Kun Ma

Report
Estimation of parameters for the simulation of
foam flow through porous media: Part 2; nonuniqueness, numerical artifacts and sensitivity
Kun Maa, Rouhi Farajzadehb,c, Jose L. Lopez-Salinasa, Clarence A. Millera,
Sibani Lisa Biswala and George J. Hirasakia*
a Department
of Chemical and Biomolecular Engineering, Rice University,
Houston, TX 77005, USA
b Shell Global Solutions International, Rijswijk, The Netherlands
c Delft University of Technology, The Netherlands
Modeling the dry-out effect of foam
flow through porous media

ug  
k rg
g


k  ( p g   g g  D)
k rgf  k rgnf  FM
1
FM 
1  fmmob  Fwater  ...
Fwater  0.5 
1.
2
2.
arctan[ epdry ( S w  fmdry )]

fmmob: the reference foam
mobility reduction factor;
fmdry: the critical water
saturation (volume fraction)
above which the maximum
foam strength is reached;
epdry: the slope which
regulates how abrupt the
foam dries out neary fmdry.
Ashoori E, Heijden TLM, Rossen WR (2010) Fractional-Flow Theory of Foam Displacements With Oil. SPE
Journal 15:pp. 260-273
Computer Modeling Group (2007) STARSTM User's Guide. Calgary, Alberta, Canada
Findings in Part 1 of this series of
work
Hybrid contour plot method
3
1.
Fit to steady-state data
Ma, K.; Lopez-Salinas, J.L.; Puerto, M.C.; Miller, C.A.; Biswal, S.L.; Hirasaki, G.J., Energy & Fuels
(in press).
Findings in Part 1 of this series of
work
Fit to steady-state data with different epdry
4
1.
Fit to transient experimental data
Ma, K.; Lopez-Salinas, J.L.; Puerto, M.C.; Miller, C.A.; Biswal, S.L.; Hirasaki, G.J., Energy & Fuels
(in press).
Non-graphical solution to match the
transition foam viscosity
 foam,app (S wt , fmm ob, fmdry)  max  foam,app (S w , fmm ob, fmdry)
Sw
 tfoam,app 
t
w
k rw (S )
w

1
k (S , fmmob, fmdry)
f
rg
t
w
g
1
f gt 
1
t
w
k rw ( S )
w

g
k rgf ( S wt , fm m ob, fm dry)
2
  tfoam,app   tfoam,app (m easured)   f gt  f gt (m easured) 
 

min Fun1 ( fm m ob, fm dry)  
t

  f t (m easured) 

(
m
easured
)
foam
,
app
g

 

5
min Fun2 (Sw )   foam,app (Sw )
2
Non-unique solutions to match the
transition foam viscosity
421cp
Hybrid contour
plot method
421cp
If we apply the
constraint
fmdry<Swt …
6
Non-unique solutions to match the
transition foam viscosity
7
1.
Ma, K.; Farajzadeh, R.; Lopez-Salinas, J.L.; Miller, C.A.; Biswal, S.L.; Hirasaki, G.J., SPE 165263 for
presentation at the SPE Enhanced Oil Recovery Conference held in Kuala Lumpur, Malaysia, 2-4 July 2013.
Non-unique solutions to match the
transition foam viscosity
Capillary Pressure, Pc
Swt=0.1037
Unstable
fg
P c*
Sw*
Liquid Saturation, Sw
1
Khatib, Z. I., G. J. Hirasaki, et al. (1988);
Farajzadeh, R., A. Andrianov, et al.
(2012)
8
1.
fmdry=0.1006
(model fit 1)
fmdry<Swt
fmdry=0.1216
(model fit 2)
fmdry>Swt
Ma, K.; Farajzadeh, R.; Lopez-Salinas, J.L.; Miller, C.A.; Biswal, S.L.; Hirasaki, G.J., SPE 165263 for
presentation at the SPE Enhanced Oil Recovery Conference held in Kuala Lumpur, Malaysia, 2-4 July 2013.
Multi-variable multi-dimensional
search (fit to all experimental data)

  foam,i ,measured
min f ( fm m ob, fm dry, epdry)   i  foam,i ,calculated

 foam,i ,measured
i 1

n
s.t.




