Mavko G., Mukerji T., Dvorkin J. The Rock Physics Handbook

Подождите немного. Документ загружается.

Units and symbols in the tables are:

UCS (MPa): unconfined compressive strength

Dt (ms/ft): P-wave sonic transit time

(m/s): P-wave velocity

r (kg/m

): bulk density

E (GPa): Young’s modulus

f (fraction): porosity

clay

(fraction): clay fraction

GR (API): gamma-ray log value

In addition to UCS, another material parameter of interest in estimating rock strength

is the angle of internal friction,  ¼ tan

1

, where m is the coefficient of internal

Table 7.14.3 Empirical relations for limestone and dolomite

(Chang et al., 2004; Zoback, 2007).

UCS ¼

Equation

number

(7682/Dt)

1.82

/145 Militzer and Stoll (1973) (22)

(2.44 þ109.14/Dt)

/145 Golubev and Rabinovich (1976) (23)

13.8E

0.51

Limestone with 10 < UCS < 300 MPa; Chang et al. (2004) (24)

25.1E

0.34

Dolomite with 60 < UCS < 100 MPa; Chang et al. (2004) (25)

276(1–3f)

Korobcheyev deposit, Russia; Rzhevsky and Novik (1971) (26)

143.8 exp(–6.95f) Middle East, low- to moderate-porosity (0.05–0.2), high

UCS (30–150 MPa); Chang et al. (2004)

(27)

135.9 exp(–4.8f) low- to moderate-porosity (0.0–0.2), high UCS

(10–300 MPa); Chang et al. (2004)

(28)

Table 7.14.2 Empirical relations for shales (Chang et al ., 2004; Zoback, 2007).

UCS ¼ Equation number

0.77(304.8/Dt)

2.93

North Sea, high-porosity Tertiary shales; Horsrud (2001) (12)

0.43(304.8/Dt)

3.2

Gulf of Mexico, Pliocene and younger shales; Chang

et al. (2004)

(13)

1.35(304.8/Dt)

2.6

Worldwide; Chang et al. (2004) (14)

0.5(304.8/Dt)

Gulf of Mexico; Chang et al. (2004) (15)

10(304.8/Dt  1) North Sea, high-porosity Tertiary shales; Lal (1999) (16)

7.97E

0.91

North Sea, high-porosity Tertiary shales; Horsrud (2001) (17)

7.22E

0.712

Strong, compacted shales; Chang et al. (2004) (18)

1.001f

–1.143

Low-porosity (<0.1), high-strength (79 MPa) shales;

Lashkaripour and Dusseault (1993)

(19)

2.922f

–0.96

North Sea, high-porosity Tertiary shales; Horsrud (2001) (20)

0.286f

–1.762

High-porosity (>0.27) shales; Chang et al. (2004) (21)

387 7.14 Velocity–porosity–strength relations

friction. Chang et al. (2004) give the relations for y, summarized in Table 7.14.4.

These relations are of doubtful reliability, but show trends that are approximately in

agreement with observations (Wong et al., 1993; Horsrud, 2001).

Use of equation (30) in Table 7.14.4 requires the reference values of GR and m

(coefficient of internal friction) for pure shale and pure sand end-members. These

have to be assumed or calibrated from well logs.

Uses

These equations can be used to estimate rock strength from properties measurable

with geophysical well logs.

Assumptions and limitations



The relations described in this section are obtained from empirical fits that show

considerable scatter, especially for carbonates. Relations should be locally cali-

brated when possible.



Relations for well-consolidated rocks (e.g., equations (10), (18), and (19) in the

tables) should not be used for estimating strength of poorly consolidated, weak

rocks. Equations (3) and (5) provide reasonable estimates for weak sands.



Equation (11) for sandstones gives a reasonable fit to UCS for porosities >0.1, with

80% of the data within 30 MPa.



Equations (15), and (19)–(21) for shales predict shale strength fairly well. For

porosities > 0.1, equations (19)–(21) fit 90% of the data to within 10 MPa.

