Jack P. Dvorkin Dvorkin/Rockphysics 1 Preface

Dvorkin/RockPhysics

Rock Physics

Jack P. Dvorkin Dvorkin/RockPhysics 1 Preface

Interpretation of Seismic Data. The main geophysical tool for illuminating the subsurface is seismic. Seismic data yield a map of the elastic properties of the subsurface. This map is useful as long as it can be interpreted to delineate structures and, most important, quantify reservoir properties. Rock physics provides links between the sediment's elastic properties and its bulk properties (porosity, lithology) and conditions (pore pressure and pore fluid).

What is Rational Rock Physics. Rock physics’ mission is to translate seismic observables into reservoir properties, e.g., translate impedance into porosity. The simplest approach is to compile a laboratory data set, relevant to the site under investigation, where, e.g., impedance and porosity are measured on a set samples. The resulting impedance-porosity trend can be applied to seismic impedance to map it into porosity. The applicability of an empirical trend is as good as the data set it has been derived from. Extrapolation outside of the data set range is possible only if the physics is understood and theoretically generalized.

MISSION OF ROCK PHYSICS

Sediment Bulk Properties Sediment Elastic Properties (Porosity, Lithology, Permeability) (P-Impedance, Poisson’s Ratio) and Conditions (Fluid, Pressure)

Transform Function From Rock Physics •Measure •Relate •Understand

Controlled Experiment k Sw Vs Vp P Vsh φ

LAB LOGS Dvorkin/RockPhysics 1 Methods of Rock Physics Reflection and Inversion

Reflection amplitude carries information about elastic contrast in the subsurface. Inversion attempts to translate this information into elastic properties at a point in space.

Point properties are important because we are interested in absolute values of porosity and saturation at a point in space.

La Cira Norte. Courtesy Ecopetrol and Mario Gutierrez. Dvorkin/RockPhysics 1 Basics Elasticity

Stress Tensor Strain Tensor x1 x1 n T u

x2 x2

1 ∂ui ∂uj Ti = σijnj ε = ( + ) ij 2 x x x3 x3 ∂ j ∂ i i j; i j. σ ij = σ ji ≠ εij = ε ji ≠

Hooke's law: 21 independent constants c e ; c c c c , c c . σ ij = ijkl kl ijkl = jikl = ijlk = jilk ijkl = klij Isotropic Hooke's law: 2 independent constants (elastic moduli) 2 ; [(1 ) ]/E. σ ij = λδijεαα + µεij εij = + ν σij − νδijσαα λ and µ -- Lame's constants; ν -- Poisson's ratio; E -- Young's modulus.

Bulk Modulus Compressional Modulus Young’s Modulus K 2 / 3 M 2 and = λ + µ = λ + µ Poisson’s Ratio

E = µ(3λ + 2µ )/(λ + µ ) ν = 0.5λ /(λ + µ )

Young σ zz = Mε zz Compressional Z σ zz = Eε zz σ yy = λεzz = ν = −ε xx / ε zz X Shear [λ /(λ + 2µ )]σ zz = Y σ xx = σ yy = [ν /(1 − ν)]σ zz σ xy = σ xz = σ xz = 2µε xz ε xx = ε yy = σ = 0 0 yz 0 ε xx = ε yy = ε zz = ε xy = ε xy = ε xz = ε yz = Dvorkin/RockPhysics 1 Basics Dynamic and Static Elasticity

WAVE EQUATION ε ~ 10-7

u(z) Compressional σ(z) σ(z+dz) z Experiment A dz

Adzρu«« = A[σ(z + dz) − σ(z)] = Adz∂σ / ∂z ⇒ ρ∂ 2u / ∂t 2 = ∂σ / ∂z M M u / z 2u / t 2 (M / ) 2u / z2 V 2 2u / z2 σ = ε = ∂ ∂ ⇒ ∂ ∂ = ρ ∂ ∂ ≡ p∂ ∂

V = M / ρ = (K + 4G / 3)/ρ; V = G / ρ; Dynamic definitions: p s M V 2 ; G V 2 ; K (V 2 4V 2 / 3); (V 2 2V 2 ) = ρ p = ρ s = ρ p − s λ = ρ p − s

STATIC UNIAXIAL EXPERIMENT ε ~ 10-2

Sample 10156-58 70

50 Pc = 2000 psi = 14.1 MPa

Axial Stress (MPa) 30 Pc = 500 psi = 3.52 MPa

Pc = 0 20

10 Courtesy Ali Mese RADIAL AXIAL 0 -0.005 0 0.005 0.01 0.015 0.02 Strain Dvorkin/RockPhysics 2 Velocity and Saturation Gassmann’s Equations

In static (low-frequency) limit, pore fluid affects only the bulk modulus of rock

