner. Here it is noteworthy that the model only
relies on the respective fracture geometry and a globally constant stiffness parameter accounting for material
stiffness as model input. No further fitting, based on experimental results, is required or performed.
Within the scope of this study, we understand “fully coupled” hydro-mechanical processes to indicate
mechanical stresses (or stress changes) closing the fracture and fluid pressures opening the fracture to be in
equilibrium at all times. Changes in the mechanical loading or fluid injection rates influence each other so
that stress changes yield a fluid pressure response and vice versa.
Since this study solves all equations in a fully coupled approach, especially very small fracture apertures can
significantly increase the computational cost associated with the solving of the system of equations. For exam-
ple, using a minimum hydraulic aperture of 0.1 μm instead of 1 μm to compute fluid flow would increase the
range of the system of equations by 3 orders of magnitudes (due to flow being computed with the cubic law,
equation (1)).
A minimum hydraulic aperture amin of 1 μm and a maximum hydraulic aperture amax of 1 mm are therefore
enforced in the numerical models to reduce nonlinearity of the system of equations which needs to be solved.
The underlying assumption being, that values outside of this range do not alter the computed flow field and
fluid pressure response. Sensitivity analysis showed the chosen minimum hydraulic fracture aperture to influ-
ence the results for values as low as 5 to 10 μm. Minimum hydraulic aperture values below 5 μm did not alter
the pressure response significantly. Increasing the maximum hydraulic fracture aperture above 1 mm also did
not alter the computed pressure response curves. Similar lower hydraulic aperture cutoffs were also proposed
in studies focusing on solving fluid flow in rough fractures by Watanabe et al. (minimum hydraulic aperture of
1μm) and Wang et al. (minimum hydraulic aperture of 10 μm) (Wang et al., 2015; Watanabe et al., 2008, 2009).
While the presented method is not limited to the chosen minimum and maximum hydraulic aperture values,
they simply represent the values chosen for this study to achieve faster numerical convergence.
VOGLER ET AL. 5
Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
Since this study solves the contact problem for locally varying mechanical apertures with a penalty method,
the mechanical apertures need to become smaller than zero, to enforce a penalty and subsequently com-
pute a local contact stress. Naturally, a hydraulic aperture below zero is not possible and a cutoff (here the
minimum hydraulic aperture) has to be enforced, below which hydraulic and mechanical aperture cannot be
equated. A constant fracture normal stiffness value is assumed across the fracture, which can be thought of
as a globally applicable material constant. Locally, the penalty method uses a linear relation between frac-
ture normal stiffness and the normal stress at the fracture surfaces. Globally, however, the spatially variable
mechanical aperture can lead to a nonlinear fracture closure behavior, as will be demonstrated in this study.
Nonlinear fracture closure with globally constant stiffness parameters can be obtained because the resistance
of the fracture to a reduced aperture with higher loads increases as more area of the fracture comes into con-
tact and the local penalty force is enforced or increases. While lower mechanical aperture values than zero
are required to compute contact forces, values larger than zero do not alter the solution, since the fracture
is then locally not in contact. Mechanical and hydraulic apertures are therefore only assumed to be equal for
the range indicated above. For very low loads (e.g., 0.1 MPa), about 10 % of the total fracture area is at or near
the upper aperture cutoff. This percentage decreases rapidly with increasing loads.
The specimen cylinders containing a fracture were simulated with increasing axial load 𝜎Z, with compressive
stresses denoted as negative in the model. The aperture fields used for mechanics and fluid flow simulation in
the fracture were obtained from matched surface scans of the fracture. Aperture fields were derived from the
matched surface sides, by averaging the aperture (arithmetic average) over square areas with 0.1 mm edge
length. On natural fractures, surface roughness variations can be expected for length scales ranging from μm
to the long wavelength variations of fault zones. For the mesh size of the numerical model, we had to reach
a compromise between rigorous physical coupling and ease of computation, as resolving fracture surface
details down to the μm scale is generally not feasible. Since solving the contact problem of rough fractures
fully coupled with fluid flow comes at considerable computational costs, we had to choose an element edge
length that resolves large-scale (e.g., for the fracture surface in this study) roughness phenomena while still
offering fast and stable computational results. For this purpose, we chose an element edge length below the
grain sizes found in the specimens of this study (e.g., 3– 8 mm). Specifically, the specimen is meshed with
element edge lengths of about 2 mm throughout the domain. Aperture values on each node on the surface of
the fracture are subsequently derived from the aperture field, which was computed from the fracture surface
scans. Fluid flow and contact mechanics are solved on two adjacent element faces on the fracture surface and
the element nodes, respectively. This means that the mechanical aperture is enforced at individual nodes,
as the mechanical contact problem is solved by computing the penetration forces between two element
nodes. The hydraulic aperture between two element faces is computed by averaging the mechanical aper-
tures on the nodes surrounding each element face.
