SESolid EarthSESolid Earth1869-9529Copernicus PublicationsGöttingen, Germany10.5194/se-8-255-2017Numerical modeling of fluid effects on seismic properties of fractured magmatic geothermal reservoirsGrabMelchiormelchior.grab@erdw.ethz.chQuintalBeatrizhttps://orcid.org/0000-0001-6714-420XCaspariEvaMaurerHansruediGreenhalghStewartInstitute of Geophysics, ETH Zurich, Zurich 8092, SwitzerlandInstitute of Earth Science, University of Lausanne, Lausanne 1015, SwitzerlandDepartment of Geosciences, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi ArabiaMelchior Grab (melchior.grab@erdw.ethz.ch)24February20178125527914October201616November201626January20171February2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://se.copernicus.org/articles/8/255/2017/se-8-255-2017.htmlThe full text article is available as a PDF file from https://se.copernicus.org/articles/8/255/2017/se-8-255-2017.pdf
Seismic investigations of geothermal reservoirs over the last 20 years have
sought to interpret the resulting tomograms and reflection images in terms of
the degree of reservoir fracturing and fluid content. Since the former
provides the pathways and the latter acts as the medium for transporting
geothermal energy, such information is needed to evaluate the quality of the
reservoir. In conventional rock physics-based interpretations, this
hydro-mechanical information is approximated from seismic velocities computed
at the low-frequency (field-based) and high-frequency (lab-based) limits. In
this paper, we demonstrate how seismic properties of fluid-filled, fractured
reservoirs can be modeled over the full frequency spectrum using a numerical
simulation technique which has become popular in recent years. This technique
is based on Biot's theory of poroelasticity and enables the modeling of the
seismic velocity dispersion and the frequency dependent seismic attenuation
due to wave-induced fluid flow. These properties are sensitive to key
parameters such as the hydraulic permeability of fractures as well as the
compressibility and viscosity of the pore fluids. Applying the poroelastic
modeling technique to the specific case of a magmatic geothermal system under
stress due to the weight of the overlying rocks requires careful
parameterization of the model. This includes consideration of the diversity
of rock types occurring in the magmatic system and examination of the
confining-pressure dependency of each input parameter. After the evaluation
of all input parameters, we use our modeling technique to determine the
seismic attenuation factors and phase velocities of a rock containing a
complex interconnected fracture network, whose geometry is based on a
fractured geothermal reservoir in Iceland. Our results indicate that in a
magmatic geothermal reservoir the overall seismic velocity structure mainly
reflects the lithological heterogeneity of the system, whereas indicators for
reservoir permeability and fluid content are deducible from the magnitude of
seismic attenuation and the critical frequency at which the peak of
attenuation and maximum velocity dispersion occur. The study demonstrates how
numerical modeling provides a valuable tool to overcome interpretation
ambiguity and to gain a better understanding of the hydrology of geothermal
systems, which are embedded in a highly heterogeneous host medium.
Introduction
Magmatic geothermal reservoirs consist of permeable extrusive and intrusive
rock formations, situated at depths where sufficiently high temperatures
prevail. They are saturated with hot fluids, and usually heated by magma
intrusions beneath the system. Evaluating the quality of such a reservoir
requires an estimate of the fluid enthalpy and of the host rock permeability.
Seismic methods are among the most efficient exploration techniques to image
the deep subsurface. The key quantities which can be obtained from a seismic
survey are the geometry of subsurface interfaces (e.g., lithological
boundaries, faults, fracture zones), the P- and S-velocities (VP
and VS) of various rock units, and the corresponding seismic
attenuation characteristics. The latter is expressed by the inverse of the P-
and S-wave-specific quality factors QP and QS. The
challenge in seismic interpretation is to link these seismic properties with
the geological/hydrological properties of interest.
To constrain the seismic interpretation, it is recommended to measure the
elastic and anelastic rock properties of small rock specimens in the
laboratory under in situ pressure, temperature, and fluid
content conditions. However, in magmatic geothermal systems, the host rock is
often highly impermeable and the fluid transport predominately takes place
within macro-fracture networks, rather than through the matrix. Such
fractures are not present in the rock samples investigated in the laboratory,
due to their limited size. Therefore, laboratory experiments only provide the
properties of relatively intact rock and indicators for the presence or
absence of fluids need to be deduced from fluid–rock interactions at larger
scales through rock physics concepts. Various such concepts of differing
complexity have been used over the last 20 years to interpret seismic
tomograms from geothermal exploration campaigns in magmatic environments.
Perhaps the simplest and most straightforward way to infer the presence of
fluids in seismic interpretation is to recognize that VP is more
sensitive to fluid saturation effects than VS, as the presence of
liquids tends to increase VP but not significantly change
VS. Thus, it is common practice to deduce fluid saturation from
seismic tomograms by interpreting the VP/VS ratio in a
qualitative manner. For instance and
interpreted VP/VS anomalies to be indicative of the
presence of supercritical fluids in a formation of the geothermal system in
the Long Valley Caldera, California, and in the Hengill volcanic complex in
Iceland, respectively. , who conducted a time-lapse
local earthquake tomography study in The Geysers, California, over a time
period of 7 years, interpreted temporal variations in a
VP/VS anomaly during the time of observation as an
indication of water depletion resulting from reservoir operation.
For a more quantitative seismic interpretation, a priori information of
the physical properties of mineral and fluid phases occurring at depth has to
be taken into account. For instance, interpreted
VP/VS anomalies observed in The Geysers, California, in
terms of steam pressure, based on a mixing law of fluid and rock mineral
properties. A more common way to incorporate fluid properties is through
well-known fluid substitution theories, such as those of
and , together with estimates of the rock frame mechanics
e.g.,. processed local
earthquake tomography data from the Yellowstone volcanic field, Wyoming,
while carried out a comparable study of data from Campi
Flegrei, Italy. They concluded from fluid substitution calculations that
VP/VS ratio anomalies were caused by gaseous pore
fluids. acquired tomograms in the Larderello–Travale
geothermal field in Italy and used the fluid substitution theory to identify
steam bearing formations, condensation zones, and over-pressured zones.
Other advanced petrophysical models consider fluid inclusions of specific
shape, usually simplified as spheres and ellipsoids, for example those
reported by or . Such a model was
applied for seismic interpretation by , who modeled
fractured rock as fluid inclusions of ellipsoidal shape, with the fluids
having properties either of supercritical water or partial melt. Based on
these calculations, they interpreted a low VP/VS
anomaly in Hengill, southeastern Iceland, as a region containing fractures
saturated with supercritical water and excluded the presence of partial
melting in the same region. interpreted a
VP and VS anomaly below the Reykjanes Peninsula in
Iceland and delineated a region with over-pressured supercritical fluids by
fitting the observed velocities to velocities obtained from an effective
medium model. The latter was a function of crack density, crack aspect ratio
and liquid-versus-supercritical fluid content.
It is important to note, as highlighted by , that effective
medium models, as traditionally used for rock physics-based seismic
interpretation, represent the unrelaxed state (high-frequency limit), in the
cases where fluid properties are directly included in the effective medium
model. At the other extreme, where fluid saturation in originally dry rock
frames is modeled by using fluid substitution techniques, the effective
medium describes the rock mechanics in the relaxed state (low-frequency
limit). Between these high and low-frequency limits, seismic velocity shows
marked dispersion and, in addition, strong frequency-dependent seismic
attenuation is observed in reservoir rocks, as a result of energy dissipation
associated with pore fluid flow triggered by stress-induced pore pressure
gradients. This effect is referred to as wave-induced fluid flow, and is
caused by various mechanisms, depending on the frequency spectrum of the
seismic wave. Velocities at ultrasonic frequencies are affected by global
(Darcy) flow due to macroscopic wavelength-scale pressure gradients
. In the intermediate sonic frequency range,
velocities and attenuation are influenced by squirt flow from microscopic
compliant cracks into more stiff pores . At low seismic
frequencies, seismic properties are affected by localized fluid flow between
mesoscopic inhomogeneities of different compressibility .
All these effects control how pore fluids, depending on their compressibility
and viscosity, as well as the hydro-mechanics of a fractured host rock, leave their
footprint on the seismic response of the reservoir, expressed in terms of
frequency dependent VP, VS, QP, and
QS. Thus, consideration of wave-induced fluid flow has a large
potential for further improving the rock physics-based seismic
interpretation. Moreover, it needs to be recognized that seismic techniques
cover a wide range of frequencies, from less than 1 Hz for local earthquake
tomography, to more than 100 Hz for active seismic investigations, to the
tens of kilohertz range for sonic borehole tools, and up to 1 MHz for piezo-electric
pulse experiments in the laboratory. Thus, it is important to not only model
the seismic response of fractured rock at the low- and high-frequency limit,
but also at intermediate frequencies.
