1 Petroleum Engineering MSc Candidate, Center for Modeling Petroleum Reservoirs, CERENA/DECivil, Instituto
Superior Técnico, Universidade Técnica de Lisboa, Lisbon, Portugal. E-mail: [email protected]
Integration of Rock Physics Models in a Geostatistical Seismic Inversion for
Reservoir Rock Properties
Amaro C.1
Abstract: The main goal of reservoir modeling and characterization is the inference of the spatial
distribution of petrophysical properties of interest, such as facies, porosity, mineral volumes and fluids.
Usually this is a two-step approach where the petrophysical properties of interest are derived from
inverted elastic models. This sequential approach does not ensure the propagation of the uncertainty
related to the seismic inversion problem into the resulting rock property models. This problem can be
tackled by inverting the seismic reflection data directly to petrophysical properties (e.g. porosity, volume
of shale and water saturation) ensuring the propagation of the uncertainty and measurement errors into
the estimated subsurface models. The purpose of this work is to invert seismic reflection data directly to
petrophysical properties, to properly propagate the uncertainty related to the seismic inversion problem
and measurement errors into the estimated subsurface elastic models. It is presented a novel methodology
that combines rock-physics models and a stochastic inversion with global perturbation method, that can
quantify the relationship between geologic processes and the corresponding geophysical signatures. This
method has been tested on real well-log data and partially stacked seismic data. The application to a real
reservoir converges towards the real seismic data and provided realistic petrophysical models and facies
volume with the corresponding elastic models retrieved from the rock-physics modeling process.
Keywords: Geostatistical Seismic Inversion, Rock-Physics Models, Facies, Reservoir Modeling
INTRODUCTION
Reservoir’s performance is directly related to the
natural heterogeneities of the subsurface geology.
Within the exploration and production stages,
reservoir modeling plays a crucial role in the
assessment of the productive zones. The main goal of
reservoir characterization is to identify the spatial
distribution of the petrophysical properties of interest,
such as facies, porosity, mineral volumes and fluid.
Frequently, petrophysical properties are derived from
inverted elastic models in a two-step approach. When
inverting seismic reflection data directly to
petrophysical properties, the uncertainty related to the
seismic inversion problem and measurement errors, is
propagated into the estimated subsurface models. The
integration of rock-physics modeling within the
inversion loop allows linking the inverted subsurface
rock properties with the corresponding elastic
response.
The inference of petrophysical properties is based on
a perturbation technique that performs a stochastic
sequential simulation (DSS; Soares 2001) and co-
simulation with joint probability distributions (Horta
and Soares 2010) of the model parameter space,
ensuring the reproduction of the prior probability
distributions, honouring the data values at each
location, reproducing the original statistics (mean and
variance), as well as reproducing the spatial
continuity pattern imposed by the variogram model.
Contrary to Sequential Gaussian Simulation (SGS;
Deutsch and Journel 1998), the use of DSS allows the
distribution of the property to be directly simulated
without any transforms, as estimated from the
experimental data (i.e. well-log data). The selection
of the first property to simulate should consider the
quality of the available well-log data. As a best-
practice, this property should be the one associated
with a larger uncertainty and smoother.
Each set of simulated and co-simulated petrophysical
properties generate a facies volume with a Bayesian
2
classification. A key input of the inversion for
petrophysical properties and facies, are the prior
facies proportions. These proportions are normally
estimated from the available well-log data before the
inversion procedure and propagate the corresponding
uncertainty in zones away from the wells. In most
cases (e.g. Grana and Della Rossa 2010) there are
three main litho-fluid classes: shale, brine sand and
oil sand. To constrain even more the reservoir, other
sub-categories can be used, such as the stiffness of the
mineral material. In simpler cases, it is easy to
identify clusters to constrain the prior probabilities,
but when dealing with a complex reservoir more
properties can helpful. In cases where the training
data, constructed from the well-log data is statistically
representative of the reservoir conditions, Bayesian
classification is a successful application to classify
facies. On the other hand, when few wells are
available to represent the lithologies and fluid types,
a useful method is to increase the training data with
Monte Carlo Simulation. The resulting facies models
represent the link between the rock properties and the
real subsurface geology and are conditioned not only
to the available seismic reflection data but also to the
existing well-log data.