Table 7.14.4 Angles of internal friction.

y (deg) ¼

Equation

number

sin

-1

[(V

– 1000)/(V

þ1000)] Shales; Lal (1999) (29)

57.8–105f Sandstone Weingarten and Perkins

(1995)

(30)

tan

1

GR  GR

sand

ðÞ

shale

þ GR

shale

 GRðÞ

sand

shale

 GR

sand



Shaley sandstones; Chang

et al. (2004)

(31)

388 Empirical relations

Flow and diffusion

8.1 Darcy’s law

Synopsis

It was established experimentally by Darcy (1856) that the fluid flow rate is linearly

related to the pressure gradient in a fluid-saturated porous medium by the following

equation:

¼A



where Q

is the volumetric fluid flow rate in the x direction, k is the permeability of

the medium,  is the dynamic viscosity of the fluid, P is the fluid pressure, and A is

the cross-sectional area normal to the pressure gradient.

This can be expressed more generally as

Q ¼



A grad P

where Q is the vector fluid volumetric velocity field and k is the permeability tensor.

Permeability, k, has units of area (m

in SI units), but the more convenient and

traditional unit is the Darcy:

1 Darcy ¼ 0:986 923  10

12

In a water-saturated rock with a permeability of 1 Darcy, a pressure gradient of 1 bar/cm

gives a flow velocity of 1 cm/s.

The hydraulic conductivity K is related to permeability k as

K ¼

kg



where g ¼9.81 m/s

is the acceleration due to gravity. Hence, the units of hydraulic

conductivity are m/s.

For water with density 1 g/cm

¼1000 kg/m

and dynamic viscosity 1 cP s ¼0.001 Pa s,

the permeability of 1 mD ¼10

–15

translates into a hydraulic conductivity of

9.81 10

–9

m/s ¼9.81 10

–7

cm/s  10

–6

cm/s. Hydraulic conductivity is often used

in ground water flow applications.

389

If the linear (not volumetric) velocity field V is considered,

V ¼



grad P

Using this latter notation, Darcy’s law for multiphase flow of immiscible fluids in

porous media (with porosity f) is often stated as

¼



grad P

where the subscript i refers to each phase and k

is the relative permeability of phase i.

Simultaneous flow of multiphase immiscible fluids is possible only when the saturation

of each phase is greater than the irreducible saturation, and each phase is continuous

within the porous medium. The relative permeabilities depend on the saturations

and show hysteresis, for they depend on the path taken to reach a particular

saturation. The pressures P

in any two phases are related by the capillary pressure

, which itself is a function of the saturations. For a two-phase system with fluid 1 as

the wetting fluid and fluid 2 as the nonwetting fluid, P

¼P

– P

. The presence of a

residual nonwetting fluid can interfere considerably with the flow of the wetting

phase. Hence the maximum value of k

may be substantially less than 1. There are

extensions of Darcy’s law for multiphase flow (Dullien, 1992) that take into account

cross-coupling between the fluid velocity in phase i and the pressure gradient in

phase j. The cross-coupling becomes important only at very high viscosity ratios

ð

=

1Þ because of an apparent lubricating effect.

In a one-dimensional, immiscible displacement of fluid 2 by fluid 1 (e.g., water

displacing oil in a water flood), the time history of the saturation S

(x, t) is governed

by the following equation (Marle, 1981):



]





¼ 0

where

V ¼V

þ V



þ kð

 

Þg=V



þ 





þ 

where V

and V

are the Darcy fluid velocities in phases 1 and 2, respectively, and g is

the acceleration due to gravity. The requirement that the two phases completely fill

the pore space implies S

þ S

¼1. Neglecting the effects of capillary pressure gives

the Buckley–Leverett equation for immiscible displacement:

390 Flow and diffusion





¼ 0

This represents a saturation wavefront traveling with velocity (V/f)(dx

/dS

Desbrandes (1985) presented the following empirical equations that link the

relative permeability for water (k

) and oil ( k

) to water saturation (S

), irreducible

water saturation (S

), and residual oil saturation (S

 S

1  S



1  S

 S

1  S

 S



where saturation can vary from 0 to 1. These relative permeability curves are plotted

in Figure 8.1.1.