Different analytical approaches exist to account for velocity dispersion and
attenuation due to wave induced fluid flow. For instance
describe the relaxation of fluid pressure between fluid inclusions, where the
compliance of the inclusions is obtained from 's
theory (). By contrast, and
used 's theory
() of poroelasticity, whereas used the
theory of viscoelasticity to consider fluid flow in a double-porosity medium.
Such theoretical models are based on some simplifying assumptions such as low
fracture density or small elastic property contrasts, together with idealized
geometries of heterogeneities. Motivated by this, numerical modeling
approaches, based on the theory of poroelasticity as in the case of
, , , and
, have become popular during the last decade to complement
analytical models.
Comparison of natural rock with its conceptual representations.
(a) Natural fractured rock with microscopic and macroscopic
fractures of complex shape. (b) Poroelastic medium representation,
where intersecting macro-fractures and the background medium are both
parameterized as isotropic poroelastic media on a finite-element grid.
(c) Effective medium representation, consisting of well-separated
macroscopic elliptic voids embedded into an isotropic elastic medium.
In this study, we use a numerical modeling technique, which is similar to
those proposed by and , to compute the
seismic phase velocities and the frequency dependent wave attenuation in
fluid-saturated fractured reservoirs. The reservoir is embedded in a
magmatic-type environment, as it is typical for Iceland. We first define the
physical properties of intact rocks based on the results of laboratory
experiments reported in the literature. We take into account the diversity of
typical rock types, which are shown to exhibit a large variability in
hydro-mechanical properties. Then, for the up-scaling to the dimensions of
macro-fractures, we study the properties of individual fractures in
dependence of the hosting intact rock using a semi-analytical effective
medium approach, which is based on 's elastic field
theory (). Once the parameters which describe the
physics of fractured rock volumes are defined for ambient confining
pressures under which the fractures are considered to be open, we study how
each of these parameters depends on lithostatic stress, under which fractures
close gradually. After this parameterization study, we finally apply the
numerical model to a fractured geothermal reservoir in Iceland, as described
in the structural geology literature. We examine how the frequency-dependent
seismic properties of a rock containing a fracture network are affected by
its saturating fluid, as well as how the observed fluid effects differ, depending on
the hosting lithology and on the effective lithostatic stress.
To study the effects of fluids on the seismic properties of fractured rock,
we use a numerical modeling technique which is based on the work of
. It primarily involves 's theory
(e.g., ) of poroelasticity and the principle of
conservation of linear momentum:
∇⋅σ=0,
where σ is the stress tensor, whose components in 2-D are
related to the corresponding elements of the strain tensor
ϵ by the constitutive law
σij=2Gdϵij+λϵ11+ϵ22δij-αPporeδij.
Here α=1-Kd/Ks is the Biot–Willis
coefficient, λ=Kd-2/3Gd is Lamé's
constant, and δij is the Kronecker delta. The quantity
Kd is the drained frame bulk modulus, Gd is the
drained frame shear modulus, and Ks is the bulk modulus of the
solid (grain) phase of the porous rock. The drained state is equivalent to no
fluid in the pores. The first two terms on the right-hand side of
Eq. () are consistent with Hooke's law of linear elasticity, whereas
the additional term αPporeδij accounts for the
stiffening of the rock in response to a pore pressure Ppore.
completed his theory by adding the conservation of fluid
mass, under the assumption of fluid incompressibility. This requires that the
flow rate into or out of an element of rock, described by Darcy's law for a
global flow of liquid in a porous medium, is equal to the temporal change in
fluid volume due to the deformation of the rock mass and due to the change in
pore pressure. Transforming the mathematical formulation used by
into the space–frequency domain, the fluid transport
equation is given by
-kηf∇2Ppore+iωαϵ11+ϵ22+iωϕKf+α-ϕKs-1Ppore=0,
where the imaginary quantity i and the angular frequency ω=2πf
represent the frequency domain equivalent of the time derivatives. Quantity
k is the hydraulic permeability, ηf is the fluid
viscosity, Kf is the fluid bulk modulus, and ϕ is the
effective porosity.
To compute the poroelastic response of the medium, we simultaneously solve
Eqs. () to () for the stress relaxation resulting from
an imposed strain, using the COMSOL
Multiphysics® finite-element solver. In its
poroelastic representation on a finite-element grid, a fractured rock as
observed in nature, containing micro-fractures and macro-fractures of complex
shape (Fig. a), is defined in a simplified manner (conceptual
representation) by a composite of two poroelastic phases – the intact-rock
domain and the fracture domain (Fig. b). The rock domain
represents the parts of the rock which are intact, apart from microscopic
cracks which are not discretized individually, and it will be referred to
hereafter as the intact rock. The fracture domain comprises all the
macroscopic fractures, which are in the model individually represented by
smooth elliptic structures. We simply refer to them as fractures in
what follows. As evident from Eqs. () and (), the
hydro-mechanical behavior of each of these two media depends on a set of
parameters, which are Kd, Gd, and
Ks, ϕ, and k for the solid phase of intact rock and
fractures, and ηf and Kf for the saturating
fluid phase. To distinguish between properties of the two media, we will mark
intact-rock properties with a hat superscript (“∧”) and the fracture
properties with a tilde superscript (“∼”) throughout the text.
The model domain has undrained boundaries, meaning that there is no fluid
flow across them. To conduct an oscillatory compressibility test, we simulate
a vertical normal stress by a displacement disturbance Δu in the
x1 direction to the top boundary, when referring to the coordinate frame
in Fig. a, and we suppress any displacements in the x2 direction at
the left and right boundaries, as well as any displacement in the x1 direction at the
bottom boundary, as defined in Eq. () in Appendix A. The
stress–strain ratio resulting under these conditions yields the complex
P-wave modulus, which is for a P-wave propagating towards the x1 direction
defined by
Mc(ω)=σ11ϵ11.
For an oscillatory shear test, we apply a displacement Δu in the
x2 direction to the top boundary, and suppress any displacement in the
x2 direction at the bottom boundary, while particles on the left and right
boundaries are free to move in both directions x1 and x2, as summarized
in Appendix A by Eq. (). From the stress–strain relation calculated
by this shear test, we obtain the frequency-dependent complex shear-wave
modulus for a S-wave propagating towards the x2 direction from the
relation
Gc(ω)=12σ12ϵ12.
(a) Two-dimensional numerical modeling scheme for a rock containing
randomly oriented, well-separated fractures, to which a normal or shear
stress is applied at the top boundary. (b) Scheme for 3-D effective
medium modeling for a rock containing ellipsoidal fractures, randomly
oriented in the x1–x2 plane. Consistent with the numerical model, the
applied normal stress σ11 and shear stress σ12 is
orthogonal to the long ellipsoid axis a3, whose orientation is held
constant.
The angle brackets in Eqs. () and
() denote the average over the entire modeling domain. Knowing the
bulk density of the rock ρb, seismic phase velocities can be
obtained from the complex elastic moduli by e.g.,VP(ω)=ReρbMc(ω)-1
and
VS(ω)=ReρbGc(ω)-1.
The attenuation factors are defined as the inverse P- and S-wave quality
factors by e.g.,QP-1(ω)=ImMc(ω)ReMc(ω) and QS-1(ω)=ImGc(ω)ReGc(ω).
These key seismic properties resulting from numerical poroelastic modeling
have incorporated the dispersive nature of propagation due to the
frequency-dependent interplay between the elastic deformation of the
fractured rock and the viscous fluid flow in the pores and fractures, as
described by Eq. (). It involves different mechanisms such as
localized fluid flow in porous background and squirt-type flow in fractures.
The price for getting such a detailed characterization of seismic properties
is that the method requires the determination of a large set of parameters.
In this study, we will give a detailed overview of typical values and likely
ranges for each of these parameters, while accounting for the large diversity
of rock types in magmatic geothermal systems. Furthermore, we will study how
these values depend on lithostatic stress. A problematic feature with this
approach is that parameterizing individual fractures by a homogeneous
poroelastic medium and not by fluid-filled cavities is a more conceptual
rather than a direct physical representation. In particular, the definition
of the fracture stiffness by intrinsic specific elastic moduli
K̃d and G̃d neglects the fact that,
in reality, the elasticity of an open fracture is a complex interplay between
the geometry of the void and the elasticity of the surrounding intact rock,
which also involves a changing behavior of the intact rock due to the
presence of the fracture, as has been described by .
