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On Two Flash Methods for CompositionalReservoir Simulations: Table Look-up and

Reduced Variables

Wei Yan, Michael L. Michelsen, Erling H. Stenby, Abdelkrim Belkadi

Center for Energy Resources Engineering (CERE)Technical University of Denmark

October 18, 2011

32ndIEA EOR Annual Symposium & Workshop

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Introduction

Flash: for a mixture of compositionz, will it split into two (or more)phases at specified Tand P and what are the phase compositionsand phase amounts?

A summary of two recent studies:

CSAT(table look-up):

Belkadi et al., Comparison of two methods for speeding up

flash calculations in compositional simulations, SPE 142132

compared with the shadow region method

Reduced variables/reduction methods:

Michelsen, M.L., Reduced variables - revisited, CEREDiscussion Meeting 2011

compared with the conventional flash

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Blind calculations without a prioriinformation

Two steps

Stability analysis: whether the feed splits into two phases?

Phase split: calculate the equilibrium compositions using theinitial estimates from the first step

Old but robust, virtually no convergence problems

More on safety than speed

The phase of composition z is stable at the specified (T,P)if and only if the tangent plane distance (TPD)

Conventional flash

Michelsen, M. L. (1982a & b) Fluid Phase Equilibria 9: 119 & 21-40.

Michelsen and Mollerup (2007) Thermodynamic models: Fundamentals and Computational Aspects

( )( )( ) ln ln ( ) ln ln 0i i i i ii

tpd w w z = + w w z

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Compositional simulations where information from previouscalculations may be utilized (NF,x, tpd, )

Distinction between different regions by TPD

Shadow region method

A. Unstable: one or twonegative TPD

B. Just stable: one trivial andone non trivial TPD=0

C. Single phase: one trivial andone non trivial TPD>0

D. Single phase: only trivialsolutions

Shadowregion

Rasmussen et al. (2006) SPE Res Eval & Eng 9: 32 38.

Michelsen and Mollerup (2007) Thermodynamic models: Fundamentals and Computational Aspects.

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CSP/CST/CSAT

inspired by the 1D analytical solution of gas injectiona

few key tie-lines in the solution path. CSP based table look-up approach to replace stability

test/phase split

Procedure

Tie-line tables constructed either in advance or adaptively For a new feed z

Compositional Space Adaptive

Tabulation (CSAT) method

Voskov and Tchelepi (2007) SPE 106029

Voskov and Tchelepi (2008) Transport in Porous Media 75: 111128.

2

( (1 ) )k k

i i i

iz y x +

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Only for phase split step to approximate flash results in two-phase region

A unique distance for each tie-line

Shortest distance ( ) from the feed z to tie-line k

The corresponding is readily obtained as

If ek for all the Mtie-lines in the table, flash thecomposition and update the tie-line table if it is two-phase.

Tie-line Table Look-up (TTL)

our implementation of CSAT

( )( )( ) = = + 2

2( ) min 1k k k k i i ii

e d z y x dk tie-line distance

( ) ( )

( ) ( )

Tk k k

Tk k k k

=

z x y x

y x y x

=k kd e

Eq.(5)

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Gas injection systems tested

System 1 System 2 System 3 System 4

Oil 13-component oil Zick Oil 1* Zick Oil 2* Zick Oil 3*

Gas 0.8 CO2+ 0.2 C1 Zick Gas 1* Zick Gas 2* Zick Gas 3*

T(K) 375.00 358.15 358.15 358.15

P (atm) 300 140 200 230

EoS used SRK PR PR PR

* 12-component fluid description from Jessen (2000) Ph.D. thesis or Orr (2007) GasInjection Processes.

Tested with 1-D slimtube simulation with 500 cells

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Analysis of CSAT using System 1 The influence of number of tie-lines Mand the tolerance on

simulation time and %skips of flash calculations

Larger Mincreases simulation time and %skips Smaller increases simulation time but decreases %skips Sorting tie-lines gives limited help

=10-4 =10-5 =10-6 =10-7

M= 100 Time (sec) 4.2 7.0 7.1 7.1

% skips 41% 0.1% 0.0% 0.0%

M= 500 Time (sec) 2.0 18.1 21.2 21.5% skips 99.9% 10% 0.3% 0.2%

M= 1000 Time (sec) 2.0 28.8 37.3 38.3

% skips 99.9% 18% 0.9% 0.4%

M= 5000 Time (sec) 2.0 66.4 134.8 157.2% skips 99.9% 64.7% 25.8% 7%

Decreasing

In

creasing

M

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Analysis of CSAT using System 1

=10-4 not accurate; =10-6 or 10-7 too few skips. Higher Mrequires even smaller

=10-4 =10-5 =10-6 =10-7

M= 1000 Time (sec) 2.0 28.8 37.3 38.3

% skips 99.9% 18% 0.9% 0.4%

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TTL with pre-calculated tie-lines

(TTL-PRE) The tie-line table can be calculated in advance to reduce

simulation time

Use M = 20000 and = 10-8 to find the most frequently

used tie-lines during the simulation.

