Development of a 1D isothermal surfactant flooding simulator

1. Background
2. Literature Review
3. Methodology
4. Research Progress
5. Summary

Adsorption and Dispersion in EOS Compositional
Flow

Akmal Aulia, G01059

EOR Centre, Petroleum Engineering, UT Petronas
Supervisor: Prof. Dr. Noaman El-Khatib

December 20th , 2010

Akmal Aulia, G01059 Adsorption and Dispersion in EOS Compositional Flow

1. Background
3. Methodology
5. Summary

Outline

Background


1. Background
3. Methodology
5. Summary

Outline

Background
Literature Review


1. Background
3. Methodology
5. Summary

Outline

Background
Literature Review
Methodology


1. Background
3. Methodology
5. Summary

Outline

Background
Literature Review
Methodology
Extension


1. Background
3. Methodology
5. Summary

Outline

Background
Literature Review
Methodology
Extension
Research Progress


1. Background
3. Methodology
5. Summary

Outline

Background
Literature Review
Methodology
Extension
Research Progress
Summary


1. Background
2. Literature Review 1.1. Problem Description
3. Methodology 1.2. Objective
4. Research Progress 1.3. Scope of Study
5. Summary

Background

Is my surfactant ﬂooding project economical?
Loss of surfactants due to adsorption


1. Background
5. Summary

Background

Is my surfactant ﬂooding project economical?
Loss of surfactants due to adsorption
Loss of slug stability due to dispersion


1. Background
5. Summary

Problem Description

Based on given ﬂuid and rock properties, is the project
economical?


1. Background
5. Summary

Problem Description

Based on given ﬂuid and rock properties, is the project
economical?
How can I evaluate the economical feasibilities? - simulation,
other quantitative methods?


1. Background
5. Summary

Problem Description

Many uses Compositional Models to simulate Chemical
Flooding processes


1. Background
5. Summary

Problem Description

Many uses Compositional Models to simulate Chemical
Flooding processes
IFT, Mobility, Relative Permeability, Residual Saturations, are
aﬀected by compositions


1. Background
5. Summary

Objective of the Study

To investigate the eﬀects of adsorption and dispersion on
compositional dynamics in surfactant ﬂooding processes.


1. Background
5. Summary

Objective of the Study

To investigate the eﬀects of adsorption and dispersion on
compositional dynamics in surfactant ﬂooding processes.
(possible extension?) To explore compositional paths under
various heterogeneity distributions.


1. Background
5. Summary

Scope of Study

1-dimensional


1. Background
5. Summary

Scope of Study

1-dimensional
isothermal


1. Background
5. Summary

Scope of Study

1-dimensional
isothermal
core scale


1. Background
5. Summary

Scope of Study

1-dimensional
isothermal
core scale
capillary pressure neglected


1. Background
5. Summary

Scope of Study

1-dimensional
isothermal
core scale
2 phase (aqueous, oleic), 3 components (surfactant, water,
oil)


1. Background
5. Summary

Scope of Study

1-dimensional
isothermal
core scale
oil)
no gas


1. Background
5. Summary

Scope of Study

1-dimensional
isothermal
core scale
oil)
no gas
homogenous, heterogeneous (possible extension)


2.1. Progress in Compositional Simulation
1. Background
2.1. Finite Diﬀerence Method
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary
2.5. Compositional Model

Progress in Compositional Simulation

AU YR AD DP EOS DIM Gas? PHS
√ √
Nolen 1973 - √ LE-RK 3D √ -
Pope 1978 Lang. - 1D √ -
Coats 1980 √- -
√ RK 3D 3
El-Khatib 1985 - 1D - 2
Porcelli 1994 - - -
√ 1D -
√ 2
Branco 1995 -
√ -
√ 1D 3
Bidner 1996 - 1D -
√ 2
Wang 1997 - - PR 3D √ -
Coats 1998 - - PR, SRK 3D √ 3
Coats 2000 -
√ -
√ ﬂash 1D,3D √ 3
UTCHEM 2000 √ √ - 3D 2,3
Bidner 2002 √- 1D - 2
GPAS 2005 Lang.
√ -
√ 3D -
√ 3
Chen 2007 Akmal Aulia, G01059 PR and Dispersion in EOS Compositional Flow
Adsorption 3D 3