The generated set of models composed by water
saturation, porosity, volume of shale and facies
volume are used as input in an ensemble of facies-
dependent rock-physics models (RPMs) to predict the
seismic velocities (upper and lower bounds of seismic
velocities) of a rock and/or facies. RPMs are used to
link data from different domains, i.e to improve
coherency between the subsurface rock properties
and elastic properties. They can be represented by a
simple regression based on well-log data or a complex
physical model, with a number of elastic parameters
to be estimated e.g., elastic moduli of matrix and fluid
components, critical porosity, aspect ratio, and/or
coordination number (Mavko et al. 2009).
There are two fundamental tasks for this method to be
consistent: a well-log calibration of the rock-physics
models and quality control. Then, the main procedure
is to ajust a theoretical model to a trend in the data.
The solid phase, is the mineral part of a composite
made of the mineral frame and the pore fluid
(Dvorkin et al. 2014). Granular media models
describe the rock as a collection of separate gains that
contact between them with a certain stiffness. This
model is usually applicable to sandstones (Simm and
Bacon 2014) and based on Hertz-Mindlin contact
theory (Dvorkin et al. 1994). A very effective
approach starts by the definition of the elastic
properties the end-members. At zero porosity, the
rock must have the properties of the mineral, and at
the high porosity, the elastic contact theory
determines the elastic properties. The interpolation of
this two end-members is based on upper and lower
Hashin-Shtrikman bounds. The upper bound is
usually associated with the contact cement (stiff-sand
model) and the uncemented or low cemented rock
(soft-sand model) is represented by the lower bound
(Mavko et al. 2009). The effect of pore fluid is
accounted by using Gassmanns’ equation (1951)
which is the most common model, within this setting,
to predict fluid substitution effects at low seismic
frequencies (Mavko et al. 2009).
The soft- and stiff-sand models for VP and VS can be
computed for a wet rock, by first calculating the
models for a room-dry grain pack to posteriorly use
them in Gassmanns’ fluid substitution equation.
Densities for the matrix and fluid can be computed
using Woods (1955) formula. The facies-dependent
rock-physics models allow the calculation of
velocities and densities for each facies individually,
to posteriorly generate synthetic seismograms
following Shueys’ (1985) 3-terms approximation.
The quality of the inversion results are assessed based
on the match between the calculated synthetic seismic
data and the real seismic reflection data with a
correlation coefficient. Within global inversion
approaches, the highest correlation coefficient
portions of the generated models are selected and
used along with the corresponding local correlation
coefficients, to produce new sets of rock property
models in a co-simulation. At each iteration, it is
performed an update of the set of rock properties
based on the trace-by-trace match between the real
and synthetic seismic data.
3
METHODOLOGY
The proposed iterative geostatistical seismic
inversion methodology (Figure 1) integrates rock-
physics modelling contains the following main steps.
First, DSS simulates water saturation models using
the available well-log as experimental data and a
spatial continuity pattern as revealed by a variogram
model. Then, co-DSS with joint probability
distributions is used to co-simulate porosity models
given the previously simulated water saturation
models and the available well-log data. The
simulation of volume of shale models also recurs to
co-DSS with joint probability distributions, given the
water saturation previously inverted model and the
available well-log data.
Before applying a classification algorithm, a training
data is generated from the well-log data. The facies of
interest are identified in a petrophysical domain such
as water saturation versus porosity and porosity
versus volume of shale, resulting in five facies: stiff-
shale, soft-shale, stiff-brine sand, soft-brine sand and
soft-oil sand. Then a Bayesian classification
algorithm uses the previously simulated models and
the training data to create a facies volume. A facies
dependent rock-physics modelling uses the set of
three simulated models (water saturation, porosity,
volume of shale) and the facies volume to compute
𝐕𝐏, 𝐕𝐒 and density. The resulting P- and S- velocity
models along with density are used to compute angle-
dependent reflection coefficients which are then
convolved with angle-dependent wavelets to generate
synthetic seismic data, following Shueys' linear
approximation After generating the partially stacked
synthetic seismic reflection data each synthetic
seismic trace is individually compared against the
corresponding real seismic trace in terms of
correlation coefficient. At
each iteration, and from the
ensemble of rock
properties generated
during the first step of the
proposed algorithm, the
portions of these models
that ensure the highest
correlation coefficient
between real and synthetic
seismic for all angles are
simultaneously selected
with the correlation
coefficients. The selection
procedure is based on cross-over genetic algorithm
where the best genes (portions of the petrophysical
models from different realizations that ensure the
highest correlation coefficient) of each iteration are
then used as seed for the generation of a new family
of models during the next iterations. Iterate the entire
procedure until a given global correlation coefficient
between the angle-dependent synthetic and real
seismic data is above a certain threshold.