Experiment sets investigating mechanical closure behavior of heterogeneous fractures, contact stresses
during loading and hydro-mechanical effects during loading were approached as follows.
1. Bulk mechanical behavior of the fracture. Mechanical closure behavior of the heterogeneous fractures was
simulated dry. The aperture field measured before any testing was used as model input for the aperture
distribution. Estimates of the joint normal stiffness are derived from average joint normal stiffness values
obtained for increments of the fracture closure data measured during experiment Set I.
2. Hydro-mechanical effects during loading. The aperture field obtained after conclusion of all experiments is
used for the HM-coupled simulations (Figure 2). Fracture aperture at the inlet in the center of the fracture
is set to 1 mm to provide a sufficiently large aperture for fluid injection through the piping. As mentioned
before, sensitivity analysis showed that larger maximum hydraulic apertures did not significantly alter the
result but came at considerable computational cost. The constant injection rate used for the experiments
(5 mL/min) is distributed evenly across the elements in the inlet. The boundary of the fracture is set to
1 mm aperture and atmospheric pressure conditions. Estimates of the joint normal stiffness are derived
from average joint normal stiffness values obtained for increments of the fracture closure data measured
during experiment Set ii.
3. Results
The aperture fields derived from matching both surface scans of the fractures before and after testing are
shown in Figures 2a and 2c. Black regions are almost in contact, deep red regions signify small, and blue
VOGLER ET AL. 6
Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
Figure 2. Aperture fields in the fracture as seen from above. Shown are Specimens A (left column), B (middle column), and C (right column). Aperture magnitude
is given from 0 (black color) to 1 mm (blue color). (a) Aperture fields before testing; (b) histogram of aperture value distribution before testing; (c) aperture field
after testing; and (d) histogram of aperture value distribution after testing.
VOGLER ET AL. 7
Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
Figure 3. Experiment Set I— Specimen B. (a) Displacement between the sensor (green marker) is measured in the
experiment. The theoretical elastic deformation of the intact rock mass (red marker) can be derived with the static
Young’s modulus. Fracture closure (magenta marker) consists of the difference between sensor measurements and
elastic deformation. The deformation curves for increasing axial load are marked (▴) and for decreasing axial load are
marked (▾). (b) Mechanical displacement between sensor location as measured in the experiment (green marker)
and numerical simulations (blue marker) is plotted versus axial load 𝜎Z. (c) Simulated mean aperture versus axial
load 𝜎Zapplied to the top of the cylinder. (d) Simulated histogram evolution of the aperture field over axial load 𝜎Z.
The frequency distribution of aperture values is shifted with axial load 𝜎Z, to enable visual comparison of the shift
of the histogram toward smaller aperture values with increasing 𝜎Z.
regions large apertures. Since most aperture regions are below 500 μm and low aperture regions are most
crucial for contact behavior, the maximum aperture displayed is 1 mm for all aperture fields for ease of com-
parison. The histogram of the aperture distribution is given below the respective aperture fields in Figures 2b
and 2d.
3.1. Experiment Set I—Bulk Mechanical Behavior of the Fracture
The measured properties during experiment Set II was the mechanical deformation (displacement) of the
specimens as recorded by the displacement transducer. The displacement transducer measures change in
distance at two points on the specimen surface. Here two displacement transducers measure displacement
on diagonally opposing sides of the specimen. If the base length between the two points is known, this dis-
placement measurement can be expressed as strain. Representative behavior of fracture closure for specimen
B is shown in Figure 3 for both experimental and numerical results. Figure 3a depicts experiment Set I with
the respective sensor displacement (green), elastic deformation (red), and fracture displacement (magenta).
