Development of a 1D isothermal surfactant flooding simulator

1. Background2. Literature Review

3. Methodology4. Research Progress

5. Summary

Adsorption and Dispersion in EOS CompositionalFlow

Akmal Aulia, G01059

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

December 20th, 2010

Akmal Aulia, G01059 Adsorption and Dispersion in EOS Compositional Flow



5. Summary

Outline

Background

Literature Review

Methodology

Extension

Research Progress

Summary




5. Summary

Outline

Background

Literature Review

Methodology

Extension

Research Progress

Summary




5. Summary

Outline

Background

Literature Review

Methodology

Extension

Research Progress

Summary




5. Summary

Outline

Background

Literature Review

Methodology

Extension

Research Progress

Summary




5. Summary

Outline

Background

Literature Review

Methodology

Extension

Research Progress

Summary




5. Summary

Outline

Background

Literature Review

Methodology

Extension

Research Progress

Summary




5. Summary

1.1. Problem Description1.2. Objective1.3. Scope of Study

Background

Is my surfactant flooding project economical?

Loss of surfactants due to adsorption

Loss of slug stability due to dispersion




5. Summary


Background

Is my surfactant flooding project economical?

Loss of surfactants due to adsorption

Loss of slug stability due to dispersion




5. Summary


Problem Description

Based on given fluid and rock properties, is the projecteconomical?

How can I evaluate the economical feasibilities? - simulation,other quantitative methods?




5. Summary


Problem Description

Based on given fluid and rock properties, is the projecteconomical?

How can I evaluate the economical feasibilities? - simulation,other quantitative methods?




5. Summary


Problem Description

Many uses Compositional Models to simulate ChemicalFlooding processes

IFT, Mobility, Relative Permeability, Residual Saturations, areaffected by compositions




5. Summary


Problem Description

Many uses Compositional Models to simulate ChemicalFlooding processes

IFT, Mobility, Relative Permeability, Residual Saturations, areaffected by compositions




5. Summary


Objective of the Study

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

(possible extension?) To explore compositional paths undervarious heterogeneity distributions.




5. Summary


Objective of the Study

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

(possible extension?) To explore compositional paths undervarious heterogeneity distributions.




5. Summary


Scope of Study

1-dimensional

isothermal

core scale

capillary pressure neglected

2 phase (aqueous, oleic), 3 components (surfactant, water,oil)

no gas

homogenous, heterogeneous (possible extension)




5. Summary


Scope of Study

1-dimensional

isothermal

core scale



no gas





5. Summary


Scope of Study

1-dimensional

isothermal

core scale



no gas





5. Summary


Scope of Study

1-dimensional

isothermal

core scale



no gas





5. Summary


Scope of Study

1-dimensional

isothermal

core scale



no gas





5. Summary


Scope of Study

1-dimensional

isothermal

core scale



no gas





5. Summary


Scope of Study

1-dimensional

isothermal

core scale



no gas





5. Summary

2.1. Progress in Compositional Simulation2.1. Finite Difference Method2.2. Explicit FDM2.3. Implicit FDM2.4. Newton-Raphson2.5. 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√

3El-Khatib 1985

√ √- 1D - 2

Porcelli 1994 - - - 1D - 2Branco 1995 - -

√1D

√3

Bidner 1996√ √

- 1D - 2Wang 1997 - - PR 3D

√-

Coats 1998 - - PR, SRK 3D√

3Coats 2000 - - flash 1D,3D

√3

UTCHEM 2000√ √

- 3D√

2,3Bidner 2002

√ √- 1D - 2

GPAS 2005 Lang. -√

3D - 3Chen 2007

√ √PR 3D

√3

Najafabadi 2009√ √

PR 3D - 3Hustad 2009 - - - 1D,2D,3D

√3




5. Summary


A glimpse on the Finite Difference Method (FDM)

f : x → u, x ∈ <. Let h = x − a and u′(x) = ddx (u(x)). The Taylor

expansion of u(x + h) and u(x − h) for 2nd order,

u(x + h) = u(x) + hu′(x) +h2

2!u′′(x) + O(h3) (1)

u(x − h) = u(x)− hu′(x) +h2

2!u′′(x)− O(h3) (2)

can yield the approximations of u′(x)

u′(x) =u(x + h)− u(x − h)

