Fluid Substitution with 261
5 4.5 ρ fluid S w ρ w + S o ρ o + S g ρ g Vp (km/s) 4 3.5 K fluid S w K w + S o K o + S g K g Patchy Saturation Drainage 3 2.5 2 Fine-scale mixing 1 = S w + S o + S g K fluid K w K o K g 0 0.2 0.4 0.6 0.8 1 Sw (fraction) Imbibition Knight and Nolen-Hoeksema (GRL, 1990) found saturation hysteresis at ultrasonic frequencies. We know now that velocities depend, not just on saturation, but also on the scales at which the phases are mixed. The curve labeled imbibition is typical when phases are mixed at a fine scale. The curve labeled drainage is typical when the phases are mixed at a coarse scale -- which we call patchy. K.1 262
Vp (km/s) 2.45 2.4 2.35 2.3 2.25 2.2 2.15 sandstone porosity = 30% patchy homogeneous 0 0.2 0.4 0.6 0.8 1 Oil Saturation 263
Increasing water saturation a. b. c. d. Endres and Knight (The Log Analyst, 1989) modeled different microdistributions of pore fluids and gas in the stiff and soft portions of the pore space. They concluded that the scale and distribution of fluids influence velocities. K.4 264
Patchy Saturation High frequency response Isolated patches some stiff, some softer Overall a higher effective velocity Very low frequency response Gassmann behavior with a single "effective fluid" 1 K eff.fl = Σ i S i K i Overall the softest, lowest velocity Critical frequency: f visc = kk f l 2 η 265
Patchy Diffusion Scales Characteristic diffusion time for a pressure disturbance with length scale L to relax: 1 f = τ L2 4D Inverting this, we can find the characteristic diffusion length over which pressure differences can relax at seismic frequency f L 4κK fl ηf D is the hydraulic diffusivity, K is the fluid bulk modulus, κ is the permeability, and η is the viscosity. f 10 Hz 1000 md 100 md 10 md 1 md 1000 md 100 md 100 Hz 10 md 1 md 10 5 Hz κ 1000 md 100 md 10 md 1 md.1 md L 1 m.3 m.1 m.03 m.3 m.1 m.03 m.01 m.01 m.003 m.001 m.0003 m.0001 m 266
K.5 Thierry Cadoret studied velocity vs. saturation using the resonant bar and found the coarse-scale and fine-scale behavior. 267
Estaillades Limestone Cadoret s velocity and attenuation vs. saturation. The fine scale distribution gives relaxed viscoelastic behavior, and the coarse scale gives unrelaxed. Therefore, we expect the largest attenuation when the velocity dispersion is largest. Hence, we get the important result that P-wave attenuation in a partially saturated rock can be much larger than in the dry or fully saturated case. 268
The problem that we address is the nonunique response of seismic velocity to fluid saturation. What are the physical conditions that cause patchy behavior? When do we use the patchy model and when do we use the homogeneous model? Our approach is to use flow simulation to study the parameters that control fluid distributions at a fine scale. Vp (km/s) 2.45 2.4 2.35 2.3 2.25 2.2 2.15 sandstone porosity = 30% patchy homogeneous 0 0.2 0.4 0.6 0.8 1 Oil Saturation study by Madhumita Sengupta and G. Mavko 269
Typical Eclipse Cell 10m x 10m Our cells: 1m x 1m ~ L c Our approach is simply to run flow simulations at a fine scale to help discover the reservoir and fluid properties that control the saturation scale. The fine scales are chosen to be approximately the critical diffusion length, so that any mix of fluids within the cell can be represented as a Reuss average fluid. 270
Porosity and permeability models for flow simulation 271
We will consider two important cases: water flood into oil, and gas flood into oil. The parameters that we consider are: relative permeability wettability density contrast permeability heterogeneity capillary pressure 272
Water Injection in Oil Relative Permeability Curves for Oil and Water 1 Rel Perm 0.8 0.6 0.4 oil oil water water 0.2 0 0 0.2 0.4 0.6 0.8 1 Sw 273
Saturations obtained from flow simulations using the dashed (top) and solid (bottom) relative permeability curves. The irreducible saturations are critical controls on the saturation extremes. Sw Sw Sw Sw 274
The patchy and uniform saturation curves are upper and lower bounds. They describe the range of velocity signatures that we can achieve by mixing the end members. Finite irreducible saturations drastically narrow the range of uncertainty. 275
Wettability The saturation distribution depends on the wettability of the rock. Most sandstone reservoirs are water wet and most carbonate reservoirs are oil wet. Water saturation map and histogram of saturation in an oil wet rock 276
Wettability Vp Oil Wet (Drainage) * * * **** Water Wet (Imbibition) Sw Wettability tends to determine whether the velocities fall high or low in the allowable range. 277
Mobility Ratios MR = k max wr η w / 2500 k or max η o Low Mobility Ratio 0.8 2000 20 40 20 40 60 80 100 0.6 0.4 0.2 1500 1000 500 0 0 0.5 1 1500 High Mobility Ratio 0.8 20 40 20 40 60 80 100 0.6 0.4 0.2 1000 500 0 0 0.5 1 278
3300 3250 V P V p 3150 Low MR Low MR 3200 3100 High MR 3050 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Water Saturation S w High MR 279
The uniform saturation model is good enough for waterflood oil-water cases. Exceptions: when the irreducible oil is very low in an oil wet rock. The main control is the finite irreducible saturations. 280
Gas Injection Into Oil Sg Sg Sg Sg Sg Sg 281
Gas Injection Effect of Mobility 20 40 (a) Low Mobility Ratio 20 40 60 80 100 Sg 0.6 0.4 0.2 3000 2000 1000 20 40 (b) High Mobility Ratio 20 40 60 80 100 0 0.6 Sg 0.4 0.2 2500 2000 1500 1000 0 0 0.5 1 500 Sg 0 0 0 0.2 0.4 0.6 0.8 Sg 282
3150 3100 Vp 3050 3000 High Mobility Ratio 2950 2900 2850 Low Mobility Ratio 2800 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 So 283
Heterogeneity of Perm Perm Models Saturation Satn Histograms 101 100.5 100 99.5 99 0.6 0.4 0.2 0 0 0.5 1 120 100 80 60 40 20 1000 800 600 400 200 0.6 0.4 0.2 0 0.6 0.4 0.2 0 0 0.5 1 0 0.5 1 284
3150 3100 3050 3000 Vp 2950 Large Scale Heterogeneities 2900 2850 Small Scale Heterogeneities 2800 0 0.2 0.4 0.6 0.8 1 So 285
Summary of Mixing Rules K Voigt = S w K w + S o K o + S g K g K Brie = ( K liquid K g )( 1 S g ) e + K g 1/K Reuss = S w / K w + S o / K o + S g / K g Brie, et al.: SPE 30595 286
Conclusions Reservoirs with gas are very likely to show patchy behavior. The uniform saturation model may be good enough for reservoirs with only oil and water The main mechanism that causes patchy behavior at the field scale is gravity. 287