6.shalysands

Stanford Rock Physics Laboratory - Gary Mavko

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Velocity, Porosity, Clay Relations


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Courtesy Per Avseth

What Controls Amplitude over thisNorth Sea Turbidite?

Lithology, porosity, pore fluids, stresses… but also sedimentation and diagenesis


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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.

L.1

“Life Story” of a Clastic Sediment

Deposition

Burial


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We observe that the clastic sand-clay system is divided intotwo distinct domains, separated by a critical porosity φc.Above φc, the sediments are suspensions. Below φc , thesediments are load-bearing.

Critical Porosity

L.1


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Critical Porosity

Traditionally, bounding methods have been considered notvery useful for quantitative predictions of velocity-porosityrelationships, because the upper and lower bounds are sofar apart when the end members are pure quartz and purewater.

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

• φ > φc, fluid-bearing suspensions. In the suspensiondomain the velocities are described quite well by the Reussaverage (iso-stress condition).

• φ < φc, load-bearing frame. Here the situation appears tobe more complicated. But again, there is a relatively simplepattern, and we will see that the Voigt average is useful.


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The first thing to note is that the clean (clay free) materialsfall along a remarkably narrow trend. These range fromvery low porosity, highly consolidated sandstones, to highporosity loose sand.

(Data from Yin et al., 1988; Han et al., 1986. Compiled andplotted by Marion, D., 1990, Ph.D. dissertation, StanfordUniversity.

L.2


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Amos Nur discovered that this narrow trend can bedescribed accurately with a modified Voigt bound. Recallthat bounds give a way to use the properties of the “pure”end members to predict the properties in between. The trickhere is to recognize that the critical porosity marks the limitsof the domain of consolidated sediments, and redefine theright end member to be the suspension of solids and fluids atthe critical porosity.

L.3

Critical “Mush”


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The Modified Voigt Bound

Velocity in rocks

The usual Voigt estimate of modulus

Modified Voigt estimate of modulus

€

VP =M ρ

€

ρ = 1−φ( )ρmineral +φρfluid

€

M = 1− φ( )Mmineral + φMfluid

€

M = 1− φ ( )Mmineral + φ Mcritical"mush"

€

φ =φφc

€

0 ≤ φ ≤ φc 0 ≤ φ ≤ 1


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L.4

Example of critical porosity behavior in sandstones.


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


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L.5

Chalks


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L.6


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Han’s Laboratory Study on Effects ofPorosity and Clay in Sandstones

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

L.7


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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

Clean sand line

C=.05.15.25

.35

C=.05.15

.25.35

Vp = (5.6-2.1C) - 6.9φ

Vs = (3.5-1.9C) - 4.9φ

Han’s Study on Phi-Clay in Sandstones


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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 – 8.06φVS = 4.06 – 6.28φ

VP = 5.02 – 5.63φVS = 3.03 – 3.78φ

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

VP = 5.41 – 6.35φ – 2.87CVS = 3.57 – 4.57φ – 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


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Han’s water-saturated ultrasonic velocity data at40 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 clayvolume fractions.

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

Shaly Sandstones (determined from 70 samples)

Water saturated40 MPa Vp = 5.59 - 6.93φ - 2.18C Vs = 3.52 - 4.91φ - 1.89C30 MPa Vp = 5.55 - 6.96φ - 2.18C Vs = 3.47 - 4.84φ - 1.87C20 MPa Vp = 5.49 - 6.94φ - 2.17C Vs = 3.39 - 4.73φ - 1.81C10 MPa Vp = 5.39 - 7.08φ - 2.13C Vs = 3.29 - 4.73φ - 1.74C5 MPa Vp = 5.26 - 7.08φ - 2.02C Vs = 3.16 - 4.77φ - 1.64C

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

L.9


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

L.12


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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

L.13

Unconsolidatedmixes of sandand kaolinite

Mixtures have a minimum in porosity that isless than either the sand or clay

observed

modeled


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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

L.14

Unconsolidatedmixes of sandand kaolinite

Mixtures have a maximum in velocity

observed

modeled


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

L.15

Dispersed sand-clay mixes tend toform “V”-shape in various domains


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

Density vs. Neutron Porosity PoorlyConsolidated Shaly Sands

Laminar ClayModel

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

Dispersed Clay Model

Dispersed clay “V”-shape in nphi-rhobdomain


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0 0.2 0.4 0.6 0.8 11000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