2h(3)

u′(x) =u(x + h)− u(x)

h(4)

u′(x) =u(x)− u(x − h)

h(5)




5. Summary


A glimpse on the Finite Difference Method (FDM)

In terms of grids, (du

dx

)i

=ui+1 − ui−1

2h(6)(

du

dx

)i

=ui+1 − ui

h(7)(

du

dx

)i

=ui − ui−1

h(8)




5. Summary


Explicit FDM

Let,dP

dt=∂2P

∂x2(9)

or,Pt = Pxx (10)

for short. Thus, discretized EXPLICITLY as:

Pn+1i − Pn

i

4t=

Pni+1 − 2Pn

i + Pni−1

(4x)2(11)




5. Summary


Explicit FDM

Solve EXPLICITLY discretized equation as,

Pn+1i = Pn

i +4t

(4x)2(Pn

i+1 − 2Pni + Pn

i−1) (12)

Therefore, use PAST information to obtain FUTURE information.




5. Summary


Implicit FDM

Recall,Pt = Pxx (13)

IMPLICIT discretization reads,

Pn+1i − Pn

i

4t=

Pn+1i+1 − 2Pn+1

i + Pn+1i−1

(4x)2(14)

Therefore, use FUTURE and PAST information to obtainFUTURE information.




5. Summary


Implicit FDM

To solve for the IMPLICIT scheme, is to solve A*x=b such that Ais a matrix and x,b are vectors. Example,

1 + r − r2 0 0

− r2 1 + r − r

2 00 − r

2 1 + r − r2

0 0 − r2 1 + r

P2

P3

P4

P5

=

f1 − kI

f2f3

f4 − kB

Tools needed for solving: Thomas algorithm, Choleskydecomposition, Conjugate Gradient, Preconditioned ConjugateGradient




5. Summary


Newton-Raphson

For single variable,

x = xold −f

f ′(15)

For multiple variables,

x = xold + p (16)

J · p = −f (17)

where (for example, 2 variables),

J =

[∂f1/∂x1 ∂f1/∂x2

∂f2/∂x1 ∂f2/∂x2

]




5. Summary


The Compositional Model: Bidner et al, 1994-2002

Nomenclature:

S j =volume of phase j

pore volume(18)

c li =

volume of component i in phase j

volume of phase l(19)

Ci =∑

j

S lc ji [=]

total volume of component i

pore volume(20)

Γi =adsorbed volume of component i

pore volume(21)

Kl = dispersion coefficient of phase l (22)




5. Summary



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

φ∂Ci

∂t+

∂

∂x

∑l∈L

c li u

l − ∂

∂x

∑l∈L

S lKl ∂c li

∂x= −∂Γi

∂t(23)

The adsorption expression is,

Γc = φαLapc (24)




5. Summary



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

∂

∂x

(λ∂Pa

∂x

)=

∂

∂t

(∑i∈C

Γi

)− ∂

∂x

(λo ∂PC

∂x

)(25)

Note: Dispersion terms collapses by summation. The dispersionterm reads,

KDm

=1

Fφ+ 1.75

Udp

Do(26)




5. Summary



Restriction relations:For i ∈ {p, c},

Ci =∑l∈L

S lc li (27)

For l ∈ {o, a}, ∑i∈C

c li = 1 (28)

For all i and l, ∑l∈L

S l = 1 (29)∑i∈C

Ci = 1 (30)




5. Summary



Supporting expressions:For l = a only,

ul = −Kk lr

µl

∂P l

∂x(31)

For all i and l ,

u = −λ∂Pa

∂x− λo ∂PC

∂x(32)

PC = Po − Pa (33)

u = uo + ua (34)




5. Summary



The unknown variables are,

ul = 2 (35)

u = 1 (36)

P l = 2 (37)

S l = 2 (38)

c li = 6 (39)

Ci = 3 (40)

TOTAL UNKNOWNS = 16 (41)

Note: 16 Unknowns vs 13 Equations !!!




