1

Effective-Medium Models for Granular Rocks

GP170/2001 #5

THE END-POINT CONCEPT

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Bu

lk M

odu

lus

(GPa)

Porosity

SOLID GLASS

Fra

me-

Su

ppor

ted F

oam

Dis

inte

grate

d F

oam Honeycomb

Structure

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

M/M

s

Porosity

OpalineRocks

Chalks

M = Vp2ρ

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

G/G

s

Porosity

OpalineRocks

Chalks

G = Vs2ρ

GLASS FOAM OPAL and CHALK

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

P-W

ave

Vel

ocit

y (k

m/s)

S2

OG5OG4

OG1

KY1

Light TuffBenchtop

Room-Dry

Porosity

VOLCANIC ROCK Solid

Hi-PhiEndpoint

Diagenetic Path

UpperBound

LowerBound

Porosity

Ela

stic

Mod

ulu

s

2

Granular Rocks -- Critical Porosity Endpoint

Idealizing Pore Structure

Steel Beads

Sand

Grain-to-Grain Contacts

GP170/2001 #5

3

Evolution of Grain Contacts -- Diagenesis

Various Diagenetic Paths

Contact Cement

Friable Sand -- Lower Modified Hashin-Shtrikman

0.30 0.35 0.40E

last

ic M

odu

lus

Porosity

ContactCement

InitialSandPack

Friable

ConstantCement Schematic depiction of three

effective-medium models forhigh-porosity sands in theelastic-modulus-porosity planeand corresponding diagenetictransformations. The elasticmodulus may me compressional,bulk, or shear.

0.30 0.35 0.40

Ela

stic

Mod

ulu

s

Porositya

ContactCementModel

0.30 0.35 0.40Porosityb

Friable SandModel

0.30 0.35 0.40Porosity

ConstantCementModel

c 3.0

3.5

4.0

0.2 0.3

Vp (km

/s)

Porosity

Quartz GrainsQuartz Cement

Quartz GrainsClay Cement

a

2

3

0.25 0.30 0.35

Vp (km

/s)

Porosity

Clean Sandw/Water

b

Quartz GrainsQuartz Cement

Shale

φ < φc φ = φc φ >φc φ =1φ =0

Increasing Porosity

1

2

3

0.2 0.3 0.4

Vel

ocit

y (k

m/s)

Porositya

Vp

Vs

2

3

0.30 0.35 0.40

Vp (km

/s)

Porosityb

GP170/2001 #5

4

Equations

Contact Cement Model

The effective bulk ( Kdry ) and shear (Gdry ) moduli of dry rock are:

Kdry = n(1 − φc )McSn / 6, Gdry = 3Kdry / 5 + 3n(1 − φc )GcSτ / 20,

where φc is critical porosity; Ks and Gs are the bulk and shear moduli ofthe grain material, respectively; Kc and Gc are the bulk and shear

moduli of the cement material, respectively; Mc = Kc + 4Gc / 3 is thecompressional modulus of the cement; and n is the coordination number

-- average number of contacts per grain (8-9). Sn and Sτ are:

Sn = An(Λn )α2 + Bn (Λn )α + Cn (Λn), An (Λn ) = −0.024153 ⋅ Λn

−1.3646,Bn (Λn ) = 0.20405 ⋅ Λn

−0.89008 , Cn (Λn ) = 0.00024649 ⋅ Λn−1.9864 ;

Sτ = Aτ (Λτ , νs )α2 + Bτ (Λτ , νs )α + Cτ (Λτ , νs ),

Aτ(Λτ ,νs ) = −10−2 ⋅ (2.26νs

2 + 2.07νs + 2.3) ⋅ Λτ0.079 νs 2 + 0.1754νs −1.342 ,

Bτ (Λτ ,νs ) = (0.0573 νs2 + 0.0937νs + 0.202) ⋅ Λτ

0.0274 ν s2 +0.0529 νs − 0.8765,

Cτ (Λτ ,νs ) = 10−4 ⋅(9.654 νs

2 + 4.945νs + 3.1) ⋅ Λτ0.01867 νs 2 + 0.4011ν s −1.8186;

Λn = 2Gc (1− νs )(1− νc) / [πGs (1− 2νc )], Λτ = Gc / (πGs );α = [(2 / 3)(φc − φ ) / (1− φc )]

0.5 ;νc = 0.5(Kc / Gc − 2 / 3 ) /(Kc / Gc +1 / 3);νs = 0.5(Ks / Gs − 2 / 3 ) /(Ks / Gs +1 / 3).