Semi-analytical effective medium modeling
To study the properties of individual fractures under dry conditions, as
required to determine the dry frame elastic moduli K̃d
and G̃d, we use the semi-analytical solution provided by
the effective medium theory. The effective elasticity of a composite
material, consisting of an isotropic elastic intact rock containing
ellipsoidal inclusions (schematically shown in Fig. c) which are
filled with an isotropic elastic material (or empty as in our case), is
calculated with the Mori–Tanaka method . An expression for
the effective elastic tensor for the case where the ellipsoids are randomly
distributed in a plane (holding one axis of the ellipsoids fixed, whereas the
other two are randomly oriented as shown in Fig. b), is given by
as
CEff-1=I+cfACm-1.
Here, Cm is the elasticity tensor (in Voigt's matrix
notation) of the intact rock, cf the volumetric concentration
of inclusions, I the identity matrix, and A the
eigenstrain concentration tensor, describing the strain under zero stress.
The latter quantity is defined by to be
A=-QI+cfP-1,
where
P=I-SCf-CmS+Cm-1Cf-Cm
and
Q=Cf-CmS+Cm-1Cf-Cm.
In Eqs. () and (), angle brackets ⋅ denote the orientational average of the corresponding tensor,
given in Appendix B. Quantity Cf is the elasticity
tensor of the fracture filling-fluid phase and S is the
tensor, whose components for ellipsoidal inclusions can
be calculated e.g., from the aspect ratio of the
ellipsoids and the elastic properties of the intact rock, i.e., from
K^d and G^d. Due to the random
orientation of the ellipsoids in the x1–x2 plane, the resulting
effective elasticity tensor is transversely isotropic and the velocities of
P- and S-waves propagating along the x1 axis are calculated from the
corresponding components of the elasticity tensor CEff
by
VP=C11ρb=C22ρb and VS=C44ρb=C55ρb.
The effective medium theory is subject to several limitations in terms of the
geometrical representation of fractured rock. The underlying theory is exact
only for non-interacting fractures , and is consistent
with the upper and lower Voigt–Reuss bounds only in the case of low
volumetric fracture density . Furthermore, as stated by
, the assumptions of non-interacting fractures and of
small fracture density are not equivalent, since for non-randomly located
fractures, the interaction might still be strong even for a dilute fracture
density. For these reasons, fractures are considered to be well separated
from each other and randomly located in space.
Dry fracture elasticity estimation
Compared with the benefits and drawbacks of the numerical modeling technique,
the semi-analytical effective medium theory has complementary cons and pros.
The effective medium is limited to non-interactive fractures, while the
poroelastic theory implemented on a finite-element gird allows modeling the
hydro-mechanical interaction of complex fracture networks. On the other hand,
the effective medium theory implicitly includes the stiffness of the
fractures, depending on the intact-rock elasticity and the geometry of the
fractures, whereas the numerical technique requires parameterizing individual
fractures by a poroelastic medium, where K̃d and
G̃d are treated as fracture intrinsic material
properties.
In the parameterization part of this paper, we will combine the two
techniques to obtain appropriate values for the dry frame fracture
stiffnesses K̃d and G̃d by varying
K̃d and G̃d until the stiffness of
the overall fractured rock resulting from the poroelastic numerical modeling
is consistent with that from the effective medium theory. To ensure that the
2-D numerical fractured rock model satisfies the requirements of the
effective medium model, we generate models of randomly located, randomly
oriented and well-separated fractures of thin elliptic shape (black lines in
Fig. a). The volumetric concentration of fractures is below 1 %,
which is below the limit for the low-fracture density assumption of 10–20 %
determined by . We define an ellipse-shaped fracture-free
area around each fracture (dotted ellipses in Fig. a), and to
guarantee that the fracture orientation is not biased by neighboring
fractures, we successively place fractures within a circular area (dashed
circle in Fig. a), which allows rotating the fracture by
360∘ independently from the orientation of neighboring fractures. As
the numerical modeling is in 2-D, we assume the fractures to extend
continuously in the out of plane direction over distances that are long
compared to the in-plane fracture dimensions. Therefore, the 3-D effective
medium model (Fig. b) contains fractures with semi-major axis
a3 being much longer than semi-minor axes a1 and a2, whereby the
solution of the effective medium theory with ellipsoidal inclusions converts
to one from a composite containing elliptic cylinders. The aspect ratio
a1/a2, the fracture density cf, and the intact-rock
properties are chosen to match those of the numerical model.
When estimating values of the stiffness K̃d and
G̃d of dry fractures, no fluids are involved and the
non-interaction condition is also satisfied for the numerical model in terms
of fluid flow between fractures. Once appropriate values of
K̃d and G̃d are found, we will
extend the complexity of the numerical fractured rock model beyond the
capability of the effective medium theory, giving an example of modeling the
seismic response of rocks containing fluid-saturated interconnected fracture
networks. Also, for this case, the volumetric fracture density is still below
1 % and thus the low fracture density condition is still fulfilled. Here,
uncertainty arises from the fact that the surrounding material of individual
fractures also includes a small fraction of weaker material as fractures
intersect. Uncertainties related to this effect can be reduced by using a
more comprehensive effective medium theory than the one presented here, such
as the self-consistent approach p. 185, which introduces
a fractured background medium in an iterative manner.
Typical stratigraphy of the Icelandic crust. Pyroclastic rocks, lava
flows, shallow dykes and sills, and intracrustal magma chambers are
characterized in our study as potential host rocks for geothermal
reservoirs.
Lithological classification (a), physical
properties (b–e) of dry rocks versus dry frame bulk modulus, and
hydraulic permeability versus bulk density (f), as reported in the
literature for low confining pressures. Bulk moduli distributions for the
lithologies are given as boxes indicating the 25th and 75th percentiles with
outliers indicated by the dots. Red lines are best-fit functions obtained
from regression analysis, using dry samples only. Bold blue, orange, yellow
and purple dots are the values used to parameterize models for lithologies A,
B, C, and D.
Geology
In the present study, we focus on Icelandic geothermal systems. Iceland is a
large subaerial part of the worldwide system of mid-ocean ridges. Thus, the
crust is to some degree of oceanic type, but anomalously thick with a maximum
Moho depth of around 20 to 40 km . The crustal sequence
has been compared with the classical oceanic ophiolite sequence
but is of larger structural and chemical
complexity . In a brief summary, the stratigraphy of
the upper few kilometers of the Icelandic crust can be subdivided into four
lithological units (Fig. ). At the shallowest depths, extrusive
rocks dominate, forming interlayered sequences of pyroclastic deposits
(hyaloclastites, tuffs, scoria, etc.) and lavaflow deposits (dense and
vesicular basalts). In lower regions, dyke and sheet intrusions (dolerite)
become more and more abundant, which reach down to depths were intra-crustal
crystallized magma chambers (gabbro bodies) exist, which are found at depths
as shallow as 1–2 km, but typically they occur
at greater depth.
The physical properties of these rock types depend to some degree on their
chemistry, which is predominantly of basaltic composition but, to a lesser
extent, also magmatic rocks crystallized from intermediate and acid magmas
exist . Depending on the temperatures and the
intensity of fluid circulation through the formations, the chemistry of the
rocks is modified by hydrothermal alteration, which additionally increases
the variety of minerals, each with potentially different physical properties.
However, it is not only the chemistry that influences the physical properties of the
rocks; the rock texture also has a strong influence. Dense gabbros are different
from, for example, vesicular basalts or a highly porous tuff, independently of their
chemical compositions. This also results in a large variability in the
seismic properties as has been reported, for example, by for
pyroclastic rocks and by for basalts, dolerites and gabbros.
Accordingly, a large variability is expected for the properties of the intact
rock in our models, and a large volume of data is required to provide
well-grounded estimates. Pyroclastic deposits are a typical feature of
on-land volcanism. Lavaflow deposits, dykes and sheets, and magma chambers
can also be found at submarine mid-ocean ridges. Therefore, we can include the
database of the ocean drilling programs for determining their physical
properties.
Model parameterization for ambient lithostatic stress
The poroelastic model of fractured rock consists of two subdomains, intact
rock and fractures. Their solid matrix is described by the same type of
parameters. For the intact rock these are the dry frame bulk modulus
K^d, dry frame shear modulus G^d, grain
bulk modulus K^s, dry bulk density
ρ^b, effective porosity ϕ^, and hydraulic
permeability k^. The analogous parameters for the fractures are
K̃d, G̃d, K̃s,
ρ̃b, ϕ̃, and k̃.
Intact-rock properties
We defined intact rock as those parts of the rock embedding the macroscopic
fractures. Due to their limited size, rock samples investigated in the
laboratory typically are free of such macro-fractures, which is why we will
refer to laboratory studies to determine intact-rock properties. Here we
present a compilation of published results, which include more than 500 rock
samples in total of diverse types from on-land volcanic systems as well as
samples included in the database of the Deep Sea Drilling Program and the
Ocean Drilling Program.