1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary

A glimpse on the Finite Diﬀerence Method (FDM)
d
f : x → u, x ∈ . Let h = x − a and u (x) = dx (u(x)). The Taylor
expansion of u(x + h) and u(x − h) for 2nd order,
h2
u(x + h) = u(x) + hu (x) + u (x) + O(h3 ) (1)
2!
h2
u(x − h) = u(x) − hu (x) + u (x) − O(h3 ) (2)
2!
can yield the approximations of u (x)
u(x + h) − u(x − h)
u (x) = (3)
2h
u(x + h) − u(x)
u (x) = (4)
h
u(x) − u(x − h)
u (x) = (5)
h

1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary

A glimpse on the Finite Diﬀerence Method (FDM)

In terms of grids,

du ui+1 − ui−1
= (6)
dx i 2h
du ui+1 − ui
= (7)
dx i h
du ui − ui−1
= (8)
dx i h


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary

Explicit FDM

Let,
dP ∂2P
= (9)
dt ∂x 2
or,
Pt = Pxx (10)
for short. Thus, discretized EXPLICITLY as:

Pin+1 − Pin P n − 2Pin + Pi−1
n
= i+1 (11)
t ( x)2


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary

Explicit FDM

Solve EXPLICITLY discretized equation as,
t
Pin+1 = Pin + (P n − 2Pin + Pi−1 )
n
(12)
( x)2 i+1

Therefore, use PAST information to obtain FUTURE information.


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary

Implicit FDM

Recall,
Pt = Pxx (13)
IMPLICIT discretization reads,

Pin+1 − Pin P n+1 − 2Pin+1 + Pi−1
n+1
= i+1 (14)
t ( x)2

Therefore, use FUTURE and PAST information to obtain
FUTURE information.


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary

Implicit FDM

To solve for the IMPLICIT scheme, is to solve A*x=b such that A
is a matrix and x,b are vectors. Example,
 r    
1 + r −2 0 0 P2 f1 − k I
 −r r
 2 1 +r r − 2 0  P3   f2 
r
  =  
 0 − 2 1 + r − 2  P4   f3 
r
0 0 −2 1 + r P5 f4 − k B

Tools needed for solving: Thomas algorithm, Cholesky
decomposition, Conjugate Gradient, Preconditioned Conjugate
Gradient


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary

Newton-Raphson

For single variable,
f
x = xold − (15)
f
For multiple variables,

x = xold + p (16)
J · p = −f (17)

where (for example, 2 variables),

∂f1 /∂x1 ∂f1 /∂x2
J=
∂f2 /∂x1 ∂f2 /∂x2


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary

The Compositional Model: Bidner et al, 1994-2002

Nomenclature:
volume of phase j
Sj = (18)
pore volume
volume of component i in phase j
cil = (19)
volume of phase l
j total volume of component i
Ci = S l ci [=] (20)
pore volume
j
adsorbed volume of component i
Γi = (21)
pore volume
Kl = dispersion coeﬃcient of phase l (22)


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary


For i ∈ {p, c} the continuity equations for each species read,

∂Ci ∂ ∂ ∂cil ∂Γi
φ + cil u l − S l Kl =− (23)
∂t ∂x ∂x ∂x ∂t
l∈L l∈L

The adsorption expression is,

Γc = φαLa
pc (24)


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary


Summing the continuity equations for all i ∈ C
yields the Overall Continuity Equation (pressure equation)

∂ ∂P a ∂ ∂ ∂PC
λ = Γi − λo (25)
∂x ∂x ∂t ∂x ∂x
i∈C

Note: Dispersion terms collapses by summation. The dispersion
term reads,
K 1 Udp
= + 1.75 (26)
Dm Fφ Do


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary

Restriction relations:
For i ∈ {p, c},
Ci = S l cil (27)
l∈L

For l ∈ {o, a},
cil = 1 (28)
i∈C
For all i and l,

Sl = 1 (29)
l∈L

Ci = 1 (30)
i∈C


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary


Supporting expressions:
For l = a only,
krl ∂P l
u l = −K (31)
µl ∂x
For all i and l,
∂P a ∂PC
u = −λ − λo (32)
∂x ∂x
PC = Po − Pa (33)
o a
u = u +u (34)


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary


The unknown variables are,

ul = 2 (35)
u = 1 (36)
l
P = 2 (37)
l
S = 2 (38)
cil = 6 (39)
Ci = 3 (40)
TOTAL UNKNOWNS = 16 (41)

Note: 16 Unknowns vs 13 Equations !!!


