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 Ks - KSat Ks - KDry f (Ks - K 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 3.0 r Sat = r Dry + fr Fluid > r 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 = - [VSwDP / Kw + V(1 - Sw )DP/ K g ]. The bulk modulus of the 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 f K f Kw Kg 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.