3 tie-lines are identified, accounting for 88% of hits

Gas saturation Recovery

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System 1: simulation timesPVI=0.5 PVI=1.2

Time

(sec)

Direct

approximation in

two-phase*

Time

(sec)

Direct

approximation

in two-phase*

Conventional/

Full stability

47.4 163.3

TTL

M=100, =10-5 7.0 0.1% 28.0 0.02%

M=500,

=10-6

21.2 0.3% 91.6 0.06%M=1000, =10-6 37.3 0.9% 166.0 0.18%

M=5000, =10-7 157.2 7% 731.5 1.5%

TTL-PRE

(three tielines)

=10-4

2.5 49% 6.4 63%=10-5 2.6 46% 6.5 61%

=10-6 2.7 45% 8.5 60%

=10-7 2.8 37% 10.1 22%

Shadow region 3.2 10.9

* Reported numbers are percentages of total flashes in two-phase region

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Just compare one tie-line in the same cell from a previousrigorous flash using tie-line distance.

Procedure

Tie-line Distance Based Approximation

(TDBA)an alternative and simpler

Calculate ek as before (only one)

If e>, do new flash, and update the tie-line if it is two-phase

If ee>10-4, use the previous results with adjustment

( )1i

i

i i old

y

y x

=

+ ,i i new iv z = ( ), 1i i new il z =

Option 1 (TDBA1): use old K values to solve Rachford-Rice Eq.

Option 2 (TDBA2): use vapor split factors i to adjust directly

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System 1: simulation times

* Reported numbers are percentages of total flashes in two-phase region

PVI=0.5 PVI=1.2

Time

(sec)

Approx. with

adjustment

in two-phase*

Direct

approximation

in two-phase*

Time

(sec)

Approx. with

adjustment

in two-phase*

Direct

approximation

in two-phase*

Conventional/

Full stability

47.4 163.3

TTL-PRE

(three tielines)

=10-4 2.5 49% 6.4 63%

=10

-7

2.8 37% 10.1 22%TDBA1

=10-4 1.5 84% 11% 3.5 86% 12%

=10-5 1.7 76% 11% 3.9 84% 11%

=10-6 2.0 68% 8% 4.7 79% 10%

=10-7 2.3 58% 7% 5.6 72% 10%

TDBA2

=10-4 1.4 84% 11% 3.2 85% 13%=10-5 1.7 77% 10% 3.7 83% 11%

=10-6 2.0 67% 9% 4.4 78% 11%

=10-7 2.3 59% 6% 5.3 73% 9%

Shadow region 3.2 10.9

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Accurate solution

TDBA1 =10-6

TDBA1 =10-4

TDBA1 =10-5

TDBA1 =10-7

Accurate solution

TDBA1 =10-6

TDBA1 =10-4

TDBA1 =10-5

TDBA1 =10-7

Cell#

Gassa

turation

PVI

Recovery(%)

TDBA1 results for System 1

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6-component gas injection simulated by PC-SAFT and SRK

Speed-up 1: SPE 142995 (soliddashed )

Speed-up 2: TDBA1 (dasheddash-dot)

TDBAs potential : speeding upcomplicated EoSs

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Summary on approximation methods

CSAT/TTL saves the time for rigorous flash but managing the tie-line table can be a significant overhead.

The simulation time increases dramatically with the number

of tie-lines used. Big tolerances lead to inaccurate results.

TTL-PRE is better but gives limited speeding-up compared withthe shadow region algorithm.

TDBA is simpler and cuts the simulation time by 1/3 to 1/2.

The approximation methods may have potential to speed upsimulation with complicated EoSs.

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Reduced variables methodsbasics Solution procedure for equilibrium calculations with a cubic EoS

where the matrix of BIPs is of low rank

If all BIPs are zero, and

Consequently, the vector of can be written as a linear combination of3 pre-calculated vectors, with ith elements 1, and bi. Sameapplies to the lnKi.