REAL DATA EXAMPLE
The available dataset comprises four partial angle
stacks, with mean reflection angles of 9º, 15º, 21º and
27º and their corresponding dependent wavelets and
a set of four well-logs composed by porosity, volume
of shale, water saturation and P- and S-Impedance.
The inversion grid has 159x419x128 cells in i, j and
k directions respectively. The grid cell size was
defined to reproduce the original inline and cross-line
spacing and the original seismic sampling interval, 2
ms. The joint distributions between water saturation
versus porosity and porosity versus volume of shale
are used as conditioning data in the direct sequential
co-simulation with joint probability distributions
(Horta and Soares 2010). In this way, the inverted
petro-elastic models mimic the real reservoir
conditions by reproducing the relationships between
the primary and secondary variables (Figure 3). The
spatial continuity pattern of each property
individually is imposed by a variogram model,
estimated for all properties in study for the vertical
direction. These variograms are part of the
geostatistical inversion procedure and are used as
Figure 1 - Schematic representation of the general framework of the proposed methodology.
4
conditioning data for the stochastic sequential
simulation of the petro-elastic properties of interest.
As a geostatistical inversion procedure, each model
reproduces the well-log data at its locations, the
variogram model imposed during the stochastic
sequential simulation and the marginal and joint
probability distributions as inferred from the
available well-log data.
The rock property models (water saturation, porosity
and volume of shale; Figure 2) agree with the main
structures as interpreted from the original seismic
data and from previous inversion studies over this
reservoir. The interpretation of the mean model of
the ensemble of petrophysical models simulated in
the last iteration (Figure 2) is one way of interpreting
the results of geostatistical seismic inversion.
However, it is important to highlight that the
individual realizations have higher variability
presenting small- and large-scale details. In the
inverted models of water saturation, porosity and
volume of shale, the well-log data values are
reproduced at well locations, as well as the
histograms of each property. The proposed
methodology also allows the assessment of the
uncertainty of each property individually by for
example, the variance model at a given iteration.
Notice that assessing the uncertainty related with
each property individually is of great importance for
better reservoir modeling. Usually, these areas are
associated with low signal-to-noise ratio, i.e., areas
with higher uncertainty, seismic and the inverted
models do not match with the seismic. Areas with
high variability are related to more uncertainty
regarding the model parameters, and usually
correspond to areas far from the wells, where fewer
data is available. Before the inversion of
petrophysical properties (continuous quantities),
facies (discrete quantities) were classified in four
facies. Each facies must be defined such as it captures
the physics of the reservoir’s geology. The facies of
interest are identified in a petrophysical domain such
as water saturation versus porosity and porosity
versus volume of shale. First, three main groups are
classified based on the well-log data rock properties
(Figure 5a and Figure 5c): shale, brine sand and oil
sand. This classification is based on assumptions
related to the known geology of the reservoir, i.e.
a) Water Saturation b) Porosity c) Volume of Shale
Figure 2 - Vertical section mean models and standard deviation computed from the last 32 models retrieved from the last
iteration. From top to bottom: water saturation, porosity and volume of shale.
a)
d)
b)
c)
Figure 3 - Joint distributions from the well-logs (on the left)
and from the best-fit models of iteration 6 (on the right) of
water saturation versus porosity (a;c) and porosity versus
volume of shale (b;d). The joint distributions reproduce the
ones estimated from the well-log data.
5
facies shale has a shale content threshold of 0.3, while
the fluid factor was added by water saturation with a
threshold of maximum 0.8 for the oil sands. Then, to
isolate the sand reservoir, another classification is
performed considering rock-physics. This
classification is based on the stiffness of the rock and
it splits each shale, brine sand and oil sand into two
categories, stiff and soft depending on the porosity
and shale content. This classification assumes that
shale facies belong to the granular media models,
such as sands. According to the available well-log
data and the geological setting, the reservoir only has
one type of oil sand is present, soft oil sand. A
Bayesian classification algorithm is then used to
classify all four facies for the entire inversion grid
considering the previously simulated petrophysical
properties. The resulting facies model reproduce the
distributions of water saturation, porosity and volume
of shale used as training data (Figure 5b and Figure
5d). The resulting facies volume (Figure 4a) is
consistent with the petrophysical models (Figure 2),
as well as the structures interpreted in the real seismic.
The ensemble of facies models generated during the
last iteration may be used to derived facies probability
for each facies (Figure 4b), showing the abundance of
stiff brine sands where the in the same location
where the stiff shale has lower probability of
occurrence.
