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