Gassmann's Equations -- Basis of Fluid Substitution

Bulk Modulus of Bulk Modulus of Dry Rock Pore Fluid Bulk Modulus of Rock w/Fluid K K K Sat = Dry + f K K K K (K K ) s − Sat s − Dry φ s − f Bulk Modulus of Mineral Phase Porosity

Shear Modulus of Shear Modulus of Rock w/Fluid GSat = GDry Dry Rock

The bulk modulus of rock saturated with a fluid is related to the bulk modulus of the dry rock and vice versa

φKDry − (1 + φ)K f KDry / Ks + K f KSat = Ks (1 − φ)K f + φKs − K f KDry / Ks

1 − (1 − φ)KSat / Ks − φKSat / K f KDry = Ks 1 + φ − φKs / K f − KSat / Ks

Velocity depends on the elastic moduli and density

4 V = (K + G )/ρ p Sat 3 Dry Sat

Vs = GDry / ρSat

ρ Sat = ρDry + φρFluid > ρDry Dvorkin/RockPhysics 3

Velocity and Porosity Summary of Theories Rock Physics Models: Velocity-Porosity

Wyllie et al. (1956) Time Average (Empirical) Total Porosity 1 φ 1 − φ = + Recommended Parameters P-wave Velocity V p VFluid VSolid in Solid Phase Rock Type Vsolid (km/s) Sandstone 5.480 to 5.950 P-wave Velocity Limestone 6.400 to 7.000 Sound Speed Dolomite 7.000 to 7.925 In Pore Fluid

Raymer et al. (1980) Equations (Empirical) 2 V p = (1 − φ) VSolid + φVFluid ⇐ φ < 0.37

6 6 Raymer’s equation is more Consolidated CLEAN accurate than Wyllie’s. SANDSTONES Sands SANDSTONES w/CLAY Neither one of the two 5 Raymer should be used to model 5 soft slow sands. Wyllie 4 Fast Sands

Vp (km/s) 4 Examples of applying 3 Wyllie’s and Raymer’s equation to sandstone Soft Slow Vp Raymer Predicted (km/s) Sands lab data 2 3 0 0.1 0.2 0.3 0.4 3456 Porosity Vp Measured (km/s)

Han (1986) Equations for Consolidated Sandstones (Empirical, Ultrasonic Lab Measurements)

Pressure Saturation Equations Comments

40 MPa 100% Water Vp=6.08-8.06φ Vs=4.06-6.28φ Clean Rock

40 MPa 100% Water Vp=5.59-6.93φ-2.18CVs=3.52-4.91φ-1.89C Rock w/Clay 30 MPa 100% Water Vp=5.55-6.96φ-2.18CVs=3.47-4.84φ-1.87C Rock w/Clay 20 MPa 100% Water Vp=5.49-6.94φ-2.17CVs=3.39-4.73φ-1.81C Rock w/Clay 10 MPa 100% Water Vp=5.39-7.08φ-2.13CVs=3.29-4.73φ-1.74C Rock w/Clay 5 MPa 100% Water Vp=5.26-7.08φ-2.02CVs=3.16-4.77φ-1.64C Rock w/Clay

40 MPa Room-Dry Vp=5.41-6.35φ-2.87CVs=3.57-4.57φ-1.83C Rock w/Clay

Vp and Vs are in km/s; the total porosity φ is in fractions; volumetric clay content in the whole rock (not in the solid phase) C is in fractions. Tosaya (1982) Equations for Shaley Sandstones (Empirical, Ultrasonic Lab Measurements)

Pressure Saturation Equations Comments

40 MPa 100% Water Vp=5.8-8.6φ-2.4CVs=3.7-6.3φ-2.1C Rock w/Clay

Vp and Vs are in km/s; the total porosity φ is in fractions; volumetric clay content in the whole rock (not in the solid phase) C is in fractions.

Eberhart-Phillips (1989) Equations for Shaley Sandstones (Empirical, Based on Han’s Data)

100% Water V p = 5.77 − 6.94φ −1. 73 C + 0.446[P − exp(−16.7P)] Saturation Vs = 3. 70 − 4.94φ −1. 57 C + 0.361[P − exp(−16.7P)]

Vp and Vs are in km/s; differential pressure P is in kilobars. 1 kb = 100 MPa.