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary

DOF=3. How to make it 0 ?

Bidner et al use equilibrium ratios,
a
cp
La
pc = a
(42)
cc
o
cw
Lo
wc = o
(43)
cc
o
cc
Kc = a
(44)
cc

- EOS can yield more accurate compositions (Chen, 2006, 2007).
- Recent work (Roshafenkr, Li, and Johns 2008) describe a few
experimental eﬀorts for phase behavior – a most likely feasible
option.


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary

A note about Roshafenker, Li, and Johns (UT Austin)

Yinghui Li wrote a thesis about the method


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary


Roshafenkr, Li, and Johns wrote a paper (2008) on the
method


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary


method
They said that the future study is to include their phase
behavior modeling methods on compositional simulators.


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary


method
This is how this research can contribute; a continuation of
their research.


1. Background
2.2. Explicit FDM
3. Methodology
2.3. Implicit FDM
2.4. Newton-Raphson
5. Summary


method
This is how this research can contribute; a continuation of
their research.
At a glance; method basically requires few experimental
samplings.


1. Background
2. Literature Review 3.1. On IMPECS
3. Methodology 3.1. IMPECS algorithm
4. Research Progress 3.2. Mobility calculations
5. Summary

A glimpse on IMPECS

Using FDM,
- IMplicit Pressure
- Explicit Concentrations + Saturations


1. Background
5. Summary

The IMPECS algorithm: Bidner (2002)
For each time step, for all gridblocks:
STEP 1: Calculate P a IMPLICITLY.


1. Background
5. Summary

STEP 2: Calculate P o .


1. Background
5. Summary

STEP 3: Calculate u, u a , u o .


1. Background
5. Summary

STEP 4: Calculate Cc , Cp via continuity equations,
EXPLICITLY.


1. Background
5. Summary

EXPLICITLY.
STEP 5: Calculate Cw , cij , S j via restriction relations.


1. Background
5. Summary

EXPLICITLY.
STEP 5: Calculate Cw , cij , S j via restriction relations.
STEP 6: Evaluate errors:
M
|(Ci )k+1 − (Ci )k |
m m (45)
m=1


1. Background
5. Summary

Mobility calculations

σ for Type II (-) can be described as a function of compositions,
√P o a 2
1 − e− i (ci −ci )
F = √
1 − e− 2
G1
log σ = log F + (1 − La ) log σ H +
pc La ; La ≤ 1
G1 + G2 pc pc
G1
log σ = log F + ; La > 1
(1 + La G2 ) pc
pc


1. Background
5. Summary


Once σ is found, we can compute the capillary number:

µaH u IN
Nvc = (46)
σ
Which leads to the residual saturations as a function of Nvc ,
j j

1, if Nvc < 10(1/T1 )−T2 ;
jr

S 
j j j j j

jrH
= T1 log(Nvc ) + T2 , if 10(1/T1 )−T2 ≤ Nvc ≤ 10−T2 ;
S  j
if Nvc > 10−T2 .

0,


1. Background
5. Summary


el
S l − S lr
krl = krl0 ;l = l (47)
1 − S lr − S l r

Sl r
krl0 = (1 − krl0H ) 1 − + krl0H (48)
S l rH
Sl r
el = (1 − e lH ) 1 − + e lH (49)
S l rH


1. Background
5. Summary


along with, (my supervisor suggested other averaging method)
l l l l l l
µl = cw µaH e α1 (cp +cs ) + cp µoH e α1 (cw +cs ) + cs α3 e α2 (cw +cp )
l l l
(50)

we can write mobility as,
Kkro Kk a
λ = λo + λa = + ar (51)
µo µ


1. Background
2. Literature Review 4.1. Checkpoints
3. Methodology 4.2. Recent Publications
4. Research Progress 4.3. Gantt Chart
5. Summary

