Mavko Notebook

INTRODUCTION TO ROCK PHYSICS

GARY MAVKO

Rock Physics Laboratory Stanford University

Hosted by Rock Solid Images

Preface

Because of the increasing importance of oil recovery, the growing complexity of recently discovered oil fields, and the growing realization that reservoirs and recovery are more heterogeneous than assumed in the past, a major shift in the use of seismic methods has taken place during the past decade. One of the central aspects of this shift involves the need to better understand the relation between the seismic properties of reservoir rocks and their production properties (porosity, permeability) and state (mineralogy, saturation, pore pressure, etc.). Some obvious applications are the evaluation of stratigraphic traps, fracture detection, and the spatial distribution of porosity and permeability.

Reservoir complexity is typically related to significant spatial heterogeneity in porosity, permeability, clay content, fracture density, and other properties. A direct consequence of this heterogeneity is the complexity of reservoir recovery processes, ranging from migration of the gas cap in reservoirs with discontinuous shales, overpressured zones, and the tracking of injected water, steam, or temperature during recovery in reservoirs with large spatial variations of permeability. This spatial variability cannot be inferred at any level of detail from well testing data, logs, or cores. It can only be obtained from remote geophysical measurements, especially seismic measurements -- and only when these seismic measurements can be understood in terms of reservoir properties.

Over the last three decades enormous strides have been made to understand the relations between the physical properties of reservoir rocks and geophysical observables -- the science now known as Rock Physics. We have gradually discovered more and more order in relations that once appeared disappointingly scattered, for example, velocity vs. porosity, porosity vs. permeability, Vp/Vs vs. saturation and lithology. Some of the keys have been to explore effects of pore pressure, stress, temperature, clay content, compaction, fluid type, and saturation.

This course covers fundamentals of Rock Physics, ranging from basic laboratory and theoretical results to practical “recipes” that can be immediately applied in the field. We will present qualitative and quantitative tools for understanding and predicting the effects of lithology, pore fluid types and saturation, stress and pore pressure, fractures, and temperature on seismic velocity and attenuation. We will present case studies and strategies for seismic interpretation, upscaling seismic and rock properties from the lab to borehole to reservoir scales, suggestions for more effectively employing seismic-to-rock properties transforms in geostatistical methods, and especially emphasize subsurface fluid detection and recovery monitoring.

Stanford Rock Physics Laboratory - Gary Mavko

1

Units of Stress

1 bar = 106 dyne/cm2 = 14.50 psi

10 bar = 1 MPa = 106 N/m2

Mudweight to Pressure Gradient

1 psi/ft = 144 lb/ft3

= 19.24 lb/gal

= 22.5 kPa/m

1 lb/gal = 0.052 psi/ft


2

Summary:

whereρρρρ densityK bulk modulusµµµµ shear modulusλλλλ Lamé's coefficientE Young's modulusνννν Poisson's ratio

P wave velocity

S wave velocity

E wave velocity

In terms of Poisson's ratio we can also write:

Relating various velocities:

Moduli from velocities:

VS = µµµµρρρρ

VP = K + (4/3)µµµµ

ρρρρ = λλλλ + 2µµµµρρρρ

VE = Eρρρρ

VP2

VS2 = 2(1–νννν)

(1–2νννν)

VE2

VP2 = (1+νννν)(1–2νννν)

(1–νννν)

νννν = VP2 – 2VS

2

2 VP2 – VS

2 = VE2 – 2VS

2

2VS2

VP

2

VS2 =

4 – VE2

VS2

3 – VE2

VS2

VE2

VS2 =

3Vp2

VS2 – 4

VP2

VS2 – 1

µµµµ = ρρρρVS2

νννν = VP2 – 2VS

2

2 VP2 – VS

2

K = ρρρρ VP2 – 4

3 VS2

E = ρρρρVE2


Parameters That Influence Seismic Velocity

3

Typical rock velocities, from Bourbié, Coussy, andZinszner, Acoustics of Porous Media, Gulf Publishing.

Type of formation P wavevelocity

(m/s)

S wavevelocity

(m/s)

Density(g/cm3)

Density ofconstituent

crystal(g/cm3)

Scree, vegetal soil 300-700 100-300 1.7-2.4 -Dry sands 400-1200 100-500 1.5-1.7 2.65 quartzWet sands 1500-2000 400-600 1.9-2.1 2.65 quartzSaturated shales and clays 1100-2500 200-800 2.0-2.4 -Marls 2000-3000 750-1500 2.1-2.6 -Saturated shale and sand sections 1500-2200 500-750 2.1-2.4 -Porous and saturated sandstones 2000-3500 800-1800 2.1-2.4 2.65 quartzLimestones 3500-6000 2000-3300 2.4-2.7 2.71 calciteChalk 2300-2600 1100-1300 1.8-3.1 2.71 calciteSalt 4500-5500 2500-3100 2.1-2.3 2.1 haliteAnhydrite 4000-5500 2200-3100 2.9-3.0 -Dolomite 3500-6500 1900-3600 2.5-2.9 (Ca, Mg)

CO32.8-2.9Granite 4500-6000 2500-3300 2.5-2.7 -Basalt 5000-6000 2800-3400 2.7-3.1 -Gneiss 4400-5200 2700-3200 2.5-2.7 -Coal 2200-2700 1000-1400 1.3-1.8 -Water 1450-1500 - 1.0 -Ice 3400-3800 1700-1900 0.9 -Oil 1200-1250 - 0.6-0.9 -



4

1

2

3

4

5

6

0 100 200 300

Bedford Limestone

Velo

city

(km

/s)

Effective Pressure (bars)

Sat.

Dry

Dry

Sat.

VP

V S

2

3

4

5

6

7

0 100 200 300

Westerly Granite

Velo

city

(km

/s)


Dry

Sat.

VP

VS

Dry

Sat.

2

3

4

5

6

0 100 200

Solenhofen limestone

Velo

city

(km

/s)


VS

VPSat. and Dry

Sat. and Dry

3

4

5

6

7

0 100 200 300

Webatuck dolomite

Velo

city

(km

/s)


Sat.

Sat.

Dry

Dry

VP

VS

The Saturation and Pressure Dependence of P- and S-wave Velocities.

F.1

Peffective = Pconfining – Ppore



5

Fundamental Observations of Rock Physics

• Velocities almost always increase with effectivepressure. For reservoir rocks they often tend toward a flat, high pressure asymptote.

• To first order, only the difference between confining pressure and pore pressure matters, not the absolute levels of each -- ”effective pressure law.”

• The pressure dependence results from the closing of cracks, flaws, and grain boundaries, which elastically stiffens the rock mineral frame.

• The only way to know the pressure dependence of velocities for a particular rock is to measure it.

• Make ultrasonic measurements on dry cores; fluid-related dispersion will mask pressure effects.

• The amount of velocity change with pressure is a measure of the number of cracks; the pressure range needed to reach the high pressure asymptote is a measure of crack shape (e.g. aspect ratio).

• Velocities tend to be sensitive to the pore fluid content. Usually the P-wave velocity is most sensitive and the S-wave velocity is less sensitive.

• Saturation dependence tends to be larger for soft (low velocity) rocks.



6

Pressure-Dependence of Velocities

It is customary to determine the pressure dependence of velocities from core measurements. A convenient way to quantify the dependence is to normalize the velocities foreach sample by the high pressure value as shown here. This causes the curves to cluster at the high pressure point. Then we fit an average trend through the cloud, as shown. The velocity change between any two effective pressuresP1 and P2 can be conveniently written as:

Remember to recalibrate this equation to your own cores!

F29

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 5 10 15 20 25 30 35 40

P-Velocity Pressure Dependence

Vp /

Vp(4

0)

Effective Pressure (MPa)

Average:V

P /V

P(40) = 1.0-0.38*exp(-P

eff /12)

Remember:Dry Cores!

V(P2)V(P1) = 1.0 – 0.38 exp(–P2 /12)

1.0 – 0.38 exp(–P1 /12)



7

Effects of Pore Fluid on P-wave Velocity (Low Frequency)

10

10.5

11

11.5

12

12.5

13

13.5

0 10 20 30 40 50

P Im

peda

nce

(km

/s)-(

gm/c

m3 )


dry

oil

water

4.4

4.6

4.8

5

5.2

5.4

0 10 20 30 40 50

Velo

city

(km

/s)


dry

oil

water

16

18

20

22

24

26

28

30

0 10 20 30 40 50

Bulk

Mod

ulus

(GPa

)


dry

oil

water

9

9.5

10

10.5

11

11.5

0 10 20 30 40 50 60

P Im

peda

nce

(km

/s)-(

gm/c

m3 )


dry

oil

water

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

0 10 20 30 40 50 60

Velo

city

(km

/s)


dry

oilwater

161718192021222324

0 10 20 30 40 50 60

Bulk

Mod

ulus

(GPa

)Effective Pressure (MPa)

dry

oil

water

Beaver Sandstone6% porosity

Fontainebleau Sandstone15% porosity

Calculations made from dry velocities, using Gassmann relation,Kmin = 36 GPa, Kwater = 2.2, Koil = 1.

F.2

Density Effect!

Density does not lead to ambiguitywhen Impedance is measured.

Imp = ρρρρV = ρρρρ modulus

V = modulusρρρρ



8

Effects of Pore Fluid on P-wave Velocity (Low Frequency)Ottawa Sand

1

2345

678

9

0 10 20 30 40 50 60

Bulk Modulus - Ottawa SandBu

lk M

odul

us (G

Pa)


dry

oil

water

1

1.5

2

2.5

0 10 20 30 40 50 60

Vp - Ottawa Sand

Velo

city

(km

/s)


dry

oil

water

0

1

2

3

4

5

6

0 10 20 30 40 50 60

P Impedance - Ottawa Sand

P Im

peda

nce

(km

/s)-(

gm/c

m3 )


dry

oil

water

F.3



9


Effects of Pore Fluid on P-wave Velocity (Low Frequency)Fontainebleau Sandstone

15% porosity

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 10 20 30 40 50

Beaver Poisson's Ratio

Pois

son'

s R

atio


dry

oil

water

1.42

1.44

1.46

1.48

1.5

1.52

1.54

0 10 20 30 40 50

Beaver Vp/Vs

Vp/V

s

P MPa

water

oil

dry

4.4

4.6

4.8

5

5.2

5.4

0 10 20 30 40 50

Beaver Vp

Velo

city

(km

/s)


dry

oil

water

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

0 10 20 30 40 50 60

Fontainebleau Vp

Velo

city

(km

/s)


dry

oil

water

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0 10 20 30 40 50 60

Fontainebleau Poisson's Ratio

Pois

son'

s R

atio


dry

oil

water

1.5

1.51

1.52

1.53

1.54

1.55

1.56

1.57

1.58

0 10 20 30 40 50 60

Fontainebleau Vp/Vs

Vp/V

s

P MPa

water

oil

dry

F.4



10

Effects of Pore Fluid on P-wave Velocity (Low Frequency)Ottawa Sand

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50 60

Ottawa Poisson's Ratio

Pois

son'

s R

atio


dry

oil

water

1.5

2

2.5

3

0 10 20 30 40 50 60

Ottawa Vp/Vs

Vp/V

s

P MPa

water

oil

dry

1

1.5

2

2.5

0 10 20 30 40 50 60

Vp - Ottawa SandVe

loci

ty (k

m/s

)


dry

oil

water

F.5



11

Velocities depend on fluid modulus and density

• When going from a dry to water saturated rock, sometimes the P-velocity increases; sometimes it decreases.

• The rock elastic bulk modulus almost always stiffens with a stiffer (less compressible) pore fluid.

• The stiffening effect of fluid on rock modulus is largest for a soft (low velocity) rock.

• The bulk density also increases when going from a dry to water-saturated rock.

• Because velocity depends on the ratio of elastic modulus to density, the modulus and density effects “fight” each other; sometimes the velocity goes up; sometimes down.

• Measures of modulus ( ), impedance ( ), and don’t have the density effect “ambiguity.”

• Be careful of ultrasonic data! At high frequencies, the elastic-stiffening effect is exaggerated for both bulk and shear moduli; so we don’t often see the density effect in the lab and the velocities will be contaminated by fluid-related dispersion.

Imp = ρρρρV = ρρρρ modulus VP / Vs

M = ρρρρV2



12


Effect of Pore Pressure

Effect of pore pressure on velocity, calculated assuming effective pressure law is valid, and assuming a fixed confining pressure of 40MPa (low frequency calculations using Gassmannrelation).

4.4

4.6

4.8

5

5.2

5.4

0 5 10 15 20 25 30 35 40

Velo

city

(km

/s)

Pore Pressure (MPa)

dry

oil

water

F.6



13

Ways that Pore Pressure Impacts Velocities

• Increasing pore pressure softens the elastic mineral frame by opening cracks and flaws, tending to lower velocities.

• Increasing pore pressure tends to make the pore fluid or gas less compressible, tending to increase velocities.

• Changing pore pressure can change the saturation as gas goes in and out of solution. Velocities can be sensitive to saturation.

• High pore pressure persisting over long periods of time can inhibit diagenesis and preserve porosity, tending to keep velocities low.



14

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

0 10 20 30 40 50 60

Fontainebleau

P-Ve

loci

ty (k

m/s

)

Effective Stress (MPa)

Sat.

Dry

15

15.2

15.4

15.6

15.8

0 10 20 30 40 50 60

Poro

sity

(%)


0

0.2

0.4

0.6

0.8

0 10 20 30 40 50 60

Soft

Poro

sity

(%)


2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

0 10 20 30 40 50 60

Fontainebleau

S-Ve

loci

ty (k

m/s

)


DrySat.

F.7

Ultrasonic velocities and porosity in Fontainebleausandstone (Han, 1986). Note the large change in velocity with a very small fractional change in porosity. This is another indicator that pressure opens and closes very thin cracks and flaws.

Pressure Dependence of Velocities



15

Dry shaly sandstone data from Han (1986). Each vertical “streak” plotted with the same symbol is a single rock at different pressures. Note the large change in modulus with little change in porosity -- another illustration that cracks and flaws have a large change on velocity, even though they contribute very little to porosity.

Only the values at high pressure and with Han's empirical clay correction applied.

0

5

10

15

20

25

30

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Bul

k M

odul

us (G

Pa)

Porosity

Pressure

0

5

10

15

20

25

30

35

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Sh

ear M

odul

us (G

Pa)

Porosity

Pressure

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4

Shea

r Mod

ulus

(GPa

)

Equivalent Porosity

asymptotic stiffness

0

5

10

15

20

25

30

0 0.1 0.2 0.3 0.4

Bul

k M

odul

us (G

Pa)

Equivalent Porosity

asymptotic stiffness

F.8



16

The Information in a Rock'sVelocity-Pressure Curve

1. High pressure limiting velocity is a function ofporosity

2. The amount of velocity change with pressureindicates the amount of soft, crack-like pore space

3. The range of the greatest pressure sensitivityindicates the shape or aspect ratios of the crack-likepore space

Effective Pressure

Velo

city

Vmineral

indicates crack shape or aspect ratio

indicates "soft" or

crack porosity

indicates porosity



17

Soft, Crack-Like Porosity

1. Includes micro and macrofractures and compliant grain boundaries.

2. Soft Porosity:

• Decreases both P and S-wave velocities

• Increases velocity dispersion and attenuation

• Creates pressure dependence of V and Q

• Creates stress-induced anisotropy of V and Q

• Enhances sensitivity to fluid changes (sensitivity to hydrocarbon indicators)

3. High confining pressure (depth) and cementation, tend to decrease the soft porosity, and thereforedecreases these effects.

4. High pore pressure tends to increase the softporosity and therefore increases these effects.



18

4

4.5

5

5.5

6

0 10 20 30 40

Vp (k

m/s

)


Font D

BeaverFont CFont E

Font A

Font GPeter A

Peter CFont H

Increasing Pore Pressure

0 0.05 0.1 0.15 0.2 0.25 0.3

Porosity

A

BDiagenesis

Loading

Transient Unloading

Font D

BeaverFont C

Font E

Font A

Font GPeter A

Peter C

Font HIncreasing Pore Pressure

Seismic Velocity and Overpressure

F.25

Curves on the left show the typical increase of velocity with effective pressure. For each sample the velocity change is associated with the opening and closing of cracks and flaws. These are typical when rapid changes in effective pressure occur, such as during production.

Curves on the right show the same data projected on the velocity-porosity plane. Younger, high porosity sediments tend to fall on the lower right. Diagenesis and cementation tend to move samples to the upper left (lower porosity, higher velocity). One effect of over-pressure is to inhibit diagenesis, preserving porosity and slowing progress from lower right to upper left. This is called “loading” type overpressure. Rapid, late stage development of overpressure canopen cracks and grain boundaries, resembling the curves on the left. This is sometimes called “transient” or “unloading” overpressure. In both cases, high pressure leads to lower velocities, but along

different trends.



19

7200

8000

8800

9600

10400

11200

2000 2500 3000 3500 4000 4500 5000

dept

h (ft

)

Vp (m/s)

Normal trend

Low velocity

Seismic Velocity and Overpressure

A typical approach to overpressure analysis is to look for low velocity deviations from normal depth trends. Caution: when overpressure is “late stage,” estimates of pressure can be too low.

F31



20

Experiments that illustrate the effective pressure law. In the first part of the experiment, effective pressure is increased by increasingconfining pressure from 0 to 80 MPa, while keeping pore pressure zero (solid dots). Then, effective pressure is decreased by keeping confining pressure fixed at 80 MPa, but pumping up the pore pressure from 0 to nearly 80 MPa (open circles). (Jones,1983.)

The curves trace approximately (but not exactly) the same trend.There is some hysteresis, probably associated with frictional adjustment of crack faces and grain boundaries. For most purposes, the hysteresis is small compared to more serious difficulties measuring velocities, so we assume that the effective pressure law can be applied. This is a tremendous convenience, since most

laboratory measurements are made with pore pressure equal 0.

2.5

3

3.5

4

4.5

5

0 20 40 60 80

St. Peter sandstoneVe

loci

ty (k

m/s

)


pc=80 MPa

pc=80 MPa

pp=0

pp=0

VP

VS

2.6

2.8

3

3.2

3.4

3.6

0 20 40 60 80

Sierra White granite

Velo

city

(km

/s)


pp=0

pc=80 MPa

VS

F.9



21

Effective pressure relationships, from Wyllie et al., 1958; see also Bourbié, Coussy, and Zinszner, 1987, Acoustics of Porous Media.

2.5

3

3.5

4

0 20 40 60 80

Berea sandstone

Velo

city

(km

/s)

Confining Pressure (MPa)

peff

=pc

peff

=28 MPap

eff=21 MPa

peff

=14 MPap

eff=7 MPa

peff

=0

F.10

Effective Pressure = Confining Pressure - Pore Pressure



22

0.5

1

1.5

2

2.5

3

3.5

4

0 50 100 150 200

Velo

city

(km

/s)


pp=1MPa

pp=41 MPa

pp=0

pp=0

pp=1MPa

VP

V S

Sat.

Sat.

Dry

Dry

pp=41 MPa

Pierre shale (ultrasonic), from Tosaya, 1982.

F.11

For shales, we also often see an increase of velocity with effective pressure. The rapid increase of velocity at low pressures is somewhat elastic, analogous to the closing of cracks and grain boundaries that we expect in sandstones.

The high pressure asymptotic behavior shows a continued increase in velocity rather than a flat limit. This is probably due to permanent plastic deformation of the shale.



23

Cotton Valley shale (ultrasonic), from Tosaya, 1982.

F27

2

2.5

3

3.5

4

4.5

5

0 200 400 600 800 1000 1200 1400

Cotton Valley Shale

Velo

city

(km

/s)

Pc (bars)

VpPp = 10

Pp = 400DRY

Vs Pp = 10

Pp = 400

DRY



24

Chalk (ultrasonic), from Gregory, 1976, replotted fromBourbié, Coussy, and Zinszner, 1987, Gulf Publishing Co.

1.5

2

2.5

3

3.5

0 10 20 30 40 50

Chalk

Velo

city

(km

/s)


Sw

100

0

5102040

80

60

VP

VS

5

φ = 30.6%

F.12

For chalks, we also see an increase of velocity with effective pressure. The rapid increase of velocity at low pressures is somewhat elastic and reversible.

The high pressure asymptotic behavior shows a continued increase in velocity rather than a flat limit. This is probably due to permanent crushing of the fragile pore space.



25

Velocity vs. pressure at various temperatures in water-saturated Westerly granite, from Nur (1980).

4.8

5

5.2

5.4

5.6

5.8

0 100 200 300 400 500

P-Ve

loci

ty (k

m/s

)


200 oC

200 oC

300 oC

300 oC

400 oC

400 oC

Dry

WetP

p = P

c (lower effective pressure than the dry)

F.13



26

Velocity vs. temperature in Berea Sandstone. (a) dryultrasonic (Mobarek, 1971), (b,c) saturated resonant bar

(Jones, 1983), replotted from Bourbié, et al., 1987.

1.55

1.6

1.65

0 20 40 60 80 100 120

Berea sandstoneWater saturated

Exte

nsio

nal V

eloc

ity (k

m/s

)

Temperature (degrees C)

pp

=4.5 MPa

pc=10 MPa

V E

2.8

3

3.2

3.4

3.6

3.8

4

0 20 40 60 80 100

P-W

ave

Velo

city

(km

/s)


20 oC100 oC200 oC

Berea Sandstone Dry

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20

Berea sandstoneWater saturated

22 oC 0.95 cp 53 oC 0.52 cp 84 oC 0.34 cp110 oC 0.26 cp

S-W

ave

Velo

city


Temp. η

pp=0.5 MPa

F.15

(a)

(b)

(c)



27

Influence of temperature on oil saturated samples, from Tosaya, et al. (1985).

2

2.5

3

3.5

0 50 100 150 200

Kern River sand

P Ve

loci

ty (k

m/s

)

Temperature oC

pe=10 MPa

100% oil

50% oil50% gas

100% gas

2

2.4

2.8

3.2

3.6

0 50 100 150 200

Venezuelan Oil Sand

P Ve

loci

ty (k

m/s

)

Temperature oC

100% oil

50% oil50% brine

100% brine

Pe = 10 MPa

F.16

We observe experimentally that velocities are most sensitive to temperature when the rocks contain liquid hydrocarbons (oil). We believe that this results from an increase of the oil compressibility and a decrease of the oil viscosity as the temperature goes up.

