Rock Acoustics

8/3/2019 Rock Acoustics

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Lecture Notes

TPG4170

Rock Acoustics

Rune M. Holt

NTNU, Trondheim, 2004

\\Boss\avd33\Medarbeidere\RMH\NTNU\Forelesningsnotater\TPG4170 Rock Acoustics.doc\E\1\29/01/2004


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Table of Contents:

1. Summary of poroelasticity. ...................................................................................3

2. Bounds for elastic moduli ......................................................................................63. Critical porosity model ..........................................................................................9

4. Effective medium theory .....................................................................................12

5. Sound velocities in rocks and other materials. ..................................................17

6. Wave velocities vs. porosity.................................................................................19

7. Sound velocities vs. lithology...............................................................................23

8. Effect of clay on velocities of sandstone .............................................................26

9. Sound velocities vs. fluid saturation. ..................................................................29

10. Temperature dependent wave velocities ............................................................38

11. Sound Velocities vs. Stresses. ..............................................................................39

12. Frequency dependent wave velocities....................................................................45References and Suggested Readings ............................................................................46



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1. Summary of poroelasticity.

The elastic behaviour of porous media (such as reservoir rocks) is described by so-

called poro-elastic theory. The prime developer of this theory was Maurice A. Biot, and

it is therefore often referred to as Biot theory.

The main difference between poroelastic and standard solid elastic theory is that

because of the two material phases (solid s; fluid f), one needs to account for

2 stresses: The external stress ijThe pore pressure pf

2 strains: The strain of a volume element attached to the rocks framework;v=usThe so-called increment of fluid content;

=(us-uf)

Biot introduced the zeta-parameter because it is convenient for describing fluid flow in

a porous medium. The time derivative of is directly related to the flow rate in Darcyslaw. is a measure of the ratio of displaced fluid volume to total volume; i.e.

(f p

p f

V V

V V K

= = )fp

(1.1)

The displaced fluid volume is resulting from the change in pore volume (as indicated by

the subscript p), and the compressibility (1/Kf) of the pore fluid.

The relationship between stress and strain in linear elasticity is linear. The simplest

linear form that includes both stress and strain parameters above is:

v

f v

K C

p C M

=

=

(1.2)

This is Biot Hookes law for isotropic stress conditions. Let us now consider two

basic tests as examples:

Example 1: Drained (hydrostatic) loading.

In this case, the pore pressure is kept constant (or zero; or the sample is tested in dry

conditions). Since the stresses in Eq. (1.2) have an absolute meaning, whereas the

strains are only relative, one can write differentials in and pfwhile keeping v and

; i.e. for the drained test



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2

0

( )

f

v fr

v

p

CK

M

C

vK

=

= =

=

(1.3)

In the first equation we define the framework bulk modulusKfras the drainedbulk

modulus, and relate it to the coefficientsK, C& M. The latter equation means that the

ratio C/M controls the relation between pore and bulk volume change in this

experiment:

(1.4)pC

V V VM

= =

pV V <

0

v

f

K

Cp

K

= =

=

' f fr vC

p KM

= =

'

ij ij f ijp =

The parameteris called the Biot coefficient. If only the pore space deforms and the

solid grains are considered incompressible, then the volume change and the pore

volume changes are equal, and =1. If the grains are also compressible, then

and


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ij is the Kronecker- =1 when i=j and 0 when ij. In practice it means that we can useHookes law as in elasticity of solids, but the stress changes must be effective stress

changes, and the elastic moduli must the framework moduli.Kis the bulk modulus of

the undrained rock (no fluid expelled; =0), while Cand Mare other poroelasticcoefficients. These parameters can be related to the elastic properties of the ingredientsof the porous medium, plus the porosity. These relationships can be derived from simple

thought experiments, which are presented in textbooks like Fjret al. (1992). The

results are:

(1.8)

2(1 )

1 (1 )

(1 )

1 (1 )

1

1

fr

f sfr

f fr

s s

fr

f s

f fr

s s

fr

s

K

K KK K

K K

K K

KK K

CK K

K K

M CK

K

= +

+

=

+

=

1fr

s

KC

K= =

( )

fr f

s s fr s f

K KK

K K K K K K = +

Thus;

(1.9)

Kf: Bulk modulus of pore fluid

Ks: Bulk modulus of solid grains

: Porosity

Kfr: Bulk modulus of the drained rock

(dry rock framework)

Gfr: Shear modulus of the rock framework

The upper of the equations above is known as the Biot-Gassmann equation. It can also

be written as

(1.10)

Biot hypothesised that the shear modulus is not influenced by the presence of the pore

fluid; i.e.:

G(undrained) = G(drained) (1.11)

Notice that the framework moduliKfrand Gfrdepend on the microstructure of the rock,in particular on the porosity. Various models exist for these relationships, from



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empirical to theoretical effective medium approaches. We will discuss such models in

Chapters 2 and 3.

Wave velocities:

The velocities of P- and S-waves in a poroelastic material (in the low frequency limit)

are alsoexpressed in the same way as for a solid material, except that the bulk modulus

now is the undrained bulk modulus K as given by Eqs. (1.8) or(1.10):

(1.12)

4

3fr

P

frS

K G

v

Gv

+=

=

The bulk density is found by adding the fluid and the solid contributions:

(1 )f s = +

1

1 N i

iR iM

=

=

(1.13)

If the material is air saturated at room conditions, then K Kfrin the Biot-Gassmann

equation (since the bulk modulus of air is negligible compared to that of the rock

framework). The density is also reduced, because the gas density is negligible compared

to the solid density. As a result of this, the S-wave velocity decreases slightly when adry sample is saturated with fluid, while the P-wave velocity normally (but not

necessarily!) increases.

2. Bounds for elastic moduli

For a composite material consisting of N components, the elastic moduli M (M = K or

G) are limited by the upper and lower bounds represented by the so called Voigts and

Reussmodels:

The Reuss model assumes a uniform state of stress, so that the strains of each

component are added, i.e.

(2.1)

where i is the volume concentration and Mi is the elastic (bulk or shear) modulus of

component i. A well-known example where the Reuss model gives a correct prediction



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is a suspension of particles (solid; s) in a fluid (f). The undrained bulk modulus of the

suspension is given by

(2.2)1 1R f sK K K

= +

1

N

V i i

i

M M =

=

(1 )V sK K=

( )

( ) ( )

4

3

413

s s s f

HS s

s s s f

K G K K

K K

K G K K

+

+ =

+

where is the porosity (fluid volume divided by total volume). This is also known as

Woods equation.

In general, the Reuss modulus MR gives a lower bound for the elastic moduli of the

composite. The Reuss bound for the shear modulus GR=0 because the fluid does not

have any shear modulus. Likewise, the drained (frame) Reuss bulk modulus is zero,since Kf=0 for evacuated pores.