One end point is at critical porosity. The elastic moduli of the dry rock at

that point are assumed to be the same as of an el astic sphere pack

subject to confining pressure. These moduli are given by the Hertz-

Mindlin (Mindlin, 1949) theory:

KHM = [n2 (1− φc)

2 G2

18π 2(1− ν)2P]

1

3 , GHM =5 − 4ν

5(2 − ν )[3n2 (1− φc)

2 G2

2π 2(1− ν)2P]

1

3 ;

where KHM and GHM are the bulk and shear moduli at critical porosity

φc , respectively; P i s the differential pressure; K , G , and ν ar e thebulk and shear moduli of the solid phase, and its Poisson's ratio,

respectively; and n is the coordination number.

The other end-point is at zero porosity and has the bulk ( K ) and

shear (G ) m oduli of the pure solid phase. These two points are

connected with the curves that have the algebraic expressions of the

lower Hashin-Shtrikman bound for the mixture of two components: the

pure solid phase and the phase that is the sphere pack.

At porosity φ the concentration of the pure solid phase (added to thesphere pack to decrease porosity) in the rock is 1 − φ / φc and that of thesphere-pack phase is φ / φc . Then the bulk ( KDry ) and shear (GDry )moduli of the dry frame are:

KDry = [φ / φc

KHM + 43 GHM+ 1 − φ / φc

K + 43 GHM]−1 − 4

3GHM ,

GDry = [φ / φc

GHM + z+ 1 − φ / φc

G + z]−1 − z, z = GHM

69KHM + 8GHMKHM + 2GHM

.

Friable Sand Model

Constant Cement ModelThe constant-cement model assumes that the initial porosity reduction from

critical porosity is due to the contact cement deposition. At some high porosity,

this diagenetic process stops and after that porosity reduces due to the

deposition of the solid phase away from the grain contacts as in the friable sand

model. This model is mathematically analogous to the friable sand model except

that the high-porosity end point bulk and shear moduli ( Kb and Gb ,

respectively) are calculated at some "cemented" porosity φb from the contact-cement model. Then the dry-rock bulk and shear moduli are:

Kdry = (φ / φb

Kb + 4Gb / 3+

1 − φ / φbKs + 4Gb / 3

)−1 − 4Gb / 3,

Gdry = (φ / φbGb + z

+1 − φ / φb

Gs + z)−1 − z, z =

Gb6

9Kb + 8GbKb + 2Gb

.

3

4

0.2 0.3 0.4

Vp (km

/s)

Porositya

Cement Quartz

CementClay

FriableSand

2.5

3.0

0.3 0.35 0.4

Vp (km

/s)

Porosityb

ContactCement

ConstantCement

FriableSand

GP170/2001 #5

5

ROCK PHYSICS DIAGNOSTIC

Often, an earth volume under examination has to be described by more that one rock physics law: differentdepth intervals may have distinctively different velocity-porosity trends due to variations in depositional anddiagenetic history. When building a rock physics model, one has to single out various velocity-porositytrends from the entire volume of data and assign these separate trends to appropriate depth intervals anddepositional sequences. This procedure is called rock physics diagnostic. Rock physics diagnostic istypically conducted on well log and core data.

20 40 60 80

.3

.4

.5

GR (API)

DE

PTH

(km

)

2 3 4 5

Vp (km/s)

Log Data

2 3 4 5

.6

.7

.8

Vp (km/s)

Dep

th (km

)

Common Saturation

Attribute a

trend to the

interval

Use for

forward/synthetic

modeling ...

0.25 0.3 0.35

Total Porosity

Cross-Plot

0.25 0.3 0.35

Total Porosity

10

15

20

25

M (G

Pa)

10

15

20

25

M (G

Pa)

Diagnose

GP170/2001 #5

6

Rock Physics Diagnostic in Sleipner -- Case Study

A vertical well, 15/9-16, penetrates the North Sea Sleipner gas/condensate reservoir comprised of Paleocene turbiditic sand. Porosity and permeability are availablefrom about 60 plugs that evenly cover the interval from 2380 to 2460 m. The vertical and horizontal permeabilities are practically identical. The sandstone is verywell sorted. The grains are predominantly quartz (average 80%) with the rest being feldspar (average 14%), mica (average 2.3%), and clay, mostly chlorite, (average2.2%). Traces of calcite and other minerals are also present. The contact cement in these rocks is quartz. The upper part of the well is saturated with gas, with thegas-water contact at 2430 m.