Displacement per load increment is larger for lower stresses and starts to converge to a finite displacement
for higher stress levels. Convergent fracture closure can be explained with increasing contact area in the frac-
ture as load increases until the fracture aperture reaches a residual value and further increase in stress is taken
up by the surrounding rock mass. For maximum axial load stresses of 10 MPa, a displacement of 0.26 mm is
observed for specimen B (Specimen A; 0.16 mm, Specimen C; 0.34 mm). After reaching the maximum axial
load stress of 10 MPa, the specimen is unloaded, during which irreversible displacement becomes appar-
ent. After fully unloading the sample, irreversible displacements of 0.175 mm (Specimen B) were recorded
(Specimen A; 0.055 mm, Specimen C; 0.24 mm). This is largely attributed to irreversible fracture closure,
VOGLER ET AL. 8
Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
frictional effects, and increased mating of the two fracture surfaces, since the intact rock mass is expected to
behave predominantly elastic in this stress regime.
We compare the mechanical fracture closure results obtained from numerical simulations with the exper-
imentally observed values in Figure 3b. Despite using a constant global normal stiffness estimated from
experimental measurements, a nonlinear response for fracture closure is observed due to the heterogeneous
character of the initial aperture field on the fracture surface. The mechanical displacement during specimen
loading is modeled with a maximum error of about 12% (roughly 0.03 mm or 30 μm) for large 𝜎Z(Figure 3b).
The mean aperture decreases strongly for axial stress up to 𝜎Z=2MPa and then converges toward 0.17 mm
mean aperture (Figure 3c). The aperture histogram evolution with increasing 𝜎Zcharacterizes the flow field
response to fracture closure and is shown in Figure 3d. The aperture frequency distribution is fairly homoge-
neous between 0.1 and 0.5 mm for 𝜎Z=0MPa and starts rapidly compressing with increasing 𝜎Z.From4to
10 MPa, most aperture values are below 0.1 mm with only small further aperture decrease, as also observed
in the mean aperture (Figure 3c).
The modeled stress distribution within the specimen and the increasing fracture closure with load are also
displayed in Figure 4, which shows vertical stress distribution (𝜎Z), the aperture field used as model input (for
zero vertical load) and the aperture field for given load stages. Figure 4 shows the increasing stress concen-
trations around the fracture, where contact area leads to large vertical stresses while other regions near an
open fracture show lower than average vertical stresses. The computed aperture field for a given applied load
increasingly deviates from the original aperture field used as model input, even for vertical loads as small as
0.25 MPa. This shows the phenomena also observed in Figures 3b and 3c, where the largest changes of the
fracture closure can be observed for early load stages where the fracture contacts offer the highest compli-
ance. As can be seen in the top right parts of Figures 4a– 4c, initial contact area is sparsely distributed across
the fracture plane for low loads, but contact area increases with vertical load.
3.2. Experiment Set II—Hydro-Mechanical Effects
The measured properties during experiment Set II were the mechanical deformation (displacement) of the
specimens as recorded by the displacement transducer and the fluid pressure necessary to sustain the given
flow rates as measured at the pump. During increasing axial load 𝜎Z, fluid inlet pressures generally increased,
the intact rock mass compressed, and the fracture closed.
Experimental data and numerical simulations are compared for the first load cycle of specimen B in Figures 5a
and 5b. Displacement computed in a numerical simulation compares well to experimental data for 𝜎Zup to
2 MPa. For higher loads, the simulation behaves stiffer, with a maximum deviation from experimental data of
0.03 mm for 𝜎Zof 10 MPa. When comparing experiment Sets I and II, the maximum deviations of simulated
and experimentally measured fracture closure are quite similar (around 0.03 mm). At low loads, the computed
injection pressure is lower than measured data. For the whole load range, injection pressure predicts the
experimentally observed trend while consistently underestimating the measured pressures by 0.2 to 0.5 MPa.
