1 Pore Fluid and Elastic Properties of Rock

GP170/2001 #2

Pore Fluid and Elastic Properties of Rock

Change in Elastic Properties -- Han's Data

.30 HAN 40 MPa HAN 40 MPa 14 .25 WATER WATER

12 .20

10 .15 OIL: K = 0.5 GPa Poisson's Ratio RHO = 0.8 g/cc 8 WATER: .10 DRY Saturated-Rock P-Impedance K = 2.5 GPa OIL RHO = 1 g/cc OIL .05 8 10 12 14 6 8 10 12 14 Dry-Rock P-Impedance P-Impedance

Change in Elastic Properties -- Soft Sand

Upper 0.4 Shale

Reservoir w/Water

0.3 Poisson's Ratio

0.2 Reservoir w/Hydrocarbons

4 6 8 P-Impedance 1 GP170/2001 #2

Physics of Pore Fluid Effect on Elastic Properties

Hooke’s Law of Linear Isotropic Elasticity (Compression Corresponds to Positive Stress and Strain)

s ij = ld ij eaa + 2Geij Þ eij = [(1 + n )s ij - nd ijs aa ]/E Þ 1 1 1 K: Bulk Modulus; G: Shear Modulus eij = (s ij - d ij s aa ) + d ij s aa . 2G 3 9K

Adding Pore Pressure: Pore pressure only affects volumetric deformation 1 1 1 1 eij = (s ij - d ij s aa ) + d ij s aa - dij Pp 2G 3 9K 3H

Pc Confining Volumetric Deformation (Hydrostatic) Pressure Pore Pressure q º eaa º e11 + e22 + e33 = Pc / K - Pp / H = (P - a P )/K Pp c p P =s = s = s a = K H c 11 22 33 ; / .

K a = 1 - Dry K Solid

In static (low-frequency) approximation, pore fluid interacts with rock through pore pressure

Effective Pressure and Stress Def Def e 1 e 1 e 1 e Pe Pe = Pc - a Pp ; s ij = s ij - ad ij Pp Þ eij = (s ij - d ijs aa ) + d ijs aa Þ q = 2G 3 9 K K

2 GP170/2001 #2

Physics of Pore Fluid Effect on Elastic Properties

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

Gassmann's Equation -- 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 f (K - K ) s Sat s Dry s f Bulk Modulus of Mineral Phase Porosity

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

f K - (1 + f )K K / K + K 1 - (1 - f )K / K - f K / K K = K Dry f Dry s f K = K Sat s Sat f Sat s (1 - f )K + f K - K K / K Dry s 1 + f - f K / K - K / K f s f Dry s s f Sat s

4 V = (K + G )/r p Sat 3 Dry Sat

Vs = GDry / r Sat r = r + fr > r Sat Dry Fluid Dry 3 GP170/2001 #2

Fluid Effect on Velocity, Impedance, and Modulus

North Sea Log Data 3.4 Fluid-Substituted 4 Top Vp = (K Sat + GDry )/ r Sat Pay 3 3.2 Bottom Vs = GDry / r Sat r = r + fr > r 3.0 Sat Dry Fluid Dry

2.8 Vp (km/s) Han's Laboratory Data 2.6

HAN 40 MPa HAN 40 MPa 2.4 4.5 5

WATER WATER 4.0 7 4 OIL: OIL: K = 0.5 GPa K = 0.5 GPa RHO = 0.8 g/cc RHO = 0.8 g/cc WATER: WATER: Saturated-Rock Vp (km/s) Saturated-Rock Vp (km/s) 3.5 OIL K = 2.5 GPa OIL K = 2.5 GPa RHO = 1 g/cc RHO = 1 g/cc P-Impedance 3 6 3 4 5 3.5 4.0 4.5 Dry-Rock Vp (km/s) Dry-Rock Vp (km/s)

3 1.8 HAN 40 MPa WATER Fluid-Substituted 25

1.7 WATER HAN 40 MPa

OIL: 1.6 OIL: 20 K = 0.5 GPa K = 0.5 GPa RHO = 0.8 g/cc RHO = 0.8 g/cc WATER: Saturated-Rock Vp/Vs WATER: Saturated-Rock Vs (km/s) OIL K = 2.5 GPa 1.5 K = 2.5 GPa OIL M-Modulus (GPa) 2 RHO = 1 g/cc RHO = 1 g/cc

2 3 1.5 1.6 1.7 1.8 15 Dry-Rock Vs (km/s) Dry-Rock Vp/Vs

0.1 0.2 0.3 0.1 0.2 0.3 Density-Porosity Density-Porosity 4 GP170/2001 #2

Approximate Fluid Substitution Equations -- Vp Only

M is the compressional modulus 4 M = r V 2 = K + G b p 3 f M - (1 + f )K M / M + K 1 - (1 - f )M / M - f M / K M = M Dry f Dry s f M = M Sat s Sat f Sat s (1 - f )K + f M - K M / M Dry s 1 + f - f M / K - M / M f s f Dry s s f Sat s