In field situations other factors can occur. For example, gas might come out of solution as the temperature goes up.



28

0.5

0.6

0.7

0.8

0.9

1

1.1

-4 -2 0 2 4 6 8 10

Barre granite

µ/µµ/µµ/µµ/µοοοο

Log viscosity (poise)

Bedford limestone

Velocity vs. viscosity in glycerol saturated samples, from Nur (1980).

0.7

0.8

0.9

1

-4 -2 0 2 4 6 8 10

Bedford limestone

V/Vo

Log viscosity (poise)

VP

VS

F.17

Temperature

Temperature

In this experiment the pore fluid is glycerol, whose viscosity is extremely sensitive to temperature. The data show a classical viscoelastic behavior with lower velocity at low viscosity and higher velocity at higher viscosity. Viscosity is one of several pore fluid properties that are sensitive to temperature.



29

Compressional velocities in the n-Alkanes vs. temperature. A drastic decrease of velocity with temperature! The numbers in the figure represent carbon numbers. From Wang, 1988, Ph.D. dissertation, Stanford University.

700

800

900

1000

1100

1200

1300

1400

-20 0 20 40 60 80 100 120 140

Alkanes

Com

pres

sion

al V

eloc

ity (m

/s)

Temperature (ÞC)

6

7

8

1011

1214 15161822

2836

F.18



30

800

900

1000

1100

1200

1300

1400

0 0.05 0.1 0.15 0.2

Alkanes

Com

pres

sion

al V

eloc

ity (m

/s)

1/Carbon Number

110 ÞC

75 ÞC

50 ÞC

22 ÞC

Compressional velocities in the n-Alkanes vs. inverse of the carbon numbers, at different temperatures. From Wang, 1988, Ph.D. dissertation, Stanford University.

F.19



31

Batzle-Wang formulas for fluid density and bulk modulus (Geophysics, Nov. 1992).

0

100

200

300

400

500

600

0 50 100 150 200

Gas Density

Gas gravity = 0.6Gas gravity = 1.2

Gas

Den

sity

(kg/

m3)


P = 0.1 MPa

P = 25 MPa

P = 50 MPa

P = 25 MPa

P = 50 MPa

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 50 100 150 200

Gas Bulk Modulus

Gas gravity = 0.6Gas gravity = 1.2

Gas

Bul

k M

odul

us (G

Pa)


0.1 MPa25 MPa

P = 50 MPa

25 MPa

P = 50 MPa

0.1 MPa

F30



32

600

650

700

750

800

850

900

950

0 50 100 150 200

Oil Density

API gravity = 10API gravity = 50

Oil

Den

sity

(kg/

m3)


0.1 MPa25 MPa

P = 50 MPa

25 MPaP = 50 MPa

0.1 MPaGOR = 100

0

0.5

1

1.5

2

2.5

0 50 100 150 200

Oil Bulk Modulus

API gravity = 10API gravity = 50

Oil

Bulk

Mod

ulus

(GPa

)


0.1 MPa25 MPa

P = 50 MPa

25 MPa

P = 50 MPa

0.1 MPa

F30



33

1

1.5

2

2.5

3

3.5

4

0 50 100 150 200

Brine Bulk Modulus

Salinity = 0Salinity = 240000 ppm

Brin

e Bu

lk M

odul

us (G

Pa)


0.1 MPa

25 MPaP = 50 MPa

25 MPaP = 50 MPa

0.1 MPa

F30

850

900

950

1000

1050

1100

1150

1200

1250

0 50 100 150 200

Brine Density

Salinity = 0Salinity = 240000 ppm

Brin

e D

ensi

ty (k

g/m

3)


0.1 MPa25 MPa

P = 50 MPa

25 MPaP = 50 MPa

0.1 MPa



34

Fluid Properties

• The density and bulk modulus of most reservoir fluids increase as pore pressure increases.

• The density and bulk modulus of most reservoir fluids decrease as temperature increases.

• The Batzle-Wang formulas describe the empirical dependence of gas, oil, and brine properties on temperature, pressure, and composition.

• The Batzle-Wang bulk moduli are the adiabaticmoduli, which we believe are appropriate for wave propagation.

• In contrast, standard PVT data are isothermal. Isothermal moduli can be ~20% too low for oil, and a factor of 2 too low for gas. For brine, the two don’t differ much.



35

Stress-Induced Velocity Anisotropy -- the Historical Basis for Seismic Fracture Detection

Stress-induced velocity anisotropy in Barre Granite (Nur, 1969). In this classic experiment, Nur manipulated the crack alignment by applying uniaxial stress. Initially the rock is isotropic, indicating an isotropic distribution of cracks. Cracks normal to the stress (or nearly so) closed, creating crack alignment and the associated anisotropy.

3.6

3.8

4

4.2

4.4

4.6

4.8

5

0 20 40 60 80

Vp (k

m/s

ec)

angle from stress axis (degrees)

stress (bars)

0

100

200

300

2.6

2.7

2.8

2.9

3

3.1

3.2

0 20 40 60 80

Vs (S

H) (

km/s

ec)


stress (bars)

0

100

200

300

400

Measurement angle related to theuniaxial stress direction

F.20

2.6

2.7

2.8

2.9

3

3.1

3.2

0 20 40 60 80

Vs (S

V) (k

m/s

ec)


stress (bars)

0

100

200

300

400

P and S waves

θ

Axial Stress



36

Stress-Induced Velocity AnisotropyDue to Crack Opening Near Failure

Uniaxial stress-induced velocity anisotropy in WesterlyGranite (Lockner, et al. 1977).

P

PSource

Source

Source

S

S S

SII

II

Frac

ture

IIAS

0.6

0.7

0.8

0.9

1

1.1

1.2

0 20 40 60 80 100

PS⊥⊥⊥⊥SIISIIA

V/V

o

Percent Failure Strength

T

T

F.21



37

Seismic Anisotropy Due to Rock Fabric

Isotropic mixtureslight alignment

layered

102030405060708090

0 0.2 0.4 0.6 0.8 1

ρVp2

fabric anisotropy

vertical (c33 )

horizontal (c11 )

Virtually any rock that has a visual layering or fabric at a scale finerthan the seismic wavelength will be elastically and seismicallyanisotropic. Sources can include elongated and aligned grains andpores, cracks, and fine scale layering. Velocities are usually faster

for propagation along the layering.



38

Velocity Anisotropy Due to Fabric

Anisotropic velocities vs. pressure. (a) and (b) Jones (1983), (c) Tosaya (1982).

3.5

4

4.5

5

5.5

6

0 20 40 60 80 100 120

GNEISS - Dry

ll foliation ⊥⊥⊥⊥ foliationll foliation

Com

pres

sion

al V

eloc

ity (k

m/s

)


5.2

5.4

5.6

5.8

6

6.2

6.4

0 20 40 60 80 100 120

MYLONITE - Dry

⊥⊥⊥⊥ foliation, ll lineation⊥⊥⊥⊥ foliation⊥⊥⊥⊥ foliation, ⊥⊥⊥⊥ lineation

Com

pres

sion

al V

eloc

ity (k

m/s

)


V33

2

1

V31 Vp 45Þ

V22 = V11

V21

V23 = V13 = V31

3

2

2.5

3

3.5

4

4.5

5

5.5

6

0 20 40 60 80 100 120 140

Pierre shaleSw = 100%

Velo

city

(km

/s)


V11Vp 45Þ

V33

Vs V12

V13

V31

F.22



39

2

2.5

3

3.5

4

4.5

5

5.5

0 200 400 600 800 1000 1200

Cotton Valley Shale - Water Saturated

Velo

city

(km

/s)

Peff (bars)

V P

V S

V11

Vp 45 o

V33

V12

V13

V31

Velocity Anisotropy in Shale

Cotton Valley shale (ultrasonic), from Tosaya, 1982.



40

Velocity Anisotropy Resulting FromThinly Layered Kerogen

P-wave anisotropy in shales (from Vernik, 1990): (1) Bakken black shales, (2) Bakken dolomitic siltstone, (3) Bakken shaly dolomite, (4) Chicopee shale (Lo, et al, 1985).

Vernik found that kerogen-bearing shales can have verylarge anisotropy, easily 50%.

1

1.1

1.2

1.3

1.4

1.5

0 10 20 30 40 50 60 70 80

P-w

ave

Ani

sotr

opy


1

1

1

432

C=13%

22.4%

30%

<0.5%

F.23



41

Velocities in kerogen-rich Bakken shales (Vernik, 1990)and other low porosity argillaceous rocks (Lo et al., 1985; Tosaya, 1982; Vernik et al., 1987). Compiled by

Vernik, 1990.

1

2

3

4

0 20 40 60 80

S-w

ave

Velo

city

Vs(

0), k

m/s

Confining Pressure, MPa

Kola Phyllite, C=2-3%

Chicopee Shale, C<1%

Cotton Valley Shale, C<1%

Bakken Black Shales, C=13-30%

2

3

4

5

6

0 20 40 60 80

P-w

ave

Velo

city

Vp(0

), km

/s

Confining Pressure, MPa

Kola Phyllite, C=2-3%

Chicopee Shale, C<1%

Cotton Valley Shale, C<1%

Bakken Black Shales, C=13-30%

F.24


Effective Medium Theories

42

Bounding Methods for Estimating Effective Elastic Moduli

For many reasons we would like to be able to model or estimate the effective elastic moduli of rocks in terms of the properties of the various constituent minerals and pore fluids. To do it precisely one must incorporate

• the individual elastic moduli of the constituents

• the volume fractions of the constituents

• geometric details of how the various constituents

are arranged

The geometric details are the most difficult to know or measure. If we ignore (or don’t know) the details of geometry, then the best we can do is estimate upper and lower bounds on the moduli or velocities.

The bounds are powerful and robust tools. They give rigorous upper and lower limits on the moduli, given the composition. If you find that your measurements fall outside the bounds, then you have made a mistake - in velocity, volume fractions, or composition!



43

Voigt and Reuss Bounds

On a strictly empirical basis one can imagine defining a power law average of the constituents

where

Special cases are the Voigt average (an upper bound):

and the Reuss average (a lower bound):

Since these are upper and lower bounds, an estimate of the actual value is sometimes taken as the average of the two, known as the Voigt-Reuss-Hill average:

Mαααα = f1M1αααα + f2M2

αααα + f3M3αααα + ...

MV = f1M1 + f2M2

MR–1 = f1M1

–1 + f2M2–1

MVRH = MV + MR2

= the effective modulus of the composite

Mi = the modulus of the ith constituent

fi = the volume fraction of the ith constituent

αααα = a constant, generally between -1 and +1

M



44

The Voigt and Reuss averages are interpreted as the ratio of average stress and average strain within the composite.

The stress and strain are generally unknown in the composite and are expected to be nonuniform. The upper bound (Voigt) is found assuming that the strain is everywhere uniform. The lower bound (Reuss) is found assuming that the stress is everywhere uniform.

Geometric interpretations:

E = σσσσεεεε = σσσσ

ΣΣΣΣfiεεεεi

= σσσσfi σσσσ

EiΣΣΣΣ

E = σσσσ

εεεε = ΣΣΣΣfiσσσσ iεεεε

= fi εεεεEiΣΣΣΣεεεε

Voigt iso-strain model Reuss iso-stress model

E = ΣΣΣΣfiEi

1E

= ΣΣΣΣ fiEi

Since the Reuss average describes an isostress situation,it applies perfectly to suspensions and fluid mixtures.



45

Backus Average for Thinly Layered MediaBackus (1962) showed that in the long wavelength limit a stratified medium made up of thin layers is effectively anisotropic. It becomes transversely isotropic, with symmetry axis normal to the strata. The elastic constants (see next page) are given by:

where

M = 12 A – B

A B F 0 0 0B A F 0 0 0F F C 0 0 00 0 0 D 0 00 0 0 0 D 00 0 0 0 0 M

,

A = 4µµµµ λλλλ + µµµµ

λλλλ + 2µµµµ + 1λλλλ + 2µµµµ

–1 λλλλλλλλ + 2µµµµ

2

B = 2µµµµλλλλλλλλ + 2µµµµ + 1

λλλλ + 2µµµµ–1 λλλλ

λλλλ + 2µµµµ2

C = 1λλλλ + 2µµµµ

–1

F = 1λλλλ + 2µµµµ

–1 λλλλλλλλ + 2µµµµ

D = 1µµµµ

–1

M = µµµµ

are the isotropic elastic constants of the individual layers. The brackets indicate averages of the enclosed properties, weighted by their volumetric proportions. This is often called the Backus average.

λλλλ, µµµµ



46

Hooke’s law relating stress and strain in a linear elasticmedium can be written as

elastic stiffnesses (moduli) elastic compliancesA standard shorthand is to write the stress and strain as vectors:

T =

σ 1= σ 11σ 2= σ 22σ 3= σ 33σ 4= σ 23σ 5= σ 13σ 6= σ 12

E =

e1= ε11e2= ε22e3= ε33e4=2ε23e5=2ε13e6=2ε12

σ 1σ 2σ 3σ 4σ 5σ 6

=

A B F 0 0 0B A F 0 0 0F F C 0 0 00 0 0 D 0 00 0 0 0 D 00 0 0 0 0 M

e1e2e3e4e5e6

Note the factor of 2 in the definition of strains.

The elastic constants are similarly written in abreviated form, and the Backus average constants shown on the previous page now have the meaning:

σ ij = cijkl εklΣkl

ε ij = Sijkl σ klΣkl



47

Velocity-porosity relationship in clastic sediments compared with the Voigt and Reuss bounds. Virtually all of the points indeed fall between the bounds. Furthermore, the suspensions, which are isostress materials (points with porosity > 40%) fall very close to the Reuss bound.

Data from Hamilton (1956), Yin et al. (1988), Han et al. (1986). Compiled by Marion, D., 1990, Ph.D. dissertation, Stanford Univ.

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100suspensionssand-clay mixt.sandclay-free sandstoneclay-bearing sandstone

P-Ve

loci

ty (m

/s)

Porosity (%)

Voigt Avg.

Reuss Avg.(Wood's Relation)

G.1



48

Hashin-Shtrikman Bounds

Interpretation of bulk modulus:

where subscript 1 = shell, 2 = sphere. f1 and f2 are volume fractions.

These give upper bounds when stiff material is K1, µµµµ1(shell) and lower bounds when soft material is K1, µµµµ1.

KHS± = K1 + f2

K2 – K1–1 + f1 K1 + 4

3 µµµµ1–1

µµµµHS± = µµµµ1 + f2µµµµ2 – µµµµ1

–1 + 2f1 K1 + 2µµµµ1

5µµµµ1 K1 + 43µµµµ1

The narrowest possible bounds on moduli that we can estimate for an isotropic material, knowing only the volume fractions of the constituents, are the Hashin-Shtrikman bounds. (The Voigt-Reuss bounds are wider.) For a mixture of 2 materials:



49

Hashin-Shtrikman Bounds

A more general form that applies when more than two phases are being mixed (Berryman, 1993):

where

indicates volume average over the spatially varying K(r), µµµµ(r) of the constituents.

KHS+ = ΛΛΛΛ µµµµ max , KHS– = ΛΛΛΛ µµµµ min

µµµµHS+ = ΓΓΓΓ ζζζζ Kmax,µµµµmax , µµµµ HS– = ΓΓΓΓ ζζζζ Kmin,µµµµmin

ΛΛΛΛ z = 1K r + 4

3z–1

– 43z

ΓΓΓΓ z = 1µµµµ r + z

–1– z

ζζζζ K,µµµµ = µµµµ6

9K + 8µµµµK + 2µµµµ



50

Bounds on the bulk and shear moduli of two-phase isotropic aggregates. Also shown is the Voigt-Reuss-Hill average (VRH). We observe that the upper and lower bounds are far apart when the materials being mixed are dissimilar, but close when materials are similar. Hence, we can often use the upper and lower bounds (or the VRH average) to model a mixture of minerals. But bounds are often not so useful when mixing minerals and pore-filling materials.From Watt, Davies, and O'Connell, 1976, Reviews of Geophysics and Space Physics, 14, 541-563.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

K*/K

MgO

Volume Fraction AgCl

Voigt

HS+

HS-

VRH

Reuss

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

K*/K

stis

hovi

te

Volume Fraction MgO

Voigt

HS+, HS-

VRH

Reuss

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

µµ µµ */µµ µµ

MgO

Volume Fraction AgCl

Voigt

HS+

HS-

VRH

Reuss

0.996

0.997

0.998

0.999

1

0 0.2 0.4 0.6 0.8 1

µµ µµ */µµ µµ

stis

hovi

te

Volume Fraction MgO

Voigt, HS+, VRH, HS-, Reuss

K*/KMgO µµµµ*/µµµµMgO

K*/Kstishovite µµµµ*/µµµµstishovite

G.2



51

Distance between bounds depends on similarity/difference of end-member constituents. Here we see that a mixture of calcite and water gives widely spaced bounds, but a mixture of calcite and dolomite gives very narrow bounds.

697071727374757677

0 0.2 0.4 0.6 0.8 1

Calcite + Dolomite

Bul

k M

odul

us (G

Pa)

fraction of dolomite

HS-, HS+, Reuss, Voigt, VRH

01020304050607080

0 0.2 0.4 0.6 0.8 1

Calcite + Water

Bul

k M

odul

us (G

Pa)

porosity

HS+

HS-, Reuss

Voigt

VRH

G13



52

Wyllie Time Average

1

2

3

d1

d2

d3

D

Wyllie et al. (1956, 1958, 1962) found that travel time through water saturated consolidated rocks could be approximately described as the volume weighted average of the travel time through the constituents:

t = DV

1V

= f1V1

+ f2V2

+ f3V3

1V = d1 /D

V1+ d2 /D

V2+ d3 /D

V3

DV

= d1V1

+ d2V2

+ d3V3

t = t1 + t2 + t3



53

Limitations:

• rock is isotropic• rock must be fluid-saturated• rock should be at high effective pressure• works best with primary porosity• works best at intermediate porosity• must be careful of mixed mineralogy (clay)

The time-average equation is heuristic and cannot be justified theoretically. It is based on ray theory which requies that (1) the wavelength is smaller than the grain and pore size, and (2) the minerals and pores are arranged in flat layers.

Note the problem for shear waves where one of the phases in a fluid, Vs-fluid →→→→ 0!

Wyllie’s generally works best for

• water-saturated rocks• consolidated rocks• high effective pressures



54

Modification of Wyllie's proposed by Raymer

Still a strictly empirical relation.

This relation recognizes that at large porosities (φφφφ > 47%) the sediment behaves as a suspension, with the Reuss average of the P-wave modulus, M = ρρρρVp2.

V = (1 – φφφφ)2 Vmineral + φφφφ Vfluid φφφφ < 37%

1ρρρρV2 = φφφφ

ρρρρfluidVfluid2 + 1 – φφφφ

ρρρρmineralVmineral2 φφφφ > 47%

1V = 0.47 – φφφφ

0.101

V37+ φφφφ – 0.37

0.101

V4737% < φφφφ < 47%



55

Comparison of Wyllie's time average equationand the Raymer equations with Marion's compilation of shaly-sand velocities from Hamilton (1956), Yin et al. (1988), Han et al. 1986).

1000

2000

3000

4000

5000

6000


P-Ve

loci

ty (m

/s)

Porosity (%)

Voigt Avg.

Wood's Relation(Reuss Avg.)

Wyllie

Raymer

G.3



56

Seismic Fluid Substitution

Pore fluids, pore stiffness,and their interaction



57

Typical Problem: Analyze how rock properties, logs, and seismic change, when pore fluids change.

Flood sandyintervals with brine

Example: We observe Vp, Vs, and density at a well and compute a synthetic seismic trace, as usual. Predict how the seismic will change if the fluid changes -- either over time at the same position, or if we move laterally away from the welland encounter different fluids in roughly the same rocks.



58

Effective moduli for specific pore and grain geometries

Imagine a single linear elastic body. We do two separate experiments--apply stresses σσσσ1 and observe displacements u1, then apply stresses σσσσ2 and observe displacements u2.

The Betti-Rayleigh reciprocity theorem states that the work done by the first set of forces acting through the second set of displacements is equal to the work done by the second set of forces acting through the first set of displacements.

σσσσij(1), u(1)

∆σ∆σ∆σ∆σ

σσσσij(2), u(2)




59

Estimate of Dry Compressibility

Applying the reciprocity theorem we can write:

Assumptions• minerals behave elastically• friction and viscosity not important• assumes a single average mineral

∆σ∆σ∆σ∆σ∆σ∆σ∆σ∆σVbulkKdry

– ∆σ∆∆σ∆∆σ∆∆σ∆vpore = ∆σ∆σ∆σ∆σ∆σ∆σ∆σ∆σVbulkKmineral

limit as

∆σ∆σ∆σ∆σ→→→→0

1Kdry

= 1Kmineral

+ 1Vbulk

∂∂∂∂vpore∂∂∂∂σσσσ


∆σ∆σ∆σ∆σ∆σ∆σ∆σ∆σ





60

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Kdr

y/K

min

eral

Porosity

Modified Voigt

Voigt

Reuss

Kφφφφ/Kmineral =

.4

.3

.2

.1

φφφφc

.5

P

porous glassss - high pressss - variable press

Relation of Rock Moduli to Pore Space Compressibility -- Dry Rock

where

G.4

A fairly general and rigorous relation between dry rock bulk modulus and porosity is

1K dry

= 1K mineral

+ φK φ

1K φ

= 1vpore

ŽvporeŽσ

is the pore space stiffness. This is a new concept that quantifies the stiffness of a pore shape.

K φ



61

What is a “Dry Rock”?

Many rock models incorporate the concept of a dry rock or the dry rock frame. This includes the work byBiot, Gassmann, Kuster and Toksoz, etc, etc.

Caution: “Dry rock” is not the same as gas-saturated rock. The dry frame modulus in these models refers to the incremental bulk deformation resulting from an increment of applied confining pressure, with pore pressure held constant. This corresponds to a “drained” experiment in which pore fluids can flow freely in or out of the sample to insure constant pore pressure. Alternatively, it can correspond to an undrained experiment in which the pore fluid has zero bulk modulus, so that pore compressions do not induce changes in pore pressure – this is approximately the case for an air-filled sample at standard temperature and pressure. However, at reservoir conditions (high pore pressure), gas takes on a non-negligible bulk modulus, and should be treated as a saturating fluid.



62

Relation of Rock Moduli to Pore Space Compressibility -- Saturated Rock

where Pore spacecompressibilitymodified by fluids.