Nur’s (1998) Critical Porosity Concept (Heuristic)

Solid-Phase Total Porosity Solid-Phase Critical Porosity Bulk Modulus Shear Modulus

KDry = KSolid (1 − φ / φc ) GDry = GSolid (1 − φ / φc )

Dry-Rock Dry-Rock Bulk Modulus Shear Modulus

Critical Porosity of Various Rocks

Sandstones 36% - 40% 1 Limestones 36% - 40% Basalt Dolomites 36% - 40% Dolomite Pumice 80% Chalks 55% - 65% Foam 0.8 Rock Salt 36% - 40% Glass Beads Cracked Igneous Rocks 3% - 6% Limestone Oceanic Basalts 20% Rock Salt Sintered Glass Beads 36% - 40% Glass Foam 85% - 90% 0.6 Clean Sandstone

Cracked 80 0.4 Igneous Rocks with Percolating 60 Cracks

Elastic Modulus / Mineral 0.2 40 Pumice with Honeycomb 20 0 Structure 0 0.2 0.4 0.6 0.8 1 Porosity/Critical Porosity Compressional Modulus (GPa) 0 0 0.2 0.4 0.6 0.8 1 Examples of using Critical Porosity Concept Porosity to mimic lab data Contact Cement Model (Theoretical)

Cement's Cement's Critical Porosity Compressional Shear Coordination Modulus Modulus Number

n(1 − φ )M S 3KDry 3n(1 − φ )G S K = c c n G = + c c τ Dry 6 Dry 5 20 Dry-Rock Dry-Rock Bulk Modulus Shear Modulus

2 −1.3646 Sn = An (Λn )α + Bn (Λn )α + Cn (Λn ), An (Λn ) = −0.024153⋅Λn , −0.89008 −1.9864 Bn (Λn ) = 0.20405⋅Λn , Cn (Λn ) = 0.00024649 ⋅Λn ; Contact 2 Cement Sτ = Aτ (Λτ ,νs )α + Bτ (Λτ ,νs )α + Cτ (Λτ ,νs ), Model 2 −2 2 0.079ν s +0.1754ν s −1.342 Aτ (Λτ ,νs ) = −10 ⋅(2.26νs + 2.07νs + 2.3) ⋅Λτ , 2 2 0.0274ν s +0.0529ν s −0.8765 Bτ (Λτ ,νs ) = (0.0573νs + 0.0937νs + 0.202) ⋅Λτ , 4 2 0.01867 2 0.4011 1.8186 − ν s + ν s − Elastic Modulus Cτ (Λτ ,νs ) = 10 ⋅(9.654νs + 4.945νs + 3.1) ⋅Λτ ;

Λn = 2Gc (1 − νs )(1 − νc )/[πGs (1 − 2νc )], Λτ = Gc /(πGs ); 0.5 α = [(2 / 3)(φ − φ)/(1− φ )] ; 0.30 0.35 0.40 c c Porosity

νc = 0.5(Kc / Gc − 2/3)/(Kc / Gc +1/3);

νs = 0.5(Ks / Gs − 2/3)/(Ks / Gs +1/3).

Uncemented Sand Model or Modified Lower Hashin-Shtrikman (Theoretical)

Uncemented Sand φ / φ 1 − φ / φ −1 4 K [ c c ] G , Model Dry = 4 + 4 − HM KHM + 3 GHM K + 3 GHM 3

φ / φc 1 − φ / φc −1 GHM  9KHM + 8GHM  GDry = [ + ] − z, z =  ; GHM + z G + z 6  KHM + 2GHM 

n2 (1 − φ )2 G2 1 5 − 4ν 3n2 (1 − φ )2 G2 1 Elastic Modulus K = [ c P]3 , G = [ c P]3 . HM 18π 2 (1 − ν)2 HM 5(2 − ν) 2π 2 (1 − ν)2 0.30 0.35 0.40 Porosity

In the above equations, K stands for bulk modulus and G stands for shear modulus. ν is Poisson’s ratio. Subscript ”c” with a modulus means “cement” and subscript ”s” means grain material. φ is the total porosity, and φc is critical porosity. P is differential pressure. All units have to be consistent. Constant Cement Model (Theoretical) Constant Cement φ / φb 1 − φ / φb −1 Model Kdry = ( + ) − 4Gb /3, Kb + 4Gb /3 Ks + 4Gb /3 φ / φ 1 − φ / φ G 9K + 8G G = ( b + b )−1 − z, z = b b b . dry G + z G + z 6 K + 2G b s b b Elastic Modulus

φb is porosity (smaller than φc) at which contact cement trend turns into constant φ cement trend. Elastic moduli with subscript “b” are the moduli at porosity φb. b These moduli are calculated from the contact cement theory with φ = φb. 0.30 0.35 0.40 Porosity

Model for Marine Sediments (Theoretical) 60

(1 − φ)/(1− φ ) (φ − φ )/(1− φ ) 4 80 K [ c c c ]−1 G , Dry = 4 + 4 − HM

K G G 3 Depth (mbsf) HM + 3 HM 3 HM 100

(1 − φ)/(1− φc ) (φ − φc )/(1− φc ) −1 G [ ] z, Data Dry = + − 120 G + z z This Model HM Suspension