The Voigt model assumes a uniform state of strain, so that the associated stresses are

additive. Thus,

(2.3)

where M again can be K or G. The Voigt modulus represents an upper bound for the

elastic moduli of a composite. It is used to describe the elastic properties of e.g. a

polycrystalline material. The drained (frame) Voigt bulk modulus is

(2.4)

Hashin and Shtrikman calculated bounds based on elastic energy considerations,

assuming the composite medium is a mechanical mixture of isotropic and homogeneous

elastic phases. One may visualise their model as packing of spheres of one material

coated by spheres of the second phase. The lower limit for the elastic moduli is found

when the softer material is coating the stiff one, while the upper limit represents a

coating of stiff material on the softer. For a two-phase (solid and fluid) medium, their

results are for the upper (K+) and lower (K-) bounds on the bulk modulus:

(2.5)



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(2.6)( )

( )1

s f

HS R

s f

K KK K

K K

= +

( ) ( )

45

3

45 2 1 2

3

s s s

HS s

s s s s

K G G

G G

K G K G

+

+ =

+ +

0HSG =

,

1 1

p p s f V V V

= +

For the shear modulus, they found:

; (2.7)

The potential use of the bounds is to find the permitted intervals for elastic properties of

the composite. Adding more information about the structure of the medium, bounds

may be found which are sufficiently close together that one may use them to actually

predict properties of the effective medium. One assumption which has been made

without actually incorporating microstructural knowledge, is the so-called Hills

average e.g. that elastic moduli may be estimated in a rough way as the average value

between the upper (e.g. Voigt) and lower (e.g. Reuss) bounds.

One model for wave velocity vs. porosity which has been (and still is) used quite often

by the oil industry is the time-average equation. It relates the P-wave velocity vp to

porosity and P-wave velocities of the solid (vp,s) and fluid (vf) phases is an empirical

equation:

(2.8)

The theoretical basis for this equation is to assume that the sound wave shares its time

passing through the rock in volumetric proportion in the solid and pore fluid. This is

strictly valid only if the wavelength is much shorter than the grain and pore size, i.e. inthe limit of very high frequency. The fact that dispersion has been found normally to be

quite low in porous rocks may explain the success of this empirically based approach.



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3. Critical porosity model

Sedimentary rocks can be considered as packages of grains which somehow are

cemented together. In between the grains, there is a pore fluid. If the porosity is high, so

that the grains do not touch each other, the undrained bulk modulus K of the porous

medium can be calculated as for a suspension (Reuss average); i.e. according to Eq.(2.2)

. The drained bulk modulus as well as the shear modulus G of a suspension is zero

(since Gf=0). These relationships are valid until the porosity gets below a limiting

critical porosity c. Then the grain skeleton starts to carry load. This limit is given by

the shape and size distribution of grains. The loosest possible packing of spherical,

equally sized grains gives a porosity of 0.476 (simple cubic packing), while for a

random loose packing one finds porosities between 0.4 and 0.45. For typical sands, the

critical porosity seems to be around 0.40. In chalk, where the pores are more like

spherical in shape, the critical porosity is higher (0.60-0.70).

When the porosity is lower than c, there will be a fintite framework bulk modulus and

a finite shear modulus. The actual values will depend on porosity, and on the structural

details of the rock, e.g. on the degree of cementation between the grains. With no

further knowledge, a simple assumption is that the framework stiffnesses will varylinearly from zero at the critical porosity to the values of the solid constituent at zero

porosity, i.e.


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The critical porosity model is illustrated in .Figure 3.1

The weaknesses of this model are that the critical porosity is not a universal constant -

not even for a given class of rocks, such as sandstones, and that the linear relationship

between moduli and porosity is a very rough approximation. The strength is that the

model is very simple, and does not require any other assumptions about microstructure

than the knowledge of a critical porosity. We will use it further in the course to gain

insight into e.g. effects of lithology, effects of clay content etc. One should however be

careful to use it uncritically in field data analysis.

(1 )fr sc

M M

=

A modification that can be made to account for the observed diversity of behaviours is

to modify Kfrand Gfrin Eqs. (3.1) and (3.2) by introducing an exponent :


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Figure 3.1 Undrained bulk modulus vs. porosity for Reuss, Voigts and NursCritical Porosity model.

0

5

10

15

20

25

30

35

40

0.00 0.10 0.20 0.30 0.40 0.50

Porosity

Bu

lkmo

du

lus

[GPa

]

=2

=1

=1/2

Figure 3.2: Modified critical porosity model using Eq.(3.5)



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4. Effective medium theory

Several more microscopically based theories exist in the literature. They can be divided

into two main groups:

Inclusion models, where pores or cracks are treated as voids or inclusions in a

solid matrix (Swiss cheese (or Jarlsberg, if preferred) model).

Grain pack models, where solid particles represent the building elements.

While the inclusion models appear most realistic for hard rocks with low porosity, they

find significant application within sedimentary rock physics. One reason for this is that

they explicitly express elastic moduli as function of porosity, while porosity plays a

more implicit role in the grain pack models.

Figure 4.1

Figure 4.1 Sketch of a solid sphere of volume V containing a spherical cavity of

volume Vc.

shows the basic idea of an inclusion model: The elastic properties of a sphere

of solid material containing a spherical void are calculated. Then, as a next step, the

effect of adding several inclusions is calculated by simply adding their influence

independently. This impies that the voids are sufficiently far apart from each other notto interact; i.e. that the inclusion density is small. For spherical pores in a solid host

material, it has been found:

Vc

Ks, Gs

V



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31 11 1

* 4

s

s s

K

K K G

= + +

(4.1)

(4.2)15 201 1

1* 9 8

s S

s S S

K G

G G K G

+

c

a=

3N aV

= < >

43

C =

+= +

These expressions are strictly valid only for low porosity. One way to make them more

general is to make the effective medium self consistent: this is done by replacing

(mathematically) the solid material surrounding the inclusion with the resulting

effective medium. The solutions can not in general be expressed analytically. They do

however have a form that leads the elastic moduli to 0 at a certain porosity., hence,

not unlike the critical porosity model.

In most cases the spherical porosity model overestimates the elastic stiffness of real

rocks. The main reason for this is that the pore shapes found in natural rocks deviate

strongly from spherical. Rather, a significant amount of low aspect ratio pores normally

exists. Such pores are much more compliant than spherical pores, and play a more

important role in controlling the wave velocities. Although there are models that may

incorporate inclusions of various shapes, we will here only mention the so-called crack

models: A penny-shaped crack is defined as an ellipsoid with a short axis = 2c and a

long axis = 2a, so that the crack aspect ratio is . Cracks affect the elastic moduli

through a parameter called crack density ():

(4.3)

The crack porosity is related to the crack density as

(4.4)

Since the aspect ratio for a crack is small, the crack porosity is usually also small. It

does however, as said above, have a significant impact on the elastic moduli. For non-

interacting cracks with a random spatial and orientational distribution, one finds:



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2a

2c

Figure 4.2: A penny shaped crack.

(4.5)2

1 1 16 11

* 9 1 2

s

s sK K

( ) ( )32 1 51 11

* 45 2

s s

s sG G

21161

9 1 2

fr

fr s

fr

K K

( ) ( )1 5

32145 2

fr fr fr s

fr

G G

= +

(4.6)( )

= +

In the self-consistent approach (Budiansky & OConnell, 1976):

(4.7)

=

(4.8)( ) =

where

161

9fr s

(1 )fr s p p cM a b

(4.9)

We see that the elastic moduli 0 as the crack density becomes large enough. For such

large crack densities, one would however not expect the basic assumption behind the

theory to be valid anymore: Cracks tend to coalesce and not be randomly distributed inspace when a material approaches the failure limit.

We may add the effects of pores and cracks in the case of reasonably low porosities and

cracke densities, i.e.

(4.10)

Mrepresents bulk or shear modulus, ap and bc are constants (depending on the solid host

material properties), and p is the spherical (equant) porosity. Since cracks are much

more compliant than spherical pores, the last term gives rise to stress dependency.



