Peng S., Zhang J., Engineering Geology for Underground Rocks

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1.2 Porosity 5

fma

bma



(1.4)

For clean sandstones at high pressures and with moderate porosities,

porosity is often estimated by the empirical time average formula (Wyllie

et al. 1956).

maf

map

/1/1



(1.5)

where v

is the formation or rock compressional velocity; v

is the grain

or matrix compressional velocity; and v

is the fluid velocity. The equation

can be rewritten by using transit time terminology in the following form:

maf

''

(1.6)

where 't is the acoustic transit time; 't

is the acoustic transit time of the

rock matrix; and 't

is the acoustic transit time of interstitial fluids.

The Wyllie equation represents consolidated and compacted formations,

generally for porosity of less than 0.25 in sandstones. Unconsolidated

sandstones, such as those in U.S. Gulf Coast, Nigeria, and Venezuela, of-

ten have much higher porosity (0.28 to 0.50). If the Wyllie equation is used

in unconsolidated sandstones, correction for compaction is necessary.

Generally, such sandstones can be identified by observing 't in adjoining

shale beds. Formations not subjected to sufficient overburden stress do not

have enough compaction and rigidity for proper transmission of an acous-

tic wave, and as a result, with decreasing overburden pressure, 't through

unconsolidated formations increases disproportionately (Baker Atlas

2002). An empirical equation can be used to calculate porosity in uncon-

solidated sandstones:

pmaf

Ctt

''

(1.7)

where the compaction correction factor can be expressed as (Baker Atlas

2002):

100

6 1 Rock properties and mechanical behaviors

where 't

is the specific acoustic transit time in adjacent shales (Psec/ft);

C is the shale compaction coefficient, ranging from 0.8 to 1.3.

The Raymer-Hunt-Gardner correlation is another empirical fit of inde-

pendently determined porosity to observe interval times (Raymer et al.

1980). It is expressed as:

''

(1.8)

where c is a constant and varies typically from 0.625 to 0.7.

Field tests and lab experiments have shown that rock porosity decreases

as the burial depth increases as shown in Fig. 1.2. The following equation

is the most commonly used to describe porosity and depth relationship:



(1.9)

where a and b are constants; Z is the depth. In the Northern North Sea, po-

rosity and depth have the following relationship (Schön 1996):

For sandstone

107.2

49.0



u

(1.10)

For shale

101.5

803.0



u

(1.11)

where Z is the depth in meters.

Similarly, the relationship between porosity and effective stress can be

expressed as the follows:



(1.12)

where a, b are the constant;

is the effective compressive stress.

It can be seen that porosity decreases as compressive effective stress (

)

increases, causing formation more compacted and consolidated.

1.2 Porosity 7

-600

-500

-400

-300

-200

-100

30 40 50 60 70 80

Porosity (%)

th (

)

Fig. 1.2. Porosity decrease with depth (starting from the sea floor) in the central

deepwater Gulf of Mexico (plotted from the core test data presented by Ostermeier

et al. 2001).

Lab experimental tests show that rock porosity decreases as the applied

stress increases. Figure 1.3 presents porosity and axial stress relationships

under uniaxial strain conditions for two sandstone specimens cored 1000 m

below the sea floor. The experiments show that the stress and porosity

have negative exponential relationships, i.e.:

0023.0

6.33



(1.13)

0027.0

7.36



(1.14)

where

is the porosity (%);

is the axial stress (MPa).

The stress and porosity can be expressed as the following general form:



(1.15)

where

is the initial porosity; c is the constant; and ı is the axial stress.

8 1 Rock properties and mechanical behaviors

0 5 10 15 20 25

Axial stress (MPa)

Porosity (percent)

Sand 1

Sand 2

Fig. 1.3. Porosity decreases with the increase of the axial stress in uniaxial strain

tests for sand specimens.

