Quan%fica%on of Permeability Heterogeneity for
Reservoir Uncertainty Quan%fica%on
Bilal Rashid1, Ann Muggeridge1, Glyn Williams2
1Imperial College London, 2BP
Outline • Introduc=on – Impact of heterogeneity on recovery
• Overview of theory – Vor=city and shear-‐strain rate
• Applica=ons – Shear-‐strain rate as a measure of heterogeneity
• Test against previous measures (Dykstra & Parsons, Schmalz & Rahme, Shook et al.)
• Conclusions
Impact of Heterogeneity on Recovery
From Tyler, N. and Finley, R. 1997. Geological Characteriza=on of Heterogeneous Reservoirs. Workshop Caracas, Venezuela Feb. 1997
Styles of reservoir heterogeneity
Why quantify heterogeneity?
Heterogeneity Measures
Quan=fy the impact of heterogeneity on flow
Exis=ng heterogeneity indices:
Sta%c
Dykstra-‐Parsons coefficient
Lorenz coefficient
Dynamic
Streamline measures
Koval’s H factor
• Rule of thumb • Rela=ve • Tested with geosta=s=cal distribu=ons
• Breakthrough =me • Sweep efficiency an absolute measure
Overview of Theory
Decompose
Two measures? Vor=city Shear Rate
€
v(x,y) = v(xIJ ,yIJ ) +12
2 ∂u∂x
∂u∂y + ∂v
∂x∂u∂y + ∂v
∂x 2 ∂v∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
Rate of Strain Tensor
+12
0 ∂u∂y −
∂v∂x
∂v∂x −
∂u∂y 0
⎛
⎝ ⎜
⎞
⎠ ⎟
∝ Vorticity
Overview of Theory
€
v(x,y) = v(xIJ ,yIJ ) +12
2 ∂u∂x
∂u∂y + ∂v
∂x∂u∂y + ∂v
∂x 2 ∂v∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
Rate of Strain Tensor
+12
0 ∂u∂y −
∂v∂x
∂v∂x −
∂u∂y 0
⎛
⎝ ⎜
⎞
⎠ ⎟
∝ Vorticity
Cauchy Stokes Decomposi=on Theorem: From Mahani et al. (2009)
Overview of Theory
Calculated at Ver=ces
€
ΔvΔx + Δu
Δy
€
ΔvΔx −
ΔuΔy
Shear-‐rate
Vor=city
Applica=on: Heterogeneity Index
€
cv (Shear) =Standard Deviation
Mean
€
J =12
2 ∂u∂x
∂u∂y + ∂v
∂x∂u∂y + ∂v
∂x 2 ∂v∂y
⎛
⎝ ⎜
⎞
⎠ ⎟ +12
0 ∂u∂y −
∂v∂x
∂v∂x −
∂u∂y 0
⎛
⎝ ⎜
⎞
⎠ ⎟
Propose: The varia=on in shear is a robust measure of heterogeneity.
Applica=on: Heterogeneity Index Test heterogeneity indices using all 85 layers from SPE 10 Model 2
Miscible – Immiscible M=1/10/100 Diff. Well Paeerns
SPE Model 2 Permeability Map
L20
L81
Heterogeneity Indices Sta%c
Dykstra-‐Parsons coefficient (Jensen & Currie 1990)
Dynamic Streamline simula=ons
Lorenz Coefficient (Shook et al. 2009) Varia=on of =me of flight distribu=on
Methodology
Single phase displacement simula%on (Finite volume)
Calculate shear-‐rate field
Calculate CV of shear-‐rate
Generate streamlines (tracer flow) 3DSL
Use streamline data to calculate:
Dynamic Lorenz coefficient CV of TOF
Compare with normalised breakthrough %me for:
Miscible Immiscible Line Drive Q5 Spot
Results – Base Line
Dykstra-‐Parsons
coefficient
Results – Base Line
Dynamic Lorenz
coefficient
Dykstra-‐Parsons
coefficient
Poor Sensi=vity
Miscible Immiscible
Results -‐ TOF
Cv(TOF)
TOF
Miscible Immiscible
Results -‐ Shear
Cv(Shear) Line Drive
Miscible Immiscible
Results -‐ Shear
Cv(Shear) Line Drive
Miscible Immiscible
Quarter 5 Spot
Conclusions
• Cv(Shear-‐Strain rate) is propor=onal to breakthrough =me & recovery
• Allows:
– Rapid evalua=on of the impact of heterogeneity on breakthrough =me – Reliable for:
• Realis%c geological models
• Range of mobility ra%os
• Different well paZerns
• May be used to both rank realisa=ons & es=mate recovery