EXAMPLE

A soft sand sample of 35% porosity Solid: The dry-rock density is 1.722 g/cm3 (a) Pure quartz -- K = 36.6 GPa; G = 45 GPa Fluid bulk modulus 2.5 GPa; density 1 g/cm3 (b) 70% quartz + 30% clay -- K = 30 GPa; G = 25.5 GPa

Dry-rock lab data -- velocity versus pressure Water-Substituted

8 2.2 2.6 Vp-Only Vp-Only 7 2.0 Vp M-Modulus 2.4 Gassmann 6 Gassmann 1.8 Dry Sandstone Dry Sandstone 35% Porosity 5 35% Porosity 2.2 1.6 4 Vp (km/s)

Velocity (km/s) K 2.0 1.4 Elastic Moduli (GPa) G Dry Dry Vs 3 70% Quartz 1.8 30% Clay 1.2 2 100% Quartz

10 20 30 10 20 30 10 20 30 10 20 30 Pressure (MPa) Pressure (MPa) Pressure (MPa) Pressure (MPa)

5 GP170/2001 #2

Partial Saturation -- Fluid's Bulk Modulus

If pore pressure increment is DP then the volume change of water is - VS DP K , and the volume change of w / w gas is - V(1 - S )DP/ K , where K and K are the bulk moduli of water and gas, respectively. SOLID w g w g

WATER The total change of volume is then DV = - [VS DP / K + V(1 - S )DP/ K ]. The bulk modulus of the w w w g water-gas mixture K f can be now calculated from this total volume change and pressure increment: GAS DV DP 1 S 1 - S = - Þ = w + w . V K K K K f f w g

For any number N of fluid phases of saturation Si , the effective bulk modulus is the Reuss low bound of their bulk moduli:

1 N S = i . K å K f i=1 i

Typical Gassmann Effect -- Partial Gas Saturation

2 OTTAWA SAND 2.0 Water 2.25 GPa Water 2.25 GPa Gas 0.005 GPa Gas 0.005 GPa

1.5 1 Vp Velocity (km/s) Mixture Bulk Modulus (GPa) 1.0 Vs 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Sw Sw 6 GP170/2001 #2

Partial Saturation -- Various Results of Fluid Substitution

Velocity Poisson's Ratio

5.0 5.0

0.30 4.5 4.5

Water/Gas Water/Oil Water/Gas 25% Porosity 25% Porosity 25% Porosity

4.0

4.0 Poisson's Ratiop Vp (km/s) 3.5 0.25 Vp (km/s)

3.5

3.0 Rock Softening

3.0 2.5 0.20

2.5 2.0 0 0.5 1 0 0.5 1 0 0.5 1 Sw Sw Sw

7 GP170/2001 #2

Recovery Monitoring Principles

Vp Vp 3 2.5 BRINE BRINE

2 Vs Vs

Velocity (km/s) OIL 1 OIL 0 10 20 30 0 10 20 30

Vp (km/s) 2.0 Diff. Pressure (MPa) Pore Pressure (MPa) GAS GAS

Phi = 0.35 Phi = 0.35

5 10 15 20 25 305 10 15 20 25 30 Effective Pressure (MPa) Pore Pressure (MPa)

2.5 Water Flood BRINE .4 BRINE

.3 Pore

Vp (km/s) Gas Pressure Pore Injection Pressure 2.0 OIL .2 OIL Gas out of

Solution GAS Poisson's Ratio GAS .1 5 10 15 20 25 30 NORTH SEA Pore Pressure (MPa) SAND Pore 0 Pressure

Such diagrams can be used to produce synthetic 3 4 5 well logs, based on production scenarios, and then P-Impedance (km/s g/cc) produce synthetic seismic. 8 GP170/2001 #2

Forward Modeling of Saturation -- Well Logs

2.2 S_26

S_26 Top Bottom .35 2.3

Pay Brine .30 TVD (km)

2.4 Poisson's Ratio

Pay .25 Oil Nu 3.1 Core Ip 3.1

2.5 5 6 7 8 9 P-Impedance

50 100 150 200 1 10 100 0 0.5 1 0.1 0.2 0.3 6 7 0.25 0.3 0.35 GR Resistivity SwT Porosity Ip Poisson's Ratio

9 GP170/2001 #2

Effect of Saturation and Tool -- Heavy Oil

2.3

Dipole

2.2Vp (km/s) Monopole PhiRHO

2.1

2.0 Depth (ft) 200 ft

1.9

Dipole

1.8 0.30 0.35 0.40 Porosity

Mono NPHI

40 60 80 100 120 20 40 60 80 100 2.0 2.1 2.2 2.3 0.2 0.3 0.4 0.5 1.8 2.0 2.2 2.4 2.6 0.6 0.8 1 0.40 0.45 GR Resistivity RHOB Porosity Vp (km/s) Vs (km/s) Poisson's Ratio