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When an external stress is applied, the lowest aspect ratio cracks are closed first, and

then subsequently higher aspect ratio cracks are closed as the stress increases. This leads

to a relatively rapid increase in moduli until all cracks are closed. From then on, further

stress increase will give rise to increase in moduli by reduction of porosity.

The second class of effective medium theories for porous media is grain pack

descriptions. The basic element in these models is the elastic contact between two

grains in contact. For two equally sized spheres in contact, the elastic contact law has

been derived by Hertz and Mindlin.

Figure 4.3: Sketch of Hertzian grain contact.

Figure 4.3 shows a sketch of such a grain contact. The two spheres have radius a. The

contact area between the spheres is a circle with radius b.s represents the displacement

as a result of a compressive forceFapplied to the pair of spheres. Following the

arguments of deGennes (1996) (Note: This is not a mathematical proof, buthandwaving

arguments of a Nobel prize winner in physics), the contact stress is then of the order

(F/b2). The contact strain is of the order (s/b). From geometrical considerations,s/b

b/a. Thus;

(4.11)2

3 1 3

22 2 2

;

( )

F s

b b

sF sb s a a

a

3

2

The global stress is of the order (F/a2) , and the global strain is of the order (s/a).

Thus;

(4.12)



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We see that this is a nonlinear elastic element, since stress is not directly proportional to

strain. The stiffness is

(4.13)11

32dMd

1

6

1

2 2 2 3

2 2

(1 )

18 (1 )

s

frs

n G

K

=

1

2 2 2 3

; 2 2

12(1 )1

10 (1 )

sfr smooth

s

n GG

=

1

2 2 2 3

, 2 2

5 4 3(1 )

5(2 ) 2 (1 )

s sfr rough

s s

n GG

=

Hence, the wave velocities, which are proportional to the square root ofM, should

increase with external stress as . For a random packing of equally sized spheres, the

bulk modulus for a porosity (given by the packing) can be calculated as

(4.14)

This result is often referred as the Hertz-Mindlin model, or the Walton model. n is the

coordination number, being defined as the average number of spheres in contact with a

given sphere. Typically n decreases with increasing porosity, from around 12 at closest

packing (porosity 26%) to 6 at most open packing (porosity 48%).

The shear modulus depends on wether the spheres are considered rough or smooth. The

results are:

(4.15)

(4.16)



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5. Sound velocities in rocks and other materials.

Table 5.1

Table 5.1: P- and S-wave velocities in rocks and some common materials.

shows measured P- and S-wave velocities in rock and rock-like, plus a few

other common materials.

Material vp (m/s) vs (m/s) (g/cm3) CommentsDry loose sand 100-500 10-300 1.8-2.1

Saturated loose

sand

1500-2000 10-300 2.0-2.4

Weak sandstone 1000-2500 500-1500 1.8-2.1

Competent

sandstone

2500-4500 1500-2500 2.0-2.3

Berea sandstone 3800-4000 2300-2400 2.2 water sat.,

unloaded

Weak Chalk 2000-2600 1000-1300 1.8-2.3

Limestone 3500-6000 2000-3500 2.4-2.7

High porosityshales

1700-2500 500-1000 1.8-2.0

Low porosity

shales

2500-5000 1000-2000 2.0-2.6 strongly

anisotropic

Coal 2200-2700 1000-1400 1.3-1.8

Gneiss 4400-5200 2700-3200 2.5-2.7

Basalt 5000-6000 2800-3400 2.5-2.7

Dolomite 3500-6500 1900-3600 2.5-2.9

Air 330 - 1.310-3

Water 1470 - 1.0

Ice 3400-3800 1700-1900 0.9

Oil 1100-1300 - 0.6-0.9

Steel 5940 3230 7.9

Plexiglass 2550 1280 1.2

-Quartz 6030 4040 2.65

Geitost (Norw.

brown cheese)

1830 ? 1.2



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A main message to be learnt from these numbers is that there are large variations within

each class of rocks. Natural rocks like sandstone, shale or limestone should not be stated

to have a typical P- or S-wave velocity. Rather the velocities depend on a number of

factors, like the the composition and the microstructure of the rock itself (lithology,porosity, degree of cementation, fluid saturation, saturating fluid, etc.), and the

boundary conditions under which the measurement is performed (w.r.t. stress, pore

pressure, temperature). In the following chapters we will describe observed wave

velocities in different rocks and see how these factors influence on the velocities. The

results will be compared to theoretical models where available. We will use the Biot

Gassmann model as our macroscopic basis, and because of its simplicity, we will use

the critical porosity model to illustrate some of the observed relationships.

The reader is also referred to the more extensive Rock Physics Handbook by Mavko et

al. (1998).



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6. Wave velocities vs. porosity.

Lucretius in the 1st century BC wrote that the more vacuum a thing contains, the morereadily it yields. He was followed by Vitrivius (10 AD) who stated that cracks make ..

bricks weak. The contents of these citations (Kendall, 1984) are essential in order to

understand how wave velocities in porous rock behave: In all rocks one finds that

velocities decrease with increasing porosity. This is intuitive if one thinks of a wave

velocity measurement as a stiffness measurement. Vitrivius statement implies that the

geometry of the pore space is also relevant, and the effect of thin cracks is particularly

large not only on the strength of rocks, but also on wave velocities and in particular

their stress sensitivity.

Figure 6.1 shows measured P- and S-wave moduli in dry, clean sandstones (afterMurphy et al., 1993). They show a nice almost linear trend, and were probably

instrumental in guiding the Stanford Rock Physics group led by Amos Nur to suggest

their critical porosity model (see also Chapter 3). Based on such a model, we would

write for the wave velocities in a dry porous medium

(6.1)

4( )(1 )

3

(1 )

s s

cp

s

K G

v

+ =

(1 )

(1 )

s

cs

s

G

v

=

(6.2)

where Ks and Gs are the solid mineral bulk and shear modulus, s is solid mineraldensity, is porosity, and c is critical porosity. The exponent inserted here is usuallytaken = 1, but there is evidence from laboratory data that weak rocks would have > 1,and strong rocks have < 1. Critical porosity in sandstone seems to be 0.4, whereas inchalk it is 0.6 0.7. In shale, nobody knows (yet).

An example of calculated velocities in dry rock for different values of is showninFigure 6.2.



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Figure 6.1: High pressure laboratory measurements of the bulk and shear frame

moduli (i.e. of dry rock) for clean quartz sandstones (from Murphy et al.,1993).

=1 =2

=1

0

1000

2000

3000

4000

5000

6000

7000

0.00 0.10 0.20 0.30 0.40 0.50

Porosity

Ve

loc

ities

[m/s]

vp

vs

=2

=1/2

=1/2

Figure 6.2: P- and S-wave velocities in dry sandstone with a critical porosity of 0.40,

modelled for different values of the exponent.

Clearly, the porosity dependence will change with fluid saturation. We will discuss the

effects of a saturating fluid in a subsequent chapter, but the general effect is to increase

i) the bulk modulus, according to the Biot-Gassmann theory, and ii) the density, by

adding a contribution f. Normally this leads to an increase in P-wave velocity,because the bulk modulus increases more with fluid saturation than the density. The



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result is also a different porosity dependence of vp than would be predicted from Eq.

(6.1). This is seen in , which also contains traditional empirical relations used

by the industry. The most common of these is the socalled time-average (Wyllie's)

equation, which simply states that the travel time through a rock is a volume weighted

average of the travel times in the solid and in the fluid phase:

Figure 6.3

Figure 6.3: P-wave velocity vs. porosity in water saturated sandstone; calculated

using the Biot-Gassmann model with a critical porosity; with the time-average, and with Raymers equation.