A similar general relation between saturated rock bulk modulus and porosity is

1K sat

= 1K mineral

+ φK φ

K φ = K φ + K mineralK fluid

K mineral – K fluid ≈ K φ + K fluid

So we see that changing the pore fluid has the effect ofchanging the pore space compressibility of the rock. Thefluid modulus term is always just added to K φ

When we have a stiff rock with high velocity, then its valueof is large, and changes in do not have much effect. But a soft rock with small velocity will have a small

and changes in will have a much larger effect.

K φ

K φ

K fluid

K fluid



63

Bulk Modulus For a Saturated Rock

cracks

pores

≈ 1 cracks≈ 0.2 pores

1Ksat

= 1Kmineral

+ φφφφKφφφφ

– φφφφKmineral

∂∂∂∂P∂∂∂∂σσσσ

∂∂∂∂P∂∂∂∂σσσσ = 1

1 + Kφφφφ1

Kfluid– 1

Kmineral

≈≈≈≈ 11 +

KφφφφKfluid

1Kdry

> 1Ksat

≈≈≈≈ 1Kmineral

1Kdry

> 1Ksat

> 1Kmineral



64

Gassmann's Relations

KsatKmineral – Ksat

=Kdry

Kmineral – Kdry+ Kfluid

φφφφ Kmineral – Kfluid

1µµµµsat

= 1µµµµdry

These are Transformations! Pore space geometry and stiffness are incorporated automatically by measurements of Vp, Vs. Gassmann (1951) derived this general relation between the dry rock moduli and the saturated rock moduli. It is quite general and valid for all pore geometries, but there are several important assumptions:

• the rock is isotropic• the mineral moduli are homogeneous• the frequency is low

“Dry rock” is not the same as gas saturated rock.

Be careful of high frequencies, high viscosity, and clay.

Useful for Fluid Substitution problem:gas

oilwater



65

Some Other Forms of Gassmann

Ksat =φ 1

Kmin– 1

K fluid+ 1

K min– 1

Kdryφ

Kdry

1Kmin

– 1K fluid

+ 1Kmin

1K min

– 1Kdry

K sat = K dry +1 –

K dryKmin

2

φK fluid

+ 1 – φK min

–K dry

Kmin2

1Ksat

= 1Kmin

+ φ

K φ +K minK fluid

K min – K fluid

K dry =Ksat

φK minK fluid

+ 1 – φ – Kmin

φKminK fluid

+ KsatK min

– 1 – φ



66

1. Extract Moduli from Velocities measured with fluid 1:

2. Transform the bulk modulus using Gassmann

where K1, K2 are dynamic rock moduli with fluids 1, 2

Kfl 1, Kfl 2 bulk moduli of fluids 1, 2ρρρρ1, ρρρρ2 density of rock with fluids 1, 2Kmin, φφφφ mineral modulus and porosityρρρρfl 1, ρρρρfl 2 density of fluids 1, 2

3. µµµµ2 = µµµµ1 shear modulus stays the same

4. Transform density

5. Reassemble the velocities

K2Kmin – K2

– Kfl 2φφφφ Kmin – Kfl 2

= K1Kmin – K1

– Kfl 1φφφφ Kmin – Kfl 1

Fluid Substitution Recipe

VP =K2 + 4

3 µµµµ2ρρρρ2

VS = µµµµ2ρρρρ2

ρρρρ 2 = 1 – φφφφ ρρρρmin + φφφφρρρρ fl 2 = ρρρρ1 + φφφφ ρρρρ fl 2 – ρρρρ fl 1

K1 = ρρρρ1(VP2 – 4

3VS2), µµµµ1 = ρρρρ1VS

2



67

Why is the shear modulus unaffected by fluids in Gassmann’s relations?

Imagine first an isotropic sample of rock with a hypothetical spherical pore. Under “pure shear”loading there is no volume change of the rock sampleor the pore -- only shape changes. Since it is easy tochange the shape of a fluid, the rock stiffness is notaffected by the type of fluid in the pore.



68

Why do the Gassmann relationsonly work at low frequencies?

Imagine an isotropic sample of rock with cracks at all orientations. Under “pure shear” loading there is no volume change of the rocksample or the pore space, because some cracks open while others close. If the frequency is too high, there is a tendency for local pore pressures to increase in some pores and decrease in others: hence the rock stiffness depends on the fluid compressibility.

However, if the frequency is low enough, the fluid has time to flow and adjust: there is no net pore volume change and therefore the rock stiffness is independent of the fluids.

This crack decreases involume. Its pore pressurelocally increases if the fluid cannot flow out of the crack.

This crack increases involume. Its pore pressurelocally decreases if the fluidcannot flow into the crack.



69

Graphical Interpretation of Gassmann's Relations

1. Plot known effective modulus K, with initial fluid.

2. Compute change in fluid term:

3. Jump vertically up or down that number of contours.

Example: for quartz and water ~ 3 contours.

∆∆∆∆KfluidKmineral

≈≈≈≈ .06

∆∆∆∆KmineralKfluid

Kmineral – Kfluid≈≈≈≈ ∆∆∆∆Kfluid

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

K/K

min

eral

Porosity

K φφφφ/K mineral = .1

.4

.3

.2

.5

K flu

id/K

min

eral

( φφ φφ =

1 o

nly)

A

A'

B

B'

~

G.6



70

Graphical Interpretation of Gassmann's Relations

1. Plot the known modulus with initial fluid (point A).2. Identify Reuss averages for initial and final fluids.3. Draw straight line through through A to initial Reuss

curve.4. Move up or down to new Reuss Curve and draw

new straight line.5. Read modulus with new fluid (point A').

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

K/K

min

eral

Porosity

K φφφφ/K mineral = .1

.4

.3

.2

.5

K flu

id/K

min

eral

(φφ φφ

= 1

only

)A

A'

Reuss (water)

Reuss (air) water

φφφφ c

~

air

G.7



71

Approximate Gassmann Relationwhen Shear Velocity is Unknown

Normally, to apply Gassmann's relations, we needto know both Vp and Vs so that we can extract thebulk and shear moduli:

and then compute the change of bulk modulus withfluids using the usual expression:

The problem is that we usually don't know Vs.

One approach is to guess Vs, and then proceed.

We have also found that a reasonably good approximation to Gassmann is

where M is the P-wave modulus:

K = ρρρρ VP

2 – 43 VS

2 µµµµ = ρρρρ VS2

KsatKmineral - Ksat

=Kdry

Kmineral - Kdry+

Kfluidφφφφ(Kmineral - Kfluid)

MsatMmineral - Msat

≈≈≈≈Mdry

Mmineral - Mdry+

Mfluidφφφφ(Mmineral - Mfluid)

M = ρρρρ VP2



72

Approximate Gassmann RelationWhen Shear Velocity is Unknown

Vp sat From Exact Gassmann’s Equation

2.5

3

3.5

4

4.5

5

5.5

6

2.5 3 3.5 4 4.5 5 5.5 6

Vsat - 40 Full

Vsat - 30

Vsat - 20

Vsat - 10

Vsat - 5

x = y

Predictions of saturated rock Vp from dry rock Vp are virtually the same for the approximate and exact forms of Gassmann’s relations.



73

Gassmann's is a Low Frequency Relation

Vp Sat Measured (high frequency)

2.5

3

3.5

4

4.5

5

5.5

6

2.5 3 3.5 4 4.5 5 5.5 6

5 MPa Wet M1

5 MPa Wet M2

5 MPa Wet Exact

40 MPa Wet M1

40 MPa Wet M2

40 MPa Wet Exact

x = y

It is important to remember that Gassmann’s relations assume low frequencies. Measured ultrasonic Vp in saturated rocks is almost always faster than saturated Vp predicted from dry rock Vp using Gassmann. Data here are for shaly sandstones (Han, 1986).

VelocityDispersion



74

Water Flood Example: Pore PressureIncrease and Change From Oil to Brine

Calculated using Gassmann from dry lab data from Troll (Blangy, 1992).Virgin condition taken as low frequency, oil saturated at Peff=30 MPa.Pressure drop to Peff=10 MPa, then fluid substitution to brine.Koil = 1., Kbrine = 2.2

G.12

2 2.5

1250

1300

1350

Brine Flood into Oil

Vp (km/s)

dept

h (m

)

Pressure

oil to water

original oil

oil atincreased Pp

brine atincreased Pp

1.61.8

22.22.42.62.8

0 5 10 15 20 25 30 35

Vp (k

m/s

)

Eff pressure (MPa)

brineoilgas

One typical depth point

• effect of pressure on frame• effect of pressure on fluids• frame+fluid: fluid substitution



75

Gas Flood Example: Pore PressureIncrease and Change From Oil to Gas

Calculated using Gassmann from dry lab data from Troll (Blangy, 1992).Virgin condition taken as low frequency, oil saturated at Peff=30 MPa.Pressure drop to Peff=10 MPa, then fluid substitution to gas.Koil = 1., Kbrine = 2.2

1.61.8

22.22.42.62.8

0 5 10 15 20 25 30 35

Vp (k

m/s

)

Eff pressure (MPa)

brineoilgas

G.12

1.8 2.4

1250

1300

1350

Gas Flood into Oil

Vp (km/s)

dept

h (m

)

Pressure

oil to gas

original oil

oil atincreased Pp

gas atincreased Pp




76

Brine Flood Example: Pore PressureDecrease and Change From Oil to Brine

Calculated using Gassmann from dry lab data from Troll (Blangy, 1992).Virgin condition taken as low frequency, oil saturated at Peff=25 MPa.Pore pressure drop to Peff=30 MPa, then fluid substitution to brine.Koil = 1., Kbrine = 2.2

1.8 2.4 3

1250

1300

1350

Brine Flood with Pressure Decline

Vp (km/s)

dept

h (m

)

original oil

oil atdecreased Pp

brine atdecreased Pp

frame effectdecreased Peff


1.61.8

22.22.42.62.8

0 5 10 15 20 25 30 35

Vp (k

m/s

)

Eff pressure (MPa)

brine

oil

gas



77

Calculated using Gassmann from dry lab data from Troll (Blangy, 1992).Koil = 1., Kbrine = 2.2

High Effective Pressure Stiffens the Rocks and Decreases Sensitivity to Pore Fluids

1.5 2 2.5 3

1250

1300

1350

Peff = 30 MPa

Vp (km/s)

dept

h (m

)

wateroil

gas

1.5 2 2.5 3

1250

1300

1350

Peff = 10 MPa

Vp (km/s)

dept

h (m

)

wateroil

gas



78

Velocity Dispersion Decreases ApparentSensitivity to Pore Fluids

From lab data on Troll sandstones (Blangy, 1992). Low frequency Vp calculated from lab, water-saturated, dispersive data using Gassmann. Illustrates that low frequency seismic might see larger differences than suggested by high frequency data.

1.5 2 2.5 3

1250

1300

1350

Vp

Vp (km/s)

dept

h (m

)

water

oil

lo f hi f

0 0.1 0.2 0.3

1250

1300

1350

Vp (brine) - Vp (oil)

delta Vp (km/s)

dept

h (m

)hi f lo f



79

Fluid Substitution in Anisotropic Rocks: Brown and Korringa’s Relations

where

effective elastic compliance tensor of dry rock

effective elastic compliance tensor of rock saturated with pore fluid

effective elastic compliance tensor of mineral

compressibility of pore fluid

compressibility of mineral material =

porosity

Sijkl(dry)

Sijkl(sat)

Sijkl0

ββββ fl

ββββ 0

φφφφ

Sααββααββααββααββ0

This is analogous to Gassmann’s relations. To apply it,one must measure enough velocities to extract the fulltensor of elastic constants. Then invert these for thecompliances, and apply the relation as shown.

Sijkl(dry) – Sijkl

(sat) =Sijαααααααα

(dry) – Sijαααααααα0 Sklαααααααα

(dry) – Sklαααααααα0

Sααααααααββ(dry) – Sααααααααββ

0 + β fl – β 0 φ



80

3000

2500

2000

1500

1000 500

0.3

0.1

0.03 0.01

0.00

320

0040

0050

0030

00

Vs sat

Vp

sat

AV

O p

aram

eter

²νν νν/

(1- νν νν

)2

3000

2500

2000

1500

1000 500

2000

4000

5000

3000

Vs sat

Vp

sat

P-P

Ref

lect

ivity

0.4

0.25

0.15

0.1

0.06

3000

2500

2000

1500

1000 500

2000

4000

5000

3000

Vs sat

Vp

sat

0.8

0.6

0.4

0.2

0

AV

O p

aram

eter

(Vp

sat -

Vp

dry)

/Vp

ave

3000

2500

2000

1500

1000 500

2000

4000

5000

3000

Vs sat

Vp

sat

0.1

0.05

0 -0.0

5

-0.1

AV

O p

aram

eter

gra

dien

t ter

m



81

Marion (1990) discovered a simple, semi-empirical way to solve the fluid substitution problem. The Hashin-Shtrikman bounds define the range ofvelocities possible for a given volume mix of two phases, either liquid or solid. The vertical position within the bounds, d/D, is a measure of the relative geometry of the two phases. For a given rock, the bounds can becomputed for any two pore phases, 0 and 1. If we assume that d/D remains constant with a change of fluids, then a measured velocity with one fluid will determine d/D, which can be used to predict the velocity relative to the bounds for any other pore phase.

Bounding Average Method (BAM)

05

10152025303540

0 0.2 0.4 0.6 0.8 1

Fluid 2 (oil)

Bul

k M

odul

us (G

Pa)

Porosity

d'

D'

H-S Upper bound (M 2 + )

H-S Lower bound (M 2 - )

05

10152025303540

0 0.2 0.4 0.6 0.8 1

Fluid 1 (water)B

ulk

Mod

ulus

(GPa

)

Porosity

H-S Upper bound (M 1 + )

H-S Lower bound (M 1 - )d

D



82

Velocity in Massilon sandstone saturated with parowax. Data from Wang(1988). Wax saturated velocities were predicted using BAM, from Wang'smeasured velocities in the dry rock and in wax (from Marion, 1990)

2800

3000

3200

3400

3600

3800

4000

4200

0 20 40 60 80 100 120 140

Massillon Light Sandstone

P-Ve

loci

ty (m

/s)

Temperature ( °°°°C)

measured parowax

BAMcalculatedparowax

measured dry

G.8

An Example of the Bam Method. The wax saturated velocities are predicted from the dry

rock velocities.



83

Velocity in dry and saturated Westerly granite. Data from Nur andSimmons (1969). Saturated velocities were predicted using BAM, from measured velocities in the dry rock (from Marion, 1990)

3000

3500

4000

4500

5000

5500

6000

6500

0 50 100 150 200 250 300

Westerley Granite

dry measuredwater sat. measuredBAM predicted

P-ve

loci

ty (m

/s)

Pressure (MPa)

HS+

HS-

G.9



84

Comparison of BAM and Gassmann predictions of saturated moduli in 10 different clay-free sandstones from measured dry velocities. BAM generally estimates the dispersion better than Gassmann (from Marion, 1990).

15000

20000

25000

30000

35000

40000

5 10 15 20 25

Bulk Modulus

water sat. measureddry measuredGassmann predictedBAM predicted

Bul

k M

odul

us (M

Pa)

Porosity (%)

-15

-10

-5

0

5

10

15

5 10 15 20 25

BAMGassmann

(Km

eas

- Kca

l)/K

mea

s (%

)

Porosity (%) G.10



85

bc

Ellipsoidal Models for Pore Deformation

Most deterministic models for effective moduli assume a specific idealized pore geometry in order to estimate the pore space compressibility:

The usual one is a 2-dimensional or 3-dimensional ellipsoidal inclusion or pore.

The quantity αααα = b/c is called the aspect ratio.

1

K φ=

1vpore

Žvpore

Žσ

Recall the general expression for the dry rock modulus:

1K dry

= 1K mineral

+ φK φ

Gassmann’s relation is a transformation, allowing us to predict how measured velocities areperturbed by changing the pore fluid. Now wediscuss a different approach in which we try to model the moduli “from scratch”.



86

Estimating the Dry Rock Modulus

Mathematicians have worked out in great detail the 3-D deformation field U, of an oblate spheroid (penny-shaped crack) under applied stress. For example, the displacement of the crack face is:

We can easily integrate to get the pore volume changeand the dry modulus:

bc

σσσσ

σσσσ

An externally applied compression tends to narrow thecrack, with the faces displacing toward each other.

U(r) = σc

K mineral

4 1 – ν 2

3π 1 – 2ν 1 – rc

2

1K dry

= 1K mineral

+ 16 1 – ν 2

9 1 – 2ν1

K mineral

Nc 3

Vbulk



87

"Crack density parameter"

Dry Rock Bulk Modulus

Modulus depends directly on crack density. Crackgeometry or stiffness must be specified to get adependence on porosity.

1

Kdry= 1

Kmineral+ 16 1 – νννν2

9 1 – 2νννν1

Kmineral

Nc3

Vbulk

1

Kdry= 1

Kmineral1 + 16 1 – νννν2

9 1 – 2ννννNc3

Vbulk

1

Kdry= 1

Kmineral1 + 16 1 – νννν2

9 1 – 2νννν ∈∈∈∈

∈∈∈∈ = NVbulk

c3

≈≈≈≈ φφφφαααα

34ππππ



88

Crack Density Parameter

In these and other theories we often encounter the quantity:

This is called the Crack Density Parameter, and has the interpretation of the number of cracks per unit volume.

Example: 2 cracks per small cell. Each crack about 2/3 the length of a cell.

εεεε = Nc3

Vbulk

L2c

v = L3

εεεε = cL

3 ≈≈≈≈ 0.07



89

Distribution of Aspect Ratios

Modulus depends on the number of cracks andtheir average lengths

An idealized ellipsoidal crack will close when theamount of deformation equals the original crackwidth:

solving gives:

We generally model rocks as having a distributionof cracks with different aspect ratios. As thepressure is increased, more and more of themclose, causing the rock to become stiffer.

U = b

1

Kdry= 1


9Kmineral 1 – 2ννννNc3Vbulk

≈≈≈≈ ααααKmineral

σσσσclose ≈≈≈≈ ααααKmineral3ππππ4

1 – 2νννν1 – νννν2



90

Kuster and Toksöz (1974) formulation based on long-wavelength, first order scattering theory (non self-consistent)

KKT

* – KmKm + 4

3 µµµµm

KKT* + 4

3µµµµm= xi Ki – Km PmiΣΣΣΣi = 1

N

µµµµKT* – µµµµm

µµµµm + ζζζζmµµµµKT

* + ζζζζm= xi µµµµi – µµµµm QmiΣΣΣΣi = 1

N

ζζζζ = µµµµ6

9K + 8µµµµK + 2µµµµ

shape Pmi Qmi

Spheres

Needles

Disks

Penny cracks

Coefficients P and Q for some specific shapes. The subscripts m and i refer to the background and inclusion materials. From Berryman (1995).

15 1 + 8µµµµm

4µµµµi + παπαπαπα µµµµm+2ββββm+ 2

Ki + 23 µµµµi + µµµµm

Ki + 43 µµµµi + παπαπαπαββββ m

Km + 43 µµµµm

Ki + 43µµµµm

Km + µµµµm + 13 µµµµi

Ki + µµµµm + 13 µµµµi

Km + 43 µµµµi

Ki + 43µµµµi

Km + 43µµµµi

Ki + 43µµµµi + παπαπαπαββββm

µµµµm + ζζζζmµµµµi + ζζζζm

15

4µµµµmµµµµm + µµµµi

+ 2 µµµµm + γγγγmµµµµi + γγγγm

+Ki + 4

3µµµµm

Ki + µµµµm + 13 µµµµi

µµµµm + ζζζζiµµµµi + ζζζζi

ββββ = µµµµ (3K+µµµµ)(3K+4µµµµ)

γγγγ = µµµµ (3K+µµµµ)

(3K+7µµµµ)


(9K+8µµµµ)(K+2µµµµ)



91

Self-Consistent EmbeddingApproximation

Walsh's expression for the moduli in terms of the porecompressibility is fairly general. However attempts toestimate the actual pore compressibility are often basedon single, isolated pores.

The self-consistent approach uses a single porein a medium with the effective modulus.

Solving for Kdry gives:

1

Kdry= 1


9Kmineral 1 – 2ννννNc3

Vbulk

1

Kdry= 1


9Kdry 1 – 2ννννNc3

Vbulk

Kdry = Kmineral 1 – 16 1 – νννν2

9 1 – 2ννννNc3

Vbulk



92

Self-Consistent Approximations

O’Connell and Budiansky (1974) model for medium with randomly oriented thin dry cracks

K and µµµµ are the bulk and shear moduli of the uncracked medium, νννν is the Poisson’s ratio, and εεεε is the crack density parameter. The calculations are simplified by the approximation:

Assumes small aspect ratios (αααα →→→→ 0).

µµµµSC*

µµµµ = 1 – 3245

1 – ννννSC* 5 – ννννSC

*

2 – ννννSC* εεεε

KSC*

K = 1 – 169

1 – ννννSC*2

1 – 2ννννSC* εεεε

εεεε = 4516

νννν – ννννSC* 2 – ννννSC

*

1 – ννννSC*2 10νννν – 3ννννννννSC

* – ννννSC* εεεε

ννννSC* ≈≈≈≈ νννν 1 – 16

9 εεεε



93

shape P*i Q*i

Spheres

Needles

Disks

Penny cracks

Coefficients P and Q for some specific shapes. The * and i refer to the unknown effective and inclusion materials. From Berryman (1995).

ββββ = µµµµ (3K+µµµµ)(3K+4µµµµ)

γγγγ = µµµµ (3K+µµµµ)

(3K+7µµµµ)


(9K+8µµµµ)(K+2µµµµ)

Self-Consistent Approximations

Berryman’s (1980) model for N-phase composites

coupled equations solved by simultaneous iteration

xi K i – K * P*i = 0ΣΣΣΣi = 1

N

xi µµµµ i – µµµµ * Q*i = 0ΣΣΣΣi = 1

N

K* + 43µµµµ*

Ki + 43µµµµ*

µµµµ* + ζζζζ*µµµµ i + ζζζζ*

K* + µµµµ* + 13 µµµµi

Ki + µµµµ* + 13 µµµµi

15

4µµµµ*µµµµ* + µµµµi

+ 2µµµµ* + γγγγ*µµµµi + γγγγ* +

Ki + 43 µµµµ*

Ki + µµµµ* + 13 µµµµi

K* + 43µµµµi

Ki + 43µµµµ i

µµµµ* + ζζζζ iµµµµ i + ζζζζi

K* + 43µµµµ i

Ki + 43 µµµµi + παπαπαπαββββ*

15 1 + 8µµµµ*

4µµµµi + παπαπαπα µµµµ*+2ββββ* + 2Ki + 2

3 µµµµ i + µµµµ*

Ki + 43 µµµµi + παπαπαπαββββ*



94

Comparison of Han's (1986) sandstone data with modelsof idealized pore shapes. At high pressure (40-50 MPa),there seems to be some equivalent pore shape that is

more compliant than any of the convex circular or spherical models.