Simulation data for aperture, flow rate, and pressure distribution for axial loads of 0.25, 2, and 10 MPa are
shown in Figures 6a– 6c. In Figure 6a, we plot the aperture distribution on the fracture predicted by the numer-
ical simulation. To allow a comparison of all load stages, the color bar indicates apertures between 0 and
100 μm. For low axial loads of 𝜎Z=0.25 MPa, the aperture field is propped open by a few contact points (see
Figure 6a), which significantly increase for 𝜎Z= 2 MPa (as seen in Figure 6b). Here larger connected regions
close to or in contact emerge. At 𝜎Zof 10 MPa, the majority of the fracture aperture is below 30 μm, with large
aperture regions only sparsely connected. Specifically, roughly 10 % of the fracture area has aperture values
below 10 μm and only 17% of the fracture has aperture values above 500 μm.
Fluid flow for low 𝜎Zin Figure 6a is distributed across the whole fracture, which could be expected, as the
aperture in the fracture is larger than 50 μm except for a few contact points. As is apparent in the aperture field
evolution (Figures 6a– 6c), regions of contact or very small hydraulic aperture increase with loading, which
leads to more pronounced flow channeling for 𝜎Zof 2 MPa (Figures 6b and 6c). At 𝜎Zof 10 MPa, two large
and a few smaller flow channels emerge along regions of remaining high aperture, while large areas of the
fracture do not experience significant flow.
Due to the small scale of the specimen, pressure diffusion causes an almost radially symmetric pressure field
distribution across the fracture (Figures 6a– 6c). Here the natural logarithm of the pressure is plotted with a
maximum of 1.5 MPa to allow for comparison of pressures between all load stages. The pressure is largest
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Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
Figure 4. Experiment Set I— Specimen B. Top: Vertical stress 𝜎Zdistribution in the whole specimen (left) and the lower
half of the specimen, separated at the fracture (right). Bottom: Fracture aperture field for unloaded state (zero stress),
used as model input (left) and fracture aperture field for given load stage (right). Shown are axial loading states for
(a) 𝜎Z=0.25 MPa; (b) 𝜎Z=6.0MPa; and (c) 𝜎Z=10.0MPa. The specimen height is 225 mm, with a fracture plane
diameter of 122 mm.
in the fracture center at the injection borehole and then rapidly declines over the first few centimeters, after
which pressure gradients become significantly smaller. The pressure field is radial symmetric for low loads and
starts to reflect preferential flow channels in the top left and bottom right for 𝜎Zof 10 MPa. Injection pressures
show a stronger response for load changes near maximum 𝜎Z, although the flow rate fields at 2 and 10 MPa
are already in close resemblance.
4. Discussion
4.1. Experiment Set I—Mechanical Fracture Closure
The characteristic nonlinear shape of the displacement curve in this experiment (Figure 3) is captured by using
the aperture field obtained from scanned surfaces during testing (Figure 2) and does not rely on numerical
fitting. This means that neither the aperture field used as model input nor other physical parameters used to
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Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
Figure 5. Experiment Set II— Specimen B. Comparison of experimental and simulation data for (a) 𝜎Zversus sensor
displacement and (b) fluid pressure at the inlet versus 𝜎Z.
compute the hydro-mechanically coupled behavior of the specimen wereadjusted to improve the accuracy of
the results. The observed convergent displacement with increasing axial load is also reported in the literature
for other experiments on fractures (Bandis et al., 1983; Barton et al., 1985; Zangerl et al., 2008). This nonlinear
fracture closure during loading is linked to the increase in contact area (Figure 4), as the two rock surfaces
experience more resistance against compression with increasing contact area.
For this experiment series, artificial tensile fractures with pronounced surface roughness and highly variable
aperture fields were used. Fractures with less surface roughness (e.g., saw cut fractures) would be expected
to display a much less pronounced nonlinearity in their closure behavior (see also Figure 3).