1.3 Permeability

1.3.1 Darcy’s law and hydraulic conductivity

In 1856, Henry Darcy investigated the flow of water in vertical homogene-

ous sand filters in connection with the fountains of the city of Dijon. From

his experiments, Darcy concluded that the rate of flow q is proportional to

the constant cross-sectional area A, proportional to the difference of hy-

draulic head (h

 h

), and inversely proportional to the distance of the two

hydraulic heads L. When combined, these conclusions give the famous

Darcy formula:

KAq



(1.16)

where K is the hydraulic conductivity. Hydraulic conductivity is com-

monly used in hydrogeology (Bear 1972) and is a measure in how easily a

particular fluid passes through a particular earth material. The following

1.3 Permeability 9

equation gives permeability and hydraulic conductivity relationship

(Zhang et al. 1997):

(1.17)

where

is the fluid density; µ is the dynamic viscosity of fluid; and k is

the permeability.

1.3.2 Permeability tensor

Permeability is the most important physical property of a porous medium.

Permeability measures quantitatively the ability of a porous medium to

conduct fluid flow. Permeability has dimensions of an area, and it is

measured in units of Darcy (1 Darcy = 0.98 u10

-12

), or more commonly

the milli-darcy or md.

Permeability of the rock depends largely on the connectedness of the

void spaces, the grain size of the rock, and cementation between rock

grains. A rock could be extremely porous, but if each pore is isolated from

the others, the rock would be impermeable. If rock grain size is small, then

void spaces of the rock are small. Therefore, the surface film of the fluid

can actually choke the movement of additional fluids through the small

spaces. This is why clays are so impermeable, even though their porosities

can be as high as 50%.

Permeability is a tensor in a manner similar to stress tensor, and in Car-

tesian coordinates it can be expressed as:

333231

232221

131211

kkk

(1.18)

More often the permeability is isotropic in the direction of the bedding

plane but anisotropic in the direction of perpendicular to the bedding plane.

In this case, if two coordinates are in the plane of the bedding, the perme-

ability tensor is diagonal with the coefficients in the plane of the bedding

having equal values and denoted as horizontal permeability, k

, and the co-

efficient in the direction perpendicular to the bedding denoted as vertical

permeability, k

. In this case the permeability tensor can be expressed as

follows:

10 1 Rock properties and mechanical behaviors

(1.19)

In the sedimentary rocks, usually horizontal permeability has a large

value than vertical permeability, depending on porosity and grain size and

grain packing. For example, a rock consisted of large flat grains has hori-

zontal permeability of 2000 md and vertical permeability of 800 md. How-

ever, for a rock with small flat grains, it has horizontal permeability of 800

md and vertical permeability of 50 md (Baker Atlas 2002).

1.3.3 Permeability and porosity correlation

For clastic rocks general correlation between permeability and porosity

follows a power law, i.e.:

(1.20)

where a and b are empirical constants.

Permeability often increases with porosity. However, rocks with very

low porosity have exhibited high permeability, and some high porosity

rocks have very low matrix permeability (Baker Atlas 2002). Permeability

can be determined by several means (Peng and Wang 2001); e.g., well test,

wireline formation tests, or core tests.

Permeability also shows strong correlations with burial depth, stress,

and rock properties (Peng et al. 2000). The in-situ stress and stress changes

caused by underground engineering have important effects on permeability

(Wang et al. 2001, Zhang et al. 2007). More detail discussions in stress-

dependent permeability will be given in Chap. 8.

1.4 Elastic modulus

Elastic modulus (also called Young’s modulus) is an important parameter

to describe stress and strain relationship. For most rocks, the uniaxial

stress-strain curve before failure takes approximately the linear form (Fig.

1.4). This can be presented by (Jaeger and Cook 1979):

(1.21)

1.4 Elastic modulus 11

where

is the stress;

is the strain; the constant, E, is called elastic

modulus. A material is linearly elastic if the above relation holds accu-

rately.

0.00.20.40.60.81.01.2

Strain (%)

Stress (MPa)

0.00.20.40.60.81.01.2

Strain (%)

Stress (MPa)

Strain (%)

0.00.20.40.60.81.01.2

Strain (%)

Stress (MPa)

0.00.20.40.60.81.01.2

Strain (%)

Stress (MPa)

Strain (%)

Fig. 1.4. Elastic modulus and stress-strain relationship for a medium-grained

sandstone under uniaxial compressive test.

Elastic modulus describes the capacity of rock deformation, or the stiff-

ness of a rock. For a high elastic modulus rock, it is less deformable (i.e.

stiff). The initial part of the complete stress-strain curve will be steep. For

a low elastic modulus (soft) rock, it is more deformable, and the initial part

of the complete stress-strain curve will be gentle (Hudson and Harrison

1997).