Diffusivity

If the fluid and the matrix containing it are compressible and elastic, the saturated

system can take on the behavior of the diffusion equation. If we combine Darcy’s law

with the equation of mass conservation given by

rðVÞþ

]ðÞ

¼ 0

plus Hooke’s law expressing the compressibility of the fluid, b

, and of the pore volume,

, and drop nonlinear terms in pressure, we obtain the classical diffusion equation:

P ¼

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

Relative permeability

1–S

Oil

Water

Figure 8.1.1 Relative permeability for water (the rising curve) and oil (falling curve) according

to equations of Desbrandes (1985) for an irreducible water saturation of 0.2 and a residual oil

saturation of 0.2.

391 8.1 Darcy’s law

where D is the diffusivity:

D ¼

ðb

þ b

One-dimensional diffusion

Consider the one-dimensional diffusion that follows an initial pressure pulse:

P ¼ P

dðxÞ

We obtain the standard result, illustrated in Figure 8.1.2, that

Pðx; tÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4pDt

x

=4Dt

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4pDt

t=t

where the characteristic time depends on the length scale x and the diffusivity:

t ¼

Sinusoidal pressure disturbance

Consider an instantaneous sinusoidal pore-pressure disturbance in the saturated

system, as shown in Figure 8.1.3.

The disturbance will decay approximately as e

–t/t

, where the diffusion time is again

related to the length and diffusivity by

It is interesting to ask, when is the diffusion time equal to the period of the seismic

wave causing such a disturbance? The seismic period is

Pressure

Figure 8.1.2 Pressure curves for one-dimensional diffusion.

392 Flow and diffusion

Equating t

to t

gives

¼ t

)

which finally gives

t ¼

For a rock with a permeability of 1 milliDarcy (1 mD), the critical frequency (1/t)is

10 MHz.

Assumptions and limitations

The following considerations apply to the use of Darcy’s law:



Darcy’s law applies to a representative elementary volume much larger than the

grain or pore scale;



Darcy’s law is applicable when inertial forces are negligible in comparison to pressure

gradient and viscous forces, and the Reynolds number Re is small (Re  1–10). The

Reynolds number for porous media is given by Re ¼ vl=; where r is the fluid

density,  is the fluid viscosity, v is the fluid velocity, and l is a characteristic length of

fluid flow determined by pore dimensions. At high Re, inertial forces can no longer be

neglected in comparison with viscous forces, and Darcy’s law breaks down;



some authors mention a minimum threshold pressure gradient below which there is very

little flow (Bear, 1972). Non-Darcy behavior below the threshold pressure gradient has

been attributed to streaming potentials in fine-grained soils, immobile adsorbed water

layers, and clay–water interaction, giving rise to non-Newtonian fluid viscosity; and



when the mean free path of gas molecules is comparable to or larger than the

dimensions of the pore space, the continuum description of gas flow becomes

invalid. In these cases, measured permeability to gas is larger than the permeability

to liquid. This is sometimes thought of as the increase in apparent gas permeability

Pressure

Figure 8.1.3 Decay of an instantaneous sinusoidal pore-pressure disturbance in a saturated system.

393 8.1 Darcy’s law

caused by slip at the gas–mineral interface. This is known as the Klinkenberg effect

(Bear, 1972). The Klinkenberg correction is given by

¼ k

1 þ

4cl



¼ k

1 þ



where k

is the gas permeability, k

is the liquid permeability, l is the mean free

path of gas molecules at the pressure P at which k

is measured, c ≈ 1isa

proportionality factor, b is an empirical parameter that is best determined by

measuring k

at several pressures, and r is the radius of the capillary.