Figure illustrates, in the form of cross plots, values for all
solid-constituent parameters as they have been reported in the literature.
Assigning each rock sample to one of the main lithologies introduced in
Sect. 3, we can study typical physical properties for each of these
lithologies. This is depicted in Fig. a for the drained bulk
modulus, with the boxes indicating the 25th and 75th
percentiles, and dots are values outside these percentiles.
Values for the dry frame elastic moduli, K^d and
G^d, are obtained from velocities VP and
VS, measured in the laboratory at ultrasonic frequencies, from
the relations
K^d=VP2ρ^-43VS2ρ^ and G^d=VS2ρ^,
provided the velocities were measured on dried rock specimens under drained
conditions. In Fig. b, K^d is plotted against
G^d, as they have been determined at the lowest confining
pressure of each dataset. As for all other parameters, we seek to establish
regression relationships using appropriate functions in order to get representative
values to parameterize the fractured rock models. The regression analysis
included diverse exponential, logarithmic, and power functions, of which the
one with the best fit was selected. For the dry frame elastic moduli, the
best-fit relationship was found to be
G^d=p1K^dp2+p3,
with p1=1.4×105, p2=0.5117, and p3=-11.5×109Pa.
The grain bulk moduli K^s were estimated using the fluid
substitution theory of
, which uses K^s together with
K^d to predict the bulk modulus of the saturated rock,
whereas G^d is assumed to be independent of liquid
saturation conditions. There is evidence for the validity of this supposition
in the data shown in Fig. , where the bulk moduli of saturated
rocks (panel b) tend to be higher than those of dry rocks (panel a), and no significant
increase is observed for the shear moduli.
(a) Dry frame bulk moduli versus dry frame shear moduli,
identical to Fig. b. (b) Saturated bulk moduli versus
shear moduli, reported in the literature (references given in the legend of
Fig. ). Red lines are best-fit functions obtained from regression
analysis on dry samples (Eq. ). The dashed line represents the
result from the fluid substitution analysis.
Applying the fluid substitution theory to all rock samples for which the
seismic velocities were measured under dry and saturated conditions, we
investigated what values of K^s are needed to predict the
velocities of saturated rocks from those of dry rocks. Whilst velocities of
dry rocks are expected to be frequency independent, strong velocity
dispersion is often observed for saturated rocks, especially at low confining
pressures, where compliant micro-cracks are open. To minimize errors due to
frequency effects, seismic velocities measured at high confining pressures
were used, resulting in values of K^s as shown in
Fig. c. From the regression analysis, we find
K^s=p4expp5K^d+p6expp7K^d,
with p4=5.82×1010Pa,
p5=3.86×10-12Pa-1, p6=8.22×1008Pa, and p7=3.99×10-11Pa-1. To test the validity of these estimates, we use
K^s together with the effective porosity ϕ^
obtained from Eq. () introduced below in order to predict the saturated
bulk moduli from the dry bulk moduli by fluid substitution. The resulting
saturated bulk moduli are indicated by the dashed line in Fig. b,
which agrees well with average values of the observed saturated bulk moduli,
which includes numerous samples not being used for the
K^s-estimation.
Most researchers who measured seismic velocities also documented the density
and the porosity of the rock samples in their publications. Densities are
plotted against K^d in Fig. d, and an
exponential relationship is indicated, which yields the following best-fit function from the regression analysis:
ρ^b=p8expp9K^d+p10expp11K^d,
with p8=2628kgm-3, p9=1.72×10-12Pa-1,
p10=-2898kgm-3, and p11=-1.38×10-10Pa-1. Values for the effective
porosities are shown in Fig. e, and the regression analysis yields the best-fit relationship
ϕ^=p12expp13K^d,
with p12=0.85, p13=-1.13×10-10Pa-1.
Since only a few authors have measured the hydraulic permeability k^ and
seismic velocities together, we refer to different publications to estimate
values for k^. The measured permeabilities are plotted against the bulk density in Fig. f and the best fit was obtained with relation
log10k^=p14expp15ρ^b+p16expp17ρ^b,
yielding the decimal logarithm of the permeability in m2, with
p14=-11.80, p15=4.26×10-05m3kg-1, p16=-3.60×10-03, and p17=2.51×10-03m3kg-1.
Intact-rock properties for the four lithologies A, B, C, and D.
Elastic moduli K^d, G^d, and
K^s are given in GPa, porosity ϕ^ in
%, permeability k^ in m2, and bulk density
ρ^b in kgm-3.
Equations ()–() fully describe the solid frame of the
intact rock as a poroelastic medium. As evident from Fig. a, it
covers a wide range of different lithologies. For the following modeling of
the seismic properties of fractured rock, we select parameters for
four characteristic models, which we will refer to as lithology A–D. They
were defined by their dry frame elastic moduli K^d in
order to cover the wide range of K^d values. Since there
is a substantial overlap for many properties of the different magmatic rock
types, lithology classes A–D cannot uniquely be assigned to one of these
rock types, but they show a tendency towards some of them, as shown in
Fig. a in terms of K^d. Lithology A resembles
most a typical pyroclastic rock (blue dots in Fig. ), lithology B
a relatively light lava flow deposit (red dots), lithology C a relatively
dense dyke or sheet intrusive (yellow dots), and lithology D a dense gabbro
body (purple dots). Corresponding parameters for these four models are shown
in Table . For readers who are interested in lithology-specific
intact-rock properties, statistical values similar tho those shown in
Fig. a are listed in Table for K^d,
G^d, K^sat, ρ^, and
ϕ^. For intact-rock permeability k^, the data coverage is
too sparse for the igneous rocks and we refer to the rock physics literature
for more detailed information.
Fracture properties
At ambient stress, fractures are assumed to be completely open, meaning that
fracture walls are not in contact with each other and they can be represented
by open ellipses (Fig. c). They are considered to be empty, i.e.,
containing no fault gauge; thus, we set the fracture porosity to a high value,
ϕ̃=90 %. Furthermore, the mineral composition is assumed to be
homogeneous across both the intact rock and the fracture subdomains, with the
grain bulk moduli of the two subdomains being identical,
K̃s=K^s. This also has the
consequence that the mineral density is the same for both subdomains and the
dry bulk density of the fracture is defined as ρ̃b=(1-ϕ̃)ρs, where the density of the mineral phase is
ρs=ρ^b/(1-ϕ^).
Intact-rock properties sorted by lithology listed for the mean value
(Mean), the minimum and maximum outliers (Min., Max.), and the 25th and 75th
percentiles (25th p., 75th p.). Elastic moduli
K^d, K^sat, and G^d
are in GPa, bulk density ρ^b in
kgm-3, and porosity ϕ^ in %.
Comparison of the fractured rock P-wave modulus (a) and
shear wave modulus (b) resulting from numerical modeling for varying
fracture elastic moduli (colored surface) with the solution resulting form
the effective medium theory (red lines) shown by way of example for lithology
B with fracture aspect ratios a1/a2=400. (c) Resulting RMS
deviations between the numerical modeling results and the effective medium
solution with the minimum (under the assumption of
K̃d/G̃d≈1.5) indicated by the
red dot. (d, e)K̃d and
G̃d for all lithologies A, B, C, and D and for the dry
frame bulk and shear moduli, respectively.
Estimates for the dry frame elastic moduli of fractures,
K̃d and G̃d, are obtained by
testing what values of K̃d and G̃d
are needed to obtain the same overall fractured rock stiffnesses
MNum and GNum from numerical modeling as the
values MEff and GEff calculated using the
effective medium theory. This test is conducted for a fractured rock model
with well-separated non-interacting fractures, which allows comparing the
results from the numerical modeling with those of the effective medium theory
as discussed above in Sect. 2.3. As an example, MNum and
GNum calculated for a model with intact-rock properties
corresponding to lithology B, and for fractures with an aspect ratio a1/a2=400, are shown in Fig. a and b by the colored surface for
varying values of K̃d and G̃d. The
intersection of this surface with the effective medium solution
MEff and GEff is marked with the red lines. There
is no solution where both P-wave and S-wave moduli from the numerical
modeling and effective medium theory coincide exactly. Thus, preferential
values of K̃d and G̃d are
determined by seeking the minimum in the root mean square deviation, defined
by
RMS=MNum-MEffM^d2+GNum-GEffG^d22.