Despite preloading the fractures, discrepancies between experimental and numerical results could be caused
by shear displacement, frictional effects during shearing, rotation, plastic deformation, seating, and mating
of the fractures in the experiments. The errors associated with the applied load and displacement measure-
ment are too small to explain any deviations of experimental and simulated data, as they are smaller than
the plotted marker size. The nonlinear decrease of aperture, shown in Figure 3c, goes in line with the frac-
ture closure shown in Figures 3a and 3b, as mean aperture is expected to decrease with fracture closure.
While aperture data without axial load can be obtained from surface scans (Figure 2), capturing aperture
fields under compression requires extensive laboratory equipment, such as CT scans, which are often tied to
specific resolutions or limitations regarding the mechanical or hydraulic boundary conditions which can be
applied. Instead, numerical simulations which capture the mechanical fracture behavior can be employed, to
gain insights into aperture behavior during compression (Figures 3c and 3d). Here a nonlinear fracture closure
behavior also yields a nonlinear mean aperture decrease (Figure 3c).
4.2. Experiment Set II—Hydro-Mechanical Effects
Analogous to loading the fracture without fluid injection, fracture closure is dominated by the applied 𝜎Z.
Despite fluid injection pressures increasing up to 6 MPa in the most extreme test, fluid pressure diffusion
in the model causes a rapid pressure decline over just 1 to 2 cm on the laboratory scale (Figure 6), which
causes fracture closure to remain relatively unaffected by high-fluid injection pressures. Nonlinear fracture
closure leads to only small fracture opening at high 𝜎Zif the effective normal stress is lowered by fluid injection
pressures (Figure 6). This further pronounces the small effect of fluid injection pressures on fracture opening.
The significantly reduced total fracture closure for experiment Set II (Figure 6) is related to the corresponding
fracture aperture field which was determined after testing concluded (Figure 2).
The hydro-mechanically coupled model captures the characteristic behavior of both fracture closure and fluid
injection pressure increase. Fluid flow behavior is dominated by the fracture aperture field and its response to
increasing axial load. This demonstrates the value for numerical frameworks capturing the mechanical behav-
ior of rough fractures and flow, as fluid flow is strongly dependent on the smallest aperture values which
define flow channels and act as a bottleneck (Figure 6). The strong influence of the smallest aperture regions
on fluid flow was also pointed out by Tsang (Tsang, 1984). HM-coupled numerical frameworks simulating phe-
nomena in rough fractures cannot only be employed to simulate processes on the laboratory scale, for specific
specimen, but also to derive averaged behavior and constitutive laws on the field scale, which can be derived
from fracture data on smaller scale.
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Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
Figure 6. Simulation of experiment Set II— Specimen B of aperture (top row), flow rate (middle row) and pressure
(bottom row) for 0.25 (left column); 2 (center column) and 10 MPa (right column). Discontinuities in the flow rate
magnitude can be observed along segmentation boundaries for the parallelization, which is a visualization artifact on
segmentation boundaries associated with the flow rate magnitude estimation on nodes from flow rates computed
across element edges.
It is noteworthy that none of the modeled aperture fields in this particular study resulted in flow rates domi-
nated by large aperture regions. However, this could depend on the aperture field configuration, as fracture
surfaces and fracture aperture fields are highly unique (Vogler, Amann, et al., 2016; Vogler, Walsh, Bayer, &
Amann, 2017). This study used artificial fractures obtained from tensile tests, where some areas of the fracture
surfaces experienced breakouts during splitting of the core. As this material was removed for the laboratory
experiments, the presented aperture fields might lead to different flow characteristics than the aperture field
of a natural fracture.
Injection pressures were underestimated for low loads by the numerical model, which could be caused by
(1) using locally averaged aperture values (local cubic law) or (2) small shear movements and rotations of the
experiment specimen, which are not captured in the simulations. Usage of the local cubic law could result
in locally averaged aperture values which offer little resistance to fluid flow as the fracture is mainly open
for low 𝜎Z. In the experiment, flow could encounter a small aperture bottleneck of a few μminflowpath
length, which might not be captured due to the element edge length constraints of the mesh employed in
the simulations. Matedness of the fracture could also change by shear movements and rotation of the fracture
specimens, which could result in a different aperture field than the one used as simulation input. Discrepancies
in experimental and simulated injection pressures can also be partially attributed to measurement errors.