The interval under investigation can be subdivided into a high-resistivity zone (HRZ) overlaying a low-resistivity zone (LRZ) with the transition at about 2410 m. Thereis a diagenetic change associated with this transition. HRZ has a restricted distribution of diagenetic chlorite and up to 5% quartz cement. LRZ has a slightly largercontent of chlorite and a smaller degree of cementation.

1 10

2400

2450

Rt (Ohm m)

DR

KB

(m

)

a0.20 0.25 0.30

Porosityb1 10 100 1000

Permeability (mD)c10 20 30Comp. Modulus (GPa)d

Various parameters versus depth in well 15/9-16. a. Farresistivity. Gray curve is for LRZ. b. Log-derived (graycurve) and core porosity. Open symbols are for LRZ. c.Permeability. Open symbols are for LRZ. d. Dry-rock (graycurve) and directly measured (black curve) compressionalmodulus. Symbols are from dry-rock lab measurements at30 MPa.

-1

0

1

2

3

4

0.20 0.25 0.30

Log

10 k

(m

D)

Well Log Porositya

R = 0.48

-1

0

1

2

3

4

0.20 0.25 0.30

Log

10 k

(m

D)

Core Porosityb

R = 0.47

Horizontal permeability versus log-derived (a)and core (b) porosity. Gray lines show bestlinear fits. Correlation coefficients are given inthe graphs. They are poor.

PROBLEM

GP170/2001 #5

7

Sleipner -- Permeability and Elasticity

a. Compressional modulus versus porosity for the Osebergand Troll samples. The data displayed are for room-dryrocks at 30 MPa effective pressure. The upper curve is fromthe contact cement theory and the bottom curve is from thenon-contact cement theory. b. Permeability versus porosityfor the same datasets. The cartoons schematically show thelocation of cement among grains (contact for Oseberg andnon-contact for Troll).

a. Dry-frame compressional modulus versus porosity forthe Oseberg and Troll samples at 30 MPa, and HRZ andLRZ. The HRZ and LRZ data are selected at the depths ofpermeability datapoints. The upper curve is from thecontact cement theory and the bottom curve is from thenon-contact cement theory. b. Permeability versusporosity for the same datasets. The open triangles are forOseberg and Troll. The filled circles are for HRZ and theopen circles are for LRZ. The Oseberg and Troll data areplotted versus core porosity whereas the HRZ and LRZdata are plotted versus log-derived porosity.

Observation

10

20

30

0.20 0.25 0.30 0.35

OsebergTroll

Com

pre

ssio

nal M

odu

lus

(GPa)

Porositya101

102

103

104

0.20 0.25 0.30 0.35

Per

mea

bilit

y (m

D)

Porosityb

10

20

30

0.20 0.25 0.30

HRZLRZ

Com

pre

ssio

nal M

odu

lus

(GPa)

Porositya

100

101

102

103

0.20 0.25 0.30

OsebergTroll

Per

mea

bilit

y (m

D)

Porosityb

Position ofSleipner Sands

a. Idealized picture of granularrock with contact and non-contact cement. b. Calculatingthe amount of non-contactcement. Gray symbols showsome scattered datapoints. c.Volumetric fraction of contact(solid line) and non-contact(dotted line) cement versus depthin well 15/9-16. Gray verticalbar shows the extent of HRZ.

DIAGNOSTIC

Grain

Contact Cement

Non-ContactCement

10

20

30

0.2 0.3 0.4

Com

pre

ssio

nal M

odu

lus

(GPa)

Porosity

Datapoint

Projection

Contact CementTrajectory

a b

φCEM

0 0.1

2400

2450

Volume Fraction in Rock

DR

KB

(m

)

c

ContactCement

Non-ContactCement

GP170/2001 #5

8

Sleipner -- Permeability

Permeability (a) and permeability normalized bygrain size squared (b) versus the volumetric fractionof non-contact cement. Gray lines show best linearfits. Correlation coefficients are given in the graphs.