Checkpoints

- Thomas Algorithm (with Fortran 95)
- Cholesky Decomposition (with Fortran 95)
- Crank-Nicholson Scheme (with Fortran 95) - Conjugate Gradient
- Jacobi-Preconditioned Conjugate Gradient (with Fortran 9)
- IMPECS solver (with Fortran 90, some progress on debugging)
- Multivariable Newton-Raphson (with Fortran 90)


1. Background
5. Summary

Note on the Implementing Adaptive NR Method: Part I

Recall,
a
cp
La
pc = a
(52)
cc
o
cw
Lo
wc = o
(53)
cc
o
cc
Kc = a
(54)
cc

La , Lo , and Kc are the swelling parameter, solubilization
pc wc
parameter, and equilibrium ratio between the two phases,
respectively.


1. Background
5. Summary

Note on the Implementing Adaptive NR Method: Part II

Also recall,

Cc = S j cc
l

l∈L
= S a cc + S o cc
a o

= S a cc + (1 − S a )Kc cc
a a
(55)


1. Background
5. Summary

Note on the Implementing Adaptive NR Method: Part III

Similary for Cp ,

Cp = S a c p + S o c p
a o

= S a La cc + (1 − S a )(1 − cc − cw )
pc
a o o

= S a La cc + (1 − S a )(1 − Kc cc − Lo cc )
pc
a a
wc
o

= S a La cc + (1 − S a )(1 − Kc cc − Lw c o Kc cc )
pc
a a a

= S a La cc + (1 − S a )(1 − Kc cc (1 + Lo ))
pc
a a
wc
(56)


1. Background
5. Summary

Note on the Implementing Adaptive NR Method: Part IV

Thus, we can set up a 2 equations - 2 unknown adaptive NR as,

f1 (S a , cc ) = S a cc + (1 − S a )Kc cc − Cc
a a a
(57)

f2 (S a , cc ) = S a La cc + (1 − S a ) ·
a
pc
a

(1 − Kc cc (1 + Lo )) − Cp
a
wc (58)


1. Background
5. Summary

Note on the Implementing Adaptive NR Method: Part V

→
− →
−
x k+1 = →k − J−1 f k
−
x (59)
where,
a
→ = S
−
xk
cca
k

→
− x1 f1 (S a , cc )
a
f k= =
x2 k
f2 (S a , cc )
a
k


1. Background
5. Summary

Note on the Implementing Adaptive NR Method: Part VI

and,
∂f1 ∂f1
 
 ∂x1 ∂x2 
J=
 ∂f2

∂f2 
∂x1 ∂x2 k


1. Background
5. Summary

Note on the Implementing Adaptive NR Method: Part VII
where,
∂f1 a
= cc (1 − Kc ) (60)
∂x1

∂f1
= S a (1 − Kc ) + Kc (61)
∂x2

∂f2
= cc (La + Kc (1 + Lo )) − 1
a
pc wc (62)
∂x1

∂f2
= S a (La + Kc (1 + Lo ))
pc wc
∂x2

−Kc (1 + Lo )
wc (63)

1. Background
5. Summary

Note on the Implementing Adaptive NR Method: Part VIII
However, constraints must be deﬁned → enhanced adaptivity!

!"#&
!"#$%& !"'& %(!"'&
!"#$%&

Figure: Description of the Adaptive Newton-Raphson.


1. Background
5. Summary

Note on the Implementing Adaptive NR Method: Part IX

Mathematically,
 a
 (S )k + S ar

 , if (S a )k+1 < S ar


 2

a or
(S a )k+1 = (S )k + (1 − S ) , if (S a )k+1 > (1 − S or )