(6.3),

1 1

p f p sv v v

= +

vP = (1 )2 vsolid + vfluid

vP = 5.54 8.22

Fundamentally, one should not expect this equation to be valid, except maybe at very

high frequencies (since it is a kind of a ray approximation). For shear waves, it is

obviously not valid, since there is no S-wave velocity in the fluid. In addition, let us

briefly mention Raymer's equation, which also may be applied to fluid saturated rocks:

(6.4)

Figure 6.4shows wave velocities in water saturated sandstones with different porosities.

The figure has two distinct features: There is a clear trend of decreasing velocity with

increasing porosity. The best fit by Wang & Nur (1992), from whom these data are

taken, was (velocity in [km/s]; porosity as fraction):

(6.5)

There is however also a significant amount of scatter in the data.

0

1000

2000

3000

4000

5000

6000

7000

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Porosity

P-waveve

loc

ity

[m/s] Critical porosity

Suspension (Reuss' average)

Time Average

Raymer



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Figure 6.4: Comparison of laboratory measured compressional wave velocities withthe velocities from Eqs. (6.3) - (6.5)(from Wang & Nur.).

We have focussed on sandstones in the examples above, but the models are of course

applicable to other types of rock as well. In Figure 6.5 we show the trend found in North

Sea shales. We notice that in this case, the data lie close to the prediction of the

suspension model.

0

1000

2000

3000

4000

5000

6000

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

Porosity

P-waveve

loc

ity

(norma

l)(m/s)

Figure 6.5: P-wave velocity normal to bedding vs. porosity (derived from water

content) in shales from the North Sea. The curve is a modified suspension

model, where the P-wave modulus is used for the solid modulus. The dataclouds represent measurements performed during rock mechanical testswith each sample (from Holt et al., 1997).



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7. Sound velocities vs. lithology.

Sound velocities can be used as indicators for lithology, but the range of variation(see ) is extremely large and depends on the state of the rock (cementation,

porosity, fractures) and the conditions in the Earth (stress, pore pressure, temperature).

Sound velocities alone can therefore generally not be thrusted for lithology

identification.

Table 5.1

The vp/vs ratio can in some circumstances be a better lithology indicator than the

velocities themselves. For example vp/vs is normally higher in shale than in sandstone;

thus, it is possible from seismics to distinguish between e.g. cap rock and reservoir rock.

Theoretically, using a critical porosity approach, the vp/vs ratio in a dry rock is governed

by the elastic moduli of the solid minerals, and independent of porosity:

(7.1)

4

3( )s s

p

dry

s s

K Gv

v G

+=

Typical bulk and shear modulus values for sand (quartz), chalk (calicite) and shale are

given in Table 7.1. The values for shale are extrapolated to zero porosity from

measurements with shale samples, since isolated clay minerals are hardly available for

measurement. The low values indicate that bound water is included in the values of the

solid material moduli. From these values, Eq. (7.1)predicts

vp/vs ~ 1.5 for dry sands

vp/vs ~ 1.9 for dry chalk

Dry shale does not exist at depth.

Table 7.1: Solid mineral bulk and shear modulus for sand, chalk and shale.

Density [g/cm3] Bulk modulus [GPa] Shear modulus

[GPa]

Sand (quartz) 2.65 37 45

Chalk (calcite) 2.71 65-75 28-32

Clay 2.6 20-25 5-10

If we account for fluid saturation, e.g. by using the Biot-Gassmann equation for the bulk

modulus



24/46

- 24 -

2

1 (

f

frf

s

KK K

K

K

)

= ++

(7.2)

with = , then we find that the vp/vs ratio increases slightly with porosity. In

case of a critical porosity law:

(1 )fr

s

K

K

4 1

3[ (1 )][1 ]

p fs

fs s c sc c

s c

v KK

Kv G G

K

= + ++

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.00 0.10 0.20 0.30 0.40

Porosity

vp/v

s

(7.3)

Since Kfis usually significantly smaller than Gs, the 3rd

term under the square root isusually also small; excepth when the porosity is near the critical porosity. Thus; vp/vs

will still be a good lithology indicator. Using brine as pore fluid (Kf=2.7 GPa), we find

for porosities well below c:

vp/vs ~ 1.5 1.6 for water saturated sandstone with moduli as given in Table 2

vp/vs ~ 1.9 2.0 for water saturated chalk with Ks=70; Gs=30 GPa

vp/vs ~ 2.1 2.3 for water saturated shale with Ks=22.5; Gs=7.5 GPa

Figure 7.1: vp/vs vs. porosity for a brine saturated sandstone; calculated using Eq.

(7.3). Critical porosity = 0.40.

Figure 7.1: vp/vs vs. porosity for a brine saturated sandstone; calculated using Eq.(7.3). Critical porosity = 0.40.

illustrates the vp/vs behaviour for brine saturated

sandstone.



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Data from laboratory core experiments and sonic logs are summarized by Mavko et al.

They report the following typical vp/vs values

Water-saturated sandstone; 0, 0.30: vp/vs 1.5 1.9Poorly consolidated, water-saturated sandstone; 0.20, 0.40: vp/vs 1.9 2.2Dry tight gas sandstone; 0, 0.15: vp/vs 1.4 1.7Chalk; gas-condensate saturated & water-saturated; 0, 0.30: vp/vs 1.6 1.8Limestone (water-saturated); 0, 0.30: vp/vs 1.7 2.0

In addition:

Shale / Clay (data from log and seismic); vp/vs 2-4

We have here looked at vp/vs as a main parameter for lithology identification. Another

may be linked to anisotropy. In particular, shale at depth is lithologically anisotropic.

We have however assumed in the theoretical predictions that the solid phase is mono-

mineralic. This is of course not true: Most sedimentary rocks contain many minerals,

and thus, the solid bulk and shear moduli need to be refined. A particular example is

given in the next Chapter, where we consider effects of clay on velocities of sandstone.



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8. Effect of clay on velocities of sandstone

As we have seen above, clay minerals have quite different elastic properties from say

quartz. When both clay and quartz are present at the same time, as in shaly sand orsandy shale, one needs to model a situation with more than one solid mineral phase. The

result depends on how the structure of clay and quartz minerals is built. Assume that the

rock consists of quartz (qtz), clay (cl) and fluid filled pore space. The volume fractions

of each phase add to 100%, i.e.

1cl qtz V V+ + =

(1 )s cl cl cl qtz v M v M = +

clcl

cl qtz

Vv

V V=

+

11 cl cl

f cl wK K K

= +

clcl

cl

V

V

=

+

(8.1)

If clay is part of the load bearing framework, then the clay minerals contribute to the

solid modulus Ks (and to Kfr). This may be modelled e.g. by using a Voigt average for

the solid (bulk and shear) moduli:

(8.2)

where vcl is the clay fraction of the solid material:

(8.3)

Figure 8.1 shows the predicted P-wave velocity vs. porosity for different clay fractions

according to this model. We see that increasing clay content reduces the P-wave

velocity.

On the other hand, if clay occur as pore fill, it can be modelled as part of the pore fluid,

using a suspension model.

(8.4)

where cl is the clay fraction of the porosity:

(8.5)

The result is that increase in clay content leads to an increase in P-wave velocity. Thus;

by observing trends in sonic logs for areas where an independent clay indicator exists,

one may be able to distinguish between clay as pore fill (which is detrimental to

permeability) and clay as part of the solid matrix.