Rock static elastic modulus can be obtained from lab core tests of either

uniaxial or triaxial compressive experiment (Meng et al. 2002). Rock dy-

namic elastic modulus (E

) can be solved from the following equations by

knowing the rock elastic compressional and shear wave velocities:

)21)(1(

)1(





(1.22)

)1(2



(1.23)

where v

and v

are the compressional and shear wave velocities of the

rock, respectively;

is the rock bulk density;

is the Poisson’s ratio of

the rock. Rock elastic velocities can be obtained by seismic survey or sonic

12 1 Rock properties and mechanical behaviors

well log. The dynamic elastic modulus can be converted into static

modulus by empirical correlations (Du et al. 2001). The dynamic modulus

also can be expressed as the following forms if one knows the interval

transit time:

)1(

)21)(1(





(1.24)

)1(2



(1.25)

where t

and t

are the compressional and shear interval transit time, respec-

tively.

From lab tests in over 400 core samples from the Gulf of Mexico, Lacy

(1997) derived the following correlation between rock dynamic elastic

modulus and ultrasonic compressional velocity:

04.2

265.0

(1.26)

where the dynamic modulus (E

) is measured in Mpsi; the compressional

velocity v

is in km/s and ranges from 1 to 6 km/s.

In general, the dynamic values of elastic modulus have been found to be

significantly greater than the static values. It was also noted that the dis-

crepancy was far greater for soft rocks, such as sandstone, than for hard

rocks, such as granite (Howarth 1984). The discrepancies between the dy-

namic and static moduli have been widely attributed to microcracks and

pores in the rocks.

From ultrasonic test data of 600 core samples in the Gulf of Mexico,

Lacy (1997) obtained the following correlation for sandstones:

dds

EEE 4533.00293.0



(1.27)

where E

is the static elastic modulus.

The similar correction exists for shales (Lacy 1997):

dds

EEE 2334.00428.0



(1.28)

From sandstones, shales, limestones, and dolomites, the generalized cor-

relation can be expressed as follows (Lacy 1997):

dds

EEE 422.0018.0



(1.29)

1.5 Poisson’s ratio 13

where E

and E

in Eqs. 1.27 to 1.29 are the static and dynamic elastic

moduli with units of one million psi (Mpsi).

Ohen (2003) gave the following relationship between the dynamic and

static moduli for the rocks in the Gulf of Mexico, i.e. the dynamic Young’s

modulus is about 18 times of the static Young’s modulus.

74.2

0158.0

(1.30)

where the elastic moduli E

and E

are in Mpsi.

1.5 Poisson’s ratio

Poisson’s ratio is the ratio of transverse strain to corresponding axial strain

on a material stressed along one axis. For a rock core subjected to axial

load, Poisson’s ratio (

) can be expressed as:



(1.31)

where

is the lateral strain;

is the axial strain. Therefore, Poisson’s ratio

can be determined by measuring the lateral and axial deformations of the

uniaxial compressive test in rock samples.

Poisson’s ratio can also be calculated from the velocities of the elastic

wave:

1)/(



(1.32)

where v

and v

are the compressional and shear velocities, respectively.

Figure 1.5 shows the calculated Poisson’s ratio by Eq. 1.32 from sonic

compressional and shear velocities in an offshore well at a water depth of

264 m. It shows that Poisson’s ratio depends on both lithology and depth

and is larger in shales and smaller in sandstones.

14 1 Rock properties and mechanical behaviors

2000

2500

3000

3500

4000

4500

0 0.1 0.2 0.3 0.4 0.5

Poisson's ratio

Depth from sea floor (m)

Fig.1.5. Calculated Poisson’s ratio from sonic compressional and shear velocities

obtained from wireline log in an offshore borehole.

Experimental results show that Poisson’s ratio in a rock depends on

lithology, confining stress, pore pressure, and porosity of the rock. Nor-

mally Poisson’s ratio is about 0.2 for sandstones, about 0.3 for carbonate

rocks, and greater than 0.3 for shales.

It is noticed that Poisson’s ratio varies significantly with the burial

depth. Figure 1.6 shows that Poisson’s ratio increases as the increase of the

depth below the sea floor in the Gulf of Mexico, where most formations