Rock Strengthand Sanding

Potential

The method of calculating the amount of contactand non-contact cement presented here is in facta method of diagnosing the texture of high-porosity sandstone from well-log data. Suchdiagnostic is important not only for obtaining ausable correlation for permeability but also forassessing the strength of rock and itssusceptibility to sanding.

Sanding

0

1

2

3

0 0.1

Log

10 k

(m

D)

Fraction of Non-Contact Cementa

R = 0.85

1

2

3

4

0 0.1

Log

10 k

/d

2 (m

D/m

m2 )

Fraction of Non-Contact Cementb

R = 0.86

0 0.1

2400

2450

Volume Fraction in Rock

DR

KB

(m

)

c

ContactCement

Non-ContactCement

No Sanding

GP170/2001 #5

9

4 6 8P-Impedance

w/W

ate

r

Mea

sure

d

0.25 0.30 0.35Porosity

1.8 2.0 2.2RHOB

0 0.5 1Sw

Nigeria -- Gas Sands -- Logs and Cross-Plots

25 50 75 100

4900

5000

5100

5200

5300

5400

5500

5600

GR

TV

D (ft

)

4

5

6

7

8

0.25 0.30 0.35 0.40

P-I

mped

an

ce

Porosity

Meaured

w/Water

4

5

6

7

8

0.25 0.30 0.35 0.40

P-I

mped

an

ce

Porosity

Meaured

w/Water

ABOVE GWC> 5287 ft

GP170/2001 #5

10

Nigeria Sands -- Diagnostic

5

6

7

8

0.25 0.30 0.35 0.40

P-I

mped

an

ce

Porosity

w/WaterCementQuartz

CementClay

FriableQuartz

Bioturbatedw/Illite

Shoreface

5

6

7

8

0.25 0.30 0.35 0.40

P-I

mped

an

ce

Porosity

w/Water

Clean SandsGR < 40

5

6

7

8

0.25 0.30 0.35 0.40

P-I

mped

an

ce

Porosity

w/WaterClean Sands

40 < GR < 50

Shoreface

Bioturbated

GP170/2001 #5

11

4.0 4.5Vp (km/s)

0.1 10 1000Permeability (mD)

CORE

0.10 0.15Porosity

CORE

0.10 0.15Porosity

LOG

Tight North Sea Gas Sandstone -- Diagnostic

0 0.5 1Sw

20 40 60

3980

3990

4000

4010

Gamma Ray

Dep

th (m

)

10

11

0.08 0.10 0.12 0.14 0.16

P-I

mped

an

ce

PHIF

Log Domain

0.10 0.15

1

10

100

Core Porosity

Per

mea

bilit

y (m

D)

GP170/2001 #5

12

10

11

0.08 0.10 0.12 0.14 0.16

P-I

mped

an

ce

PHIF

Log Domain

0.1 0.15

1

10

100

Core Porosity

Per

mea

bilit

y (m

D)

Below Shale Peak

Above Shale Peak

Tight North Sea Gas Sandstone -- Diagnostic

10

11

0.08 0.10 0.12 0.14 0.16P-I

mped

an

ce

PHIF

Raymer85% Quartz + 15% Clay

70% Quartz + 30% Clay

GP170/2001 #5

13

Diagnostic Applets

2 3 4 5Vp (km/s)

1 10Resistivity

Rt

Rxo

25 50 75GR

Dep

th

Marl

100 m

0.1 0.3Total Porosity

2

3

4

5

0.1 0.2 0.3 0.4

Vp (km

/s)

Total Porosity

Common Saturation

Contact Cement Unconsolidated Constant Cement Raymer

DIAGNOSTIC DISPLAY

0.20 0.25 0.30 0.35Total Porosity

Cement

Unconsolidated

ConstantCement

0.20 0.25 0.30 0.35

10

15

20

25

Total Porosity

Com

pre

ssio

nal

Mod

ulu

s

Water

Oil

Gas

4

5

6

7

8

0.2 0.3 0.4P-I

mped

an

ce

Total Porosity

ContactCementEquation

UnconsolidatedShale

Equation 4

5

6

7

8

0.2 0.3 0.4

P-I

mped

an

ce

Total Porosity

Brine

Gas

Oil

Diagnostic Curves Pore-Fluid Effect

Rock Physics Diagnostic allows the userto quickly derive rational impedance-porosity equations for variouslithologies present in the interval(Figures at right).

The diagnostic equations can beadjusted for any pore fluid for seismicidentification away from the well.

GP170/2001 #5