From theory, it is expected that the bulk and shear moduli of dry fractures
are of similar magnitude e.g.,, assuming
K̃d/G̃d→1. Experimental
results, however, indicate a ratio for dry fractures which is in fact small
but larger than 1. observed a ratio in the range 1.3≤K̃d/G̃d≤5,
reported 1.7≤K̃d/G̃d≤5, and
found the ratio to lie in the range 1.7≤K̃d/G̃d≤1.9. Therefore, we use
here a ratio of K̃d/G̃d≈1.5,
which is at the lowest end of the experimentally observed values and, thus,
closest to the theoretical prediction stated by . Under this
constraint we find a pair of K̃d and
G̃d values which results in a good agreement (low RMS
value) between the numerical modeling and effective medium result, shown by
the red dot in Fig. a, b, and c. This procedure is repeated for all
lithologies A–D and for aspect ratios varying between 100 and 600, resulting
in K̃d and G̃d values shown in
Fig. d and e and listed in Table .
Intrinsic bulk and shear moduli for the fractures for lithologies
A–D and for variable aspect ratios, given with units (GPa).
The hydraulic permeability of open fractures is defined from well-established
empirical relations reported in the hydro-mechanical literature. Based on
calculations of laminar flow between two parallel walls, the volumetric flux
through a fracture was described by the cubic law to scale with the cube of
the aperture , leading to hydraulic
permeability of the fracture defined as k̃=eh212,
where the fracture aperture (width) is given by the hydraulic aperture eh.
This was defined to be the aperture needed to explain the actually observed
flow rate through a fracture with rough fracture walls in a parallel plate
model. Thus, eh can be regarded as a parallel-wall equivalent aperture.
Based on experimental observations, a formula for calculating eh was
suggested by to be
eh=JRC2.5heh2[µm],
where h is the average mechanical aperture of the fracture. JRC
is the joint roughness coefficient, with JRC=2.5 indicating a
very smooth fracture, whereas a fracture with JRC=20 is extremely
rough . In our models, we use JRC=15, assuming
relatively rough fractures.
Model parameterization as a function of lithostatic stress
To study how the solid frame of the intact rock behaves with depth, we
analyze their dependence on the effective confining pressure P′,
which is defined as the difference between the actual confining pressure (or
lithostatic stress) and pore pressure P′=Pconf-Ppore. For the individual fractures we consider the simplest
case of an effective stress applied normal to the long fracture axis, given
as the normal effective stress σn′.
Intact-rock properties as a function of confining pressure
In the laboratory, VP and VS are usually measured
at various confining pressures to simulate the lithostatic stress conditions
as a function of depth. Such datasets allow one to study the change in the
bulk modulus and the shear modulus as a function of confining pressure. As
most drained bulk and shear moduli follow a parabolic relationship with
increasing pressure, first- and second-order derivatives were obtained from
the observed curvatures within the pressure ranges of high data coverage. The
drained elastic moduli at a given effective pressure are then given by the
polynomials
K^d(P′)=K^d,0+∂K^d∂P′P′+∂2K^d∂P′2P′2
and
G^d(P′)=G^d,0+∂G^d∂P′P′+∂2G^d∂P′2P′2,
where K^d,0 and G^d,0 are the
respective elastic moduli at zero confining pressures. To estimate values of
the first- and second-order derivatives with respect to the effective
confining pressure, we use all entries of the literature database, for which
both VP and VS were measured at varying confining
pressures, and where the pore pressure is known in order to calculate the
effective confining pressure. These data have been divided into subsets, with
rock samples which show K^d,0<20GPa
(representing lithology A), 15 GPa<K^d,0<45GPa (lithology B), 40 GPa<K^d,0<70GPa (lithology C) and 65 GPa>K^d,0
(lithology D), and average values for the derivatives with respect to P′
have been calculated for each subset. They are valid within the range of
effective pressure for which enough data coverage is provided, which is up to
70 MPa for subset A, up to 120 MPa for subset B, and up to
200 MPa for subsets C and D. The resulting bulk and shear moduli as a
function of P′ for the four lithologies A–D, approximated by substituting
the resulting derivatives into Eqs. () and (), are
shown in different colors in Fig. a–d for the bulk moduli and in
Fig. e–h for the shear moduli, together with laboratory data
indicated in gray and the individual polynomial fits in green. Values for the
derivatives are given in Table . The goodness of fit was
determined in terms of R2 values and average values above 0.93 were
observed for the bulk moduli, and above 0.97 for the shear moduli. Higher-order polynomials were also tested, which did not improve the data
fit much, which is why the second-order polynomial was chosen as the lowest-order
polynomial consistent with the data trend. The reason for the better fit of
G^d is that shear moduli can be obtained from single
VS experiments leading to higher quality data, whereas for
K^d, combined P- and S-wave experiments are required,
which leads to larger errors, especially if VP and
VS were not measured during the same confining-pressure cycle.
Drained elastic bulk modulus (a–d) and shear
modulus (e–h) versus effective confining pressure. Gray and black
curves are experimental data reported in the literature (see references in
Fig. ), green curves show fitted functions (according to
Eqs. and , with average goodness of fit as indicated
by the R2 values) and the thick color-coded graphs represent average
trends taken for lithologies A–D.
Referring to experimental studies, hydraulic permeability of intact rock as a
function of confining pressure has been described by a log–log relationship,
e.g., by ,
logk^P′=logk^0-blogP′,
with ambient-pressure permeability k^0 and the coefficient
b indicating the curvature of the function or the slope when plotting
log(k^) versus log(P′).
To determine values of b for the four cases of lithologies A–D, we analyze
the datasets which comprise permeability measurements for intact-rock cores
at varying confining pressures. They are shown in gray in Fig. a.
For each dataset, a best-fit curve according to Eq. () was
obtained. Resultant values for coefficient b are plotted against resulting
ambient-pressure permeability k^0 in Fig. b.
Since b represents the curvature of the log(k^) versus
P′ relationship, its magnitude indicates the sensitivity of the
permeability to changes in confining pressure. It is interesting to observe
that rocks with intermediate permeability are only slightly sensitive to
pressure, whereas both low and high permeability rocks show a stronger
sensitivity. Values of b chosen to represent lithologies A–D are shown in
different colors in Fig. b and listed in Table .
Resulting graphs for k^(P′) are shown color-coded in Fig. a.
Derivatives for the dry frame elastic moduli with respect to
effective confining pressure (calculated for elastic moduli in GPa and
effective pressure in MPa), and b values representing the slope of
the log–log permeability–pressure relationship.
(a) Hydraulic permeability versus effective confining
pressure and (b) values for the coefficient b versus the ambient
confining-pressure permeability. Gray graphs are the permeability data
reported in the literature and colored curves and dots indicate values used
to parameterize lithology A–D.
The change in intact-rock porosity resulting from a change in applied
effective pressure was described by . They presented an
expression for the change in porosity at a specific effective pressure
Pn′ due to an increment of effective pressure dP′, which can
be written as a function of the dry frame bulk modulus and the grain bulk
modulus:
dϕ^Pn′=-1-ϕ^Pn-1′1K^dPn-1′-1K^sdP′,
where the initial porosity and the dry frame bulk modulus are also functions
of the effective confining pressure, given at the initial pressure
Pn-1′=Pn′-dP′. Thus, the porosity at a
given effective pressure P′ can be calculated stepwise using small
pressure increments dP′ and updating ϕ^(Pn-1′)
and K^d(Pn-1′) at each step.
The grain bulk modulus K^s is assumed to be approximately
constant at varying confining pressures. The dry bulk density of the intact
rock varies according to the porosity variation, ρ^b=(1-ϕ^(P′))ρs, assuming the density of the
mineral phase ρs is constant.
Fracture properties as a function of normal stress
The closure of natural unfilled fractures under normal stress was described
by as a function of specific normal and tangential
fracture compliance. These quantities are related to the dry frame bulk and
shear moduli by 1B̃n=M̃dh=K̃d+43G̃dhand1B̃t=2G̃dh,
in the cases where B̃n and B̃t are
the compliances of dry fractures.
Referring to experimental observations, described the
fracture closure under normal stress being of hyperbolic form, becoming
asymptotic to a small nonzero residual aperture. Based on their expressions,
we calculate the fracture aperture as a function of normal stress by
hσn′=h0-dhσn′=h0-σn′σn′+aM̃d,0ah0,
where M̃d,0 is the dry frame P-wave modulus of the
fracture and h0 is the mechanical aperture, both at ambient normal stress
as indicated with the zero subscript. The coefficient a is defined as the
maximum fracture closure coefficient, being the factor relating zero stress
aperture to the maximum aperture closure at very high stress,
dhmax=ah0, which we introduced to eliminate the specific
compliance in the expressions of . Stress-dependent
apertures resulting from () are given in Fig. c for
lithologies A–D, with the maximum and minimum values in each case which
results from the aspect-ratio dependency of M̃d,0 (Table ).