0

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4

Shea

r Mod

ulus

(GPa

)

Equivalent Porosity

HS -

triangle

sphere

needle

HS +

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4

Bul

k M

odul

us (G

Pa)

Equivalent Porosity

HS -

triangle

sphere

needle

HS +

G.11



95

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100

Water-saturated Carbonates

interparticle porositymicro porositymoldic porosityintraframe porositydensely cemented

Velo

city

(m/s

)

porosity (percent)

spherical pores

tubular pores

crack-like pores(α = 0.1)

G14

Comparison of self-consistent elliptical crack modelswith carbonate data. The rocks with stiffer pore shapes are fit best by spherical pore models, whilethe rocks with thinner, more crack-like pores are fitbest by lower aspect ratio ellipsoids.

Data from Anselmetti and Eberli., 1997, in Carbonate Seismology, SEG.



96

dry rock:cracks make the rock softer.

saturated “drained” rockwhen squeezed slowly, fluid can escape and rock behaves as though it is dry.

saturated “undrained” rockwhen jacketed or squeezed quickly, rock is stiffer.



97

soft stiff

Cracks deform the most when tractions are applied to their faces--in both normal and shear.

If the cracks are oriented randomly and equally in all directions, then the rock will probably behave isotropically. But if there is a preferred orientation, then the rock will be anisotropic: stiffer in some directions than other.

Crack Anisotropy

"Ex" "Ex"

"Ey" "Ey"Isotropic Anisotropic

"Ex" = "Ey" "Ex" > "Ey"c1111 c2222

Rock Physics Laboratory - Gary Mavko

Fluid Flow

98

Viscosity

σxz = 2η εxz

εxz = 12

ŽUxŽz

σxz = ηŽUxŽz

Shear stress in the fluid is proportional to the fluidvelocity gradient.

V

Stationary

z

x

Fluid Velocity Profile

where ηηηη is the viscosity. Or in terms of the strainrate:

Units:

1 Poise = 1 dyne–seccm2 = 0.1 newton–sec

m2

water at 20oC: η ≈ .01 Poise ≈ 1 centiPoise


Fluid Flow

99

Darcy’s Law:

where

volumetric flow ratepermeability of the mediumviscosity of the fluidcross sectional area

Differential form:

Q = – κκκκηηηηA∆∆∆∆P

∆∆∆∆I

Q =κκκκ =ηηηη =A =

U = – κκκκηηηη grad P

where is the filtration velocity U

PU

∆∆∆∆ l

P + ∆∆∆∆P


Fluid Flow

100

Units

Darcy’s law:

Permeability κκκκ has dimensions of area, or m2 in SI units. But the more convenient and traditional unit is the Darcy.

1 Darcy ≅≅≅≅ 10–12 m2

In a water saturated rock with permeability of 1Darcy, a pressure gradient of 1 bar/cm gives a flow velocity of 1 cm/sec.

Q = – κκκκ

ηηηηA∆∆∆∆P∆∆∆∆I


Fluid Flow

101

Kozeny-Carman RelationThe most common permeability model is to assume that rocks have nice round pipes for pore fluids to flow.

Compare this with general Darcy’s law:

Combining the two gives the permeability of a circular pipe:

We can rewrite this permeability in terms of familiar rock paramters, giving the Kozeny-Carman equation:

where: φφφφ is the porosityS is the specific pore surface areaττττ is the tortuosity

d is a typical grain diameterB is a geometric factor

κκκκ = Bφφφφ3

ττττ2S2

κκκκ = ππππR4

8A = ππππR2

AR2

8

κκκκ = Bφφφφ3d2ττττ

Q = – ππππR4

8ηηηη∆∆∆∆P∆∆∆∆I

Q = – κκκκηηηηA∆∆∆∆P

∆∆∆∆I

The classical solution for laminar flow through a circular pipe gives:

strong scale dependence!

2R


Fluid Flow

102

Schematic porosity/permeability relationship in rocks from Bourbié,Coussy, Zinszner, 1987, Acoustics of Porous Media, Gulf Publishing Co.

10 -9

10 -7

10 -5

10 -3

10 -1

10 1

1 10

Perm

eabi

lity

(Dar

cy)

Porosity (%)

Claysand

shales

Silts

Micriticsandstones

Shalysandstones

Granularlimestones

Crystallinerocks

Tightsediments

Clean coarse-grained sandstones

H.1


Fluid Flow

103

Demonstration of Kozeny-Carman relation in sintered glass,from Bourbié, Coussy, and Zinszner, 1987,

Acoustics of Porous Media, Gulf Publishing Co.

1

10

100

1000

0 10 20 30 40 50

280 µµµµm spheres50 µµµµm spheres

κκ κκ /d

2

(x10

e-6)

Porosity (%)

Sintered Glass

H.2

Here we compare the permeability for two synthetic porous materials having very different grain sizes. When normalized by grain-size squared, the data fall on top of each other -- confirming the scale dependence.


Fluid Flow

104

Porosity/permeability relationship in Fontainebleau sandstone,from Bourbié, Coussy, and Zinszner, 1987,

Acoustics of Porous Media, Gulf Publishing Co.

H.3

1

10

100

1000

10000

2 4 6 8 10

Per

mea

bilit

y (m

D)

Porosity (%)2 30

n = 8

n = 3

κκκκ = a φφφφ n

A particularly systematic variation of permeability withporosity for Fontainebleau sandstone. Note that the slope increases at small porosity, indicating an exponent on porosity larger than the power of 3 predicted by the Kozeny-Carman relation.


Fluid Flow

105

0.00001

0.0001

0.001

0.01

0.1

Perm

eabi

lity

(mD

)

.05Porosity

.20.10

φφφφc ≈≈≈≈ 0.045

Kozeny-Carman Relation with Percolation

Hot-pressed Calcite (Bernabe et al, 1982), showing a good fit to the data using the Kozeny-Carmanrelation modified by a percolation porosity.

As porosity decreases from cementation and compaction, it is common to encounter a percolationthreshold where the remaining porosity is isolated ordisconnected. This porosity obviously does not contribute to permeability. Therefore, we suggest,purely heuristically, replacing giving

φφφφ →→→→ (φφφφ – φφφφc)

κκκκ = B (φφφφ – φφφφc)3 d2

H.4

κ = B (φ – .045)3 d2


Fluid Flow

106

Fused Glass Beads (Winkler, 1993)

0.0001

0.001

0.01

0.1

1

10

100

Perm

eabi

lity

/ D2 200 micron

Porosity

100 micron

50 micron

.05 .50.10

φφφφc ≈≈≈≈ 0.035

H.5

κ = B (φ – .035)3 d2


Fluid Flow

107

Fontainebleau Sandstone (Bourbié et al, 1987)

1

10

100

1000

10000

Perm

eabi

lity

(mD

)

Porosity.02 .30.05 .10

φφφφc ≈≈≈≈ 0.025

H.6

Here we show the same Fontainebleau sandstone dataas before with the Kozeny-Carman relation modified by a percolation porosity of 2.5%. This accounts for the increased slope at low porosities, while retaining theexponent of 3.

κ = B (φ – .025)3 d2


Fluid Flow

108

Diffusion

The stress-strain law for a fluid (Hooke’s law) is

which can be written as

combining with Darcy’s law:

gives the classical diffusion equation:

where D is the diffusivity

εεεεαααααααα = 1KP

∇∇∇∇2P = –ηηηηκκκκK

∂∂∂∂P∂∂∂∂t

∇∇∇∇2P = –1D

∂∂∂∂P∂∂∂∂t

∇∇∇∇••••U = 1K

∂∂∂∂P∂∂∂∂t

U = –κκκκηηηη ∇∇∇∇P


Fluid Flow

109

Examples of Diffusion Behavior

1-D diffusion from an initial pressure pulse

Standard result:

P = P0δδδδ x

P x,t = P0

4ππππDte

x2

–4Dt = P04ππππDt

eττττ–t

Characteristic time scale

ττττ = x2

4D


Fluid Flow

110

Examples of Diffusion Behavior

Sinusoidal pressure disturbance

Disturbance decays approximately as

ττττd = λλλλ2

4D

λλλλ


Velocity Dispersion and Q

111

(1) Seismic velocities almost always increase with frequency, and

(2) Seismic waves are always attenuated as theytravel through rocks.

These two observations are usually intimately related. Both usually increase from dry to fluid saturated conditions, and both usually decrease with increasing effective pressure.

These effects complicate the comparison of laboratory and field data, but they also reveal details about the pore space and the pore fluids it contains.



112

In most rocks and sediments, the velocity tends toincrease with frequency. This is accompanied byattenuation. Attenuation tends to be highest in frequency range where velocity is increasing mostrapidly.

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.5

1

1.5

2

-5 0 5M

odul

us (M

/M0)

log( ωωωω/ωωωωr)

1/Q

M∆∆∆∆M

M0

M�

I.1



113

These plots (Winkler, 1985) show typical features of ultrasonic laboratoryvelocities in rocks. The low frequency Gassmann theory almost always

underestimates the saturated velocities relative to the dry velocities. The discrepancy is usually greatest at low effective pressure.

3

3.2

3.4

3.6

3.8

4

4.2

0 10 20 30 40 50

P-Ve

loci

ty (k

m/s

)


dry

saturatedBiot

(high f)

Gassmann(low f)

Brine

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

0 10 20 30 40 50

S-Ve

loci

ty (k

m/s

)


dry

saturated

Biot(high f)

Gassmann(low f)

Brine

I.2



114

The difference between dry and saturated velocities and the disagreementwith the low frequency Gassmann theory often increases with fluid

viscosity. Again the differences are greatest at low pressures.Data from Winkler (1985).

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 10 20 30 40 50

S-Ve

loci

ty (k

m/s

)


saturated

dry

Oil

Biot(high f)

Gassman(low f)

3

3.2

3.4

3.6

3.8

4

4.2

0 10 20 30 40 50

P-Ve

loci

ty (k

m/s

)


saturated

dry

Oil

Biot(high f) Gassmann

(low f)

I.3



115

Failure of Gassmann's theory to predict saturated ultrasonic velocitiesrelative to dry velocities. Navajo sandstone data from Coyner (1984).

4

4.2

4.4

4.6

4.8

5

5.2

0 20 40 60 80 100 120

P-Ve

loci

ty (k

m/s

)


SatDry

low frequency(Gassmann)

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

0 20 40 60 80 100 120

S-Ve

loci

ty (k

m/s

)


SatDry


I.4



116

E1

E2

η

ViscoelasticityWe have been talking about elastic materials where stress is proportional to strain.

volumetric

shear

general

Viscoelastic materials also depend on rate or history.

Maxwell model

Voigt model

Standard linear solid

σσσσ11 + σσσσ22 + σσσσ333 = K εεεε11 + εεεε22 + εεεε33

σσσσ ij = 2µµµµεεεε ij

εεεε ij =σσσσ ij2µµµµ +

σσσσ ij2ηηηη

σσσσ ij = 2ηηηηεεεε ij + 2µµµµεεεε ij

ηηηησσσσ ij + E1 + E2 σσσσ ij = E2 ηηηηεεεεij + E1εεεεij

σσσσ ij = λλλλδδδδ ijεεεεαααααααα + 2µµµµεεεεij



117

Wave Propagating in a Viscoelastic Solid

At any point, the stress and strain are out of phase:

The ratio of stress to strain is the complex modulus.

u x,t = u0exp –αααα ωωωω x exp i ωωωωt – kx

σσσσ = σσσσ0exp i ωωωωt – kx

εεεε = εεεε0exp i ωωωωt – kx – ϕϕϕϕ



118

Quality Factor “Q”

low Q: large dissipation high Q: small dissipation

Different views of Q:energy dissipated per wave cycle

peak strain energy of the wave

velocity

frequency

phase delay

amplitude loss per cycle

1Q = ∆∆∆∆W

2ππππW

1Q = ααααV

ππππf

1Q ≈≈≈≈ 1

ππππ ln u tu t + ττττ

1Q = tan–1 ϕϕϕϕ



119

frequency

spec

tra

(db) S(x1)

S(x2)

ln ratio

Spectral Ratio MethodWe can think of Q-1 as the fractional loss per wavelength or per oscillation. Therefore over a fixed distance there is a tendency for shorter wavelengths to attenuate more:

or

If we propagate the wave

Then we can compare the amplitudes at two different distances:

1Q ≈≈≈≈ ααααV

ππππf

αααα ≈≈≈≈ ππππfVQ

u = u0exp –ααααx

ln ux2

ux1= – αααα x2 – x1

ln ux2

ux1= – ππππf

QV x2 – x1



120

εεεε = εεεε0eiωωωωt

Standard Linear Solid

If we assume sinusoidal motion

Then we can write:

with the complex, frequency-dependent modulus

In the limits of low frequency and high frequency

σσσσ = σσσσ0eiωωωωt

M ωωωω = E2 E1 + iωηωηωηωη

E1 + E2 + iωηωηωηωη =M∞∞∞∞ M0 + i ωωωω

ωωωωrM0M∞∞∞∞

M∞∞∞∞ + i ωωωωωωωωr

M0M∞∞∞∞

M0 = E2E1

E1 + E2, ωωωω →→→→ 0

M∞∞∞∞ = E2 , ωωωω →→→→ ∞∞∞∞

Re M ωωωω =

M0M∞∞∞∞ 1 + ωωωωωωωωr

2

M∞∞∞∞ + ωωωωωωωωr

2M0

σσσσ 0 = M ωωωω εεεε0



121

Standard Linear Solid

1Q = Ml ωωωω

MR ωωωω=

ωωωωωωωωr

M0M∞∞∞∞ M∞∞∞∞ – M0

M0M∞∞∞∞ 1 + ωωωωωωωωr

2

Similarly, we can write Q as a function of frequency:

where

The maximum attenuation

occurs at

1Q = E2

E1 E1 + E2

ωωωωωωωωr

1 + ωωωωωωωωr

2

ωωωωr = E1 E1 + E2ηηηη

1Q max

= 12

E2E1 E1 + E2

1Q max

= 12

M∞∞∞∞ – M0M0M∞∞∞∞

≈≈≈≈ 12

∆∆∆∆MM

ωωωω = ωωωωr



122

Standard Linear Solid Model

Attenuation and velocity dispersion tend to be mostlocalized in frequency. Attenuation is largest wherevelocity is changing most rapidly with frequency.

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.5

1

1.5

2

-5 0 5M

odul

us (M

/M0)

log( ωωωω/ωωωωr)

1/Q

M∆∆∆∆M

M0

M�

I.5



123

Finally

V(ωωωω)V(ωωωω0)

= M0 + ∆∆∆∆MM0

= 1 + ∆∆∆∆MM0

Nearly Constant Q ModelLiu, et al. (1976) considered a model in which simple attenuation mechanisms are combined such that the attenuation is nearly a constant over a finite range of frequencies.

We can then write

which relates the velocity dispersion within the band of constant Q, to the value of Q and the frequency.

We can express as:

V(ωωωω)/V(ωωωω0)

Expanding for small and substituting in:

∆∆∆∆M /M

12

∆∆∆∆MM0

≈≈≈≈ 1ππππQ log ωωωω

ωωωω0

1Q ≈≈≈≈ ππππ

log ωωωωωωωω0

12

∆∆∆∆MM0

V ωωωωV ωωωω0

= 1 + 1ππππQ log ωωωω

ωωωω0



124

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.4

0.8

1.2

1.6

2

-6 -4 -2 0 2 4 6 8M

odul

uslog( )ω

1/Q

M

nearly constant

Nearly Constant Q Model

Attenuation is nearly constant over a finite range offrequencies. It is sometimes interpreted as a super-position of individual (Standard Linear Solid) attenuation peaks. The broadening of the attenuationpeak is accompanied by a broadening of the range offrequency where velocity increases.

I.6



125

As with the nearly constant Q model, we can simplify this for large Q, giving:

Constant Q Model

Kjartansson (1979) considered a model in which Q is strictly constant. In this case the complex modulus and Q are related by:

where

M ωωωω = M0iωωωωωωωω0

2γγγγ

1Q ≈≈≈≈ ππππ

log ωωωωωωωω0

12

∆∆∆∆MM0

γγγγ = 1

ππππarctan 1Q



126

1/Q

M

ω

log( )

Constant Q Model

Attenuation is constant for all frequencies, andvelocity always increases with frequency.

I.7



127

Fontainebleau Sandstone

Just as velocity increases with effective pressure, so does Q. The strong pressure dependence is a clue that cracks are important for the physical mechanism of attenuation. From Nathalie Lucet, 1989, Ph.D. dissertation, Univ. of Paris/IFP.

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 5 10 15 20 25 30 35 40

Velocity

BR-EBR-SUS-PUS-S

Velo

city

(m/s

)

Pressure (MPa)

0

50

100

150

200

0 5 10 15 20 25 30 35 40

Attenuation

BR-EBR-SUS-PUS-S

1000

/Q

Pressure (MPa)

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 50 100 150 200 250 300

Velocity Dispersion

ExtensionTorsionCQ ModelNCQ Model

Vbr/V

us

1000/Qbr I.8



128

Some values of Q in rocks and sediments, summarizedby Bourbié, Coussy, and Zinszner, 1987, Acoustics of

Porous Media, Gulf Publishing Co.

Location Type of rock Depth (m) Measurementfrequency (Hz)

Limon (Colorado) Pierre shale 0-225 50-450 32

Gulf Coast (30 kmsouth of Houston)

Loam/sand/claySands and shalesSandy clayClay/sand

0-33-30

30-150150-300

50-40050-40050-40050-400

218175136

Offshore-Lousiana (Pleistocene)Southeast TexasSoutheast TexasSoutheast Texas

Clay/sandSands and shalesSame but more sandySandbanks, silty shaleMostly shaleSand (23%) and claySand (20%) and clayLimestone and chalkSand (45%) and claySand (24%) and clay

1170-17701770-20702070-2850900-15601560-18001800-2100600-15601590-1755660-1320

>1020

Š125Š125Š125Š80Š80Š80Š80Š80

15-4040-70

67>273

2852

>2733041

>2732855

Beaufort Sea(Canada)

549-1193945-1311

125425

OffshoreBaltimore

Siliceous chalk

Siliceous chalk withporcellanite joints

278-442

442-582

5000-15000

5000-15000

68 onave.287on

ave.S an d b an k s, silty sh ale

McDonald et al.(1958)

Tullos and Reid(1969)

Hauge (1981)

Ganley, Kansewich(1980)

Golberg (1958)

from Carmichael (1984) and Goldberg (1985)



129

Dominant fluid mechanisms affecting velocity and Q

Gassmann Model• static, zero frequency limit• no viscous/inertial effects• uniform pore pressure

Biot "global flow" Model• viscous/inertial effects• average flow only• average relative motion of fluid

and solid lead to dispersion and attenuation

Squirt "local flow" Model• viscous/inertial effects• grain-scale fluid motion leads to

dispersion and attenuation• effectively stiffens the soft porosity• superimposed on Biot/Gassmann

Patchy Saturation Model• large scale patches of saturation• patch-scale diffusion leads to

dispersion and attenuation• microscale squirt can be

superimposed



130

Biot TheoryBiot developed a macroscopic theory to attempt to model the behavior of fluid-saturated poroelasticsystems.

His generalized form of Hooke’s law:

where λλλλ and µµµµ are the dry rock moduli, and the fluid pressure P is linearly related to the normal stresses (and not the shears) by a new constant ββββ. Similarly, the increment of fluid content ξξξξ in an elementary cell of solid is linearly related to the pore pressure and the solid volumetric strain εεεεαααααααα. These describe essentially the same mechanical problem as the Gassmann theory for coupling the fluid and solid.

The equations of motion are:

where ρρρρuw describes an inertial coupling between the solid and the fluid, and χχχχ is a dissipation term.

σσσσ ij = λλλλδδδδ ijεεεεαααααααα + 2µµµµεεεεij – ββββPδδδδ ij

ξξξξ = 1MP + ββββεεεεαααααααα

∂∂∂∂σσσσ ij∂∂∂∂xj

= ρρρρ ∂∂∂∂2ui∂∂∂∂t2 + ρρρρuw

∂∂∂∂2wi∂∂∂∂t2

∂∂∂∂P∂∂∂∂xi

= ρρρρuw∂∂∂∂2ui∂∂∂∂t2 + ρρρρw

∂∂∂∂2wi∂∂∂∂t2 + 1

χχχχ∂∂∂∂wi∂∂∂∂t



131

Biot Theory

The high frequency limiting velocities and , are given by [in Geertsma and Smit’s (1961) notation]:

VS∞∞∞∞

VP∞∞∞∞

VS∞∞∞∞ = µµµµfr

ρρρρ0 1 – φφφφ + φφφφρρρρ fl 1 – a–1

12

where

Kfr, µµµµfr bulk and shear moduli of dry rock frameK0 bulk modulus of mineralKfl effective bulk modulus of pore fluidφφφφ porosityρρρρ0 mineral densityρρρρfl fluid densityρρρρ low frequency density of saturated composite:

a-1 tortuosity

The low frequency limiting velocities are the same as predicted by Gassmann’s relations.

ρρρρ = 1 – φφφφ ρρρρ0 + φφφφρρρρ fl

VP∞∞∞∞ = 1ρρρρ0 1 – φφφφ + φφφφρρρρ fl 1 – a–1 Kfr + 4

3µµµµfr +φφφφ ρρρρ

ρρρρfla–1 + 1 – Kfr

K01 – Kfr

K0– 2φφφφa–1

1 – KfrK – φφφφ 1

K + φφφφK

12



132

Biot Theory

Biot and later Stoll (1977) considered adding frame attenuation on top of fluid effects. This plot by Stollshows attenuation vs. frequency for two extreme cases and for a typical sand showing how frame losses and fluid losses combine to control the overall response.