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2000

3000

4000

5000

6000

0.00 0.10 0.20 0.30

Porosity

P-waveve

loc

ity

[m/s]

vcl = 0.05

vcl = 0.15

vcl = 0.25

vcl = 0.35

Figure 8.1: Calculated P-wave velocity vs. porosity for water saturated sandstonewith different clay contents. The plot was derived using Biot-Gassmann

and a critical porosity model, where the solid modulus is given by a

Voigt average of clay and quartz contributions.

Figure 8.2: Water-saturated ultrasonic velocities measured at 40 MPa confining

pressure for different clay fractions (taken from Mavko et al., 1998).



28/46

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Based on experimental results, Han and later Vernik (1994) distinguished between

sandstones according to their clay content C and made empirical fits based on a

classification of siliclastics, representing clean arenites, arenites, wackes, and shales:

vP = 6.07 7.97 (C< 0.02)vP = 5.52 6.91 (C0.02 0.15)

vP = 5.19 7.21 (C< 0.15 0.35)

vP = 4.93 9.03 (C= 0.35)

vP = 5.59 6.93 2.18C

vS = 3.52 4.911.89C

(8.6)

Figure 8.2 shows a presentation of such data reproduced from Hans Thesis by Mavko

et al. (1998). The resemblance with is striking.Figure 8.1

Han, Nur & Morgan derived empirical fits for water saturated sandstone (at 40 MPa

confining stress) with porosity below 30% and clay content C below 50%:

(8.7)

Marion, Nur, Yin & Han measured with samples made by mixing sand and clay

(kaolinite) powder, with clay contents ranging from 0 to 100%. They found that there is

a maximum in velocity (associated with a porosity minimum) on the transition between

shaly sand and sandy shale. This is shown in Figure 8.3.

Figure 8.3: P-wave velocity vs. clay content in sand-clay mixtures at differentstresses. From Marion et al., 1992.



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9. Sound velocities vs. fluid saturation.

When a dry rock sample is saturated with liquid, the following parameters affect the

change wave velocities:

- The bulk modulus K increases according to the Biot-Gassmann theory (cfr. Eq.(7.2)) because of the liquid contribution.

- The bulk density increases by f.

In addition, the framework moduli (Kfrand Gfr) are normally assumed unaffected by the

saturating fluid (neglecting possible chemical interactions).

4

3p

K Gv

+=

s

Gv =

0

1000

2000

3000

4000

5000

6000

7000

0.00 0.10 0.20 0.30 0.40

Porosity

Wave

Velo

cities

[m/s]

vp sat

vs sat

vp dry c-p

vs dry c-p

Thus; P-wave velocity may increase or decrease, depending on whether

the fluid contribution to K or is the larger. S-wave velocity will decrease

upon saturation, since G is assumed unaffected. At high frequencies, fluid induced

dispersion may cause an increase in both velocities.

The Biot Gassmann theory can be used to predict the effect of liquid saturation in the

low frequency limit. Figure 9.1 shows the predicted effect of brine saturation on P- and

S-wave velocities. As above, typical parameters for sandstone have been used, together

with a critical porosity model with c = 0.40. We see that the P-wave velocity increasesupon saturation at high porosities; or when the P-wave velocity is below 3500 m/s. Athigher velocities, the rock stiffness is so high that the fluid contribution to it becomes

less significant than the density increase. The S-wave velocity is seen to decrease as a

result of saturation in all cases, as expected.

Figure 9.1: Wave velocities for dry and saturated sandstone, based on Biot-Gassmann theory with a critical porosity law.



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The effect of partial saturation depends on the distribution of the fluid phases in the pore

space. If mixed at a fine scale, the fluid properties may be estimated through a Reuss

(isostress) average, i.e.

(9.1),

1 i

if f i

S

K K=

1fK kl Dt

f

,f i f i

i

K S K =

Si represents the concentrations of fluid phase i. For water containing air bubbles, the

suspension model is appropriate. Since air has a bulk modulus (1.410-4 GPa) which isnegligible compared to that of water (2.2 GPa), the fluid bulk modulus is governed by

the gas modulus, except when the water saturation approaches 1. This is also seen in

laboratory experiments: Only when the water saturation exceeds 95% or more is the P-

wave velocity of the rock found to increase. Acoustic wave measurements are therefore

good indicators of gas (shallow gas, gas zones in reservoirs).

It is worthwhile to notice that for very low water saturation levels, such as when the

rock is ovendried before measurements are performed, a strong increase in P-wave

velocity may also be seen. This observation implies that the water has a certain

softening effect on the grain contact. Since most rocks in the Earth have a certain water

saturation, this means that laboratory experiments for wave velocity measurement

should not be performed with oven dried cores.

The basis for Equation (9.1) is that the various fluid phases feel the same stress; in order

words, that the fluid pressure is in equilibrium between the different phases. This

implies that pressure diffusion may take place during the passage of the wave, say in thecourse of one period. Hence, from standard diffusion theory, the size of the patches

should be related to the time scale as

(9.2)

HereD denotes a diffusion coefficient, which depends on the permeability kand the

fluid viscosity , in addition to the bulk modulus of the pore fluid, and the porosity. For

the seismic frequency range (10 100 Hz) we find that the patches must be smaller than

10 30 cm in order for the Reuss average to be used. If this condition is not fullfileld,

other mixing rules than Eq. (9.1) are required. One obvious alternative is the Voigt

average, which gives an upper bound to the fluid bulk modulus:

(9.3)

More realistic averages require more detailed knowledge about the patchiness. If this is

not available (which is the normal case), one may for instance use the following

approach for the bulk modulus of the rock:



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- 31 -

1 1

4

3

1

frfr

K GK G

=+

(9.4)

Figure 9.2: P-wave velocity vs. water saturation in a gas-water saturated porousrock. The uniform saturation case is modelled with the Reuss average,

and the patchy case denoted by the upper limit Voigt model Laboratory

data are also shown. The figure is taken from the Rock Physics course byGary Mavko at Stanford University.

http://pangea.stanford.edu/courses/gp262/Notes.htm

Patchiness is an issure in particular in reservoirs which contain gas. It may also be

important in monitoring of EOR (Enhanced Oil Recovery) processes where one fluidphase is displaced by another.

For reservoir monitoring applications, the effect of different saturating liquids is

therefore of significant relevance. This is also the case if one wishes to distinguish e.g.

oil and water saturated zones from logs. In order to quantify such effects, one needs to

know velocities of typical fluids present in sedimentary rocks under reservoir

conditions. An extensive study was reported by Batzle & Wang (1992). A summary of

selected results is presented in .Table 9.1

The calculation leading to Figure 9.1 is performed with fluid properties typically seen atroom conditions. The density and bulk modulus of fluids (both liquid and gas) are

however sensitive to pressure as well as to temperature, and will therefore be quite

http://pangea.stanford.edu/courses/gp262/Notes.htmhttp://pangea.stanford.edu/courses/gp262/Notes.htm


32/46

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different under reservoir conditions than at atmospheric conditions. The bulk and shear

modulus of the rock framework is also stress and pore pressure dependent, so to obtain

realistic values from core measurements in the laboratory is a tedious task.

Table 9.1: Characteristic elastic properties and densities of typical pore fluidsunder different pressure and temperature conditions.