5. Summary


DOF=3. How to make it 0 ?

Bidner et al use equilibrium ratios,

Lapc =

cap

cac

(42)

Lowc =

cow

coc

(43)

Kc =coc

cac

(44)

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




5. Summary


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

Yinghui Li wrote a thesis about the method

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

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

This is how this research can contribute; a continuation oftheir research.

At a glance; method basically requires few experimentalsamplings.




5. Summary











5. Summary











5. Summary











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5. Summary

3.1. On IMPECS3.1. IMPECS algorithm3.2. Mobility calculations

A glimpse on IMPECS

Using FDM,- IMplicit Pressure- Explicit Concentrations + Saturations




5. Summary


The IMPECS algorithm: Bidner (2002)

For each time step, for all gridblocks:

STEP 1: Calculate Pa IMPLICITLY.

STEP 2: Calculate Po .

STEP 3: Calculate u, ua, uo .

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

STEP 5: Calculate Cw , cji , S

j via restriction relations.

STEP 6: Evaluate errors:

M∑m=1

|(Ci )k+1m − (Ci )

km| (45)




5. Summary











M∑m=1

|(Ci )k+1m − (Ci )

km| (45)




5. Summary











M∑m=1

|(Ci )k+1m − (Ci )

km| (45)




5. Summary











M∑m=1

|(Ci )k+1m − (Ci )

km| (45)




5. Summary











M∑m=1

|(Ci )k+1m − (Ci )

km| (45)




5. Summary











M∑m=1

|(Ci )k+1m − (Ci )

km| (45)




5. Summary


Mobility calculations

σ for Type II (-) can be described as a function of compositions,

F =1− e−

√Pi (c

oi −ca

i )2

1− e−√

2

log σ = log F + (1− Lapc) log σH +

G1

G1 + G2La

pc ; Lapc ≤ 1

log σ = log F +G1

(1 + LapcG2)

; Lapc > 1




5. Summary



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

Nvc =µaHuIN

σ(46)

Which leads to the residual saturations as a function of Nvc ,

S jr

S jrH=

1, if Nvc < 10(1/T j

1)−T j2 ;

T j1

[log(Nvc) + T j

2

], if 10(1/T j

1)−T j2 ≤ Nvc ≤ 10−T j

2 ;

0, if Nvc > 10−T j2 .




5. Summary



k lr = k l0

r

(S l − S lr

1− S lr − S l ′r

)e l

; l 6= l ′ (47)

k l0r = (1− k l0H

r )

(1− S l ′r

S l ′rH

)+ k l0H

r (48)

e l = (1− e lH)

(1− S l ′r

S l ′rH

)+ e lH (49)




5. Summary



along with, (my supervisor suggested other averaging method)

µl = c lwµ

aHeα1(c lp+c l

s) + c lpµ

oHeα1(c lw +c l

s) + c lsα3e

α2(c lw +c l

p) (50)

we can write mobility as,

λ = λo + λa =Kko

r

µo+

Kkar

µa(51)




5. Summary

4.1. Checkpoints4.2. Recent Publications4.3. Gantt Chart

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)




5. Summary


Note on the Implementing Adaptive NR Method: Part I

Recall,

Lapc =

cap

cac

(52)

Lowc =

cow

coc

(53)

Kc =coc

cac

(54)

Lapc , Lo

wc , and Kc are the swelling parameter, solubilizationparameter, and equilibrium ratio between the two phases,respectively.




5. Summary


Note on the Implementing Adaptive NR Method: Part II

Also recall,

Cc =∑l∈L

S jc lc

= Sacac + Soco

c

= Sacac + (1− Sa)Kcc

ac (55)




5. Summary


Note on the Implementing Adaptive NR Method: Part III

Similary for Cp,

Cp = Sacap + Soco

p

= SaLapcc

ac + (1− Sa)(1− co

c − cow )

= SaLapcc

ac + (1− Sa)(1− Kcc

ac − Lo

wccoc )

= SaLapcc

ac + (1− Sa)(1− Kcc

ac − LwcoKcc

ac )

= SaLapcc

ac + (1− Sa)(1− Kcc

ac (1 + Lo

wc))