10 -8

10 -6

10 -4

10 -2

10 0

10 0 10 1 10 2 10 3 10 4 10 5

Effects of Two DifferentKinds of Energy Loss

no frame lossno fluid losscomplete sediment model

Atte

nuat

ion

(nep

ers/

met

er)

Frequency (Hz) I.9



133

Squirt Flow

When a rock is compressed by the stress of a passing wave, increments of pore pressure are induced in the pore fluid. At very low frequencies there is time for the pore pressure to equilibrate throughout the pore space, and the fluid effect is described by the Gassmanntheory.

However, at high frequencies we expect that unequal pore pressures are induced on the microscale of individual pores--larger increments in the soft, crack-like porosity and smaller increments in the stiffer, equi-dimensional pores. If these do not equilibrate, the rock will be stiffer, and the velocities will be faster, than at low frequencies when they do equilibrate.

This frequency-dependent distribution of pore pressure leads to velocity dispersion, and the tendency for the fluid to flow and adjust leads to attenuation.



134

Estimating the High Frequency Squirt Flow Modulus

We compute the high frequency bulk modulus in 2 steps:1. the unrelaxed bulk modulus of the “wetted” frame where liquid is trapped in the thinnest cracks and the remaining spaceis dry, is given by

pore pressure in the ith thin crack:

Combining gives:

1

Khigh f– 1

Kmineral= 1

Kfluid– 1

Kmineralφφφφi

∂∂∂∂Pi∂∂∂∂σσσσΣΣΣΣ

low P

high P

∂∂∂∂Pi∂∂∂∂σσσσ ≈≈≈≈ 1

1 + 1Kfluid

– 1Kmineral

φφφφi /∂∂∂∂φφφφi /∂∂∂∂σσσσ dry

1Khigh f

≈≈≈≈ 1Kdry high P

+ 1Kfluid

– 1Kmineral

φφφφsoft σσσσ + ...

P4

P1

P2

P3

So trapping water in the thinnest cracks is approximately the same as closing the cracks under high pressure.

2. Finally the remaining pore space is saturated usingGassmann with Khigh f used as the “dry” rock modulus.



135

Squirt Flow

The shear modulus:

Comparing with the bulk modulus:

1

µµµµhigh f≈≈≈≈ 1

µµµµdry– 4

15∂∂∂∂φφφφi /∂∂∂∂σσσσ dry

1 + 1Kfluid

– 1Kmineral

φφφφi /∂∂∂∂φφφφi /∂∂∂∂σσσσ dryΣΣΣΣ

low P

high P

1µµµµhigh f

– 1µµµµdry

≈≈≈≈ 415

1Khigh f

– 1Kdry



136

Constructing the Unrelaxed Moduli

Confining Pressure

Bul

k M

odul

us

dry data

Biot / Gassmann

Shea

r Mod

ulus

dry data

Biot / Gassmann

saturated data

unrelaxed frame

unrelaxed frame

saturated data

I.10



137

4

4.2

4.4

4.6

4.8

5

5.2

0 20 40 60 80 100 120

Navajo

P-Ve

loci

ty (k

m/s

)


SatDry

high frequency(Mavko and Jizba)

high frequency(Biot)


2.6

2.8

3

3.2

3.4

0 20 40 60 80 100 120

Navajo

S-Ve

loci

ty (k

m/s

)


Sat

Dryhigh frequency

(Mavko and Jizba)

high frequency(Biot)


I.11

In these plots, the dry data are taken as inputs. The ultrasonic water-saturated data are compared with predictions byGassmann, the high frequency Biot limit, and the high frequency squirt limit.



138

3

3.1

3.2

3.3

3.4

3.5

0 20 40 60 80 100 120

Westerly

S-Ve

loci

ty (k

m/s

)


Sat

Dry


Biot and Gassmann

4.8

5

5.2

5.4

5.6

5.8

6

6.2

0 20 40 60 80 100 120

WesterlyP-

Velo

city

(km

/s)


Sat

Dry


Biot and Gassmann

I.12



139

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.05 0.1 0.15 0.2 0.25 0.3

Biot

local flow

1µµµµsat

– 1µµµµdry

1Ksat

– 1Kdry σσσσ

– 1Ksat

– 1Kdry highσσσσ

I.13



140

Most physical mechanisms of dispersion and attenuationhave a characteristic frequency where attenuation is largest and velocity is changing most rapidly with frequency. It also separates the low frequency “relaxed”behavior from the high frequency “unrelaxed” behavior.

It is very difficult to predict the characteristic frequencyvery accurately, because it depends on idealized modelassumptions, and details of the rock microstructure thatare not well known.

Nevertheless, here are some rough estimates:

Biot:

patchy saturation:

viscous shear in crack:

squirt:

fbiot = φηφηφηφη2ππππρρρρfk

fvisc = kKfl2ηηηη

fsquirt = K0αααα3

ηηηη

fvisc. crack = αµαµαµαµ2πηπηπηπη



141

Biot Theory

Compiled in Bourbié, Coussy, and Zinszner,1987, Acoustics of Porous Media, Gulf Publishing Co.

Parameter Porosity(%)

Permeability(mD)

Characteristic frequency

Sample Water(h = 1cP)

(4)

Normal oil(h = 10-50

cP) (4)

Heavy oil(h = 100-

500 cP) (4)

Fontainebleausandstone (1)

5 0.1 80 MHz 800-4000MHz

8-40 GHa

Fontainebleausandstone (1)

20 1000 30 kHz 300-1500kHz

3-15 MHz

Tight sand (2) 8 0.02 1 GHz 10-50 GHz 100-500GHz

CordovaCreamlimestone (2)

24.5 9 4.5 MHz 45-230MHz

450-300MHz

Sintered glass 28.3 1000 42 kHz 420-2100kHz

4.2-21 MHz

(1) Bourbié and Zinszner (1985) (2) Carmichael (1982) (3) Plona and Johnson (1980)

(4) Viscosity η is expressed in centipoises (1 cP = 1 mPa. s).



142

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.05 0.1 0.15 0.2 0.25 0.3

Bio

t Dis

pers

ion

of V

p

Porosity

Permeability vs. Seismic Dispersion?

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.1 0.2 0.3 0.4 0.5 0.6

Building SandstoneP-SandstoneGulf SandstoneClean Sandstone

Bio

t Dis

pers

ion

of V

p

Clay Content I.14



143

0

0.02

0.04

0.06

0.08

0.1

0 0.05 0.1 0.15 0.2 0.25 0.3

Non

-Bio

t Dis

pers

ion

of V

p

Porosity

Permeability vs. Seismic Dispersion?

0

0.02

0.04

0.06

0.08

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

Building SandstoneP-SandstoneGulf SandstoneClean Sandstone

Non

-Bio

t Dis

pers

ion

of V

p

Clay Content I.15


Partial Saturation

144

Knight and Nolen-Hoeksema (GRL, 1990) found saturationhysteresis at ultrasonic frequencies.

We know now that velocities depend, not just on saturation, but also on the scales at which the phases are mixed. The curve labeled “imbibition” is typical when phases are mixed at a fine scale. The curve labeled “drainage” is typical when the phases are mixed at a coarse scale -- which we call “patchy.”

2

2.5

3

3.5

4

4.5

5

0 0.2 0.4 0.6 0.8 1

Vp (k

m/s

)

Sw (fraction)

Drainage

Imbibition

K.1


Partial Saturation

145

2.15

2.2

2.25

2.3

2.35

2.4

2.45

0 0.2 0.4 0.6 0.8 1

Vp

(km

/s)

Oil Saturation

sandstoneporosity = 30%

patchy

homogeneous


Partial Saturation

146

Murphy (GRL, 1984) found that the hysteresis anddisagreement with Gassmann is primarily a problem of

high frequencies -- partially saturated dispersion.

Velocity vs. Saturationat Different Frequencies

2

2.5

3

3.5

4

4.5

5

5.5

6

0 0.2 0.4 0.6 0.8 1

Velo

city

(km

/s)

Water Saturation

200 kHz

1-2 kHzP

S

CQ

BGD

BGD

CQ

1-2 kHz

200 kHz

K.2


Partial Saturation

147

Murphy (GRL, 1984) found that ultrasonic velocitiesincrease with saturation much faster than predicted by

the low frequency Gassmann theory.

Ultrasonic Velocity vs. Pressureat Different Saturations

1500

2000

2500

3000

3500

4000

4500

5000

5500

0 10 20 30 40 50

Gro

up V

eloc

ity (m

/s)


100%

70%0%P

S

SR 6521.6Frequency 800kHz

K.3


Partial Saturation

148

Endres and Knight (The Log Analyst, 1989) modeled different microdistributions of pore fluids and gas in the stiff and soft portions of the pore space. They concluded that the scale and distribution of fluids influence velocities.

a.

b.

c.

d.

Increasing water saturation

1600

1700

1800

1900

2000

2100

2200

2300

0 0.2 0.4 0.6 0.8 1

P-W

ave

Velo

city

(m/s

)

a

b

c

d

K.4


Partial Saturation

149

Patchy Saturation

High frequency responseIsolated patches—somestiff, some softer

Overall a highereffective velocity

Very low frequency response

Gassmann behaviorwith a single "effectivefluid"

Overall the softest,lowest velocity

Critical frequency:

fvisc = kKfl2ηηηη

1Keff.fl

= SiKi

ΣΣΣΣi


Partial Saturation

150

Patchy Diffusion ScalesCharacteristic diffusion time for a pressure disturbancewith length scale L to relax:

Inverting this, we can find the characteristic diffusion length over which pressure differences can relax at seismic frequency f

1f = ττττ ≈≈≈≈ L2

4D

L ≈≈≈≈ 4κκκκKηηηηf

Hz

f L10 Hz 1000 mD

100 mD10 mD1 mD

1000 mD100 mD10 mD1 mD

1000 mD100 mD10 mD1 mD

.1 mD

1 m.3 m.1 m

.03 m.3 m.1 m

.03 m

.01 m

.01 m.003 m.001 m

.0003 m

.0001 m

κκκκ

105

100 Hz

D is the hydraulic diffusivity, K is the fluid bulk modulus, κκκκ is the permeability, and ηηηη is the viscosity.


Partial Saturation

151

Estaillades LimestoneE

wav

e Ve

loci

ty (m

/s)

Water Saturation

DryingDrying (Reuss average)Drying (Voigt average)Gassmann formula

DepressurizationDepressurization (Reuss average)Depressurization (Voigt average)

K.5

Thierry Cadoret studied velocity vs. saturation using the resonant bar and found the coarse-scale and fine-scale behavior.


Partial Saturation

152

Estaillades Limestone

2500

2550

2600

2650

2700

0.5 0.6 0.7 0.8 0.9 1

DryingDepressurizationE

wav

e Ve

loci

ty (m

/s)

Water Saturation

0

5

10

15

20

25

30

0.5 0.6 0.7 0.8 0.9 1

DryingNCQ Model

1000

/Qe

Water Saturation K.6

Cadoret’s velocity and attenuation vs. saturation. The fine scale distribution gives relaxed viscoelasticbehavior, and the coarse scale gives unrelaxed. Therefore, we expect the largest attenuation when the velocity dispersion is largest. Hence, we get the important result that P-wave attenuation in a partially saturated rock can be much larger than in the dry or fully saturated case.


Flow Simulation and Saturation Scales

153

2.15

2.2

2.25

2.3

2.35

2.4

2.45

0 0.2 0.4 0.6 0.8 1

Vp

(km

/s)

Oil Saturation

sandstoneporosity = 30%

patchy

homogeneous

The problem that we address is the nonuniqueresponse of seismic velocity to fluid saturation.What are the physical conditions that cause patchybehavior? When do we use the patchy model andwhen do we use the homogeneous model?

Our approach is to use flow simulation to study theparameters that control fluid distributions at a fine scale.

study by Madhumita Sengupta and G. Mavko



154

Porosity and permeability models for flow simulation



155

We will consider two important cases: water flood into oil, and gas flood into oil.

The parameters that we consider are:• relative permeability• wettability• density contrast• permeability heterogeneity• capillary pressure



156

Relative Permeability Curves for Oil and Water

Water Injection in Oil



157

SaturationsSaturations obtained from flow simulations using the obtained from flow simulations using the dashed (top) and solid (bottom) relative permeability dashed (top) and solid (bottom) relative permeability curves. The irreduciblecurves. The irreducible saturationssaturations are critical controlsare critical controlson the saturation extremes.on the saturation extremes.



158

The patchy and uniform saturation curves are upperand lower bounds. They describe the range of velocitysignatures that we can achieve by mixing the endmembers. Finite irreducible saturations drastically narrow the range of uncertainty.



159

Water saturation map and histogramof saturation in an oil wet rock

Wettability

The saturation distributiondepends on the wettabilityof the rock.

Most sandstone reservoirsare water wet and most carbonate reservoirsare oil wet.



160

Wettability

Wettability tends to determine whether the velocitiesfall high or low in the allowable range.

** ****

** ******

Oil WetOil Wet(Drainage)(Drainage)

Water WetWater Wet(Imbibition)(Imbibition)

Sw

Vp



161

Low Mobility Ratio

20 40 60 80 100

20

400.2

0.4

0.6

0.8

0 0.5 10

500

1000

1500

2000

2500

High Mobility Ratio

20 40 60 80 100

20

400.2

0.4

0.6

0.8

0 0.5 10

500

1000

1500

Mobility Ratios



162

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 13050

3100

3150

3200

3250

3300

Water Saturation

VP

High MR

Low MR

SSww

VVpp

Low MRLow MR

High MRHigh MR



163

Summary

• The uniform saturation model is good enough for waterflood oil-water cases.

• Exceptions: when the irreducible oil is very low in an oil wet rock.

• The main control is the finite irreduciblesaturations.



164

Gas Injection Into Oil



165

Gas InjectionEffect of Mobility

(a) Low MobilityRatio

20 40 60 80 100

20

40

0

0.2

0.4

0.6

0 0.5 10

1000

2000

3000

(b) High MobilityRatio

20 40 60 80 100

20

40

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.80

500

1000

1500

2000

2500



166

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12800

2850

2900

2950

3000

3050

3100

3150

Vp

So

High Mobility Ratio

Low Mobility Ratio



167

Heterogeneity of Perm

99

99.5

100

100.5

101

0 0.5 1

20406080100120

0

0.2

0.4

0.6

0 0.5 1

200

400

600

800

1000

0

0.2

0.4

0.6

0 0.5 1

0

0.2

0.4

0.6

Perm Models Saturation

SatnHistograms



168

0 0.2 0.4 0.6 0.8 12800

2850

2900

2950

3000

3050

3100

3150

Large ScaleHeterogeneities

Small ScaleHeterogeneities

Vp

So



169

Summary

• When there is gas in the reservoir it is quite easy to get patchy saturation.

• This patchy saturation is controlled by the irreducible saturations.



170

Conclusions

• Reservoirs with gas are very likely to show patchy behavior.

• The uniform saturation model may be good enough for reservoirs with only oil and water

• The main mechanism that causes patchy behavior at the field scale is gravity.


North Sea Turbidite

171


Shaly Sands

172

Velocity-porosity relationship in clastic sediments and rocks. Datafrom Hamilton (1956), Yin et al. (1988), Han et al. (1986). Compiled

by Marion, D., 1990, Ph.D. dissertation, Stanford Univ.

1000

2000

3000

4000

5000

6000


P-Ve

loci

ty (m

/s)

Porosity (%)

Voigt Avg.

Wood's Relation

L.1


Shaly Sands

173

We observe that the clastic sand-clay system is divided into two distinct domains, separated by a critical porosityφφφφc. Above φφφφc, the sediments are suspensions. Below φφφφc , the sediments are load-bearing.

Critical Porosity

load-bearing suspension

1000

2000

3000

4000

5000

6000


P-Ve

loci

ty (m

/s)

Porosity (%)

Voigt Avg.

Wood's Relation

L.1


Shaly Sands

174

Critical Porosity

Traditionally, bounding methods have been considered not very useful for quantitative predictions of velocity-porosity relationships, because the upper and lower bounds are so far apart when the end memebers are pure quartz and pure water.

However, the separation into two domains above and below the critical porosity helps us to recognize that the bounds are in fact useful for predictive purposes.

• φφφφ > φφφφc, fluid-bearing suspensions. In the suspension domain the velocities are described quite well by theReuss average (iso-stress condition).

• φφφφ < φφφφc, load-bearing frame. Here the situation appears to be more complicated. But again, there is a relatively simple pattern, and we will see that the Voigt average is useful.


Shaly Sands

175

The first thing to note is that the clean (clay free) materials fall along a remarkably narrow trend. These range from very low porosity, highly consolidatedsandstones, to high porosity loose sand.

(Data from Yin et al., 1988; Han et al., 1986. Compiled and plotted by Marion, D., 1990, Ph.D. dissertation, Stanford University.

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100suspensionssandclay-free sandstone

P-Ve

loci

ty (m

/s)

Porosity (%)

Voigt Avg.

Wood's Relation

L.2


Shaly Sands

176

Amos Nur discovered that this narrow trend can be described accurately with a modified Voigt bound. Recall that bounds give a way to use the properties of the “pure” end members to predict the properties in between. The trick here is to recognize that the critical porosity marks the limits of the domain of consolidated sediments, and redefine the right end member to be the suspension of solids and fluids at the critical porosity.

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100

P-Ve

loci

ty (m

/s)

Porosity (%)

Voigt

Reuss

Critical Porosity

Solid line: SaturatedDash line: Dry

L.3


Shaly Sands

177

The Modified Voigt Bound

Velocity in rocks

The usual Voigt estimate of modulus

Modified Voigt estimate of modulus

VP = Mρρρρ

φφφφ = φφφφφφφφc

0 ≤≤≤≤ φφφφ ≤≤≤≤ φφφφc 0 ≤≤≤≤ φφφφ ≤≤≤≤ 1.0

ρρρρ = 1 – φφφφ ρρρρmineral + φφφφρρρρfluid

M = 1 – φφφφ Mmineral + φφφφMfluid

M = 1 – φφφφ Mmineral + φφφφMcritical "mush"


Shaly Sands

178

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

SandstonesWater saturated

ρρ ρρVp2 (n

orm

aliz

ed)

porosity

φcReuss

average

approximatedata trend

suspensionload-bearing

L.4

Example of critical porosity behavior in sandstones.


Shaly Sands

179

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100

Water saturated carbonates


Velo

city

(m/s

)

porosity (percent)

average trend

Reuss average suspension line

L.29

Data from Anselmetti and Eberli, 1997, in Carbonate Seismology, SEG.


Shaly Sands

180

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100


Velo

city

(m/s

)

porosity (percent)

Intraframe porosity

average trend

above averagepore stiffness

(f) L.30

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100


Velo

city

(m/s

)

porosity (percent)

Moldic porosity

average trend


(e) L.31

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100


Velo

city

(m/s

)

porosity (percent)

micro porosity

average trend

below averagepore stiffness

(d) L.32

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100


Velo

city

(m/s

)

porosity (percent)

Interparticle, intercrystalline porosity

average trend

averagepore stiffness

(c) L.33

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100


Velo

city

(m/s

)porosity (percent)

densely cemented,low porosity

average trend


(b) L.34

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100

Water saturated carbonates


Velo

city

(m/s

)

porosity (percent)

average trend

Reuss average suspension line

(a)

L.29


Shaly Sands

181

L.5

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8

Chalks

Nor

mal

ized

M M

odul

us

Porosity

Calcite

Brevik

Urmos and Wilkens

Reuss Bound


Shaly Sands

182

L.6

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

Glass Foam

Nor

mal

ized

M M

odul

us

Porosity

Glass


Shaly Sands

183

Effects of Clay

Han (1986, Ph.D. dissertation, Stanford University)studied the effects of porosity and clay on 80 sandstonesamples represented here.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Building SandstoneP-SandstoneGulf SandstoneClean SandstoneTight Gas Sandstone

Cla

y C

onte

nt

PorosityL.7


Shaly Sands

184

Han (1986) found the usual result: velocities tend to decrease with porosity, but with a lot of scatter about the regressions when clay

is present (water saturated).

L.8

3

3.5

4

4.5

5

5.5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35


Vp (k

m/s

)

Porosity

1.5

2

2.5

3

3.5

4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35


Vs (k

m/s

)

Porosity


Shaly Sands

185

Han’s Relations (40 MPa)Clean sandstones (10 samples)

Clay-bearing sandstones (70 samples)

Ignoring the clay

Including a clay term

R = correlation coefficient; % = RMS

VP = 6.08 – 8.06φφφφVS = 4.06 – 6.28φφφφ

VP = 5.02 – 5.63φφφφVS = 3.03 – 3.78φφφφ

VP = 5.59 – 6.93φφφφ – 2.18CVS = 3.52 – 4.91φφφφ – 1.89C

VP = 5.41 – 6.35φφφφ – 2.87CVS = 3.57 – 4.57φφφφ – 1.83C

R = 0.99 2.1%R = 0.99 1.6%

R = 0.80 7.0%R = 0.70 10%

R = 0.98 2.1%R = 0.95 4.3%

R = 0.90R = 0.90

dry

wat

er s

atur

ated


Shaly Sands

186

22.5

33.5

44.5

55.5

6

0 0.1 0.2 0.3 0.4

Shaly sandstones

0-10%10-20%20-30%30-40%

Vp (k

m/s

)

porosity

c = .05.15.25.35

clay volume

Han’s water-saturated ultrasonic velocity data at 40 MPa compared with his empirical

relations evaluated at four different clay fractions.

Han’s empirical relations between ultrasonic Vp and Vs in km/s with porosity and clay volume fractions.

Clean Sandstones (determined from 10 samples) Water saturated40 MPa Vp = 6.08 - 8.06φφφφ Vs = 4.06 - 6.28φφφφ

Shaly Sandstones (determined from 70 samples)

Water saturated40 MPa Vp = 5.59 - 6.93φφφφ - 2.18C Vs = 3.52 - 4.91φφφφ - 1.89C30 MPa Vp = 5.55 - 6.96φφφφ - 2.18C Vs = 3.47 - 4.84φφφφ - 1.87C 20 MPa Vp = 5.49 - 6.94φφφφ - 2.17C Vs = 3.39 - 4.73φφφφ - 1.81C10 MPa Vp = 5.39 - 7.08φφφφ - 2.13C Vs = 3.29 - 4.73φφφφ - 1.74C5 MPa Vp = 5.26 - 7.08φφφφ - 2.02C Vs = 3.16 - 4.77φφφφ - 1.64C

Dry40 MPa Vp = 5.41 - 6.35φφφφ - 2.87C Vs = 3.57 - 4.57φφφφ - 1.83C

L.9


Shaly Sands

187

The critical porosity, modified Voigt bound incorporating Han's clay correction.