0.5 0.55 0.6 0.65 1.5 1.6 1.7 1.8 GHM  9KHM + 8GHM  a Neutron Porosity P-Wave Velocity (km/s) z =  ; φ > φc . 6  KHM + 2GHM 

Consolidated Sand Model or Modified Upper Hashin-Shtrikman (Theoretical)

K φ / φ 1 − φ / φ 4 60 s K [ b b ]−1 G , Dry = 4 + 4 − s Modified Upper Kb + 3 Gs Ks + 3 Gs 3 50 Hashin-Shtrikman φ / φ 1 − φ / φ G 9K + 8G G = [ b + b ]−1 − z, z = s s s . 40 Dry G G 6 K 2G b + z s + z s + s 30

20 (K Phi ) Elastic moduli with subscript “b” are the moduli at porosity φb. These moduli can b b Bulk Modulus (GPa) be calculated from the contact cement theory with = , or chosen at some initial φ φb 10 point as suggested by data. 0 0 0.1 0.2 0.3 Porosity

Dry Non-Load-Bearing Clay Clay < 35% 3% < Clay < 18% All Samples Model (Theoretical) No Clay 5 The velocity (or elastic moduli) are plotted versus the load-bearing frame porosity φF = φt + C(1- φclay) instead of Vp (km/s) 4 total porosity φt. C is the volume of clay in rock, and φclay is the internal porosity of clay. 18% < Clay < 37% A scatter collapses onto a single trend 3 (right frame). 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 Porosity Porosity Load-Bearing Frame Porosity Examples: Velocity-Porosity

Quartz Contact Cement Constant Contact 3.5 4 Cement Constant Cement Cement Cement Contact Clay Cement Cement 3.0 Friable Sorting CORE DATA #2

3 SATURATED Vp (km/s) Elastic Modulus Initial Friable Sand 2.5 P-WAVAE VELOCITY (km/s) Pack #1 Non-Contact 0.30 0.35 0.40 0.25 0.30 0.35 0.40 Cement Porosity 2 Porosity 0.2 0.3 POROSITY

Friable 30 8 Contact Cement Equation 25 Contact Cement 7 20 M-Modulus (GPa) 15 6

10 P-Impedance 5

14 Unconsolidated Shale 12 Equation4

Friable 0.2 0.3 0.4 10 Total Porosity

8 Contact 6 Cement

Shear Modulus (GPa) 4

Han Clean Han Clean 2 Han 3-8% Clay Han 3-8% Clay Han >18% Clay Han >18% Clay 0 0 0.1 0.2 0.3 0.4 5 5 Acae_8 Porosity Depth > 9.5 kft Caliper < 9.5 Vp (km/s) Vp (km/s)

4 4

Acae_7 Depth > 9.5 kft Caliper < 9.5

0.1 0.2 0.1 0.2 Density-Porosity Density-Porosity Dvorkin/RockPhysics 4 Velocity and Porosity Fluvial Sandstones -- La Cira Case Study

Interpreting Impedance Inversion

11 SHALE 10

6 P-Wave Impedance 5 SAND SAND 4 0 0.1 0.2 0.3 CHANNEL Total Porosity SHALE

ALL DATA COURTESY ECOPETROL and MARIO GUTIERREZ Dvorkin/RockPhysics 5

Vp and Vs Summary of Theories Dvorkin/RockPhysics 5

Rock Physics Models: Vp and Vs

S-wave Velocity (km/s) P-wave Velocity (km/s) Castagna et al. (1985) Mudrock -- Empirical

Sand/Shale -- SW = 100% VS = 0.862 ⋅VP −1.172

S-wave Velocity (km/s) P-wave Velocity (km/s) Castagna et al. (1993) -- Empirical

Sand/Shale -- SW = 100% VS = 0.804 ⋅VP − 0.856

Krief et al. (1990) -- “Critical Porosity’ 2 2 2 2 VP_ Saturated − VFluid VP_ Mineral − VFluid Any mineral = V 2 V 2 Any fluid S_ Saturated S_ Mineral

S-wave Velocity (km/s) P-wave Velocity (km/s) Grinberg/Castagna (1992) 4 2 4 2 1 j j −1 −1 Empirical V {[ X a V ] [ X ( a V ) ] } S = ∑ i ∑ ij P + ∑ i ∑ ij P 2 i=1 j =0 i=1 j =0 Any mineral 4 S = 100% W X 1 ∑ i = i=1

i Mineral ai2 ai1 ai0 1Sandstone0 0.80416 -0.85588 2Limestone -0.05508 1.01677 -1.03049 3Dolomite0 o.58321 -0.07775 4Shale0 0.76969 0.86735

Willimas (1990) -- Empirical S-wave Velocity (km/s) P-wave Velocity (km/s) SW = 100%

Sand VS = 0.846 ⋅VP −1. 088 Shale VS = 0.784 ⋅VP − 0.893