The reservoir – soft oil sand – has high probability
of occurrence also in areas where the progradation is
located but with less continuity when compared to
the predominant facies. It is clear the intercalation
between sand and shale facies and the predominance
of stiff facies with small-scale details
Table 1 - Rock-physics model parameters: density 𝝆,
bulk modulus K, and shear modulus 𝝁 of the matrix
and fluid components for a 𝜱𝒄 of 0.4, n of 4 and Peff of
20 Mpa.
which is a consequence of constraining the stochastic
simulation and co-simulation to the inverted models
to posteriorly generate facies volumes.
𝜌 (g/cm3) K (GPa) 𝜇 (GPa)
Sand 2.59 39.90 46.24
Shale 2.52 6.81 19.82
Oil 0.63 0.59 n.a.
Water 1.1 2.68 n.a.
a)
b)
Figure 4 -Vertical section extracted from the best-fit
facies model (top) and the probability of occurence of soft
oil sand (bottom). Where facies 1 corresponds to soft
shale, facies 2 to stiff shale, facies 3 to soft brine sand,
facies 4 to stiff brine sand and facies 5 to soft oil sand.
a)
b)
c)
d)
Figure 5 - Training data for facies modeling using the
training data from the wells (on the left); joint distributions
from the best-fit inverted models (on the right) of water
saturation versus porosity and porosity versus volume of
shale. Each joint distribution is color coded by facies.
6
a)
d)
The elastic models (Figure
6a, Figure 6b, Figure 6c)
calculated from the
petrophysical models using
the facies dependent rock
physics models calibrated
with the well-log (Table 1)
reproduce the well-log data
and the histograms are
close to the ones computed
from the original well-log
data(Figure 6d, Figure 6e,
Figure 6f). The global
correlation coefficient
(62%, Figure 8) between
the synthetic seismic elastic
models is higher than the
global correlation
coefficient achieved by the
mean model (Figure 7). The
areas associated with The
synthetic seismic reflection
data computed from the
arithmetic mean elastic
models (Figure 9) matches
successfully the real
partially stacked seismic
data.
b)
e)
c)
f)
Figure 6 - Vertical sections of the mean elastic models retrieved from the 32 simulations
of the last iteratio (on the left) along with the reproduction of the histogram of the inverted
model (filled blue) and well-log data (red). From top to bottom, P- (a) and S- wave
velocity (b) and density (c).
a) Nearstack b) Mid-nearstack c) Midstack d) Mid-farstack
Figure 7 – Correlation coefficients between the real and synthetic seismic
data of all partial angle stacks. From left to right: nearstack, mid-nearstack,
midstack and mid-farstack.
Figure 8 - Global correlation coefficient
evolution per iteration.
7
It reproduces the location of the main primary
reflections and the amplitude variations versus offset.
low degree of uncertainty (low standard deviation)
are related to areas where the synthetic seismic
converged properly towards the real one, associated
with high correlation coefficients. On the other hand,
areas with low correlation coefficients are possible
related with values not considered in the conditioning
dataset, as well as seismic data with low signal-to-
noise ratio, very common in real datasets. Within this
inverse methodology, the noisy areas are assigned a
higher uncertainty throughout the entire procedure.
DISCUSSION
After 6 iterations, an the inversion procedure reached
a global correlation coefficient between the synthetic
seismic and real seismic of 62%. The use of partially
stacked seismic data allows the inversion of seismic
directly for rock properties to better distinguish litho-
fluid facies, instead of the traditional acoustic models.
All models obtained from the entire iterative
geostatistical inversion procedure with rock physics
integration, reproduce: the values of the conditioning
data at its locations (Figure 2); the joint and marginal
distributions of water saturation, porosity and volume
of shale and the spatial continuity model of each
property imposed within the sequential simulation by
a variogram.
This method is successful to discriminate lithology
parameters and detect hydrocarbon probability of
occurrence directly from the petrophysical properties.
From the petrophysical properties itself, it is possible
to detect some geological features such as the shale
trend along the presented horizontal slice, with low
porosity and high shale content crossing the
horizontal time slice ( Figure 10c) which is also
present in Figure 10d, classified mainly as stiff shale
a)
b)
Figure 9 – Real reflection data (a) and synthetic seismic
section of the arithmetic mean of 32 simulations of the
last iteration, of the partial angle midstack.
a) Water Saturation b) Porosity c) Volume of Shale d) Facies Volume
Figure 10 – Horizontal time-slices (from left to right) of water saturation (a), porosity (b), volume of shale (c)
and facies volume (d) (k=120).