Normalized normal (a) and tangential (b)
compliances, fracture aperture (c), and hydraulic
permeability (d) versus effective normal stress. Gray graphs
in (a) and (b) are the experimental observations reported
in the literature. Colored graphs are the analytical calculations for
lithologies A–D, where for each lithology the maximum and minimum values are
given, since all properties vary depending on the fracture aspect ratio or
fracture aperture.
An expression for the dry frame elastic moduli of the fractures can also be
derived from the work of , leading to
M̃dσn′=M̃d,01-aσn′σn′+aM̃d,01-σn′σn′+aM̃d,02.
This equation gives the P-wave modulus as a function of normal stress,
ambient-pressure elasticity, and fracture closure coefficient a. Thus,
M̃d,0 is an intrinsic material property, which we can
define without requiring any information about the absolute aperture such as
h0 or h(σn′).
From Eq. (), the effective bulk and shear moduli of the fracture
can be calculated according to the relations
K̃σn′=M̃σn′(1+ν)3(1-ν),G̃σn′=M̃σn′(1-2ν)2(1-ν),
where ν is the Poisson ratio, describing the ratio between bulk and
shear modulus, ν=(3K̃d/G̃d-2)/(6K̃d/G̃d+2). We can either
assume a constant Poisson's ratio ν=ν0, as observed at low normal
stress, or use a stress-dependent Poisson's ratio ν=ν(σn′) in order to take stress-dependent effects
into account, such as the higher resistance against shear, when rough
fracture walls become interlocked during the closure at increased normal
stress.
In the literature, the fracture compliance at varying normal stress has been
experimentally investigated in terms of specific compliances
B̃n and B̃t and we cannot directly
compare it with the outcome of Eq. (), because of the scaling by
the absolute fracture aperture (Eq. ). Instead, we can compare the
stress-dependent fracture compliance normalized by the initial zero stress
compliance, where the absolute fracture aperture cancels out:
B̃nσn′B̃n,0=M̃dσn′hσd′M̃d,0h0=1-σn′σn′+aM̃d,02.
This is shown in Fig. a, where the
B̃n/B̃n,0 ratios reported in the
literature are displayed in gray. The normalized P-wave compliances for
lithologies A–D, calculated according to the right-hand side of
Eq. () and using a=0.75, are shown in color, where for each
lithology two graphs are shown for the maximum and minimum values, depending
on the aspect ratio. Using a constant Poisson's ratio, normalized S-wave
compliances are identical to the normalized P-wave compliances. They are
given in Fig. b for model A–D, again color-coded and compared with
the literature B̃n/B̃n,0 ratios in
gray. Using a fracture closure coefficient a=0.75, we observe a similar behavior of the fracture compliances as reported in the literature, such as those of with its strong decrease in fracture compliance
already at moderate values of σn′ or those of
, which only decrease relatively slowly as
σn′ is elevated to 60 MPa (Fig. a and
b).
At zero σn′, we considered the fractures being
open and we used the cubic law (Eq. ) for calculating the
hydraulic permeability of fractures. At increased
σn′, the fractures start to close and the two
fracture walls come locally into contact with each other. Due to these
contacts, as stated by , the reduction of permeability with
increasing σn′ is more rapid than the cube of the
joint closure and the cubic law is not valid anymore. For this reason,
extended the cubic law, yielding
k̃(σn′)=ehσn′212⋅1+lnehσn′eh,03ehσn′eh,032-ehσn′eh,0+k̃res.
In Eq. (), the first additional term leads to permeability
reducing faster with increasing σn′ than the cube
of the fracture closure. The last term is the residual permeability
k̃res, which incorporates the approximately constant
permeability at very high σn′, where all compliant
parts of the fractures are closed and fluid flow takes place through the
stiffest pores which remain open. The hydraulic permeability resulting from
Eq. () is shown in Fig. d for the maximum and minimum
cases of the four lithologies A–D and for σn′=10-14m2.
Assuming that the closure of the fracture is entirely compensated for by a
decrease in fracture void, whereas the area which is occupied by the material
comprising the microscopic roughness remains constant, leads to a fracture
porosity as a function of σn′, given by
ϕ̃σn′=hσn′-1-ϕ̃0h0hσn′.
The grain bulk modulus K̃s is assumed to be
approximately constant at varying confining pressures, as for the intact
rock. The dry bulk density of the fractures varies according to the porosity
variation, ρ̃b=(1-ϕ̃(σn′))ρs, assuming
that the density of the mineral phase ρs is constant.
Geometry of the fractured rock example (a), and the
statistical distribution of the orientations (b),
apertures (c), segment half-lengths (d) and aperture–length
cross plot with the gray line representing values for an aspect ratio of
a1/a2=400(e). Apertures are given as they were defined for
zero lithostatic stress.
Example: fractured rock of variable depth and lithologyModel setup
After determining the hydro-mechanical properties of the intact rock and the
fractures, the final task is to model the seismic properties of a rock mass
containing an interconnected fracture network, which is saturated with a
fluid of specific properties. Here we present a synthetic example using a
model containing a fracture network which represents a highly fractured
geothermal reservoir. The network geometry is based on the structural geology
observations of , who examined a highly fractured
paleo-geothermal field associated with the Húsavík–Flatey fault in northern
Iceland. The network is embedded in a host rock consisting of basaltic lava
flow piles, meaning that the petrography is similar to the one we presented
above as lithology B. The original depth of the system is estimated to be
approximately 1.5 km below the Earth's surface. Today, the overburden
has been largely removed by erosion and the fracture network outcrops at the
surface, preserved in the form of mineral-filled veins.
P-wave modulus (a), S-wave modulus (b), inverse
P-wave quality factor (c), and inverse S-wave quality
factor (d), modeled for a fractured rock with the geometry shown in
Fig. , and fracture and intact-rock properties corresponding to
those of lithology B at a lithostatic stress of 15 MPa.
described in detail the statistics of the network
geometry in terms of the spatial fracture frequency, as well as the
orientation, the width, and the length-to-width relationships of the fossil
fractures. This gives a complete image of the fracture network as it is
required to set up our model. This is done using a model generator, which
places fractures randomly within the model domain, incorporating the fracture
network statistics by weighting functions which are identical to the
observations from the Húsavík–Flatey fault. The resulting model is shown in
Fig. a, together with statistical distributions of the fracture
orientations (panel b), apertures (panel c), segment half-lengths (panel d),
and a cross plot of fracture aperture a2 versus fracture length a1
(panel e). The gray line in Fig. e indicates an aspect ratio
a1/a2=400, which is the average aspect ratio observed by
. The half-lengths of fracture segments were defined
as the half of the distance between intersection points. However, for
fracture segments terminating in a fracture tip, the histogram in
Fig. d accounts for the entire length, which is the relevant
quantity to estimate the diffusion length (see Sect. 6.2).
The variability in fracture aperture and aspect ratios is considered not only
in the model geometry but also when assigning the parameters of the
poroelastic media representing the fractures. Fractures of larger aperture
exhibit higher permeabilities according to Eq. (), and fractures
of larger a1/a2 aspect ratios are stiffer than those with small
a1/a2 ratios as shown in Fig. d and e. Corresponding ranges
for the (normalized) fracture compliance are plotted against lithostatic
stress in Fig. a and b, as they were computed from
Eq. () using the ambient-pressure stiffnesses listed in
Table . Ranges for the fracture permeability as a function of
lithostatic stress are computed from Eq. () and shown by the
maximum and minimum curves in Fig. d.
The seismic properties for the fractured rock model were computed by using
parameters corresponding to the four lithologies A–D, and for effective
lithostatic pressures ranging from ambient pressures up to a maximum of
120 MPa. Depending on the assumed density distribution of the
overburden, the effective confining pressure of a geothermal reservoir
situated at 3–4 km depth is around 60 MPa. The intact rock
and the fractures were saturated with liquid water of constant properties,
viz., a fluid of bulk modulus Kf=2GPa, and dynamic
viscosity ηf=0.001Pas. The P-wave and
S-wave elastic moduli, M and G, and attenuation factors,
1/QP and 1/QS, are computed according to
Eqs. ()–(), with M=VP2(ω)ρb,s and G=VS2(ω)ρb,s, and with ρb,s
being the bulk density of the saturated fractured rock. The normal and shear
stress–strain relationships are obtained by solving
Eqs. ()–() with the COMSOL
Multiphysics® finite-element solver, using
the boundary conditions given in Eq. () for the compressibility
test, and those in Eq. () for the shear test. Frequencies were
varied over a wide spectrum from 10-2 to 106Hz.
Results
Example results for the deduced seismic properties of the fractured rock are
shown in Fig. , plotted as the P-wave modulus M (panel a),
S-wave modulus G (panel b), inverse P-wave quality factors
1/QP (panel c), and inverse S-wave quality factors
1/QS (panel d) against the logarithmically scaled frequency.
