on two flash methods for compositional reservoir...

On Two Flash Methods for Compositional Reservoir 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

32nd IEA EOR Annual Symposium & Workshop

2

Introduction

� Flash: for a mixture of composition z, will it split into two (or more) phases at specified T and P and what are the phase compositions and 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”, CERE Discussion Meeting 2011

� compared with the conventional flash

3

� “Blind” calculations without a priori information

� Two steps

� Stability analysis: whether the feed splits into two phases?

� Phase split: calculate the equilibrium compositions using the initial 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: 1–19 & 21-40.

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

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

tpd w w zϕ ϕ= + − − ≥∑w w z

4

� Compositional simulations where information from previous calculations may be utilized (NF, x, tpd, …)

� Distinction between different regions by TPD

Shadow region method

A. Unstable: one or two negative TPD

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

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

D. Single phase: only trivial solutions

Shadow region

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

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

5

� CSP/CST/CSAT

� inspired by the 1D analytical solution of gas injection—a 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: 111–128.

2( (1 ) )k ki i i

i

z y xβ β ε − + −

6

� 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 β is readily obtained as

� If ekε for all the M tie-lines in the table, flash the composition and update the tie-line table if it is two-phase.

Tie-line Table Look-up (TTL)—our implementation of CSAT

( )( )( )β β= = − + −∑ 22( ) min 1k k k ki i ii

e d z y x dk tie-line distance

( ) ( )( ) ( )

Tk k k

Tk k k kβ

− −=

− −

z x y x

y x y x

=k kd e

Eq.(5)

7

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) Gas Injection Processes.

� Tested with 1-D slimtube simulation with 500 cells

8

Analysis of CSAT using System 1

� The influence of number of tie-lines M and the tolerance on simulation time and %skips of flash calculations

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

ε =10-4 ε =10-5 ε =10-6 ε =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 ε

Incr

easi

ng M

9

Analysis of CSAT using System 1

� ε=10-4 not accurate; ε =10-6 or 10-7 too few skips.� Higher M requires even smaller ε

ε =10-4 ε =10-5 ε =10-6 ε =10-7

M = 1000 Time (sec) 2.0 28.8 37.3 38.3% skips 99.9% 18% 0.9% 0.4%

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400 450 500

Ga

s sa

tura

tio

n

Cell number

Accurate solution

CSAT M=1000 eps=1E-4

CSAT M=1000 eps=1E-5

CSAT M=1000 eps=1E-6

CSAT M=1000 eps=1E-7

10

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 ε = 10-8 to find the most frequently used tie-lines during the simulation.

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

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400 450 500

Ga

s sa

tura

tio

n

Cell number

Accurate solution

CSAT-PRE eps=1E-4

CSAT-PRE eps=1E-5

CSAT-PRE eps=1E-6

CSAT-PRE eps=1E-7

0

10

20

30

40

50

60

70

80

90

100

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Re

cov

ery

(%

)

PVI

Accurate solution

CSAT-PRE eps=1E-4

CSAT-PRE eps=1E-5

CSAT-PRE eps=1E-6

CSAT-PRE eps=1E-7

Gas saturation Recovery

11

System 1: simulation times

PVI=0.5 PVI=1.2Time(sec)

Directapproximation in

two-phase*

Time(sec)

Directapproximation in two-phase*

Conventional/Full stability

47.4 163.3

TTLM=100, ε =10-5 7.0 0.1% 28.0 0.02%M=500, ε =10-6 21.2 0.3% 91.6 0.06%M=1000, ε =10-6 37.3 0.9% 166.0 0.18%M=5000, ε =10-7 157.2 7% 731.5 1.5%

TTL-PRE(three tielines)

ε =10-4 2.5 49% 6.4 63%ε =10-5 2.6 46% 6.5 61%ε =10-6 2.7 45% 8.5 60%ε =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

12

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

� Procedure

Tie-line Distance Based Approximation (TDBA)—an alternative and simpler

� Calculate ek as before (only one)

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

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

( )1i

ii i old

y

y x

βθβ β

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

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

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

13

System 1: simulation times

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

PVI=0.5 PVI=1.2Time(sec)

Approx. with adjustment

in two-phase*

Directapproximation in two-phase*

Time(sec)

Approx. with adjustment

in two-phase*

Directapproximation in two-phase*

Conventional/Full stability

47.4 163.3

TTL-PRE(three tielines)

ε =10-4 2.5 49% 6.4 63%ε =10-7 2.8 37% 10.1 22%TDBA1ε =10-4 1.5 84% 11% 3.5 86% 12%ε =10-5 1.7 76% 11% 3.9 84% 11%ε =10-6 2.0 68% 8% 4.7 79% 10%ε =10-7 2.3 58% 7% 5.6 72% 10%TDBA2ε =10-4 1.4 84% 11% 3.2 85% 13%ε =10-5 1.7 77% 10% 3.7 83% 11%ε =10-6 2.0 67% 9% 4.4 78% 11%ε =10-7 2.3 59% 6% 5.3 73% 9%