10 GP170/2001 #2 Pore Fluid and Frequency -- Velocity/Frequency Dispersion

5000 SATURATED Limestone 3.0 5MPa Pressure 4.5 Dispersion may be seen 2.8 in heavy oil rock 4000 4.0 DRY 2.6 Vp 3.5 2.4 4700

Vs (km/s) 3000 Vp (km/s)

2.2 Vp and Vs (m/s) 3.0 GASSMANN 2.0 10000 cPs 2.5 4600 1.8 2000 Vs 100 cPs 0 20 40 60 80 100 0 20 40 60 80 100 Differential Pressure (MPa) Differential Pressure (MPa) 4 5 6 Vp (m/s) 1 cPs Log Frequency (Hz) 4500

0.2 Effect of Viscosity Limestone 5MPa Pressure SQUIRT FLOW 4400

0 1 2 3 4 5 6 P Log Frequency (Hz) 0.1

LAB MEASUREMENTS HAVE TO BE CONDUCTED ON ROOM-DRY SAMPLES 1/Q (Inverse Quality Factor) S

0 4 5 6 Log Frequency (Hz) LIMESTONE

3.1 Partial Saturation

Soft thin fracture 3.0

Squirting flow between saturated Vp (km/s) 50 kHz Squirting flow between soft and stiff 2.9 pore space and a gas pocket pore space

2.8 1 kHz 11 0 0.2 0.4 0.6 0.8 1 Saturation GP170/2001 #2

Saturation and Poisson's Ratio

High-porosity sands -- lab room-dry data Poisson's ratio -- fluid substitution

4 2.5 .4 .4

Rock w/GAS Rock w/GAS Slow w/Water

FAST SS .3 .3

2.0 FAST SS 3 Fast w/Water .2 FAST .2 Vs (km/s) Vp (km/s) Poisson's Ratio SLOW SS Poisson's Ratio SLOW SS 1.5

.1 .1 2 SLOW

Rock w/GAS 1.0 0 0 0.2 0.3 0.4 0.2 0.3 0.4 0.2 0.3 0.4 0.2 0.3 0.4 Porosity Porosity Porosity Porosity

.4 Slow w/Water

.3 Fast w/Water

Seismic detection crossplot .2 Poisson's Ratio

.1 Fast w/GAS

Slow w/GAS 0 3 4 5 6 7 8 P-Impedance 12 GP170/2001 #2

Using Offset to Differentiate Sand Type

Q Forward model .4

Fast w/GAS Slow w/Water 7

.3 SHALE: Vp = 3 km/s; Poisson's Ratio = 0.35; RHOB = 2.3

6 Ip Poisson's Ratio .2 Soft Water Sand: Fast Gas Sand: PHI = 0.28; PHI = 0.276;

5 Vp = 2.81 km/s; Vp = 3.22 km/s; Slow w/Water Fast w/GAS Poisson's Ratio = 0.278; Poisson's Ratio = 0.127; RHOB = 2.19 RHOB = 1.92 .1 0.2 0.3 0.4 0.2 0.3 0.4 Porosity Porosity

Forward model 0

Slow w/Water

-0.1 Rpp

Fast w/GAS

-0.2 0 10 20 30 40 Angle of Incidence 13 GP170/2001 #2

PATCHY SATURATION CONCEPT

Low Frequency: Easy Cross-Flow High Frequency: No Cross-Flow Slight Shale Content Homogeneous Saturation Patchy Saturation Variation D

GAS LIQUID Rock w/Liquid Rock w/Liquid

Rock w/Gas Rock w/Gas

3.2 LIMESTONE 50 kHz 100 kHz 3.1 FREQUENCY 1 kHz

3.0

2.9 Drainage Vp (km/s) 2.8

2.7 Imbibition 2.6 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 Water Saturation Water Saturation Water Saturation 14 GP170/2001 #2

Patchy Saturation and Elastic Properties in Well Logs

Dry Rock Dry Rock at 20 MPa at 20 MPa 0.4 Han Ottawa+ Jizba Clay 0.4 0.3 OTTAWA SAND OTTAWA SAND 0.2 2.0 0.3 Poisson's Ratio 0.1 Patchy Strandenes Blangy Patchy 0

Vp (km/s) 0.2 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 Poisson's Ratio Porosity Porosity 6 1.5 Homogeneous Homogeneous Dry Rock Dry Rock at 20 MPa at 20 MPa 0.1 5 0 0.5 1 0 0.5 1 Brine Saturation Brine Saturation 4

3 Vp (km/s)

2

1460 1 Cut-Off0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 Porosity Porosity Shale Vs Vp from GR 1480Depth (m)

1500

Clay 1520 Core

0 0.2 0.4 0.2 0.4 0 0.5 1 1 2 0 0.2 VSHALE and Clay Porosity Sw Velocity (km/s) Poisson's Ratio 15