( )

nRT A

P V B V V B= +

C

i i

i

B b n= C C

ij i j

i j

A a n n= (1 )ij ii jj ija a a k =

ln i n a i b iC C A C b = + +

2C

i ii ij j

j

A a a n=

2

C

i ij j

ji

AA a nn

= =

*ln i n a ii b iC C a C b = + +

ln i

iia

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A brief history First - to our knowledge - used 30 years ago by Michelsen and

Heideman (1981) for critical point calculation

Suggested for flash calculations by Michelsen (1986)

Single nonzero BIP row/column, Jensen and Fredenslund (1987)

Generalized for nonzero BIPs by Hendriks (1992)

Extensively used in the generalized version for the last 20 years

Its advantages first questioned in public by Haugen and Becknerin 2011 (SPE 141399)

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Arguments against reduced variables Essentially restricted to the cubic EOS

Difficult to be formulated as unconstrained minimization problemsconsequently, less safe.

More cumbersome composition derivatives

The simple algebraic operations required to evaluateAi are today veryinexpensive (SIMD)

Our experience: Effort of the conventional approach grows

approximately linearly with C , not proportionally with C2

or C3

.

A fair comparison between minimization based reduced variablesmethod and conventional flash requires substantial coding (perhapsmodest potential for improvement)

But recent development by Nichita and Graciaa (2010) enabled anadaption to Michelsens existing code without extensive modifications!

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Reduced variables by spectral

decomposition Consider the matrix P with elements Pij=1-kij. We calculate the

spectral decomposition

where is the kth eigenvalue of P and u the correspondingeigenvector. The eigenvalues are numbered in decreasing

magnitude. Assume now that the eigenvalues are negligible for k> Mwhere M >

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Contd We then get and

and with

Net results

Vector of : Linear combination of 2m + 3 vectors

Results identical to full approach

Computational effort reduced from C2 to 2CMplus overhead!

Successive substitution

Conventional implementation, where the reduction method is onlyimplemented to calculateAi

Acceleration as usual

No effect on convergence behavior

Used for stability analysis, as well as for phase split

1

MT

k k k

k

=

= P u u1 1

M M

ij k ii ik jj jk k ik jk

k k

a a u a u e e = =

= =

1 1 1 1

2 2C M C M

i ij j k ik jk j k ik

k k j k

A a n e e n d e= = = =

= = = 1

2C

k k jk j

j

d e n=

=

ln i

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Second order Gibbs energy minimization

Independent variables c1, c2, , and cM+2 Gradient

or

where is vapor moles iand (from Rachford-Rice equation)

Hessian

Wij looks complex to calculate, but simple algebraic expressions for theelements can be derived.

iv

2

1

lnM

i l il

l

K c e+

=

= , 1 1i Me + = , 2i M ie b+ =

1

Ci

ij i j

vG G

c v c=

=

c=g Wg

c TH WHW

/ij i j

W v c=

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Procedure for the 2nd

order minimization

Calculate the K-factors from c

Solve the Rachford-Rice equation to get vi Calculate conventional gradient and Hessian

Calculate transformation matrix W

Calculate c-based gradient and Hessian

Calculate corrected c using trust-region approach

Similar procedure for Stability Analysis

How does it compare to the classic approach?

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Alternative simplification: sparse k

where

and

Uses approximately 2mCmultiplications.

1

(1 ) ( )C

ii jj ij j ii i

j

a a k n a S S =

=

1

C

jj j

j

S a n=

=

1

1

C

jj ij j

j

i m

jj ij j

j

a k n i mS

a k n i m

=

=

= >

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Test examples Example 1Modified SPE3 with 9 components. Modified such that

all kij = 0 for i > 3, j > 3. Non-zero interaction coefficients for N2,CO2 and CH4.

Example 2Removal of N2 and CO2. Only CH4 has nonzero BIPs.Phase diagram largely unmodified, as the content of the removedspecies was small. Only 5 variables in the reduced case!

In both tests, the mixture was expanded to 27 or 25 componentsby subdividing the last species.

One million flash calculations in an equidistant 1000 by 1000 grid in

Tand P. All calculations are blind. About 60% two-phase.

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Test example 1

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Summary on reduced variables Modest effect of increasing C: Linear, but less than proportional

No advantage of reduction methods over alternatives

Fastest result: utilize sparsity of BIP-matrix

Computing times are in general very implementation dependent

Other implementations of reduction methods might be more efficientthan the one used here.

To be convincing, they should be able to beat the current computingtimes.

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Acknowledgment

Danish Council for Technologyand Production Sciences