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Shaly Sandstone Data Before Clay-Corrections

modeldata (clean-sandstones)data (shaly-sandstones)

Velo

city

(km

/s)

Porosity

P-Velocity

S-Velocity

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Clay-Corrections With Empirical Relation

data (clean-sandstones)data (shaly-sandstones)critical por. models for clean sandstonecritical por. models for shaly sandstone

Velo

city

(km

/s)

Porosity

P-Velocity

S-Velocity

L.12


Shaly Sands

188

Porosity vs. clay weight fraction at various confining pressures. FromDominique Marion, 1990, Ph.D. dissertation, Stanford University. Data

are from Yin, et al., 1988.

Sand, shaley sand Shale, sandy shale

0

10

20

30

40

50

60

70

0 20 40 60 80 100

Poro

sity

(%)

Clay Content by Weight (%)

0 MPa

40 MPa

20 MPa

30 MPa

10 MPa

50 MPa

0

10

20

30

40

50

60

70

0 20 40 60 80 100

Poro

sity

(%)


0 MPa

40 MPa

20 MPa

30 MPa

10 MPa

50 MPa

L.13


Shaly Sands

189

Velocity vs. clay weight fraction at various confining pressures. FromDominique Marion, 1990, Ph.D. dissertation, Stanford University. Data

are from Yin, et al., 1988.

Sand, shaley sand Shale, sandy shale

1500

2000

2500

3000

3500

0 20 40 60 80 100

P-Ve

loci

ty (m

/s)


40 MPa

20 MPa30 MPa

10 MPa

50 MPa

1500

2000

2500

3000

3500

0 20 40 60 80 100

P-Ve

loci

ty (m

/s)


40 MPa

20 MPa30 MPa

10 MPa

50 MPa

L.14


Shaly Sands

190

Influence of clay content on velocity-porosity relationship at a constant confining pressure (50 MPa). Distinct trends for shalysand and for shale are schematically superposed on experimental data on sand-clay mixture. From Dominique Marion, 1990, Ph.D. dissertation, Stanford University. Data are from Yin, et al., 1988, and Han, 1986.

2000

2500

3000

3500

0.1 0.15 0.2 0.25 0.3 0.35

P-Ve

loci

ty (m

/s)

Porosity

shaley sands

sandshale

sandy shales

50 MPa

c = øs

L.15


Shaly Sands

191

Gulf of Mexico Well, Herron et al, 1992, SPE 24726

40

20

0

60

3

2

1

Clay (vol%)

Poro

sity

(vol

%)

Vp (k

m/s

)3780-4800 ft 3780-4800 ft

Poro

sity

(vol

%)

Clay (vol%)

Vp (k

m/s

)

3

2

1

60

40

40 40

20

20 200

00

4800-5895 ft4800-5895 ft

L.17


Shaly Sands

192

Amoco's Well in the Hastings Field (On-Shore Gulf Coast)

Density vs. Neutron Porosity Poorly Consilidated Shaly Sands

Laminar Clay Model

2.30

Marion Model

Increasing Clay Content

nphi

rhob

(g/c

m )

2.00

2.10

2.20

2.40

2.50

2.60

2.700.00 0.10 0.20 0.30 0.40 0.50

3

L.18


Shaly Sands

193

Schlumberger, 1989

Density Porosity vs. Neutron Porosity in Shaly Sands

Sho

0.5

0.4

0.3

0.1

Q

QuartzPoint

0.1

0.2

0.3 0.4 0.5

T o D r y C l a y P o i n t

T o W a t e r P o

GasSand

Sd

C

ClSh

0.2

φφφφN

φφφφD

T o W a te r P o in t

C le an W a te r S a nd s

A

B

L.19


Shaly Sands

194

Yin’s laboratory measurements on sand-clay mixtures.

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

0 20 40 60 80 100

Permeability (Gas) vs. Clay Content

Perm

eabi

lity

(mD

)

Clay Content (% by weight)

0 MPa

10 MPa

20 MPa30 MPa

40 MPa50 MPa

shaly sand

sandy shale

L.20


Shaly Sands

195

Yin’s laboratory measurements on sand-clay mixtures.

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Permeability (Gas) vs. Porosity

Perm

eabi

lity

(mD

)

Porosity

0 MPa

30 MPa

10 MPa

50 MPa 40 MPa

20 MPa

0%

5%

10%

15%20%

25%

30%

40%

50%

65%

85%

100%

% clay content by weight

L.21


Shaly Sands

196

Permeability vs. porosity data in Gulf-Coast sandstones reflect the primary influence of clay content on both permeability and porosity.Kozeny-Carman relations for pure sand and pure shale are also shown (dashed lines) to illustrate the effect of porosity on permeability. FromDominique Marion, 1990, Ph.D. dissertation, Stanford University.

0.0001

0.01

1

100

10 4

10 6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Perm

eabi

lity

(md)

Porosity

K-C sand

K-C shale

L.22


Shaly Sands

197

Yin's laboratory measurements on sand-clay mixtures.

1800

2000

2200

2400

2600

2800

1 10 100 1000 10 4

P-Velocity vs. Permeability (Gas)

P-Ve

loci

ty (m

/s)

Permeability (md)

0% (pure sand)

5%

10%15%

20%

25%

30%40%

50%

65%

85%

100% (pure clay)% clay content by weight

L.23


Shaly Sands

198

L.36

0

1000

2000

3000

4000

5000

6000

0 0.1 0.2 0.3 0.4 0.5

Varied Velocity-Porosity Trends

Porosity

Gulf of Mexico (Han)

Vp Troll

Oseberg

Cementing Trend

Han’s large data set spans a large range of depths and clearly shows the steep cementing trend, which would be favorable for mapping velocity (or impedance) to porosity. Other data sets from the Troll and Oseberg indicate much shallower trends.


Shaly Sands

199

0

1000

2000

3000

4000

5000

6000

0 0.1 0.2 0.3 0.4 0.5

Cementing vs. Sorting Trends

Porosity

Troll

Gulf of Mexico (Han)

Oseberg

Vp

Reuss Bound(Deposition)

Cementing Trend

SortingTrend

The slope of the velocity-porosity trend is controlled by the geologic process that controls variations in porosity. If porosity is controlled by diagenesis and cementing, we expect a steep slope – described well by a modified upper bound. If it is controlled by sorting and clay content (depositional) then we expect a shallower trend – described well by a modified lower bound.


Shaly Sands

200

Generalized Sandstone Model

L.36

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5

Cementing vs. Sorting Trends

Vp

Porosity

clean cementing trend

Suspension Line(Reuss Bound)

sorting trend

New Deposition

Mineral point


Shaly Sands

201

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

North Sea Clean sands

shallow oil sand deeper water sand

Vp

Total Porosity

increasing cement

Suspension Line

poor sorting

• all zones converted to brine• only clean sand, Vsh <.05

L.37


Shaly Sands

202

L.37

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

North SeaClean vs. Shaly Sands

2508-2545 m, vsh<.052508-2545 m, Vsh>.32701-2750 m, vsh<.052701-2750 m, Vsh>.3

Vp

Total Porosity

increasing cement

Suspension Line

poor sorting

all zonesconverted to brine

more clay


Shaly Sands

203

2

2.5

3

3.5

4

4.5

5

5.5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

P-w

ave

Velo

city

(km

/s)

Porosity

?Jizba 9,000 ft (75 MPa)

Han 15,000 ft (40 MPa)R = 0.70

Han 12,000 ft (40 MPa)R = 0.80

Han 10,000 ft (40 MPa)R = 0.96

Blangy 5,000 ft, Troll (30 MPa)R = 0.76

L.27


Shaly Sands

204

Xu and White (1995) developed a theoretical model for velocities in shaly sandstones. the formulation uses the Kuster-Toksöz and Differential Effective Medium theories to estimate the dry rock P and S velocities, and the low frequency saturated velocities are obtained from Gassmann’s equation. The sand-clay mixture is modeled with ellipsoidal inclusions of two different aspect ratios. The sand fraction has stiffer pores with aspect ratio αααα

≈≈≈≈

0.1 - 0.15, while the clay-related pores are more compliant with αααα

≈≈≈≈

0.02-0.05. The velocity model simulates the “V” shaped velocity-porosity relation of Marion et al. (1992) for sand-clay mixtures. The total porosity φφφφ = φφφφsand + φφφφclay where φφφφsand and φφφφclay are the porosities associated with the sand and clay fractions respectively. These are approximated by

where Vsand and Vclay denote the volumetric sand and clay content respectively. Shale volume fromlogs may be used as an estimate of Vclay. through the log derived shale volume includes silts, and overestimates clay content, results obtained by Xuand White justify its use. The properties of the solid mineral mixture are estimated by a Wyllie time average of the quartz and clay mineral velocities, and arithmetic average of t heir densities:

where subscript 0 denotes the mineral properties. these mineral properties are then used in the Kuster-Toksöz formulation along witht he porosity and clay content, to calculate dry rock moduli and velocities. The limitation of small pore concentration of the Kuster-Toksöz model is handled by incrementally adding the pores in small steps such that the non-interaction criterion is satisfied in each step. Gassmann’s equations are used to obtain low frequency saturated velocities. High frequency saturated velocities are calculated by using fluid-filled ellipsoidal inclusions in the Kuster-Toksöz model.

The model can be used to predict shear wave velocities (Xu and White, 1994). Estimates of Vs may be obtained from known mineral matrix properties and measured porosity and clay content, or from measured Vp and either porosity or clay content. Su and White recommend using measurements of P-wave sonic log since it is more reliable than estimates of shale volume and porosity.

φsand = 1 – φ – Vclay

φ1 – φ = Vsand

φ1 – φ

φclay = Vclay

φ1 – φ

1VP0

=1 – φ – Vclay

1 – φ1

VPquartz

+Vclay1 – φ

1VPclay

1VS 0

=1 – φ – Vclay

1 – φ1

VSquartz

+Vclay1 – φ

1VSclay

ρ0 =

1 – φ – Vclay1 – φ ρquartz +

Vclay1 – φρclay


Sand Models

205

Grain

Contact cement

A B

R a a

Non-contact cement Scheme 1 Scheme 2

C

Dvorkin’s Cement ModelJack Dvorkin introduced a cement model that predicts

the bulk and shear moduli of dry sand when cement is deposited at grain contacts. The model assumes that the cement is elastic and its properties may differ from those of the grains.

It assumes that the starting framework of cemented sand is a dense random pack of identical spherical grains with porosity , and the average number of contacts per grain C = 9. Adding cement reduces porosity and increases the effective elastic moduli of the aggregate. The effective dry-rock bulk and shear moduli are (Dvorkin and Nur, 1996)

where

φ0 ≈ 0.36

G c = ρ cVSc2

Keff = 16 C 1 – φ0 M c Sn

M c = ρcVPc2

Geff = 35 K eff + 3

20 C 1 – φ0 G c S τ

is the cement's density; and and are its P-and S-wave velocities. Parameters and are proportional to the normal and shear stiffness, respectively, of a cemented two-grain combination. They depend on the amount of the contact cement and on the properties of the cement and the grains. (see next page)

ρc VPc VSc

Sn S τ


Sand Models

206

where G and are the shear modulus and the Poisson's ratio of the grains, respectively; and are the shear modulus and the Poisson's ratio of the cement; a is the radius of the contact cement layer; R is the grain radius.

Dvorkin’s Cement Model

Constants in the cement model: S n = A n α2 + B n α + C n

A n = – 0.024153 Λ n–1.3646

Bn = 0.20405 Λ n–0.89008

Cn = 0.00024649 Λ n–1.9864

S τ = A τ α2 + B τ α + C τ

A τ = –10–2 2.26ν 2 + 2.07ν + 2.3 Λ τ0.079ν2 + 0.1754ν – 1.342

B τ = 0.0573ν 2 + 0.0937ν + 0.202 Λ τ0.0274ν2 + 0.0529ν – 0.8765

C τ = –10–4 9.654ν 2 + 4.945ν + 3.1 Λ τ0.01867ν2 + 0.4011ν – 1.8186

Λ n = 2G c

πG1 – ν 1 – ν c

1 – 2ν cΛ τ = G c

πG α = a

R

ννcGc


Sand Models

207

The amount of the contact cement can be expressed through the ratio of the radius of the cement layer a to the grain radius R:

α

α = a/RThe radius of the contact cement layer a is not necessarily directly related to the total amount of cement: part of the cement may be deposited away from the intergranular contacts. However by assuming that porosity reduction in sands is due to cementation only, and by adopting certain schemes of cement deposition we can relate parameter to the current porosity of cemented sand . For example, we can use Scheme 1 (see figure above) where all cement is deposited at grain contacts:

αφ

α = 2 φ0 – φ

3C 1 – φ0

0.25= 2 Sφ0

3C 1 – φ0

0.25

or we can use Scheme 2 where cement is evenly deposited on the grain surface:

α =

2 φ0 – φ3 1 – φ 0

0.5

= 2Sφ03 1 – φ0

0.5

In these formulas S is the cement saturation of the pore space - the fraction of the pore space occupied by cement.

Dvorkin’scement model

Grain

Contact cement

A B

R a a

Non-contact cement Scheme 1 Scheme 2

C


Sand Models

208

If the cement's properties are identical to those of the grains, the cementation theory gives results which are very close to those of the Digby model. The cementation theory allows one to diagnose a rock by determining what type of cement prevails. For example, it helps distinguish between quartz and clay cement. Generally, Vp predictions are much better than Vs predictions.

2

2.5

0.2 0.3

Vs (k

m/s

)

porosity

quartz-cemented

clay-cementedTHEORETICAL

CURVES

3

4

0.2 0.3

Vp (k

m/s

)

porosity

quartz-cemented

clay-cemented

THEORETICALCURVES

Predictions of Vp and Vs using the Scheme 2 model for quartz and clay cement, compared with data from quartz and clay cemented rocks from the North Sea.

Dvorkin’s Cement Model


Sand Models

209

LOAD-BEARING(CONTACT)

CEMENT

POROSITY

MO

DUL

US

NO CEMENT

NON-LOAD-BEARING

(NON-CONTACT) CEMENT

2

3

4

0.1 0.2 0.3 0.4

Vp

(km

/s)

Porosity

UncementedTrajectory

Cement-ClayTrajectory

Cement-QuartzTrajectory

a

2

3

4

0.1 0.2 0.3 0.4

TrollOseberg QuartzOseberg Clay

Vp

(km

/s)

Porosity

b

Sand models can be used to “Diagnose” sands


Sand Models

210

Dvorkin’s Uncemented Sand Model

This model predicts the bulk and shear moduli of dry sand when cement is deposited away from grain contacts. The model assumes that the starting framework of uncemented sand is a dense random pack of identical spherical grains with porosity , and the average number of contacts per grain C = 9. The contact Hertz-Mindlin theory gives the following expressions for the effective bulk ( ) and shear( ) moduli of a dry dense random pack of identical spherical grains subject to a hydrostatic pressure P:

φ0 = 0.36

KHM GHM

KHM = C2 1 – φ0

2 G2

18 π2 1 – ν 2 P1/3

GHM = 5 – 4ν

5 2 – ν3C2 1 – φ 0

2 G 2

2π2 1 – ν 2 P1/3

where is the grain Poisson's ratio and G is the grain shear modulus.

ν


Sand Models

211

Dvorkin’s Uncemented Sand ModelIn order to find the effective moduli at a different porosity, a heuristic modified Hashin-Strikman lower bound is used:

Keff = φ / φ0

K HM + 43 G HM

+ 1 – φ / φ0

K + 43 G HM

–1– 4

3 GHM

G eff = [ φ / φ0

G HM +G HM

69KHM + 8G HMK HM + 2GHM

+ 1 – φ / φ0

G +GHM

69K HM + 8GHMKHM + 2GHM

]–1

–GHM

69KHM + 8G HMKHM + 2GHM

0

20

40

60

80

100

0 0.1 0.2 0.3 0.4

M-m

odul

us (G

Pa)

porosity

solid

HERTZ-MINDLIN

increasingpressure

M = ρρρρVP

2

Illustration of the modified lower Hashin-Shtrikman bound for various effective pressures. The pressure dependence follows from the Hertz-

Mindlin theory incorporated into the right end member.


Sand Models

212

Dvorkin’s Uncemented Sand ModelThis model connects two end members: one has zero porosity and the modulus of the solid phase and the other has high porosity and a pressure-dependent modulus as given by the Hertz-Mindlin theory. This contact theory allows one to describe the noticeable pressure dependence normally observed in sands.The high-porosity end member does not necessarily have to be calculated from the Hertz-Mindlin theory. It can be measured experimentally on high-porosity sands from a given reservoir. Then, to estimate themoduli of sands of different porosities, the modifiedHashin-Strikman lower bound formulas can be used where KHM and GHM are set at the measured values. This method provides accurate estimates for velocities in uncemented sands. In the figures below the curves are from the theory.

0

1

2

3

0.2 0.3 0.4

velo

city

(km

/s)

porosity

Vp

Vs

saturatedP

eff = 5 MPa

0

1

2

3

0.2 0.3 0.4porosity

Vp

Vs

saturatedP

eff = 15 MPa

0

1

2

3

0.2 0.3 0.4porosity

Vp

Vs

saturatedP

eff = 30 MPa

Prediction of Vp and Vs using the lower Hashin-Shtrikmanbound, compared with measured velocities from

unconsolidated North Sea samples.


Sand Models

213

0.8

1.2

1.6

2

2.4

2.8

0.3 0.4 0.5

velo

city

(km

/s)

porosity

Vp

Vs

North Seasand

Ottawasand

theory

This method can also be used for estimating velocities in sands of porosities exceeding 0.36.


Sand Models

214

2 3

2.1

2.2

2.3

Vp (km/s)

Well #1

A40 80 120

GRB2 3 4

1.7

1.8

1.9

Vp (km/s)

Well #2

C

Marl

Limestone

40 80 120GRD

2.5

3

3.5

0.25 0.3 0.35 0.4

p(

)

Porosity

Contact CementLine

UnconsolidatedLine

ConstantCement Fraction (2%) Line

Well #1

Well #2

North Sea Example

Study by Per Avseth, along with J. Dvorkin, G. Mavko, and J. Rykkje


Sand Models

215

0 50 100

1730

1740

1750

1760

1770

1780

1790

1800

1810

1820

1830

1840

Dep

th(m

)

gamma ray (API)0 2 4

1730

1740

1750

1760

1770

1780

1790

1800

1810

1820

1830

1840

Vp, Vs and Vp/Vs

VpVs

Vp/Vs

1.8 2 2.2 2.4 2.6

1730

1740

1750

1760

1770

1780

1790

1800

1810

1820

1830

1840

density

Well-Log Measurements of Cemented High-Porosity Sandstone Interval

2.5 3 3.5

1780

1785

1790

1795

1800

1805

1810

1815

1820

1825

1830

Vp

Dep

th(m

)

0.3 0.35 0.4

1780

1785

1790

1795

1800

1805

1810

1815

1820

1825

1830

Porosity

Dep

th(m

)


Sand Models

216

Sorting Analysis of Thin-Sections

100 300 500 700 9000

50

100

150

Grain diameter − @1785.1m

Fre

quen

cy

100 300 500 700 9000

50

100

150

Grain diameter − @1790.1mF

requ

ency

100 300 500 700 9000

50

100

150


Fre

quen

cy

100 300 500 700 9000

50

100

150


Fre

quen

cy

0.4m m0.4m m

0.4m m0.4m m


Sand Models

217

Sorting and Rock Physics Properties

0.25 0.3 0.35 0.4

2.6

2.8

3

3.2

3.4

3.6

density porosity

Vp

0.6 0.8 1 1.22.6

2.8

3

3.2

3.4

3.6

sorting (std/mean)

Vp

0.6 0.8 1 1.2 1.40.28

0.3

0.32

0.34

0.36

0.38

sorting (std/mean)

dens

ity p

oros

ity

0.6 0.8 1 1.2 1.4

0.3

0.32

0.34

0.36

0.38

0.4

sorting (std/mean)

heliu

m p

oros

ity

0.6 0.8 1 1.21.3

1.4

1.5

1.6

1.7

1.8

sorting (std/mean)

Vs

(km

/s)

0.6 0.8 1 1.21.8

1.9

2

2.1

2.2

sorting (std/mean)

Vp/

Vs


Sand Models

218

Sorting Versus Cementation

0.25 0.3 0.35 0.42

2.5

3

3.5

4

porosity

Vp

(km

/s)

contact cement

line4%

2%

0.25 0.3 0.35 0.42

2.5

3

3.5

4

porosity

Vp

(km

/s)

2%

0.25 0.3 0.35 0.42

2.5

3

3.5

4

porosity

Vp

(km

/s)

4%

0.25 0.3 0.35 0.42

2.5

3

3.5

4

porosity

Vp

(km

/s)

2%

4%


Sand Models

219

Thin-Section and SEM Analyses

Well #2 Cemented

0.25 mm

Well #1 Uncemented

0.25 mm

SEM cathode-luminescent image: Well #2

0.1 mm0.1 mm

SEM back-scatter image: Well #2

Unconsolidated(Facies IIb)

Cemented(Facies IIa)

Back-scatter light Cathode lum. light

Qz-cement rim Qz-grain


220

A North Sea Example

PROBLEM

Can we predict and characterize reservoirs in North Sea deep-water clastic systems

using seismic data?