They were computed for lithology B, undergoing a lithostatic pressure of
15 MPa.
Comparing the elastic moduli with the corresponding attenuation graphs, we
observe the typical behavior in accordance with Kramers–Kronig dispersion
relation . At the same frequencies at which M and G are
strongly dispersive with distinct inflection points, 1/QP and
1/QS reach their local maxima. In the example shown here, these
frequencies, referred to hereafter as characteristic frequencies
fc, are fc=10-1Hz and, for the less prominent
attenuation peak, fc=104Hz for the P-wave properties. For the S-wave properties, they are fc=10-1Hz and, for the most pronounced attenuation peak, fc=105Hz, the latter having secondary peaks at around 101
and 103Hz. linked the characteristic frequency
with the diffusion length ld, over which wave-induced fluid
pressure diffusion takes place, by the relation
2πfc=Dld2,
where D is the hydraulic diffusion coefficient, D=k/ηf(ϕ/Kf+(α-ϕ)/Ks)-1M/Msat, and where M/Msat is the ratio of the dry
frame P-wave modulus and the undrained (saturated) P-wave modulus. The
spatial scales of the fluid pressure diffusion during numerical oscillation
tests can be best inferred from snap shots of the Darcy fluxes q,
which are deduced from the local permeability values k and the pore
pressure gradients through the equation
q=kηf∇Ppore.
Amplitudes of pore fluid fluxes occurring under oscillatory
compression at 10-1Hz(a) and
104Hz(b) for the fractured rock properties
corresponding to lithology B at a lithostatic stress of 15 MPa
(seismic properties for the same case are shown in Fig. a and c).
Subplots at the bottom are enlarged views of specific regions in the
top plots, marked with the red boxes.
Amplitudes of pore fluid fluxes occurring under oscillatory shear at
10-1Hz(a) and 105Hz(b), for the
fractured rock properties corresponding to lithology B at a lithostatic
stress of 15 MPa (seismic properties for the same case are shown in
Fig. b and d). Subplots at the bottom are enlarged views of
specific regions in the top plots, marked with the red boxes.
Absolute amplitudes of the fluid fluxes ||q||=q12+q22
(with qi being the ith component of the flux vector) occurring under
compressional oscillations at frequencies of 10-1 and 104Hz
are shown in Fig. a and b, respectively. Absolute fluxes arising
under shear oscillations at frequencies of 10-1 and 105Hz
are depicted in Fig. a and b, respectively.
P-wave modulus (a), S-wave modulus (b), inverse
P-wave quality factor (c), and inverse S-wave quality
factor (d) for the four lithology cases A–D, all modeled for a
lithostatic stress of 15 MPa. The red curves are identical to those
in Fig. . The dashed parts of the graphs indicate the low-frequency ranges, at which the numerical solution breaks down due to the very
low permeability and porosity of the corresponding lithologies.
At the low frequency of 10-1Hz, we observe increased fluxes
inside all fractures (see zoom plots in Figs. a and a),
as well as within large areas of the surrounding intact rock. This shows that
during one oscillation cycle, the pore fluid flows from the strongly
compressed compliant fractures deeply into the stiffer, and less permeable,
intact rock. Thus, it is a flow between heterogeneities, with the stiffness
and permeability values differing by several orders of magnitudes. It takes
place at scales larger than the pore scale but smaller than the wavelength,
which is why this dispersion mechanism is commonly called the mesoscopic flow (MF)
mechanism e.g.,. For the example shown here, fluxes are
more widespread for the compression experiment than for the shear experiment,
which is in agreement with the larger 1/QP magnitude compared
with the magnitude of 1/QS at 10-1 Hz, shown in
Fig. c and d, respectively. Furthermore, the attenuation peak due
to MF occurs as a single maximum without minor side peaks. This is because
the characteristic frequency of MF is predominantly controlled by the medium
with the lower fluid mobility k/ηf,
which is the intact-rock subdomain here, having a constant permeability
throughout the entire model domain of 7×10-18m2.
Solving Eq. () for the diffusion length using the poroelastic
parameters of the intact rock and fc=10-1Hz, we find
ld≈0.02m, which is in good agreement with the
width of regions with increased fluid fluxes in Figs. a
and a.
At increased frequencies, fluid fluxes into the intact rock become less
pronounced (Figs. b and b), because the shorter
oscillation cycles limit the pressure relaxation by fluid flow from the
fractures into the intact rock with its low permeability. In the intact rock,
fluid flow only takes place within the direct vicinity of the fractures and is
more pronounced at the tips of individual fractures. Thus, as frequency
increases, fluid flow concentrates more and more in the highly conductive
fractures. This flow is driven by pressure gradients between different
interconnected fractures, which undergo various degrees of compression,
either because they are oriented differently relative to the direction of the
applied oscillation stress or because they are of different stiffness. As we
identify fluid flow between different fractures at these higher frequencies,
the corresponding dispersion mechanism is equivalent to the squirt flow
(SF) type e.g.,, but at a larger spatial scale. The
stress for the compressibility test was oriented along the x1 axis, when
referring to the coordinate frame in Fig. a, and the stress of the
shear experiment was parallel to the x2 axis and of the dextral
form. Therefore, fluid flow dominates in different fractures for the
compressibility and for the shear test, which is why there are different
numbers of peaks in the 1/QP and 1/QS plots, and
why they are occurring at different frequencies varying between around
10-1 and 105Hz. This range can be explained by the fact that
the characteristic frequencies linearly scale with fracture permeability
according to Eq. (), which in the case shown here are within the
range of 10-8 and 10-11m2. The diffusion length required
to explain the observed characteristic frequencies is ld≈0.1m, which is similar to the half-length of most fracture segments
between fracture-intersection points in our model (Fig. d).
P-wave modulus (a), S-wave modulus (b), inverse
P-wave quality factor (c), and inverse S-wave quality
factor (d) for varying lithostatic stresses, modeled for a fractured
rock with a geometry as shown in Fig. and rock properties
corresponding to those of model B. The solid curves are identical to those in
Fig. .
Next, the seismic properties of all lithologies were modeled. The numerical
results obtained for lithostatic pressures of 15 MPa are shown in
Fig. . The red graphs, representing lithology B, are identical
with those shown in Fig. , whose characteristics were discussed
above. Comparing first the absolute magnitudes of the elastic moduli of all
four lithologies, we observe that the fractured rock models A to D become
successively stiffer, as a logical consequence of the increased stiffness of
all the rock components. The attenuation peak, which we interpreted to be of
MF type, occurs at characteristic frequencies fc≤101Hz. Comparing this peak for the four lithologies A–D, we
observe (and anticipate for the cases where fc is outside the
considered frequency range) a decrease in amplitude, AA>AB>AC>AD, which can be explained by
the intact-rock porosities being strongly decreasing in the order
ϕ^A>ϕ^B>ϕ^C>ϕ^D. Higher porosities entail larger amounts of fluid
in the saturated rocks and, hence more energy is consumed by fluid flow
leading to higher attenuation. This effect opposes and dominates over the
effect of varying stiffness contrasts, which here are of similar magnitudes
for the four lithologies but which would amplify the attenuation when the
stiffness contrasts increase. Regarding the characteristic frequency, it is
observed (and anticipated) to decrease for the four models, fc,A>fc,B>fc,C>fc,D, which is in
agreement with Eq. () and the fact that the intact-rock hydraulic
permeability of the four lithologies progressively decreases as
k^A>k^B>k^C>k^D. The opposite effect is observed for the attenuation
peaks at the higher frequencies, which can be related with the SF-type
mechanism. For the lithologies A–D the peaks occur with increasing
amplitudes AA<AB<AC<AD and at increasing characteristic frequencies
fc,A<fc,B<fc,C<fc,D.
This is because there is less resistance against the fractures closure under
lithostatic stress for the softest rock of type A compared to B, C and
subsequently D (see Fig. c). Therefore, fractures embedded in
lithology D remain the most open and retain their original permeability and
fluid-saturated pore space the most, which is why the SF-attenuation peak is
largest and occurs at the highest frequency for lithology D and subsequently
lowers for lithology C, B, and A.
Finally, to study the effect of lithostatic stress on the seismic properties
of fractured rock, the elastic moduli and the seismic attenuation are
computed for lithology B, on which an effective lithostatic stress is applied
ranging from ambient pressures up to 120 MPa. Numerical results are
shown in Fig. . We observe the elastic moduli M and G to
strongly increase with stress, which is predominantly due to the strong
stress dependence of the fracture stiffness (Fig. a and b),
together with the less dominant stiffening of the matrix (Fig. ).