(56)




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(Sa, cac ) = Saca

c + (1− Sa)Kccac − Cc (57)

f2(Sa, cac ) = SaLa

pccac + (1− Sa) ·

(1− Kccac (1 + Lo

wc))− Cp (58)




5. Summary


Note on the Implementing Adaptive NR Method: Part V

−→x k+1 = −→x k − J−1−→f k (59)

where,

−→x k =

[Sa

cac

]k

−→f k =

[x1

x2

]k

=

[f1(Sa, ca

c )

f2(Sa, cac )

]k




5. Summary


Note on the Implementing Adaptive NR Method: Part VI

and,

J =

∂f1∂x1

∂f1∂x2

∂f2∂x1

∂f2∂x2

k




5. Summary


Note on the Implementing Adaptive NR Method: Part VII

where,

∂f1∂x1

= cac (1− Kc) (60)

∂f1∂x2

= Sa(1− Kc) + Kc (61)

∂f2∂x1

= cac (La

pc + Kc(1 + Lowc))− 1 (62)

∂f2∂x2

= Sa(Lapc + Kc(1 + Lo

wc))

−Kc(1 + Lowc) (63)




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.




5. Summary


Note on the Implementing Adaptive NR Method: Part IX

Mathematically,

(Sa)k+1 =

(Sa)k + Sar

2, if (Sa)k+1 < Sar

(Sa)k + (1− Sor )

2, if (Sa)k+1 > (1− Sor )

(Sa)k+1, otherwise.




5. Summary


Note on the Implementing Adaptive NR Method: Part X

and,

(cac )k+1 =

(cac )k + 0

2, if (ca

c )k+1 < 0

(cac )k + 1

2, if (ca

c )k+1 > 1

(cac )k+1, otherwise.




5. Summary


Fortran Results Part I

Table: Compositional Simulator’s Input Parameters

Parameter Assigned Value Units Description

uIN 10−4 cm/s input flowrate

SorH, SarH 0.35 res. sat. at high IFTPIN, POUT 1 atm endpoint Pressuresφ 0.24 porosity

CINs 0.1 overall surfactant conc.

CINp 0 overall oil conc.

L 100 cm core (porous media) lengthK 0.5 Darcy permeability

ko0Hr , ka0H

r 1, 0.2 Rel. Permeability at high IFT

µoH , µaH 5, 1 cP phase viscosities




5. Summary


Fortran Results Part II

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

1 2 3 4 5

n=1

n=2

n=3

Figure: Aqueous phase pressures across grid at different timesteps.




5. Summary


Fortran Results Part III

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5

n=1

n=2

n=3

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




5. Summary


Fortran Results Part IV

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5

n=1

n=2

n=3

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




5. Summary


Fortran Results Part V

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5

n=1

n=2

n=3

Figure: Aqueous phase saturation across grid at different timesteps.




5. Summary


Fortran Results Part VI

Figure: Oleic phase saturation across grid at different timesteps.




5. Summary


Achievements - 3 semesters residency

Published 2 journal articles:Akmal Aulia and Noaman El-Khatib, ”Mathematical Description of the Implementation of the AdaptiveNewton-Raphson Method in Compositional Porous Media Flow,” International Journal of Basic andApplied 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, ”SmartOilfield Data Mining for Reservoir Analysis,” International Journal of Engineering and TechnologyIJET-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 ChemicalFlood 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




5. Summary


Gantt Chart

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

Dissertation Writing

Submit Dissertation

2009 2010Items

2 Conference Papers, 2 Journal Articles

2011

attempt 1 more journal




5. Summary

Summary

The importance of IMPECS in solving coupled PDE.

The importance of Newton-Raphson methods in many aspectof IMPECS.




5. Summary

Summary

The importance of IMPECS in solving coupled PDE.

The importance of Newton-Raphson methods in many aspectof IMPECS.




5. Summary

Thank you for coming! Questions and Comments?


Development of a 1D isothermal surfactant flooding simulator

Technology

tham boon

yinghui li

2pin pi1

solving coupled

finite dierence

surfactant

summaryfortran

compositional