2
fmmob 0
Swc  fmdry  1  S gr
epdry  0
Unconstrained search
with a penalty function

  foam,i ,measured
min f ( fm m ob, fm dry, epdry)   i  foam,i ,calculated

 foam,i ,measured
i 1

n
2

  ( fm dry S wc ) 2  


 0, when fm dry  S wc

 k , when fm dry  S wc
9
1.
Ma, K.; Farajzadeh, R.; Lopez-Salinas, J.L.; Miller, C.A.; Biswal, S.L.; Hirasaki, G.J., SPE 165263 for
presentation at the SPE Enhanced Oil Recovery Conference held in Kuala Lumpur, Malaysia, 2-4 July 2013.
Multi-variable multi-dimensional
search (fit to all experimental data)
when
 foam,i,measured   tfoam
otherwise
10
i  5
i  1
Numerical oscillation in transient
foam simulation
 foam,app 
1
k rw
w
 foam,app  
local foam apparent viscosity
g
k ( pout  pin )
(u w  u g ) L
NX  50
11

k rgf
average foam apparent viscosity
t D  0.005xD
fmmob  47196
fm dry  0.1006
epdry  500
Numerical oscillation in transient
foam simulation
NX  50
12
t D  0.005xD
fmmob  47196
fm dry  0.1006
epdry  500
Numerical oscillation in transient
foam simulation
Case 1
Case 3
13
f gt  0.5
 tfoam,app  421 cp
Case 2
Case 5
Case 4
Parameter
Case 1
Case 2
Case 3
Case 4
Case 5
ΔtD/ΔxD
NX
epdry
fmmob
fmdry
nw
0.005
50
500
47196
0.1006
1.96
0.0005
50
500
47196
0.1006
1.96
0.005
200
500
47196
0.1006
1.96
0.005
50
500
28479
0.2473
4.0
0.005
50
100
69618
0.1020
1.96
Sensitivity of foam parameters to nw
14
f gt  0.5
 tfoam,app  421 cp
Sensitivity of foam parameters to Swc
15
f gt  0.5
 tfoam,app  421 cp
Sensitivity of foam parameters to
epdry
1
16
f gt  0.5
 tfoam,app  421 cp
S wt  S wc
  w (1  f gt )  nw
 (1  S gr  S wc )  0 t
  0.1037
 k rw  foam,app 
Conclusions
1. For a preset epdry, it is found that two pairs of values of fmmob
and fmdry can sometimes match steady-state fgt and μfoamt. By
applying the constraint fmdry<Swt, one can rule out the solution
which does not match the rest of experimental data points.
2. To match all available data points using multi-dimensional multivariable search, one can use the unconstrained optimization
approach with an appropriate initial guess which is close to the
global optimum. The penalty function method for constrained
optimization can be applied for a wider range of initial guesses.
17
Conclusions
3. Finite difference simulation for the transient foam process is
generally consistent with the method of characteristics. A less
abrupt change in foam mobility in the foam displacement front
is needed to minimize the oscillation numerical artifact in the
average foam apparent viscosity history. Small epdry leads to
lower amplitude in numerical oscillation and larger apparent
viscosity when foam breaks through.
4. Foam parameters are sensitive to the parameters in relatively
permeability curves. For foam parameter estimation by
matching steady-state fgt and μfoamt, the water relative
permeability exponent nw affects the estimation of both fmmob
and fmdry, and the connate water saturation Swc is particularly
influential in estimating fmdry. An increase in epdry causes a
decrease in fmmob, but no substantial change is found in
fmdry.
18
Acknowledgements / Thank You / Questions
Abu Dhabi National Oil Company (ADNOC)
the Abu Dhabi Oil R&D Sub-Committee
Abu Dhabi Company for Onshore Oil Operations (ADCO)
Zakum Development Company (ZADCO)
Abu Dhabi Marine Operating Company (ADMA-OPCO)
the Petroleum Institute (PI), U.A.E
US Department of Energy (under Award No. DE-FE0005902)
Petróleos Mexicanos (PEMEX)
Shell Global Solutions International
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