Fluid Density [g/cm3] Sound velocity

[m/s]

Bulk modulus

[GPa]

Air at room

conditions

1.2910-3 331 1.410-4

Hydrocarbon gas at

100 C and 25 MPa0.15 0.35 550 - 650 0.05 0.15

Hydrocarbon gas at

100 C and 50 MPa

0.25 0.45 750 - 900 0.15 0.35

Hydrocarbon gas at

200 C and 25 MPa0.15 0.3 5-600 0.05 0.1

Fresh water at room

conditions

1.00 1480 2.2

3.5 % brine at room

conditions

1.05 1520 2.4

3.5 % brine at

100C and 25 MPa0.97 1625 2.6

3.5 % brine w/

dissolved gas

100C and 25 MPa0.97 1525 2.3

3.5 % brine at

200C and 50 MPa0.90 1510 2.1

Light dead oil at

room conditions

0.80 1320 1.4

Light dead oil at

100C and 25 MPa0.76 1190 1.1

Light dead oil at

200C and 50 MPa0.70 1050 0.8

Light live oil

100C and 25 MPa0.64 810 0.4

Light live oil

200C and 50 MPa0.58 890 0.4



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- 33 -

Let us now look in some detail at the fluid properties: In an ideal gas, the pressure P is

proportional to the number of molecules per volume unit (N/V), to Boltzmanns

constant k and to absolute temperature Ta (in [K]):

aPV NkT =

( )/

T

PP

V V

=

(9.5)

The elastic modulus is

(9.6)

The density is

m

N

mV=

a

m

kTv

m=

a

m

kTv

m=

aPV ZNkT =

(9.7)

where mm is the molecular mass. Thus; the wave velocity is

(9.8)

This is of course a huge oversimplification for real gases and liquids. Some realism is

added by considering the process of acoustic wave propagation as adiabatic, not

isothermal as was implicit in the derivation of Eq. (9.8). Thus the bulk modulus in Eq.(9.6) has to be multiplied by , which is equal to the ratio of the heat capacitiesmeasured under constant pressure (Cp) and constant volume (Cv) conditions:

(9.9)

is 1.4 for an ideal gas.

The ideal gas law is also not a good approximation for a real gas or liquid. An

improvement can be made by multiplying the right hand side of the ideal gas law by apressure and temperature dependent so-called compressibility factor Z;

(9.10)

In addition, the variable composition of natural gases adds further complexity. We will

here only give the following empirical relationships (from Batzle & Wang, 1992) for

natural gases under pressures and temperatures that are common in petroleum

exploration and production:

Gas density is



34/46

- 34 -

(9.11)28.8

a

GP

ZRT

3 40.03 0.00527(3.5 ) (0.642 0.007 0.52)pr pr pr pr Z T P T T E +

1.2

2 210.109(3.85 ) exp 0.45 8(0.56 )pr

pr

pr pr

PE T

T T

4.892 0.4048pr

PP

G=

94.72 170.75

apr

TT

G=

+

0

1

s

pr

pr T

PK

P Z

Z P

0 2

5.6 27.1

0.85 8.7exp 0.65( 1)( 2) ( 3.5) prpr pr PP P = + + + + +

2 5 3 2

1.5 2 2

(1170 9.6 0.055 8.5 10 2.6 0 0029 0.0476 )

(780 10 0.16 ) 1820

brine water v v S T T T P TP P

S P P S

= + + +

+ +

where G is the specific gravity, i.e. the ratio of gas density to air density at 15.6 C andatmospheric pressure. G is 0.5 0.6 for methane, and 1.5 2 for gases with heavy

components or high carbon numbers. P is given in [MPa].R is the gas constant (=8.31 J /

g mole C).

(9.12)= + +

(9.13) = +

where the pseudopressure Pprand pseudocritical temperature Tprboth should not be 0.1 or more.

(9.14)

(9.15)

The adiabatic gas bulk modulus is

(9.16)

where 0 is

(9.17)

The equations show that bulk modulus and density both increase with increasing

pressure and increasing gravity, and decrease with increasing temperature. This is

illustrated in the examples in Table 3 and further in Batzle & Wangs Paper.

The (P-) wave velocity in brine can be fitted to the following empirical relation:

(9.18)

where the velocity of fresh water is



35/46

- 35 -

4 3

0 0

i j

water ij

i j

v w T P = =

=

[ ]{ }60.668 0.44 1 10 300 2400 (80 3 3300 13 47 )brine water S S P PS T T S P PS = + + + + + +

6 2 3

2 5 3 2 2

1 1 10 ( 80 3.3 0.00175

489 2 0.016 1.3 10 0.333 0.002 )

water Y T T

P TP T P T P P TP

= + +

+ +

(9.19)

The coefficients wij are given in . When applying Eqs. (9.18) and (9.19) the

velocities are given in [m/s], S is the salinity in weight fraction [ppm/1000.000], T is

temperature in [C], and P is pressure in [MPa].

Table 9.2

Table 9.2: The coefficients wij for calculation of wave velcoity vs. pressure andtemperature in water (taken from Batzle & Wang; 1992).

In addition, the densities of brine and water (in [g/cm3]), respectively are given by:

(9.20)

(9.21)

w00 = 1402.85 w01 = 1.524 w02 = 3.43710-3 w03 = -1.19710

-5

w10 = 4.781 w11 = -0.0111 w12 = 1.73910-4 w13 = -1.62810

-6

w20 = -0.04783 w21 = 2.74710-4 w22 = -2.13510

-6 w23 = 1.23710-8

w30 = 1.48710-4

w31 = -6.50310-7

w32 = -1.45510-8

w33 = 1.32710-10

w40 = -2.19710-7 w41 = 7.98710

-10 w42 = 5.23010-11 w43 = -4.61410

-13

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 50 100 150 200 250 300

Temperature [C]

Bu

lkMo

dulus

[GPa

]

Salinity 3.5 %

Salinity 20 %

P = 1 atm.

P = 50 MPa

P = 1 atm.Bulk modulus of brine

P = 25 MPa;

brine w/gas

P = 25 MPa

Figure 9.3: Calculated bulk modulus of brine vs. temperature under different

conditions of pressure, salinity, and dissolved gas content.



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- 36 -

Predicted velocities for brines under different conditions of pressure and temperature

are shown in Figure 9.3. The following main features should be noticed:

- Modulus (and velocity) of brine increases with increasing temperature at

temperatures below 60 80 C and decreases above that.- Modulus and velocity of brine increase with increasing brine salinity.

- Modulus and velocity increase with increasing pressure.

When saturation is partial, then the effect of gas on the fluid modulus must be

estimated. If gas is dissolved, then one generally assumes a slight reduction of fluid

modulus with respect to that of the gas free pore fluid. The bulk modulus of brine with

dissolved gas is thought to follow a relationship of the form

(9.22)( )1 0.0494

brinebrine

G

KK with dissolved gas

R=

+

1.5 0.64 0.306

10 10log log (0.712 76.71 3676 ) 4 7.786 ( 17.78)GR P T P S T

= + +

112

10 20

0

2096 3.7 4.64 0.0115 4.12(1.08 1) 12.6

v T P TP

= + +

4 1.1750.972 3.81 10 ( 17.78)

P

T

=

7 3 2 4

0 0(0.00277 1.71 10 )( 1.15) 3.49 10P P P P = + +

where RG is the gas-water ratio in [l/l]. RG is a function of pressure and temperature:

(9.23)

This effect is illustrated in , where a decrease is observed as a result of

dissolved gas.