Comparing the attenuation peaks for varying stress, we observe the MF peaks
occurring at an approximately constant frequency of 10-1Hz,
which is consistent with the approximately constant permeability of the
intact rock, varying by not more than 1 order of magnitude for a confining
pressure ranging from 0 to 120 MPa. The SF-attenuation peak, however,
occurs at a characteristic frequency of fc>106Hz in the
case of zero lithostatic stress, and decreases down to fc≈103Hz for a lithostatic stress of 30 MPa. For higher
lithostatic stress the characteristic frequency decreases even more, down to
frequencies which overlap with those of the MF-type dispersion, where the
peaks start to overlap indistinguishably. For both types of attenuation
mechanism, the amplitudes decrease with increasing lithostatic stress. This
is due to the reduced porosity, and hence a reduced pore water content,
combined with a stiffening of both intact rock and fractures at elevated
lithostatic stress. This gives rise to lower amplitudes of the wave-induced
fluid pressure gradients due to the smaller compressibility contrast. As a
consequence of these two effects, the amount of fluid flow is reduced and
less energy is dissipated, leading to smaller attenuation peaks at increasing
lithostatic stress.
Discussion and conclusion
In fluid-saturated fractured rock, viscoelastic interaction between the
intact rock, the fractures, and the saturating pore fluid causes velocity
dispersion and seismic wave attenuation. The underlying mechanisms have been
studied in the past by various researchers, as summarized by
, and there is a broad consensus about how the degree of
seismic wave attenuation and the characteristic frequency at which it occurs
depends on the hydro-mechanical properties of the materials constituting the
rock. Petrophysical models which consider such viscous fluid flow are able to
link seismic quantities which are measured in geothermal exploration
campaigns with the hydrological properties. The reason why these models have
not been used routinely to date in seismic interpretation is to a large
extent because they depend on many input parameters, some of which are
difficult to quantify.
We have determined the input parameters for magmatic geothermal systems, as
required in numerical oscillation tests, and wide ranges were observed for
most properties when considering the high diversity of magmatic rock types.
Most of the input parameters also depend on lithostatic stress, which is why
we provided a compilation of functions to calculate the input parameters for
varying effective stress. Using these parameters, we computed the seismic
properties of rock volumes containing an interconnected fracture network
saturated with liquid water. Results from the numerical modeling demonstrate
how seismic velocities and attenuation factors strongly depend on the
lithology. This was already established for P- and S-wave velocities in our
earlier experimental study . Here, this is ground-truthed by
a large database extracted from the literature, which shows that the seismic
velocity structure of magmatic geothermal systems primarily reflects the
subsurface lithology. The effects of reservoir permeability and fluid content
are only minor. Interpreting seismic data in terms of hydrological target
parameters against the contrast of this background heterogeneity can be
achieved by studying the seismic attenuation, i.e., the decrease in seismic
amplitudes with increasing distance of travel, and, if available, by studying
the velocity dispersion.
Our modeling results show how the magnitudes of seismic attenuation and its
dispersion are associated with stiffness contrasts and porosity. Large
attenuation peaks were found for a rock volume containing a network of open
fractures, which decreased considerably when subjecting the fracture network
to elevated lithostatic pressures forcing the fractures to close. The
characteristic frequency, at which the attenuation reaches its peak, is
linked with the fluid mobility, which is a measure of hydraulic permeability
and fluid viscosity. At low seismic frequencies, the attenuation is observed
to be controlled by mesoscopic fluid flow from fractures into the surrounding
porous intact rock, with the characteristic frequency linearly scaling with
intact-rock permeability and fluid viscosity. At sonic to ultrasonic
frequencies, attenuation is associated with squirt flow between
interconnected mesoscopic fractures which are compressed to differing degrees
during normal and shear oscillations. Here, the characteristic frequency
linearly scales with fracture permeability and fluid viscosity.
The spread in the observed critical frequencies illustrates that fluid
effects in fractured rock can be detected with various seismic techniques
(passive, active, sonic, etc.) or in the ideal case by the combined use of
different seismic techniques to cover a broader frequency spectrum. On the
other hand, there seems to be no general rule that governs the frequencies at
which the conditions are given to assume either the relaxed state (low-frequency limit) or unrelaxed state (high-frequency limit), beyond which
traditional rock-physics concepts are strictly valid. Thus, concepts which
incorporate wave-induced fluid flow, like the one we presented in our study,
can help improve the quantitative interpretation of all kinds of seismic
data.
In the scope of this study, we modeled the influence of wave-induced fluid
flow on seismic properties for the wide range of rock properties but kept
the fluid properties constant at those of liquid water at ambient pressure
and temperature. To fully exploit the potential of the modeling technique we
presented, it can incorporate changes in fluid bulk modulus and viscosity as
they vary under phase transitions from liquid to the boiling state and
ultimately to the vapor phase. This has interesting application
possibilities, such as the interpretation of changes in the seismic
response measured by time-lapse seismic experiments conducted to monitor
changes in the fluid phase during reservoir operation. Saturating the
fractured rock with boiling fluid also raises the question of how rock
properties vary with elevated temperatures. Our modeling technique is valid
only for brittle rocks, but pronounced effects can already be expected at
temperatures below the brittle–ductile transition. Experimental
investigations on the stiffness and permeability of intact rock and fractures
at elevated temperatures are rare. In general, it is known that an increase
in temperature results in softening of the intact rock. Thus, increasing the
temperature may cause similar behavior to moving from a stiff lithology to a
more compliant one as examined in this study.
Data availability
All data used in this study are taken from the literature. Datasets are available
in the publications to which we refer in Sects. 4 and 5 and in the legends of Figs. 4, 8, and 9.
Boundary conditions
The boundaries Γ of the model domain Ω consist of undrained
boundaries. To conduct an oscillatory compressibility test, we simulate a
normal stress by a displacement disturbance Δu in the x1 direction
to the top boundary ΓT, when referring to the coordinate
frame in Fig. a, and we suppress any displacements in the
x2 direction at the left ΓL and right
ΓR boundaries, as well as any displacement towards the
x1 direction at the bottom boundary ΓB, i.e., rigid
boundaries at the right, left, and bottom, given by
u1=Δu,x1,x2∈ΓTu2=0,x1,x2∈ΓR∪ΓLu1=0,x1,x2∈ΓB.
Accordingly, we apply a displacement in the x2 direction to
ΓT for the oscillatory shear test, and suppress any
displacement towards the x2 direction at ΓB,
u1=0,x1,x2∈ΓBu2=Δu,x1,x2∈ΓT.
Meanwhile, particles on ΓT and ΓR are
free to move into both directions x1 and x2.
Orientational average
The orientational average of a fourth-order tensor is
〈X〉=1π∫0πX(θ)dθ,
where the rotation around the third principal axis x3 by the angle
θ is obtained by applying the transformation law
Xijklθ=∑p=13∑q=13∑r=13∑s=13RipRjqRkrRlsXpqrs,
with R being the corresponding entry of the rotation matrix
R=cos(θ)-sin(θ)0sin(θ)cos(θ)0001.
This leads to the following expressions for the averaged elasticity tensor
〈X〉, given in Voigt's matrix notation:
〈X11〉=〈X22〉=18(3X11+3X22+X12+X21+nX66)〈X33〉=X33〈X12〉=〈X21〉=18(X11+X22+3X12+3X21-nX66)〈X13〉=〈X23〉=12(X13+X23)〈X31〉=〈X32〉=12(X31+X32)〈X44〉=〈X55〉=n2(X44+X55)〈X66〉=n8(X11+X22-X12-X21+nX66),
where the factor n depends on the definition of the shear components
X44, X55, and X66, when transforming the fourth-rank
elasticity tensor into Voigt's matrix notation.
This work is part of the PhD project of Melchior Grab, conducted under the supervision of Hansruedi Maurer and Stewart Greenhalgh.
For the numerical modeling, they authors cooperated with Beatriz Quintal, who
provided the numerical modeling code implemented in COMSOL
Multiphysics®, and Eva Caspari. The
manuscript was prepared by Melchior Grab with contributions from all
co-authors.
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors would like to thank Céline Mallet from the Institute of Earth
Science at the University of Lausanne, Lorenz Grämiger from the Geological
Institute at ETH Zurich, and Benny Löffel from the Department of
Mathematics at ETH Zurich for their inspiring discussions. Special thanks
also go to Thomas Driesner from the Institute for Geochemistry and Petrology
at ETH Zurich for his constructive leadership of the Sinergia project
COTHERM2, which builds the overarching framework for this study. COTHERM2 is funded
by the Swiss National Science Foundation, grant number CRSII2_160757. Last
but not least, we thank the two anonymous reviewers for their constructive
comments. Edited by: U. Werban
Reviewed by: two anonymous referees
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