Porosity

Vp

Statoil B, Brine Substituted

sandy leg

shaley leg

Example for fluvial sands

Each color represents adifferent fining-upwardsequence


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Schlumberger, 1989

Density Porosity vs. NeutronPorosity in Shaly Sands

Sho

0.5

0.4

0.3

0.1

Q

QuartzPo in t

0.1

0.2

0.3 0.4 0.5

G asSand

Sd

C

ClSh

0.2

φN

φD

A

B

L.19

To wate

r poin

t

To w

ater

poi

nt

To D

ry C

lay

poin

t

Clean Wate

r Sands


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Yin’s laboratory measurements on sand-claymixtures

L.20


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Yin’s laboratory measurements onsand-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


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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.

L.22


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Yin's laboratory measurements onsand-clay mixtures.

L.23


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L.360

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 andclearly shows the steep cementing trend, which would befavorable for mapping velocity (or impedance) to porosity.Other data sets from the Troll and Oseberg indicate muchshallower trends.

Velocity-porosity trend is non-unique and is determinedby the geologic process that controls porosity


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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 thegeologic process that controls variations in porosity. Ifporosity is controlled by diagenesis and cementing, weexpect a steep slope – described well by a modified upperbound. If it is controlled by sorting and clay content(depositional) then we expect a shallower trend – describedwell by a modified lower bound.


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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


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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


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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


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0 0.1 0.2 0.3 0.4 0.50

1000

2000

3000

4000

5000

6000

Porosity

Vp

Data Before (blue) and After (red) Cementing

Cementing Trend

0 500 1000 1500 2000 2500 3000 3500 40000

1000

2000

3000

4000

5000

6000

V s

Vp

Data Before (blue) and After (red) Cementing

Cementing Trend

Decrease porosity 5% by Cementing

L39


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0 500 1000 1500 2000 2500 3000 3500 40000

1000

2000

3000

4000

5000

6000

V s

Vp

Data Before (blue) and After (red) Sorting

Sorting Trend

0 0.1 0.2 0.3 0.4 0.50

1000

2000

3000

4000

5000

6000

Porosity

Vp

Data Before (blue) and After (red) Sorting

Sorting Trend

Decrease porosity 5% by Sorting

L39


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L.37

Sand, Shale Depth Trends

What about intermediate facies?3000

2000

20 40 60

P –

Vel

ocity

Porosity (%)

Clean Sand Compaction

Shale Compaction


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0 0.2 0.4 0.6 0.8 11000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

Porosity

Vp

Statoil B, Brine Substituted

sandy leg

shaley leg

Sand-Clay “V” Mixing Law

Sandpoint

Sandpoint


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3000

2000

20 40 60

P –

Velo

city

Porosity (%)

Clean SandShale

0 MPa5 MPa

50 MPaShaleySand

Sand, Shale Depth Trends


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Clean SST

Clayey Shale

Depth Progression in a Fluvial Sequence


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Diagenetic Trend

DepositionalTrend

Porosity

Vp

Diagenetic Trend

DepositionalTrend

Reservoir quality

GR

Porosity ( Density)

Vp

Florez, Stanford University, 2002


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GrainContactcement

A B

R a a

Non-contactcement Scheme 1 Scheme 2

C

Dvorkin’s Cement Model

M c = ρcVPc2

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

ρc VPc VSc S n

S τ

Jack Dvorkin introduced a cement model that predicts thebulk and shear moduli of dry sand when cement is depositedat grain contacts. The model assumes that the cement iselastic and its properties may differ from those of the grains. It assumes that the starting framework of cemented sand isa dense random pack of identical spherical grains withporosity , and the average number of contacts pergrain C = 9. Adding cement reduces porosity and increasesthe effective elastic moduli of the aggregate. The effectivedry-rock bulk and shear moduli are (Dvorkin and Nur, 1996)

where

φ0 ≈ 0.36

€

Keff =16

C 1−φ0( )Mc

) S n

€

µeff =35

Keff +320

C 1−φ0( )µc

) S τ

€

µC = ρcVSc2


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where 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 ModelConstants in the cement model:

ν νc

€

) S n = An (Λn )α

2 + Bn (Λn )α + Cn (Λn )

