1 Petroleum Engineering MSc Candidate, Center for Modeling Petroleum Reservoirs, CERENA/DECivil, Instituto

Superior Técnico, Universidade Técnica de Lisboa, Lisbon, Portugal. E-mail: catarina.amaro@tecnico.ulisboa.pt

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

REFERENCES

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