Figure 9.3

The (P-wave) velocity in oil (Figure 9.4) is also temperature and pressure dependent.

There is a strongly decreasing trend of bulk modulus (and velocity) with increasing

temperature, and increasing modulus (and velocity) with pressure. The equations

leading to this plot are also based on no compositional changes as a function of pressure

and temperature. These equations are (for dead oil):

(9.24)

where0 is the density of the oil measured at 15.6 C and atmospheric pressure. Thedensity under pressure and temperature conditions is

(9.25)+ +

The pressure dependent part of the density is

(9.26)

Again, dissolved gas, which is present in live crude oil, reduces the bulk modulus (andvelocity), more so than in brine. This means that laboratory measurements on dead

oils can be quite misleading. Again the effect of dissolved gas is incorporated through a



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factorRG , which can be expressed as

(9.27)0

1.2054.072

((max) 0.02123 0.00377GR G Pe T

=

10

0

' (1 0.001 )GRB

= +

1.1751

20

0

0.972 0.00038 2.49 ( ) 17.8GG

B R T

= + + +

0

0

0.0012 Gliveoil

GR

B

+=

The velocity is now calculated with Eq. (9.24), but replacing the density0 with a

pseudodensity:

(9.28)

where

(9.29)

The oil density is

(9.30)

0.0

0.5

1.0

1.5

2.0

2.5

0 50 100 150 200 250 300

Temperature [C]

Bu

lkmo

du

lus

[GPa

]

Bulk Modulus of Oil

Clearly the velocity behaviour is sensitive to the composition of the oil, and the plot in

is given for one fixed composition.Figure 9.4

Figure 9.4: Calculated bulk modulus of oil vs. temperature for different conditions ofpressure. Also shown is the result for a live oil, containing dissolved gas.

P = 50 MPa

P = 1 atm.

P = 25 MPa; live oil

P = 25 MPa



38/46

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10.Temperature dependent wave velocities

Wave velocities depend on temperature, for 2 obvious reasons:

- The bulk modulus of the pore fluid is temperature dependent

- The bulk and shear moduli of the host mineral is temperature dependent

As an example, the stiffness of quartz decreases with temperature from room

temperature and upwards. This decrease is linked to the transition from low-temperature

(-quartz) to high temperature (-quartz) at 573 C. According to Rzhevsky & Noviksbook (1971), Youngs modulus of quartzite decreases from 80 to about 65 GPa as

temperature is increased from 0 to 200 C.

Figure 10.1

Figure 10.1: Calculated temperature dependence of P-wave velocity in dry, oil- andbrine-saturated sandstone.

shows predicted temperature dependence in dry and liquid saturatedsandstone using the temperature dependence found in quartz in a critical porosity law

for the frame moduli, and the temperature dependence of fluid properties described

above in the Biot-Gassmann equation. The expected trend, which is in agreement with

observations, is thus that the velocities decrease with temperature, and more so in

saturated than in dry sandstone.

2500

3000

3500

4000

0 50 100 150 200 250 300

Temperature [C]

P-wave

Ve

loc

ity

[m/s]

vp dry

vp brine

vp oil

Brine salinity: 3.5%

Pressure: 25 MPa



39/46

- 39 -

11.Sound Velocities vs. Stresses.

In most rock cores, wave velocities increase strongly with increasing stress at low stress

levels, and level off at high stresses. This is commonly observed in sandstone. Anexample is shown in Figure 11.1.

First, let it be pointed out that the often made basic assumption of linear elasticity

implies that elastic moduli are constants independent of stress. The observation of

stress dependence therefore indicates violence of the basic assumption. This may

happen through stress changes sufficiently large to alter porosity (which is assumed

constant in the linear poroelastic theory), or it may happen because of a nonlinearity in

the stress strain relationship.

The obvious physical explanation for such nonlinearities is that an external stressimproves the grain to grain contacts and / or closes microcracks / microfissures in the

rock, thereby increasing the stiffness. This means that a well cemented rock with

welded grain contacts is expected to show less stress sensitivity than a poorly cemented

rock with many Hertzian grain contacts. It also implies that a cored rock, which has

been unloaded from its in situ stress state and therefore contains microcracks, is

expected to show much stronger stress sensitivity than the same rock under virgin

conditions in situ. The effect of such core damage on wave velocities and their stress

dependence have been discussed elsewhere (e.g. Fjr & Holt, 1999; Nes et al, 2002).

2500

2550

2600

2650

2700

2750

0 5 10 15 20 25 30 35

External Stress [MPa]

P-waveve

loc

ity

[m/s]

Figure 11.1: Characteristic stress dependence of wave velocities as seen in most rock

cores.

Figure 11.1A few exceptions to the general behaviour illustrated in can be mentioned.

In high porosity shales, the stress dependence effect is often quite small. This is thought



40/46

- 40 -

to be because water always covers clay particles. Grain contacts are then often

constituted by water, which in itself is thought to be stress insensitive. In chalk,

increasing hydrostatic stress may lead to pore collapse which causes a velocity

reduction because the breaking of grains and grain bonds leads to a weakening of the

chalk structure. This is illustrated in Figure 11.2.

Figure 11.2: Measured velocities vs. stress in Red Wildmoor sandstone, Fullers Earth(claystone) and outcrop chalk. From Holt et al., 1991.



41/46

- 41 -

Velocities depend on an effective stress ', which is the external stress minus a certaincontribution from the internal (pore) pressure pf:

' = pf

2500

2600

2700

2800

2900

0 10 20 30 40 50

External Stress = Pore Pressure [MPa]

P-waveve

loc

ity

[m/s]

Dry Rock

(11.1)

The coefficient is normally assumed to be close to 1, but values < (and >) 1 may befound. From Eq. (11.1) one sees that a high pore pressure plays the same role as a low

external stress, causing the sound velocity to be reduced. This is used in the estimation

of abnormal pore pressures from logs (and may also be used in seismics). Since the

expected trend in a homogeneous formation would be a monotonous increase of

velocity with depth (because the effective stress increases with depth), an overpressured

zone shows up as a low velocity zone breaking the expected trend.

It is however important to underline that the effective stress law above is not the same

as the Biot effective stress law from linear poroelasticity. Here we consider the stress

dependence of an elastic parameter, which already means that we are beyond linear

elasticity. Further, as seen above, the pore pressure has an intrinsic effect on the bulk

modulus and density of the pore fluid, without affecting the rock framework. This is

illustrated in Figure 11.3, which shows calculated P-wave velocities for a rock saturated

with brine and with oil, for a case where the framework is modelled as stress

insensitive. If the effective stress principle applied, we would expect the same stress

sensitivity independent of fluids, and if=1 (as is conventionally thought), there wouldnot be any stress dependency of the saturated rock in this case.

Oil Saturated Rock

Brine Saturated Rock

3.5% salinity

T=150 C

Figure 11.3: Calculated P-wave velocity vs. external stress (=pore pressure) in a

sandstone where the framework is assumed stress insensitive.



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When a rock sample is stressed in anisotropic stress conditions, the wave velocities

become anisotropic. This is particularly the case near failure (Figure 11.4), where the

velocities with propagation direction and / or polarisation along the minor principal

stress direction are found to decrease substantially. This is result of opening of

microcracks (grain bond breakage) in this direction. It is also the case for a core sampleretrieved from the Earth, where stress release has caused development of an oriented

microcrack distribution, which carries a memory of the in situ stress state.