 2

 a
(S ) , otherwise.
k+1


1. Background
5. Summary

Note on the Implementing Adaptive NR Method: Part X

and,  a
 (cc )k + 0 a

 , if (cc )k+1 < 0


 2

a
a
(cc )k+1 = (cc )k + 1 , if (c a )
c k+1 > 1



 2

 a
(c ) , otherwise.
c k+1


1. Background
5. Summary

Fortran Results Part I

Table: Compositional Simulator’s Input Parameters

Parameter Assigned Value Units Description

u IN 10−4 cm/s input ﬂowrate
S orH , S arH 0.35 res. sat. at high IFT
PIN , POUT 1 atm endpoint Pressures
φ 0.24 porosity
IN
Cs 0.1 overall surfactant conc.
IN
Cp 0 overall oil conc.
L 100 cm core (porous media) length
K 0.5 Darcy permeability
o0H a0H
kr , kr 1, 0.2 Rel. Permeability at high IFT
µoH , µaH 5, 1 cP phase viscosities


1. Background
5. Summary

Fortran Results Part II
1.1 
1.09 
1.08 
1.07 
1.06 
1.05  n=1 
1.04  n=2 
1.03 
n=3 
1.02 
1.01 
1 
0.99 
1  2  3  4  5 

Figure: Aqueous phase pressures across grid at diﬀerent timesteps.


1. Background
5. Summary

Fortran Results Part III
1 
0.9 
0.8 
0.7 
0.6 
n=1 
0.5 
n=2 
0.4 
n=3 
0.3 
0.2 
0.1 
0 
1  2  3  4  5 

Figure: Surfactant phase composition (aqueous phase) across grid at
diﬀerent timesteps.

1. Background
5. Summary

Fortran Results Part IV
1 
0.9 
0.8 
0.7 
0.6 
n=1 
0.5 
n=2 
0.4 
n=3 
0.3 
0.2 
0.1 
0 
1  2  3  4  5 

Figure: Surfactant phase composition (oleic phase) across grid at
diﬀerent timesteps.

1. Background
5. Summary

Fortran Results Part V
1 
0.9 
0.8 
0.7 
0.6 
n=1 
0.5 
n=2 
0.4 
n=3 
0.3 
0.2 
0.1 
0 
1  2  3  4  5 

Figure: Aqueous phase saturation across grid at diﬀerent timesteps.


1. Background
5. Summary

Fortran Results Part VI

Figure: Oleic phase saturation across grid at diﬀerent timesteps.


1. Background
5. Summary

Achievements - 3 semesters residency

Published 2 journal articles:
Akmal Aulia and Noaman El-Khatib, ”Mathematical Description of the Implementation of the Adaptive
Newton-Raphson Method in Compositional Porous Media Flow,” International Journal of Basic and
Applied Sciences IJBAS-IJENS Vol. 10 No. 06 ISSN: 2077-1223 (accepted with minor revision).
Akmal Aulia, Tham Boon Keat, Muhammad Sanif M., Noaman El-Khatib, and Mazuin Jasamai, ”Smart
Oilﬁeld Data Mining for Reservoir Analysis,” International Journal of Engineering and Technology
IJET-IJENS Vol. 10 No. 06 ISSN: 2077-1185 (accepted).

and 2 conference papers:
Akmal Aulia and Noaman El-Khatib, ”Mathematical modeling of Adsorption and Dispersion in Chemical
Flood EOS Compositional Flow”, ICIPEG 2010, 15-17 June 2010, Kuala Lumpur, Malaysia
Akmal Aulia, Tham Boon Keat, Muhammad Sanif Bin Maulut, Noaman El-Khatib, and Mazuin Jasamai,
”Mining Data from Reservoir Simulation Results”, ICIPEG 2010, 15-17 June 2010, Kuala Lumpur, Malaysia


1. Background
5. Summary

Gantt Chart

2009 2010 2011
Items
Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Literature Review
Model Development
Model Discretization
Code Development
Debug
Analytical Solutions with MOC
Publications 2 Conference Papers, 2 Journal Articles attempt 1 more journal
Dissertation Writing
Submit Dissertation


1. Background
3. Methodology
5. Summary

Summary

The importance of IMPECS in solving coupled PDE.


1. Background
3. Methodology
5. Summary

Summary

The importance of IMPECS in solving coupled PDE.
The importance of Newton-Raphson methods in many aspect
of IMPECS.


1. Background
3. Methodology
5. Summary

Thank you for coming! Questions and Comments?


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Development of a 1D isothermal surfactant flooding simulator