8
with soft shale in some portions of that area, showing
the clear relationship of rock-physics per facies.
The variance models (Figure 2) computed between
the models generated during the last iteration show
lower variability i.e., lower spatial uncertainty in the
volume of shale inverted models, following water
saturation and porosity. Usually high values of
variance are related to areas where the seismic is
noisy and the inverted petrophysical models cannot
produce synthetic seismic data that fits the real
seismic, or the lack of real petrophysical properties.
Most of the spatial uncertainty is related with the
location of geological boundaries of facies bodies,
because of its similarities with the inverted
petrophysical models (Figure 2) and the facies
volume (Figure 4).
Also, it is important to highlight that the interpretation
of reservoir models along with their corresponding
uncertainty allows better decision making and risk
management.
CONCLUSIONS
The presented methodology was successfully
designed, implemented and applied in a real dataset
resulting in a good match between the real and the
inverted synthetic seismic data. The novel iterative
geostatistical seismic inversion methodology that
simultaneously integrates seismic reflection data,
well-log data and rock physics models can retrieve
directly from the seismic data reliable petrophysical
models such as water saturation porosity, volume of
shale and facies volumes. The results of the presented
inversion method are consistent with other seismic
inversions applied to this reservoir, but also add value
due to the identification of new geological features
and proper fluid/lithology characterization.
This approach provides a direct connection between
the seismic response and the geological
(petrophysical) properties, by the application of Rock
Physics Models allowing the propagation of the
uncertainty related to the seismic inversion in one-
step approach.
It is an efficient method to guide and improve
qualitative interpretation, as well as avoid ambiguities
in seismic interpretation related to fluids/lithology,
sand/shale and porosity/saturation. This novel
method can be applied to all reservoir where the
physical link between the elastic and petrophysical
properties can be described by a suitable rock physics
model, as well as adapt other rock properties and
litho-fluid facies.
ACKNOWLEDGEMENTS
The authors would like to thanks Schlumberger for
the donation of the academic licenses of Petrel® and
CERENA/IST for supporting this work. CA want to
thanks Professor Dario Grana from the University of
Wyoming for the valuable input to this work and the
Department of Geology and Geophysics for the three
months stay.
APPENDIX A
ROCK-PHYSICS MODELS
The soft- and stiff-sand models are based on Hertz-
Mindlin grain-contact theory and provide estimation
of the bulk and shear moduli of a dry rock, assuming
a random pack of identical spherical grains under an
effective pressure P, with a certain critical porosity
(𝜱𝒄) and a coordination number (Mavko et al. 2009).
𝐾𝐻𝑀 = √𝑛2(1 − 𝛷𝑐)2𝜇𝑚𝑎𝑡
2 𝑃
18𝜋2(1 − 𝜈)2
3
and
A - 1
𝜇𝐻𝑀 =5 − 4𝜈
10 − 5𝜈√
3𝑛2(1 − 𝛷𝑐)2𝜇𝑚𝑎𝑡2 𝑃
2𝜋2(1 − 𝜈)2
3
A - 2
where μmat is the shear modulus of the solid phase
and ν is the grain Poissons’ ratio.
The matrix moduli is calculated using Voigt-Reuss-
Hill averages for a matrix with sand and clay
materials:
9
𝐾𝑚𝑎𝑡 =1
2(𝑉𝑐𝐾𝑐 + (1 − 𝑉𝑐)𝐾𝑠 +
1
𝑉𝑐
𝐾𝑐+
𝐾𝑠
(1 − 𝑉𝑐)
)
and
A - 3
𝜇𝑚𝑎𝑡 =1
2(𝑉𝑐𝜇𝑐 + (1 − 𝑉𝑐)𝜇𝑠 +
1
𝑉𝑐
𝜇𝑐+
𝜇𝑠
(1 − 𝑉𝑐)
) A - 4
where Vc is the volume of clay, Kc, μc, are the bulk
and shear moduli, respectively of the clay and and 𝐾s,
μs is the bulk and shear moduli of the sand. The bulk
(KHM) and shear (μHM
) moduli of a room-dry rock is
estimated recurring, for example to Hertz-Mindlin
grain-contact theory, under the assumption that the
sand frame with a random pack of identical spherical
grains is under an effective pressure P, with a certain
critical porosity ( 𝛷𝑐 ) and a coordination number
(Mavko et al. 2009).