1 km1 km

SEISMIC AMPLITUDE MAP OF A NORTH SEA SUB-MARINE FAN


221

A North Sea Example

SEISMIC LITHOFACIES IN DEEP- WATER CLASTIC SYSTEMS

A seismic lithofacies is a seismically resolvable sedimentary unit characterized by its:

• lithology (clay content)

• bedding configuration (massive, interbedded, chaotic)

• petrography (grain size, clay location and cementation)

• seismic properties (seismic velocities and density)


222

A North Sea Example

I

II

III

IV

V

VI

FACIES I: Gravels and conglomerates

FACIES II:Thick-bedded sandstone

FACIES III:Interbedded sandstone-shale

FACIES IV:Silty shale and silt-laminated shale

FACIES V:Pure shale

FACIES VI: Chaotic deposits


223

A North Sea Example

1.5 2 2.5 3 3.5 4

2100

2150

2200

Vp (km/s)

Dep

th (m

)V

IV II b

II c

III

Identification of Seismic Lithofacies From Well Logs (Field A, Well A-1)

V

IV

40 60 80 100 120 140

2100

2150

2200

Gamma Ray (API)

Dep

th (m

)

V

IV II b

II c

III


224

A North Sea Example

IIa

III

IIcIIb

IVV

Overburden

Tuff

Chalk1.0 km

200 m

Depth = 2.0 km

2-D Geological Model


225

A North Sea Example

40 60 80 100 120 140

2.2

2.6

3

3.4

Gamma Ray (API)

P-w

ave

velo

city

(km

/s)

IIb

IIcIII

IVV

Rock Physics Analysis

40 60 80 100 120 140

0.2

0.3

0.4

Gamma Ray (API)

Poro

sity

IIbIIc

III

IV V


226

A North Sea Example

40 60 80 100 120 140

4.5

5.5

6.5

7.5

Gamma Ray (API)

Aco

ustic

Impe

danc

e

IIb

IIc III

IV

V

40 60 80 100 120 1401.5

2

2.5

3

Gamma Ray (API)

Vp/V

s

IIb IIc

IIIIV

V

Seismic Properties


227

A North Sea Example

Seismic Reflectivity Modeling(AVO)

R(θθθθ) = R0 + G * sin (θθθθ)

Ref

lect

ivity

0

- 0.1

0.1

II bV IIc

IIaIII

Angle of incidence ( θθθθ )

10 20 504030

Cap-rock = IV

0 60


228

A North Sea Example

Facies-Guided Reflectivity Modeling

IIb

VIV

III

IIc

Schematic facies map of sub-marine fan


229

A North Sea Example

Facies-Guided Reflectivity Modeling

Zero offset reflectivity

0.02

0.04

-0.02

0

-0.04

-0.12

-0.16

-0.08

Zero offset reflectivity+ AVO gradient

0


230

A North Sea Example

Seismic Response of Unconsolidated and Cemented High-Porosity Sands

Well #2

A B C D

Well #1

E F

Well #1 Well #2

Unconsolidated (oil) Cemented (oil)


231

A North Sea Example

Zero-offset Reflectivity, R(0)

Relatively high R(0) (blue)

Relatively low R(0) (yellow)


232

A North Sea Example

AVO Gradient, G

Relatively large negativegradient G (yellow)

Relatively smallgradient (blue)


233

A North Sea Example

Bivariate pdfs: R(0) vs. G


234

A North Sea Example

Facies classification usinglinear discriminant analysis

Sample to be classifiedFacies 1

Facies 2 Facies 3

R(0)

G

M2=(x-µi)TΣ-1(x- µi )

= Facies 2

Mahalanobis distance


235

A North Sea Example

Oil sands

Shale

Brine sands

Interbeddedsand-shales

3D seismic lithofacies prediction


236

A North Sea Example

Variability of AVO Response

Monti-Carlo, drawn from pdf, and assuming siltyshale cap rock.


237

A North Sea Example

Seismic Lithofacies PredictionLine 1 Through Well #1

Well #1 (CDP 1464)

1380 1400 1420 1440 1460 1480 1500 1520 1540 1560 1580−0.1

0

0.1

0.2

Seismic lithofacies prediction − Line #1

R(0

)

1380 1400 1420 1440 1460 1480 1500 1520 1540 1560 1580

−0.5

0

0.5

G

1380 1400 1420 1440 1460 1480 1500 1520 1540 1560 1580

2

4

6

8

Fac

ies

1380 1400 1420 1440 1460 1480 1500 1520 1540 1560 1580

2

4

6

8

Top Heimdal

Oil ?


238

A North Sea Example


Well #3 (CDP1844)

1700 1750 1800 1850 1900−0.1

0

0.1

R(0

)


1700 1750 1800 1850 1900−0.4

−0.2

0

0.2

G

1700 1750 1800 1850 1900

2

4

6

8

Fac

ies

1700 1750 1800 1850 1900

2

4

6

8

Top Heimdal


239

A North Sea Example


Well # 2Well #2 (CDP 2232)

2000 2050 2100 2150 2200 2250 2300−0.1

0

0.1

R(0

)


2000 2050 2100 2150 2200 2250 2300

−0.5

0

0.5

G

2000 2050 2100 2150 2200 2250 2300

2

4

6

8

Fac

ies

2000 2050 2100 2150 2200 2250 2300

2

4

6

8

Top Heimdal


240

A North Sea Example

Conclusions

• Seismic lithofacies link sedimentology to rockphysics.

• Paleocene deep-water clastic systems in the NorthSea show:

- Sand-shale ambiguity in acousticimpedance.

- Sand-shale discrimination in Vp/Vsratio.

• AVO analysis should be applied to predict seismiclithofacies from seismic data in our case.

• Seismic lithofacies can improve our understanding ofseismic signatures in any depositional system.


AVO

241

2

2.5

3

3.5

4

4.5

5

5.5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Vp (k

m/s

)

Porosity

2

2.5

3

3.5

4

4.5

5

5.5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Vp (k

m/s

)

Porosity

2

2.5

3

3.5

4

4.5

5

5.5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

water-saturatedgas-saturated

Vp (k

m/s

)

Porosity

Water-saturated40 MPa

Water-saturated10-40 MPa

Gas andWater-saturated10-40 MPa

L8


AVO

242

N.1

More than 400 sandstone data points, with porositiesranging over 4-39%, clay content 0-55%, effectivepressure 5-40 MPa - all water saturated.

When Vp is plotted vs. Vs, they follow a remarkablynarrow trend. Variations in porosity, clay, and pressuresimply move the points up and down the trend.

0 1 2 3 4 5 6 7 80

0.51

1.52

2.53

3.54

Shaly SandsWater Saturated

φφφφ = .22-.36φφφφ = .04-.30φφφφ = .32-.39

Vp (km/s)

Vs (k

m/s

)

(Data from Han, Blangy, Yin)

increasing φφφφ , clay, and pore pressure


AVO

243

N.2

Variations in porosity, pore pressure, and shalinessmove data along trends. Changing the pore fluid causes the trend to change.

00.5

11.5

22.5

33.5

4

0 1 2 3 4 5 6 7 8

Vs (k

m/s

)

Vp (km/s)

gas

water

saturation

increasing φφφφ , clay, and pore pressure


AVO

244

00.5

11.5

22.5

33.5

4

0 1 2 3 4 5 6 7 8

Vs (k

m/s

)

Vp (km/s)

saturation

Poor fluid discrimination.Shear not much value.

Good fluid discrimination.Shear could be valuable.

gas

waterpore

pressure

0

2

4

6

8

10

0 5 10 15 20

S-Im

peda

nce

P-Impedance

saturation

Good fluid discrimination.Shear could be valuable.

Less fluid discrimination gas

waterpore

pressure

1

1.5

2

2.5

3

0 1 2 3 4 5

Vp/V

s

Vs

gas

watersaturation

Good fluid discrimination.

Poor fluid discrimination pore pressure

Different shear-related attributes.N.2


AVO

245

Seismic Amplitudes

Many factors influence seismic amplitude:

• Source energy and spectrum• Source coupling• Source radiation pattern• Receiver response, coupling, and pattern• Scattering• Spherical divergence• Focusing• Anisotropy• Intrinsic attenuation• Reflection coefficient

S R


AVO

246

Because there are so many factors affecting amplitude, it is desireable, in almost any analysis of amplitudes, to take advantage of relative measurements, to look for redundancy or changes in time and/or space:

• Surface-consistent analysis of data• Changes in reflection character along a horizon • Differences in time during production or an EOR operation• Variations with azimuth


AVO

247

Very large amplitude reflections on a stacked section are sometimes called “Bright Spots”.

The reflection coefficient of a normally-incident P-wave on a boundary is given by:

where ρρρρV is the acoustic impedence. So that anything that causes a large contrast in impedance can cause a large reflection. Candidates include:

• changes in lithology• changes in saturation

Bright Spots

R = ρρρρ2V2 – ρρρρ1V1

ρρρρ2V2 + ρρρρ1V1


AVO

248

Example of a bright spot.

N.3


AVO

249

In an isotropic medium, a wave that is incident on a boundary will generally create two reflected waves (oneP and one S) and two transmitted waves. The total sheartraction acting on the boundary in medium 1 (due to thesummed effects of the incident an reflected waves) mustbe equal to the total shear traction acting on the boundary in medium 2 (due to the summed effects of the transmitted waves). Also the displacement of a point inmedium 1 at the boundary must be equal to the displace-ment of a point in medium 2 at the boundary.

VP1, VS1, ρ1

VP2, VS2, ρ2

θ1

φ1

θ2φ2

Reflected P-wave

Incident P-wave

Reflected S-wave

Transmitted P-wave

Transmitted S-wave

sin θθθθ1VP1

= sin θθθθ2VP2

= sin φφφφ1VS1

= sin φφφφ2VS2

N.4


AVO

250

sin θ1 cos φ1 –sin θ2 cos φ2

–cos θ1 sin φ1 –cos θ2 –sin φ2

sin 2θ1V1S1

cos 2φ1ρ2S2

2V1ρ1S1

2V2sin 2θ2 –ρ2S2V1

ρ1S12 cos 2φ2

cos 2φ1 –S1V1

sin 2φ1 –ρ2V2ρ1V1

cos 2φ2 – ρ2S2ρ1V1

sin 2φ2

ARPARSATPATS

=

–sin θ1–cos θ1sin 2θ1

–cos 2φ1

By matching the traction and displacement boundary conditions, Zoeppritz (1919) derived the equations relating the amplitudes of the P and S waves:


AVO

251

AVO - Shuey's Approximation

P-wave reflectivity versus angle:

R(θθθθ) ≈≈≈≈ R0 + ER0 + ∆ν∆ν∆ν∆ν

1–νννν 2 sin2θθθθ + 12

∆∆∆∆VPVP

tan2θθθθ – sin2θθθθ

R0 ≈≈≈≈ 12

∆∆∆∆VPVP

+ ∆ρ∆ρ∆ρ∆ρρρρρ

E = F – 2(1 + F)1 – 2νννν

1 – νννν

F = ∆∆∆∆VP /VP∆∆∆∆VP /VP + ∆ρ∆ρ∆ρ∆ρ /ρρρρ

∆∆∆∆VP = VP2 – VP1

∆ρ∆ρ∆ρ∆ρ = ρρρρ2 – ρρρρ1

VP = VP2 + VP1 /2

ρρρρ = ρρρρ2 + ρρρρ1 /2

∆∆∆∆VS = VS2 – VS1 VS = VS2 + VS1 /2


AVO

252

AVO - Aki-Richard's approximation:

P-wave reflectivity versus angle:

∆∆∆∆VP = VP2 – VP1

∆ρ∆ρ∆ρ∆ρ = ρρρρ2 – ρρρρ1

VP = VP2 + VP1 /2

ρρρρ = ρρρρ2 + ρρρρ1 /2

R(θθθθ) ≈≈≈≈ R0 + 1

2∆∆∆∆VPVP

– 2VS2

VP2

∆ρ∆ρ∆ρ∆ρρρρρ + 2∆∆∆∆VS

VSsin2θθθθ

+ 12

∆∆∆∆VPVP

tan2θθθθ – sin2θθθθ

R0 ≈≈≈≈ 1

2∆∆∆∆VPVP

+ ∆ρ∆ρ∆ρ∆ρρρρρ

∆∆∆∆VS = VS2 – VS1 VS = VS2 + VS1 /2


AVO

253

AVO Response

P-Velocity Poisson ratio AVO response contrast contrast

negative negative increasenegative positive decreasepositive negative decreasepositive positive increase


AVO

254

Vp-Vs Relations

There is a wide, and sometimes confusing, variety of published Vp-Vs relations and Vs prediction techniques, which at first appear to be quite distinct. However, most reduce to the same two simple steps:

1. Establish empirical relations among Vp, Vs, and porosity for one reference pore fluid--most often water saturated or dry.

2. Use Gassmann’s (1951) relations to map these empirical relations to other pore fluid states.

Although some of the effective medium models predict both P and S velocities assuming idealized poregeometries, the fact remains that the most reliable and most often used Vp-Vs relations are empirical fits to laboratory and/or log data. The most useful role of theoretical methods is extending these empirical relations to different pore fluids or measurement frequencies. Hence, the two steps listed above.


AVO

255

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4

SandstonesWater Saturated

Vp (k

m/s

)

Vs (km/s)

mudrockVs = .8621V p-1.1724

Castagna et al. (1993)Vs = .8042V p-.8559

Han (1986)Vs = .7936V p-.7868

(after Castagna et al., 1993)

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4

ShalesWater Saturated

Vp (k

m/s

)

Vs (km/s)

mudrockVs = .8621 V p- 1.1724

Castagna et al. (1993)V s = .8042V p-.8559

Han (1986)Vs = .7936V p-.7868


N.5


AVO

256

0

1

2

34

5

6

78

0 0.5 1 1.5 2 2.5 3 3.5 4

LimestonesWater Saturated

Vp (k

m/s

)

Vs (km/s)

V s = V p/1.9Pickett (1963)

Castagna et al. (1993)Vs = -.05508 V P2 + 1.0168 V p - 1.0305

water (after Castagna et al.,1993)

01

2

3

45

6

7

8

0 0.5 1 1.5 2 2.5 3 3.5 4

DolomiteWater Saturated

Vp (k

m/s

)

Vs (km/s)

Castagna et al. (1993)V s = .5832V p -.07776

Pickett (1963)Vs = V p/1.8


N.6


AVO

257

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3 3.5 4

Shaly SandstonesWater Saturated

Vp-sat c>.25Vp-sat c<.25

Vp (k

m/s

)

Vs (km/s)

clay > 25 % Vs=.8423Vp-1.099

clay < 25 %Vs=.7535Vp-.6566

mudrock Vs=.8621Vp-1.1724

(Data from Han, 1986)

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3 3.5 4

Shaly SandstonesWater Saturated (hf)

Vp-sat Phi>.15Vp-sat Phi<.15

Vp (k

m/s

)

Vs (km/s)

porosity > 15 % Vs = .7563Vp-.6620

porosity < 15 %Vs = .8533Vp-1.1374

mudrock Vs = .8621Vp-1.1724

(Data from Han, 1986)

N.7


AVO

258

Dry Poisson’s Ratio Assumption

02468

10121416

0 5 10 15 20 25 30 35

clay < 10%clay > 10%

Vs2

dry

Vp2 dry

ν = 0.01

ν = 0.1

ν = 0.2

ν = 0.3

ν = 0.4

Shaly Sandstones - Dry

The modified Voigt Average Predicts linear moduli-porosity relations. This is a convenient relation for use with the critical porosity model.

These are equivalent to the dry rock Vs/Vp relation and the dry rock Poisson’s ratio equal to their values for pure mineral.

The plot below illustrates the approximately constant dry rock Poisson’s ratio observed for a large set of ultrasonic sandstonevelocities (from Han, 1986) over a large rance of effective pressures (5 < Peff < 40 MPa) and clay contents (0 < C < 55% by volume).

Kdry = K0 1 – φφφφ

φφφφc

µµµµdry = µµµµ0 1 – φφφφφφφφc

VSVP dry rock

≈≈≈≈ VSVP mineral

ννννdry rock ≈≈≈≈ ννννmineral

N.8


AVO

259

Krief’s Relation (1990)

The model combines the same two elements:

1. An empirical Vp-Vs-φφφφ relation for water-saturated rocks, which is approximately the same as the critical porosity model.2. Gassmann’s relation to extend the empirical relation to other pore fluids.

Dry rock Vp-Vs-φφφφ relation:

where ββββ is Biot’s coefficient. This is equivalent to:

where

ββββ and Kφφφφ are two equivalent descriptions of the pore space stiffness. Determining ββββ vs. φφφφ or Kφφφφ vs φφφφdetermines the rock bulk modulus Kdry vs φφφφ.

Krief et al. (1990) used the data of Raymer et al. (1980) to empirically find a relation for ββββ vs φφφφ:

Kdry = Kmineral 1 – ββββ

1Kdry

= 1K0

+ φφφφKφφφφ

1Kφφφφ

= 1vp

dvpdσσσσ PP = constant

; ββββ =dvpdV PP = constant

=φφφφ Kdry

Kφφφφ

1 – ββββ = 1 – φφφφ m(φφφφ) where m(φφφφ) = 3/ 1 – φφφφ


AVO

260

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Mod

ulus

(Nor

mal

ized

)

Porosity

Critical Porosity Model

Krief Model

φφφφc

Assuming dry rock Poisson’s ratio is equal to the mineral Poisson’s ratio gives

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Bio

t Coe

ffici

ent

Porosity

Critical Porosity Model

Krief Model

φφφφc

Kdry = K0 1 – φφφφ m(φφφφ) where m(φφφφ) = 3/ 1 – φφφφ

µµµµdry = µµµµ0 1 – φφφφ m(φφφφ)

N.9


AVO

261

VP–sat

2 – Vfl2

VS–sat2 = VP0

2 – Vfl2

VS02

Expressions for any other pore fluids are obtained fromGassmann’s equations. While these are nonlinear, they suggest a simple approximation:

where VP-sat, VP0, and Vfl are the P-wave velocities of the saturated rock, the mineral, and the pore fluid; and VS-sat and VS0 are the S-wave velocities in the saturated rock and mineral. Rewriting slightly gives

where VR is the velocity of a suspension of minerals in a fluid, given by the Reuss average at the critical porosity.

This modified form of Krief’s expression is exactly equivalent to the linear (modified Voigt) K vs φφφφ and µµµµ vs φφφφrelations in the critical porosity model, with the fluid effectsgiven by Gassmann.

VP–sat

2 = Vfl2 + VS–sat

2 VP02 – Vfl

2

VS02

which is a straight line (in velocity-squared) connecting the mineral point ( ) and the fluid point ( ). A more accurate (and nearly identical) model is to recognize that velocities tend toward those of a suspension at high porosity, rather than toward a fluid, which yields the modified form

VP02 , VS0

2 Vfl2 , 0

VP–sat

2 – VR2

VS–sat2 = VP0

2 – VR2

VS02


AVO

262

0

5

10

15

20

25

30

35

0 5 10 15 20

Vp-Vs Relation in Dry andSaturated Rocks

V p2 (k

m/s

)2

Vs2 (km/s)2

saturated

dry

Sandstones mineral point

fluid point

N.10


AVO

263

N.11

0

10

20

30

40

50

0 4 8 12 16

Vp-Vs Relation in Sandstoneand Dolomite

V p2 (k

m/s

)2

Vs2 (km/s)2

Sandstone

Dolomite

mineralpoints

fluid points


AVO

264

VS = 1

2 XiΣΣΣΣi = 1

LaijVP

jΣΣΣΣj = 0

Ni

+ XiΣΣΣΣi = 1

LaijVP

jΣΣΣΣj = 0

Ni –1 –1

Greenberg and Castagna (1992) have given empirical relations for estimating Vs from Vp in multimineralic, brine-saturated rocks based on empirical, polynomial Vp-Vs relations in pure monomineralic lithologies (Castagna et al., 1992). The shear wave velocity in brine-saturated compositelithologies is approximated by a simple average of the arithmetic and harmonic means of the constituent purelithology shear velocities:

Castagna et al. (1992) gave representative polynomial regression coefficients for pure monomineralic lithologies:

Regression coefficients for pure lithologies with Vp and Vs in km/s:

Xi = 1ΣΣΣΣi = 1

L

VS = ai2VP2 + ai1VP + ai0 (Castagna et al. 1992)

whereL number of pure monomineralic lithologic constituentsXi volume fractions of lithological constituents aij

empirical regression coefficientsNi order of polynomial for constituent iVp, Vs P and S wave velocities (km/s) in composite brine-saturated,

multimineralic rock

Lithology a i2 a i1 a i0

S andstone 0 0.80416 -0.85588Limestone -0.05508 1.01677 -1.03049Dolomite 0 0.58321 -0.07775S hale 0 0.76969 -0.86735


AVO

265

1

2

3

4

5

6

7

0 1 2 3 4 5

SandstoneLimestoneDolomiteShale

Vp (k

m/s

)

Vs (km/s)

Note that the above relation is for 100% brine-saturated rocks. To estimate Vs from measured Vp for other fluid saturations,Gassmann’s equation has to be used in an iterative manner. In the following, the subscript b denotes velocities at 100% brine saturation and the subscript f denotes velocities at any other fluid saturation (e.g. this could be oil or a mixture of oil, brine, and gas). The method consists of iteratively finding a (Vp, Vs) point on the brine relation that transforms, withGassmann’s relation, to the measured Vp and the unknown Vs for the new fluid saturation. the steps are as follows:

1. Start with an initial guess for VPb.2. Calculate VSb corresponding to VPb from the empirical regression.3. Do fluid substitution using VPb and VSb in the Gassmannequation to get VSf.4. With the calculated VSf and the measured VPf, use theGassmann relation to get a new estimate of VPb. Check with previous value of VPb for convergence. If convergence criterion is met, stop; if not go back to step 2 and continue.

N.12


AVO

266

0

5

10

15

0 5 10 15 20 25 30 35

Vs2

Vp

2

dry

saturatedlow frequency

saturatedultrasonic

1

1.5

2

2.5

3

3.5

4

2 2.5 3 3.5 4 4.5 5 5.5 6

Vs

Vp

dry

saturatedlow frequency

saturatedultrasonic

Ultrasonic data from Han (1986)

N.14


AVO

267

Coe fficie n t s for t h e e q u a t ion ρb = aVp2 + bVp + c

Li t h o l o g y a b c V p Ra n g e ( Km / s )

Sh a le - .0 2 6 1 .3 7 3 1 .4 5 8 1 .5 - 5 .0Sa n d s t on e - .0 1 1 5 .2 6 1 1 .5 1 5 1 .5 - 6 .0Lim e s t on e - .0 2 9 6 .4 6 1 0 .9 6 3 3 .5 - 6 .4Dolom it e - .0 2 3 5 .3 9 0 1 .2 4 2 4 .5 - 7 .1An h y d r it e - .0 2 0 3 .3 2 1 1 .7 3 2 4 .6 - 7 .4

Coe fficie n t s for t h e e q u a t ion ρb = dVpf

Li t h o l o g y d f V p Ra n g e( Km / s )

Sh a le 1 .7 5 .2 6 5 1 .5 - 5 .0Sa n d s t on e 1 .6 6 .2 6 1 1 .5 - 6 .0Lim e s t on e 1 .5 0 .2 2 5 3 .5 - 6 .4Dolom it e 1 .7 4 .2 5 2 4 .5 - 7 .1An h y d r it e 2 .1 9 .1 6 0 4 .6 - 7 .4

1

2

3

4

5

6

1.8 2 2.2 2.4 2.6 2.8

Shales

Vp (k

m/s

)

density (g/cm 3)

ρρρρ = -.0261Vp 2 +.373Vp+1.458

ρρρρ = 1.75Vp .265

Both forms of Gardner’s relations applied to log and lab shale data, as presented by Castagna et al. (1993)

Polynomial and powerlaw forms of the Gardner et al. (1974) velocity-density relationships presented by Castagna et al. (1993). Units are km/s and g/cm3.