€

) S τ = Aτ (Λτ ,ν )α

2 + Bτ (Λτ ,ν)α + Cτ (Λτ ,ν ),

€

Aτ (Λτ ,ν ) = −10−2 ⋅ (2.26ν 2 + 2.07ν + 2.3) ⋅ Λτ0.079ν 2 +0.1754ν −1.342,

€

Bτ (Λτ ,ν ) = (0.0573ν 2 + 0.0937ν + 0.202) ⋅ Λτ0.0274ν 2 +0.0529ν −0.8765,

€

Cτ (Λτ ,ν ) =10−4 ⋅ (9.654ν 2 + 4.945ν + 3.1) ⋅ Λτ0.01867ν 2 +0.4011ν −1.8186;

€

Λn = 2µc (1−ν )(1−ν c ) /[πµ(1− 2ν c )]€

An (Λn ) = −0.024153 ⋅ Λn−1.3646,

€

Cn (Λn ) = 0.00024649 ⋅ Λn−1.9864

€

Bn (Λn ) = 0.20405 ⋅ Λn−0.89008

€

Λτ = µc /(πµ )

€

α = a /R

€

µc

€

µ


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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/R

The 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:

α φ

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 evenlydeposited on the grain surface:

α = 2 φ0 – φ

3 1 – φ0

0.5= 2Sφ0

3 1 – φ0

0.5

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

Dvorkin’scement model

GrainContactcement

A B

R a a

Non-contactcement Scheme 1 Scheme 2

C


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If the cement's properties are identical to those of the grains,the cementation theory gives results which are very close tothose of the Digby model. The cementation theory allowsone to diagnose a rock by determining what type of cementprevails. For example, it helps distinguish between quartzand clay cement. Generally, Vp predictions are much betterthan Vs predictions.

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


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Sand models can be used to “Diagnose” sands


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Dvorkin’s Uncemented Sand ModelThis model predicts the bulk and shear moduli of dry sandwhen cement is deposited away from grain contacts. Themodel assumes that the starting framework of uncementedsand is a dense random pack of identical spherical grainswith porosity , and the average number of contactsper grain C = 9. The contact Hertz-Mindlin theory gives thefollowing expressions for the effective bulk ( ) andshear ( ) moduli of a dry dense random pack ofidentical spherical grains subject to a hydrostatic pressureP:

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 – φ0

2 G 2

2π2 1 – ν 2 P1/3

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

ν


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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 HM6

9KHM + 8G HMK HM + 2GHM

+ 1 – φ / φ0

G + GHM6

9K HM + 8GHMKHM + 2GHM

]–1

– GHM6

9KHM + 8G HMKHM + 2GHM

Illustration of the modified lower Hashin-Shtrikman bound for various effectivepressures. The pressure dependence follows from the Hertz-Mindlin theory

incorporated into the right end member.


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Dvorkin’s Uncemented Sand ModelThis model connects two end members: one has zeroporosity and the modulus of the solid phase and the otherhas high porosity and a pressure-dependent modulus asgiven by the Hertz-Mindlin theory. This contact theoryallows one to describe the noticeable pressure dependencenormally observed in sands.The high-porosity end member does not necessarily have tobe calculated from the Hertz-Mindlin theory. It can bemeasured experimentally on high-porosity sands from agiven reservoir. Then, to estimate the moduli of sands ofdifferent porosities, the modified Hashin-Strikman lowerbound formulas can be used where KHM and GHM are set atthe measured values. This method provides accurateestimates for velocities in uncemented sands. In the figuresbelow the curves are from the theory.

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

unconsolidated North Sea samples.


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


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2 3

2 .1

2 .2

2 .3

Vp (km/s)

Dep

th (

km)

Well #1

A40 80 120

G RB2 3 4

1 .7

1 .8

1 .9

Vp (km/s)

Well #2

Dep

th (

km)

C

Marl

Limestone

40 80 120G RD

2 .5

3

3 .5

0.25 0 .3 0.35 0 .4

Vp

(km

/s)

Porosity

Contact CementL i n e

UnconsolidatedL i n e

ConstantCement Fraction (2%) Line

Well #1

Well #2

North Sea Example

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


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Sorting Analysis of Thin-Sections

0.4mm0.4mm

0.4mm0.4mm


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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

6.shalysands

Documents

shaly sandstones

stanford university

effects of porosity

mpa vp

porosity relationships

low porosity

clay relations

velocityporosity relationship