Extensions

When inertial forces are not negligible (large Reynolds number), Forchheimer suggested a

nonlinear relation between the fluid flux and the pressure gradient (Bear, 1972)asfollows:

¼ aV þ bV

where a and b are constants.

Another extension to Darcy’s law was proposed by Brinkman by adding a Lapla-

cian term to account for solid–fluid interface interactions (Sahimi, 1995). The

modified equation, sometimes termed the Darcy–Brinkman equation is:

rP ¼



V þ 

where the viscosity 

can, in principle, be different from the pure fluid viscosity, ,

though in Brinkman’s original paper and in many applications they are taken to be the

same (Sahimi, 1995). The Darcy–Brinkman equation is useful for modeling flow in

very high-porosity media, and also in transition zones between porous media flow

and open-channel flow.

8.2 Viscous flow

Synopsis

In a Newtonian, viscous, incompressible fluid, stresses and velocities are related by

Stokes’ law (Segel, 1987):



¼Pd

þ 2D



where s

denotes the elements of the stress tensor, v

represents the components of the

velocity vector, P is pressure, and  is dynamic viscosity.

For a simple shear flow between two walls, this law is called the Newton friction

law and is expressed as

394 Flow and diffusion

t ¼ 

;

where t is the shear stress along the flow, v is velocity along the flow, and y is the

coordinate perpendicular to the flow.

The Navier–Stokes equation for a Newtonian viscous incompressible flow is

(e.g., Segel, 1987)



þ v  grad v



¼grad P þ v

where r is density, t is time, v is vector velocity, and D is the Laplace operator.

Useful examples of viscous flow

(a) Steady two-dimensional laminar flow between two walls (Lamb, 1945):

vðyÞ¼

2

ðR

 y

Þ; Q ¼

3

where 2R is the distance between the walls and Q is the volumetric flow rate per

unit width of the slit (Figure 8.2.1).

(b) Steady two-dimensional laminar flow in a circular pipe:

vðyÞ¼

4

ðR

 y

Þ; Q ¼

8

where R is the radius of the pipe (see Figure 8.2.1).

Q ¼

4

þ b

where a and b are the semiaxes of the cross-section.

(d) Steady laminar flow in a pipe of rectangular cross-section:

Q ¼

24

abða

þ b

Þþ



n¼1

ð2n  1Þ

 a

tanh pb

2n  1



þ b

tanh pa

2n  1



where a and b are the sides of the rectangle. For a square this equation yields

v(y)

Figure 8.2.1 Steady laminar flow in a two-dimensional slot, or in a pipe

of circular cross-section.

395 8.2 Viscous flow

Q ¼0:035144



(e) Steady laminar flow in a pipe of equilateral triangular cross-section with the

length of a side b:

Q ¼

ﬃﬃﬃ

320

(f) Steady laminar flow past a sphere (pressure on the surface of the sphere depends

on the x coordinate only):

P ¼



where v is the undisturbed velocity of the viscous flow (velocity at infinity), and

the origin of the x-axis is in the center of the sphere of radius R (Figure 8.2.2).

The total resistance force is 6pvR.

The combination

Re ¼

vR



where R is the characteristic length of a flow (e.g., the pipe radius) and Re is the

Reynolds number. Flows where Re < 1 are called creeping flows. (Viscous

forces are the dominant factors in such flows.) A flow becomes turbulent (i.e.,

nonlaminar) if Re > 2000.

Uses

The equations presented in this section are used to describe viscous flow in pores.

Assumptions and limitations

The equations presented in this section assume that the fluid is incompressible and

Newtonian.

8.3 Capillary forces

Synopsis

A surface tension, g, exists at the interface between two immiscible fluids or

between a fluid and a solid (Figure 8.3.1). The surface tension acts tangentially to

Figure 8.2.2 Steady laminar flow past a sphere.

396 Flow and diffusion