N.15


AVO

268

3

3.5

4

4.5

5

5.5

6

6.5

7

2 2.2 2.4 2.6 2.8 3

Dolomite

Vp (k

m/s

)

density (g/cm 3)

ρρρρ = -.0235Vp 2 +.390Vp +1.242

ρρρρ = 1.74Vp .252

Both forms of Gardner’s relations applied to laboratory dolomite data.

N.16


AVO

269

1

2

3

4

5

6

1.8 2 2.2 2.4 2.6 2.8

SandstonesVp

(km

/s)

density (g/cm 3)

ρρρρ = -.0115Vp 2 +.261Vp+1.515

ρρρρ = 1.66Vp .261

1

2

3

4

5

6

1.8 2 2.2 2.4 2.6 2.8

Limestones

Vp (k

m/s

)

density (g/cm 3)

ρρρρ = -.0296Vp 2 +.461Vp +0.963

ρρρρ = 1.5Vp .225ρρρρ = 1.359Vp .386

Both forms of Gardner’s relations applied to laboratory limestone data. Note that the published powerlaw form does not fit as well as the polynomial. we also show a powerlaw form fit to these data, which agrees very well with the polynomial.

Both forms of Gardner’s relations applied to log and lab sandstone data, as presented by Castagna et al. (1993).

N.17


Example 4D Feasibility Study

270

Reservoir modeling

Flow simulation

Rock physics modeling of

elastic properties

Synthetic seismic imaging

Comparison with field

4D seismic

Update model Reservoir management decisions

Steps involved in monitoring

reservoir performance



271

Rock Physics 4D Modeling Steps

Model-Building• Build geologic reservoir model:

lithology, porosity, perm at each pixel• Specify reservoir fluid properties:

gravity, GOR, salinity, etc.• Assign Vp, Vs, and density to each pixel

of geologic model for reference fluid

Upscale and Flow Simulate

Rock Physics Mapping to Seismic – ateach time step and each pixel:• Simulator predicts P, T, So, Sw, Sg• Model K, ρρρρ of water, oil, gas • Compute “average” fluid properties• downscale/upscale?• Use Gassmann to update Vp, Vs, density• Adjust Vp, Vs for changes in P



272

� Assessing feasibility � Interpreting data after acquisition

Reservoir Monitoring Analysis

� Realistic synthetic data� Based on North Sea:– Braided river channels– 5 km by 5 km, 2 faults– 24 production wells and 17 water injectors

�Geophysics:Biondo Biondi, David Lumley, Gary Mavko, Tapan Mukerji, James Rickett�Petroleum Engineering:Clayton Deutsch, Runar Gundesø, Marco Thiele�SRB, SEP, SCRF, SUPRI-B�Chevron, Norsk-Hydro

Study was performed by various groups at Stanford:

Example feasibility study:



273

Reservoir Model



274

� Hierarchical object-based approach– Define stratigraphic layers– Model channel sand/shale facies– Assign porosity/permeability within facies

� 3 layers with net:gross = 0.8, 0.4, 0.6

� Within channel sands φφφφmean = 23%

� 200 million geological modelling cells

Geologic/Geostatistical Modeling



275

Hierarchical Modeling



276

� Lithologies and Fluids1. Porosity from geologic modeling2. Vp-Vs-φφφφ-permeability relations 3. Saturations and pressures from flow simulation4. Gassmann for fluid substitution

� Scaling and Gridding

Rock Physics



277

Rock Physics Relations



278

� PVT-properties and relative permeabilities from a North-Sea dataset� Faults as non-nearest neighbor connections (sealing and non-sealing)� Simulation run for 3 years: 6 months primary production 2.5 years with water injection� Computationally intensive: importance of scale-up

Flow Simulation



279

Flow Simulation Grid



280

Gas and Water Saturations



281

• Scattering using first-order Born approximation- Plane-layer methods cannot deal with

heterogeneities- Finite difference methods too slow

• Amplitudes from Rayleigh scattering

Seismic Modeling

• Survey design- Area: 2.4 km x 3 km - Wavelet: 45 Hz - Offsets: 0 km and 3 km - Completed at t = 0 and t = 3 years

• Constant offset migration preserving true relative amplitudes



282

4-D Seismic



283

4-D Seismic



284

4-D Seismic



285

4-D Seismic AVO Effects



286

� Rock/fluid properties� Scales of saturation� Frequency effects� Upscaling

Rock Physics Issues

� Survey Repeatability� Resolution� AVO information� Acquisition geometry � Differential seismic attributes

Seismic Imaging Issues

General Issues

• Different data formats andcoordinates• Integration of expertise, not

just data

Chalks

Water saturated Measurement type: sonic logs Effective pressure: Reference: Urmos J., and Wilkens R. H., 1993, In situ velocities in pelagic carbonates:

new insights from ocean drilling program leg 130, Ontong Java, Journ. Geophys. Res. 98, No. B5, pp 7903-7920.

min max mean std. dev

Vp 1.5250 4.2960 2.1571 0.3053 Vs 1.5930 2.5120 2.0304 0.3030

Vp/Vs 1.6174 1.7872 1.6700 0.0592 porosity 0.0995 0.7488 0.5026 0.0757 density 1.4296 2.5590 1.8498 0.1285

impedance 2.2973 10.9935 4.0249 0.8859

1.4

1.6

1.8

2

2.2

2.4

2.6

1.5 2 2.5 3 3.5 4 4.5

dens

ity (g

/cm

3)

Vp (km/s)

ρρρρ = 1.045 + 0.37308VpR = 0.88676

0 50 100 150 200 2501.4

1.6

1.8

2

2.2

2.4

2.6

count

dens

ity (g

/cm

3) mean: 1.8498

std. dev: 0.12845no. of points: 552

1.4

1.6

1.8

2

2.2

2.4

2.6

1.5 2 2.5 3 3.5 4 4.5

Vs (k

m/s

)

Vp (km/s)

Vs = 0.34543 + 0.49511VpR = 0.98899

1.6

1.65

1.7

1.75

1.8

2.5 3 3.5 4 4.5

Vp/V

s

Vp (km/s)

Vp/Vs = 1.3886 + 0.082698VpR = 0.84521

0

50

100

150

200

250

300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

coun

t

porosity

mean: 0.5026std. dev: 0.075654no. of points: 552

1.5

2

2.5

3

3.5

4

4.5

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Vp (k

m/s

)

porosity

Vp = 5.0598 φφφφ2 - 8.5052 φφφφ + 5.1284R = 0.92231

0 50 100 1501.5

2

2.5

3

3.5

4

4.5

count

Vp (k

m/s

) mean: 2.1571std. dev: 0.30532no. of points: 552

1.4

1.6

1.8

2

2.2

2.4

2.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Vs (k

m/s

)

porosity

Vs = 2.7665 + -2.9334 φφφφR = 0.9927

0 0.5 1 1.5 2 2.51.4

1.6

1.8

2

2.2

2.4

2.6

count

Vs (k

m/s

)


2

4

6

8

10

12

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

impe

danc

e 1

06

(kg/

m3

)(m/s

)

porosity

I = 1.678 φφφφ2 + -2.712 φφφφ + 1.333R = 0.96574

0 50 100 150 2002

4

6

8

10

12

count

impe

danc

e 1

06

(kg/

m3

)(m/s

)


Dolomite

Water saturated calculated from dry data using Gassmann equations Measurement type: ultrasonic Effective pressure Reference: Geertsma, J., 1961, Velocity-log interpretation: the effect of rock bulk

compressibility, Society of Petroleum Engineers Journal, December. Yale, D. P., and Jamieson Jr., W. H., 1994, Static and dynamic rock mechanical

properties in the Hugoton and Panoma fields, Kansas, SPE paper 27939, presented at the SPE Mid-Continent Gas symposium, Amarillo, Texas, May, 1994.


Vp 3.4068 7.0214 5.3901 0.6935 Vs 2.0116 3.6443 2.9697 0.36614

Vp/Vs 1.5916 2.0855 1.8152 0.0745 porosity 0.0030 0.3210 0.1297 0.0649 density 0 3.6443 2.5673 0.2538

impedance 0.7836 1.9302 1.4010 0.2321

2

2.2

2.4

2.6

2.8

3

3 4 5 6 7 8

dens

ity (g

/cm

3)

Vp (km/s)

ρρρρ = 1.8439 + 0.13786VpR = 0.82496

0 5 10 15 202.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

count

dens

ity (g

/cm

3)


2

2.5

3

3.5

4

3 4 5 6 7 8

Vs (k

m/s

)

Vp (km/s)

Vs = 0.25524 + 0.5036VpR = 0.95378

1.5

1.6

1.7

1.8

1.9

2

2.1

3 4 5 6 7 8

Vp/V

s

Vp (km/s)

Vp/Vs = 1.6558 + 0.029589VpR = 0.27552

0

5

10

15

20

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

coun

t

porosity


3

4

5

6

7

8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Vp (k

m/s

)

porosity

Vp = 6.6067 + -9.3808 φφφφR = 0.87762

0 5 10 15 20 253

4

5

6

7

8

count

Vp (k

m/s

)


2

2.5

3

3.5

4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Vs (k

m/s

)

porosity

Vs = 3.5817 + -4.7194 φφφφR = 0.83621

0 2 4 6 8 10 12 14 162

2.5

3

3.5

4

count

Vs (k

m/s

)


0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

impe

danc

e 1

07

(kg/

m3

)(m/s

)

porosity

I = 1.833 + -3.3081 φφφφR = 0.92461

0 5 10 15 200.6

0.8

1

1.2

1.4

1.6

1.8

2

count

impe

danc

e 1

07

(kg/

m3

)(m/s

)


Sandstones

Water saturated Measurement type: ultrasonic Effective pressure: 30, 40 MPa Reference: Han, D.-H., 1986, Effects of Porosity and Clay Content on Acoustic

Properties of Sandstones and Unconsolidated Sediments, Ph.D. dissertation, Stanford University.


Vp 3.1300 5.5200 4.0904 0.5051 Vs 1.7300 3.6000 2.4094 0.3966


impedance 6.6044 13.9656 9.7344 1.6378

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

3 3.5 4 4.5 5 5.5 6

dens

ity (g

/cm

3)

Vp (km/s)

Vp = 1.5694 + 0.19519 ρρρρR = 0.7312

0 5 10 152

2.1

2.2

2.3

2.4

2.5

2.6

2.7

count

dens

ity (g

/cm

3)


1.5

2

2.5

3

3.5

4

3 3.5 4 4.5 5 5.5 6

Vs (k

m/s

)

Vp (km/s)

Vs = -0.73327 + 0.7683VpR = 0.97843

1.2

1.3

1.41.5

1.6

1.7

1.8

1.9

2

3 3.5 4 4.5 5 5.5 6

Vp/V

s

Vp (km/s)

Vp/Vs = 2.1909 + -0.1178VpR = 0.71418

0

5

10

15

20

0 0.1 0.3 0.4

coun

t

porosity


3

3.5

4

4.5

5

5.5

6

0 0.1 0.3 0.4

Vp (k

m/s

)

porosity

Vp = 4.9442 + -5.2011 φφφφR = 0.72826

0 5 10 153

3.5

4

4.5

5

5.5

6

count

Vp (k

m/s

)


1.5

2

2.5

3

3.5

4

0 0.1 0.3 0.4

Vs (k

m/s

)

porosity

Vs = 2.9814 + -3.4843 φφφφR = 0.6213

0 2 4 6 8 10 12 14 161.5

2

2.5

3

3.5

4

count

Vs (k

m/s

)


6

8

10

12

14

0 0.05 0.1 0.15 0.2 0.25 0.3

impe

danc

e 1

06 (k

g/m

3 )(m/s

)

porosity

I = 12.954 + -19.612 φφφφR = 0.84679

0 2 4 6 8 10 12 146

8

10

12

14

count

impe

danc

e 1

06

(kg/

m3

)(m/s

)


Tight Gas Sandstones

Dry Measurement type: ultrasonic Effective pressure: 40 MPa Reference: Jizba, D.L., 1991, Mechanical and Acoustical Properties of Sandstones and

Shales, Ph.D. dissertation, Stanford University.

min max mean std. dev Vp 3.8090 5.5730 4.6688 0.3778 Vs 2.5940 3.5030 3.0603 0.2317


impedance 0.8902 1.4897 1.1744 12.5620

2.2

2.3

2.4

2.5

2.6

2.7

3.5 4 4.5 5 5.5 6

dens

ity (g

/cm

3)

Vp (km/s)

ρρρρ = 1.9617 = 0.11787VpR = 0.39424

0 1 2 3 4 5 6 7 82.2

2.3

2.4

2.5

2.6

2.7

count

dens

ity (g

m/c

m3

)


2.4

2.6

2.8

3

3.2

3.4

3.6

3.5 4 4.5 5 5.5 6

Vs (k

m/s

)

Vp (km/s)

Vs = 0.4855 + 0.55149VsR = 0.8997

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

3.5 4 4.5 5 5.5 6

Vp/V

s

Vp (km/s)

Vp/Vs = 1.2686 + 0.055098VpR = 0.38609

0

2

4

6

8

10

0 0.1 0.2

coun

t

porosity


3.5

4

4.5

5

5.5

6

0 0.1 0.2

Vp (k

m/s

)

porosity

Vp = 4.8689 + -3.8366 φφφφR = 0.37566

0 1 2 3 4 5 6 73.5

4

4.5

5

5.5

6

count

Vp (k

m/s

)mean: 4.6688std. dev: 0.3778no. of points: 42

2.4

2.6

2.8

3

3.2

3.4

3.6

0 0.1 0.2

Vs (k

m/s

)

porosity

Vs = 301491 + -1.7039 φφφφR = 0.27218

0 1 2 3 4 5 6 72.4

2.6

2.8

3

3.2

3.4

3.6

count

Vs (k

m/s

)


0.8

0.9

1.1

1.2

1.3

1.5

1.6

0 0.03 0.06 0.09 0.11 0.14 0.17

impe

danc

e 1

07

(kg/

m3

)(m/s

)

porosity

I = 1.2957 + -2.325 φφφφR = 0.68459

0 2 4 6 8 10

0.8

1

1.2

1.3

1.5

1.7

count

impe

danc

e 1

07

(kg/

m3

)(m/s

)


Limestone

Water saturated and calculated from dry data using Gassmann equations Measurement type: ultrasonic, resonant bar Effective pressure: 10, 30, 40, 50 MPa Reference: Cadoret, T., 1993, Effet de la Saturation Eau/Gaz sur les Propriétés

Acoustiques des Roches, Ph.D. dissertation, University of Paris, VII. Lucet, N., 1989, Vitesse et attenuation des ondes elastiques soniques et ultrasoniques

dans les roches sous pression de confinement, Ph.D. dissertation, University of Paris. Yale, D. P., and Jamieson Jr., W. H., 1994, Static and dynamic rock mechanical

properties in the Hugoton and Panoma fields, Kansas, SPE paper 27939, presented at the SPE Mid-Continent Gas symposium, Amarillo, Texas, May, 1994.



impedance 0.6871 1.5080 1.4320 0.21978

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

3 3.5 4 4.5 5 5.5 6

dens

ity (g

/cm

3)

Vp (km/s)

ρρρρ = 1.5138 + 0.20238VpR = 0.89241

0 2 4 6 8 102

2.1

2.2

2.3

2.4

2.5

2.6

2.7

count

dens

ity (g

/cm

3)


1.6

1.8

2

2.2

2.4

2.62.8

3

3.2

3 3.5 4 4.5 5 5.5 6

Vs (k

m/s

)

Vp (km/s)

Vs = -0.039208 + 0.5443VpR = 0.9586

1.6

1.7

1.8

1.9

2

2.1

3 3.5 4 4.5 5 5.5 6

Vp/V

s

Vp (km/s)

Vp/Vs = 1.9511 + -0.017304VpR = 0.14147

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5

coun

t

porosity


3

3.5

4

4.5

5

5.5

6

0 0.1 0.2 0.3 0.4 0.5

Vp (k

m/s

)

porosity

Vp = 5.6248 + -6.65 φφφφR = 0.88643

0 2 4 6 8 10 123

3.5

4

4.5

5

5.5

6

count

Vp (k

m/s

)


1.6

1.8

2

2.2

2.4

2.62.8

3

3.2

0 0.1 0.2 0.3 0.4 0.5

Vs (k

m/s

)

porosity

Vp = 3.053 + -3.8664 φφφφR = 0.9179

0 2 4 6 8 10 12 141.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

count

Vs (k

m/s

)


0.6

0.8

1

1.2

1.4

1.6

0 0.08 0.15 0.22 0.3 0.37 0.45

impe

danc

e 1

07

(kg/

m3

)(m/s

)

porosity

I = 1.4922 + -2.3319 φφφφR = 0.93198

0 2 4 6 8 100.6

0.8

1

1.2

1.4

1.6

1.8

count

impe

danc

e 1

07

(kg/

m3

)(m/s

)


High Porosity Sandstones

Water saturated Measurement type: ultrasonic Effective pressure: 35, 40 MPa Reference: Strandenes, Sverre, 1991, Rock Physics Analysis of the Brent Group

Reservoir in the Oseberg Field, SRB special volume.



impedance 7.5747 9.9798 8.5693 0.6717

2.1

2.2

2.3

2.4

2.5

2.6

2.7

3.4 3.6 3.8 4 4.2 4.4

dens

ity (g

/cm

3)

Vp (km/s)

ρρρρ = 1.4507 + 0.2192VpR = 0.47861

0 2 4 6 8 10 12 142.1

2.2

2.3

2.4

2.5

2.6

2.7

count

dens

ity (g

/cm

3)


1.9

2

2.1

2.2

2.3

2.42.5

2.6

2.7

3.4 3.6 3.8 4 4.2 4.4 4.6 4.8

Vs (k

m/s

)

Vp (km/s)

Vs = 0.0068626 + 0.56839VpR = 0.9438

1.6

1.65

1.7

1.75

1.8

1.85

1.9

3.4 3.6 3.8 4 4.2 4.4 4.6 4.8

Vp/V

s

Vp (km/s)

Vp/Vs = 1.7691 + -0.0037876VpR = 0.023229

0

5

10

15

20

0 0.1 0.3 0.4

coun

t

porosity


3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

0 0.1 0.3 0.4

Vp (k

m/s

)

porosity

Vp = 4.3038 + -2.2279 φφφφR = 0.51345

0 5 10 15 203.4

3.6

3.8

4

4.2

4.4

4.6

4.8

count

Vp (k

m/s

)


1.9

2

2.1

2.2

2.3

2.42.5

2.6

2.7

0 0.1 0.3 0.4

Vs (k

m/s

)

porosity

Vs = 2.4868 + -1.6268 φφφφR = 0.54027

0 2 4 6 8 101.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

count

Vs (k

m/s

)


6.7

7.5

8.3

9

9.8

10.5

0.13 0.17 0.21 0.25 0.29 0.33

impe

danc

e 1

06

(kg/

m3

)(m/s

)

porosity

I = 10.63 + -8.912 φφφφR = 0.71932

0 2 4 6 8 10 127.5

8

8.5

9

9.5

10

10.5

count

impe

danc

e 1

06

(kg/

m3

)(m/s

)


Poorly Consolidated Sandstones

Water saturated Measurement type: ultrasonic Effective pressure: 30 MPa Reference: Blangy, J.P., 1992, Integrated Seismic Lithologic Interpretation: The

Petrophysical Basis, Ph.D. dissertation, Stanford University.



impedance 4.8892 7.0157 5.7684 0.5118

2

2.1

2.2

2.3

2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2

dens

ity (g

/cm

3)

Vp (km/s)

Vs = 2.1003 + -2.4245 φφφφR = 0.81711

0 2 4 6 8 102

2.1

2.2

2.3

count

dens

ity (g

/cm

3) mean: 2.1101

std. dev: 0.053682no. of points: 38

1.2

1.3

1.4

1.5

1.6

1.7

2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2

Vs (k

m/s

)

Vp (km/s)

Vs = -0.31459 + 0.61088VpR = 0.90174

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2

Vp/V

s

Vp (km/s)

Vp/Vs = 2.4319 + -0.14998VpR = 0.32147

0

1

2

3

4

5

6

7

0.2 0.3 0.4

coun

t

porosity


2.4

2.5

2.6

2.7

2.8

2.93

3.1

3.2

0.2 0.3 0.4

Vp (k

m/s

)

porosity

Vp = 3.77420 + -3.4144 φφφφR = 0.75854

0 1 2 3 4 5 62.4

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

count

Vp (k

m/s

) mean: 2.7303std. dev: 0.18232no. of points: 38

1.2

1.3

1.4

1.5

1.6

1.7

0.2 0.3 0.4

Vs (k

m/s

)

porosity

Vs = 2.1003 + -2.4245 φφφφR = 0.81711

0 1 2 3 4 5 61.2

1.3

1.4

1.5

1.6

1.7

count

Vs (k

m/s

)


4

5

6

7

8

0.2 0.23 0.27 0.3 0.33 0.37 0.4

impe

danc

e 1

06

(kg/

m3

)(m/s

)

porosity

I = 8.9745 + -10.485 φφφφR = 0.82938

0 2 4 6 8 10

5

6

7

8

count

impe

danc

e 1

06

(kg/

m3

)(m/s

)


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Geophysics, 60, 1750-1755. Milholland, P., Manghnani, M.H., Schlanger, S.O., and Sutton, G.H., 1980, Geoacoustic modeling of deep-sea

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Mavko Notebook

Documents

rock properties

seismic measurements

fundamentals of rock

seismic velocity

terms of reservoir properties

seismic interpretation

upscaling seismic

reservoir complexity