𝐾𝐻𝑀 = √𝑛2(1 − 𝛷𝑐)2𝜇𝑚𝑎𝑡
2 𝑃
18𝜋2(1 − 𝜈)2
3
and
A - 5
𝜇𝐻𝑀
=5 − 4𝜈
10 − 5𝜈√
3𝑛2(1 − 𝛷𝑐)2𝜇𝑚𝑎𝑡2 𝑃
2𝜋2(1 − 𝜈)2
3
A - 6
where μmat is the shear modulus of the solid phase
and ν is the grain Poissons’ ratio.
For the effective porosity values between zero and the
critical porosity, this model that best fits the data
(modified lower and/or upper Hashin-Shtrikman
bound), interpolates the the two end-members, i.e. the
solid-phase elastic moduli (𝐾𝑚𝑎𝑡 and 𝜇𝑚𝑎𝑡
) and the
elastic moduli of the dry rock (𝐾𝐻𝑀 and 𝜇𝐻𝑀
). At any
porosity 𝛷 < 𝛷𝑐 , the main point of the “soft”
connector (modified lower Hashin-Shtrikman bound)
is given by the following equations:
𝐾𝑠𝑜𝑓𝑡 = [𝛷 𝛷𝑐⁄
𝐾𝐻𝑀 +43
𝜇𝐻𝑀
+1 − 𝛷 𝛷𝑐⁄
𝐾𝑚𝑎𝑡 +43
𝜇𝐻𝑀
]
−1
−4
3𝜇𝐻𝑀
A - 7
𝝁𝒔𝒐𝒇𝒕 = [𝜱 𝜱𝒄⁄
𝝁𝑯𝑴 +𝟒𝟑
𝝃 𝝁𝑯𝑴
+𝟏 − 𝜱 𝜱𝒄⁄
𝝁𝒎𝒂𝒕 +𝟒𝟑
𝝃 𝝁𝑯𝑴
]
−𝟏
−𝟏
𝟔𝝃𝑯𝑴 𝝁𝑯𝑴,
A - 8
𝜉𝐻𝑀 =9𝐾𝐻𝑀 + 8𝜇𝐻𝑀
𝐾𝐻𝑀 + 2𝜇𝐻𝑀
For for any porosity 𝛷 > 𝛷𝑐 the modified upper
Hashin-Shtrikman bound, or the “stiff” connector
is given by:
𝐾𝑠𝑡𝑖𝑓𝑓 = [𝛷 𝛷𝑐⁄
𝐾𝐻𝑀 +43
𝜇𝑚𝑎𝑡
+1 − 𝛷 𝛷𝑐⁄
𝐾𝑚𝑎𝑡 +43
𝜇𝑚𝑎𝑡
]
−1
−4
3𝜇𝑚𝑎𝑡
A - 9
𝑲𝒔𝒕𝒊𝒇𝒇 = [𝜱 𝜱𝒄⁄
𝑲𝑯𝑴 +𝟒𝟑
𝝁𝒎𝒂𝒕
+𝟏 − 𝜱 𝜱𝒄⁄
𝑲𝒎𝒂𝒕 +𝟒𝟑
𝝁𝒎𝒂𝒕
]
−𝟏
−𝟒
𝟑𝝁𝒎𝒂𝒕
A - 10
𝜉 =9𝐾𝑚𝑎𝑡 + 8𝜇𝑚𝑎𝑡
𝐾𝑚𝑎𝑡 + 2𝜇𝑚𝑎𝑡
While density (𝜌) is simply the arithmetic average of
the various solid and fluid components of the rock
(weighted according to their volume fractions)
velocity is sensitive (Simm and Bacon 2014).
Gassmanns’ equation (1951) is used to model fluid
substitution effects at low seismic frequencies
(Mavko et al. 2009). P- and S-wave velocities are
estimated using matrix and fluid properties:
Ksat = Kdry +(1 −
Kdry
Kmat)
2
𝛷Kfl
+1 − 𝛷Kmat
−Kdry
𝐾𝑚𝑎𝑡2
and
A - 11
𝜇𝑠𝑎𝑡 = 𝜇𝑑𝑟𝑦 A - 12
From the saturated-rock elastic moduli, velocities can
be obtained by;
10
V𝑃 = √𝐾𝑠𝑎𝑡 +
43
𝜇𝑠𝑎𝑡
𝜌,
and
A - 13
V𝑆 = √𝜇𝑠𝑎𝑡
𝜌. A - 14
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