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Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
However,experimental measurements show a smooth response to the fracture deformation (Figure 5), and no
oscillations or other measurement fluctuations could be observed. Given the smooth fluid pressure response,
and since fluid flow into the fracture cannot be induced with negative pump pressures, the error associated
with injection pressures is most likely significantly lower than the maximum error associated with maximum
injection pressures of the pump (at around 50 MPa). Nonetheless, mechanical and hydro-mechanical effects
during axial loading of the specimen were captured in the numerical model while using the aperture field
under zero stress state as model input and not performing numerical fitting.
It should be noted that the presented experiments were performed on granodiorite specimens, which have
a low permeability, and do not display a pronounced poroelastic effect for the applied fluid pressures.
However, other rock types such as carbonates or clays may display a stronger poroelastic response during
fluid pressurization, such as pressure solution or swelling (Fang et al., 2017; Yasuhara et al., 2004).
5. Conclusions
The presented study utilizes a fully hydro-mechanically coupled model to further the understanding of cou-
pled processes of fluid flow in heterogeneous fractures. Results demonstrate that fracture closure behavior
and corresponding fluid pressure response observed in laboratory experiments can be captured with a local
cubic law model with spatially variable mechanical and hydraulic aperture.
Based on aperture field input, the model computes aperture field changes, contact areas, stress distributions
within the specimen, and changes of fluid flow distributions with increasing vertical loading. For loading of a
dry specimen, the model captures the nonlinear fracture closure increase with loading which is observed in
experiments of this work and previous studies. During fluid injection under fracture compression, the numer-
ical framework reproduces the nonlinear fracture behaviors with maximum deviations of 10% for fracture
closure and 0.2 to 0.5 MPa for the fluid injection pressure response.
Through these experimental and numerical investigations, the possibility of scaling fracture roughness effects
for both mechanical and hydraulic properties is possible in the future. While this work compared experi-
ments and simulations in a single fracture, further investigations are needed to study the hydro-mechanical
interactions between multiple rough fractures.
Notation
aaperture (m);
ahyd hydraulic aperture (m);
amax maximum hydraulic aperture (m);
amech mechanical aperture (m);
amin minimum hydraulic aperture (m);
bibody force (gravity) (m2∕s);
ispatial direction (−);
kcontact stiffness (Pa/m);
nface normal vector (−);
pfluid pressure (Pa);
pefluid pressure at edge e(Pa);
prfluid pressure on face r(Pa);
pinj fluid injection pressure (Pa);
Φfinite element shape function (−);
qin
rsource or sink term (m3/s);
Qflowrate(m
3/s);
̂
Qnormalized flow rate (−);
tiexternally applied traction (Pa);
Tij Cauchy stress tensor (Pa);
udisplacement across the fracture surface (m);
xjump in displacement across the fracture (m);
Γexternal boundary of solid body (−);
𝜇dynamic viscosity of water (Pa s);
VOGLER ET AL. 13
Journal of Geophysical Research: Solid Earth 10.1002/2017JB015057
Ωsolid body (−);
𝜌msolid body density (kg/m3);
𝜌wfluid density (kg3/m3);
̄𝜌er fluxed volume into the cell rfrom edge e(m3/s);
𝜎Zstress in the zdirection (Pa).
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Acknowled gments
The authors want to thank the chair of
geosensors and engineering geodesy
at ETH Zurich and especially Robert
Presl for their support with the
photogrammetry scanner. The authors
gratefully acknowledge R. Seifert for
assistance with laboratory equipment.
This work was partially supported by
the GEOTHERM II project, which is
funded by the Competence Center
Environment and Sustainability of the
ETH Domain. We would like to thank
the Editor André Revil and Associate
Editor Ludmila Adam for handling the
manuscript and the reviewer Mark
McClure and an anonymous reviewer
for their time and valuable feedback.
Data employed in this study can be
found in the presented figures and by
contacting the corresponding author
Daniel Vogler at davogler@ethz.ch.
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