The presence of fractures or cracks smaller than the wavelength will significantly

reduce sound velocities. This essentially happens because the stiffness is strongly

reduced by the presence of cracks. The stress dependence discussed above is related to

closing of microcracks. If the fractures are larger than the wavelength, they will act as

reflectors.

Figure 11.4: P- velocities along (axial) and prependicular (radial) to the major

principal stress during a triaxial test with sandstone (failure strain:

0.015).



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Let us finally look at the stress changes that may occur in a depleting reservoir. Clearly,

these stress changes may produce effects that can be seen in time-lapse seismics. For a

reservoir which is relatively large in extent compared to thickness, one may assume that

(i) there is no horizontal strain during depletion; and (ii) the reservoir feels the full

weight of the overburden at all times (no shielding; or stress arching). Hookes lawmay now be used to estimate the stress change, as well as the compaction and porosity

change associated with depletion. From Hookes law we write

(11.2)

1

1

x x y z

z x y z

E E E

E E E

=

= +

' ' '

' ' '

10

1

fr fr

h h h v

fr fr fr

fr fr

z h h v

fr fr fr

E E E

h

h E E E

= =

= = +

'

v fp

' ' ( )

1 1

fr fr

h v f

fr fr

p

= =

We now associate the z-direction with the in situ vertical (v) direction, and the x- and y-directions with the in situ horizontal (h) directions. Here we assume the in situ stress to

be isotropic (=h) in the horizontal plane, but that assumption does not affect the resultsbelow)According to poroelasticity theory, we need to replace the stresses with the

effective stress changes occurring during depletion. The elastic parameters to be used

are those of the drained rock framework. Hence:

(11.3)

Since the total vertical stress is not changing, the effective vertical stress change in the

reservoir is directly given by the pore pressure change:

= (11.4)

From the first of the two equations under(11.3) we then find the horizontal effective

stress change as

(11.5)

For a Poisson ratio of 0.25, the effective horizontal stress increase will be 1/3 of the

effective vertical stress increase. Since the total vertical stress is constant, the total

horizontal stress is actually decreasing during depletion. Notice that depletion implies

that the pore pressure change pf is negative.

The second equation under(11.3)permits us to compute the compaction h (in meters),given the initial thickness h of the reservoir. Combining with Eq. (11.5) we find that

(1 )(1 2 ) 1

( ) ( ) 4(1 )

3

fr fr

f ffr fr

fr fr

h p h p hEK G

+

= = +(11.6)



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The porosity change during depletion can also be calculated (more tedious). The result

is:

(11.7)1

1 ( )f

fr

pK

=

1 22

1

fr

v h f

fr

p

= + =

is the mean stress change during depletion, given as

(11.8)

The porosity changes during depletion are usually small, and the direct effect of a

porosity change on wave velocities will normally also be small. The stress changes may

however be sufficiently large to exceed the elastic limits, so that new faults or fractures

are made within the reservoir (or in the overburden!), or old faults are activated. Inthose cases one would expect a significant effect of the stress change on the 4D seismic

response. One would also expect a significant 4D effect when the reservoir is very soft

(Hertzian contacts) or when it is fractured.



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12. Frequency dependent wave velocities

Sound velocities are slightly frequency dependent, increasing with frequency from

seismic to ultrasonic frequencies. Typically the total increase in velocity is a few %. Thetransition between low and high frequency behaviour depends on pore fluid viscosity ,and on the dispersion mechanism. If dispersion is caused by global viscous flow of pore

fluid (Biot flow), then

cf

1cf

(12.1)

If dispersion is caused by local flow (squirt flow) between pores and cracks / pore

throats, then

(12.2)

The latter mechanism is often thought be the most important in sedimentary rocks.

For most rocks, the seismic frequency range is thought to be well below the transition

frequencies given above. Thus; the seismic velocities are assumed frequency

independent, although they may be smaller than laboratory measured (ultrasonic)

velocities, and also possibly smaller than those measured with a sonic logging tool.

A final note at the end: When computing elastic stiffnesses from wave velocities, onegenerally finds that the dynamic elastic moduli are larger than thos measured in a static

experiment. Although part of that can be related to the frwequency dependence

discussed here, the major contribution to the discrepancy comes from the fact that the

amplitude of an elastic wave is too small to trigger non-elastic processes, such as crack

sliding and grain contact plastification. Such processes do however usually contribute to

the static deformation of rock. Thus; for instance, using the P-wave modulus of a dry

rock directly to estimate reservoir compaction (Eq. (11.6)) gives a lower limit to the

actual compaction that will occur in the field during depletion.



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References and Suggested Readings

Batzle, M. & Wang, Z. (1992) Seismic properties of pore fluids. Geophysics 57, 1396-

1408.

Biot, M.A. (1962) Mechanics of deformation and acoustic propagation in porous media. J.

Appl. Phys. 33, 1482-98.

Fjr, E., Holt, R.M., Horsrud, P., Raaen, A.M., Risnes, R. (1992) Petroleum Related

Rock Mechanics. Elsevier, 338 pp.

Fjr, E. & Holt, R.M. (1999) Stress and stress release effects on acoustic velocities from

cores, logs and seismics. Proc. SPWLA Ann. Int. Mtg., Oslo; 12pp.

Holt, R.M., Fjr, E., Raaen, A.M., Ringstad, C. (1991) Influence of stress stae and stress

history on acoustic wave propagation in sedimentary rocks.In Shear Waves in Marine

Sediments; eds. J.M.Hovem et al.; Kluwer; pp. 167-174.

Holt, R.M., Snsteb, E.F., Horsrud, P. (1997) Acoustic velocities of North Sea shales.EAGE; Amsterdam NL, June97; 2pp.

Kendall, K. (1984) Connection between structure and strength of porous solids. InPhys.

& Chem. of Porous Media; eds. D.L.Johnson & P.N.Sen; Am. Inst. Phys. Proc. 107;

pp.78-88.

Marion, D., Nur, A., Yin, H., Han, D. (1992) Compressional velocity and porosity in

sand-clay mixtures. Geophysics 57, 554-563.

Mavko, G., Mukerji, T., Dvorkin, J. (1998) The Rock Physics Handbook. Cambridge Un.

Press; 329 pp.

Murphy, W., Reischer, A., Hsu, K. (1993) Modulus decomposition of compressional and

shear velocities in sand bodies. Geophysics 58, 227-239.

Nes, O._M., Holt, R.M., Fjr, E. (2002) The reliability of core data as input to seismic

reservoir monitoring studies. SPE Res. Eng. & Eval. ; Febr. 2002; pp. 79-86.

Nur, A. & Wang, Z. (1988) Seismic and acosutic velocities in reservoir rocks. Vol.I:

Experimental studies. SEG Geophysics Reprint Series No.10.

Rzhevsky, V. & Novik, G. (1971) The Physics of Rocks. MIR Publ.; 320pp.

Vernik, L. (1994) Predicting lithology and transport properties from acoustic velocities

based on petrophysical classification of siliclastics. Geophysics 59, 420-427.

Wang, Z. and Nur., A. (1992) ) Seismic and acosutic velocities in reservoir rocks. Vol.II:

Theoretical and model studies. SEG Geophysics Reprint Series No.10.

White, J.E. (1983) Underground sound. Elsevier.

Wyllie, M.R.J., Gregory, A.R., Gardner, G.H.F. (1957) An experimental investigation of

factors affecting elastic wave velocities in porous media. Geophysics 23, 459-493.

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