Shadow region 3.2 10.9

14

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400 450 500

Ga

s sa

tura

tio

n

Cell number

Accurate solution

TDBA1 eps=1E-4

TDBA1 eps=1E-5

TDBA1 eps=1E-6

TDBA1 eps=1E-7

0

10

20

30

40

50

60

70

80

90

100

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Re

cov

ery

(%

)

PVI

Accurate solution

TDBA1 eps=1E-4

TDBA1 eps=1E-5

TDBA1 eps=1E-6

TDBA1 eps=1E-7

Accurate solution

TDBA1 ε=10-6

TDBA1 ε=10-4

TDBA1 ε=10-5

TDBA1 ε=10-7

Accurate solution

TDBA1 ε=10-6

TDBA1 ε=10-4

TDBA1 ε=10-5

TDBA1 ε=10-7

Cell#

Gas saturation

PVI

Recovery (%)

TDBA1 results for System 1

15

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

� Speed-up 1: SPE 142995 (solid�dashed )

� Speed-up 2: TDBA1 (dashed�dash-dot)

TDBA’s potential : speeding up complicated EoS’s

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0 5 10 15 20 25 30

Sim

ula

tio

n t

ime

ra

tio

Number of components

CPA

PC-SAFT

CPA new

PC-SAFT new

CPA+TDBA w.r.t. SRK+TDBA

PC-SAFT+TDBA w.r.t. SRK+TDBA

CPA+TDBA w.r.t. SRK

PC-SAFT+TDBA w.r.t. SRK

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30

Sim

ula

tio

n t

ime

(se

c)

Number of components

SRK

CPA

PC-SAFT

CPA new

PC-SAFT new

SRK+TDBA

CPA+TDBA

PC-SAFT+TDBA

16

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 with the 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 up simulation with complicated EoS’s.

17

Reduced variables methods—basics

� Solution procedure for equilibrium calculations with a cubic EoSwhere the matrix of BIP’s is of low rank

If all BIPs are zero, and

Consequently, the vector of can be written as a linear combination of 3 pre-calculated vectors, with i’th elements 1, and bi. Same applies to the lnKi.

( )

nRT AP

V B V V B= −

− +C

i ii

B b n=∑C C

ij i ji j

A a n n=∑∑ (1 )ij ii jj ija a a k= −

ˆln i n a i b iC C A C bϕ = + +

2C

i ii ij jj

A a a n= ∑

2C

i ij jji

AA a n

n

∂= =∂ ∑

*ˆln i n a ii b iC C a C bϕ = + +

ˆln iϕiia

18

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 Beckner in 2011 (SPE 141399)

19

Arguments against reduced variables

� Essentially restricted to the cubic EOS

� Difficult to be formulated as unconstrained minimization problems—consequently, less safe.

� More cumbersome composition derivatives

� The simple algebraic operations required to evaluate Ai are today very inexpensive (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 variables method and conventional flash requires substantial coding (perhaps modest potential for improvement)

� But recent development by Nichita and Graciaa (2010) enabled an adaption to Michelsen’s existing code without extensive modifications!

20

Reduced variables by spectral decomposition

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

where is the k’th eigenvalue of P and u the corresponding eigenvector. The eigenvalues are numbered in decreasing magnitude. Assume now that the eigenvalues are negligible for k> M where M >

21

Cont’d

� 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 2CM plus overhead!

� Successive substitution

� Conventional implementation, where the reduction method is only implemented to calculate Ai

� Acceleration as usual

� No effect on convergence behavior

� Used for stability analysis, as well as for phase split

1

MT

k k kk

λ=

=∑P u u1 1

M M

ij k ii ik jj jk k ik jkk k

a a u a u e eλ λ= =

= =∑ ∑

1 1 1 1

2 2C M C M

i ij j k ik jk j k ikk k j k

A a n e e n d eλ= = = =

= = =∑ ∑ ∑ ∑1

2C

k k jk jj

d e nλ=

= ∑

ˆln iϕ

22

Second order Gibbs energy minimization

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

or

where is vapor moles i and (from Rachford-Rice equation)

� Hessian

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

iv

2

1

lnM

i l ill

K c e+

=

= ∑ , 1 1i Me + = , 2i M ie b+ =

1

Ci

ij i j

vG G

c v c=

∂∂ ∂=∂ ∂ ∂∑

c =g Wg

c T≈H WHW

/ij i jW v c= ∂ ∂

23

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?

24

Alternative simplification: sparse k

where

and

� Uses approximately 2mC multiplications.

1

(1 ) ( )C

ii jj ij j ii ij

a a k n a S S=

− = −∑

1

C

jj jj

S a n=

=∑

1

1

C

jj ij jj

i m

jj ij jj

a k n i m

S

a k n i m

=

=

≤= >

∑

∑

25

Test examples

� Example 1—Modified 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 2—Removal of N2 and CO2. Only CH4 has nonzero BIPs. Phase diagram largely unmodified, as the content of the removed species was small. Only 5 variables in the reduced case!

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

� One million flash calculations in an equidistant 1000 by 1000 grid in T and P. All calculations are blind. About 60% two-phase.

26

Test example 1

27

Test example 2

28

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 efficient than the one used here.

� To be convincing, they should be able to beat the current computing times.

29

Acknowledgment

Danish Council for Technology and Production Sciences