On Two Flash Methods for Compositional Reservoir Simulations: Table Look-up and Reduced Variables

Transcription

1 32 nd IEA EOR Annual Symposium & Workshop 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

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 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 2

3 Conventional flash Blind calculations without a priori information Two steps Stability analysis: whether the feed splits into two phases? The phase of composition z is stable at the specified (T,P) if and only if the tangent plane distance (TPD) ( ϕ ϕ ( )) tpd( w) = w ln w + ln ( w) ln z ln z 0 i i i i i i 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 Michelsen, M. L. (1982a & b) Fluid Phase Equilibria 9: 1 19 & Michelsen and Mollerup (2007) Thermodynamic models: Fundamentals and Computational Aspects 3

4 Shadow region method Compositional simulations where information from previous calculations may be utilized (N F, x, tpd, ) Distinction between different regions by TPD Shadow region 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 Rasmussen et al. (2006) SPE Res Eval & Eng 9: Michelsen and Mollerup (2007) Thermodynamic models: Fundamentals and Computational Aspects. 4

5 Compositional Space Adaptive Tabulation (CSAT) method CSP/CST/CSAT 5 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 Vapor fraction Criterion Voskov and Tchelepi (2007) SPE k z j x j β = k k y j x j k k 2 zi ( β yi + (1 β ) xi ) < ε i Voskov and Tchelepi (2008) Transport in Porous Media 75: k tieline index j component index arbitrarily chosen if one of the stored tie-lines satisfies these criteria for the composition of interest, the table is used to look up the flash results. Otherwise, a standard EOS based phase behavior procedure is employed.

6 Tie-line Table Look-up (TTL) our implementation of CSAT Only for phase split step to approximate flash results in twophase region A unique distance for each tie-line k k Shortest distance ( d = e ) from the feed z to tie-line k ( ( β i i ( β ) i )) 2 e k = ( d k ) = min z y k + 1 x k d k tie-line distance i The corresponding β is readily obtained as β = T ( z x k ) ( y k x k ) T ( y k x k ) ( y k x k ) If e k <ε, accept tie-line k as flash solution, and β calculated from Eq.(5) If e k >ε for all the M tie-lines in the table, flash the composition and update the tie-line table if it is two-phase. 2 Eq.(5) 6

7 Gas injection systems tested Tested with 1-D slimtube simulation with 500 cells System 1 System 2 System 3 System 4 Oil 13-component oil Zick Oil 1* Zick Oil 2* Zick Oil 3* Gas 0.8 CO C 1 Zick Gas 1* Zick Gas 2* Zick Gas 3* T (K) P (atm) EoS used SRK PR PR PR * 12-component fluid description from Jessen (2000) Ph.D. thesis or Orr (2007) Gas Injection Processes. 7

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 Decreasing ε Increasing M ε =10-4 ε =10-5 ε =10-6 ε =10-7 M = 100 Time (sec) % skips 41% 0.1% 0.0% 0.0% M = 500 Time (sec) % skips 99.9% 10% 0.3% 0.2% M = 1000 Time (sec) % skips 99.9% 18% 0.9% 0.4% M = 5000 Time (sec) % skips 99.9% 64.7% 25.8% 7% Larger M increases simulation time and %skips Smaller ε increases simulation time but decreases %skips Sorting tie-lines gives limited help 8

9 Analysis of CSAT using System 1 1 Accurate solution 0.8 CSAT M=1000 eps=1e-4 CSAT M=1000 eps=1e-5 Gas saturation CSAT M=1000 eps=1e-6 CSAT M=1000 eps=1e Cell number ε =10-4 ε =10-5 ε =10-6 ε =10-7 M = 1000 Time (sec) % skips 99.9% 18% 0.9% 0.4% ε=10-4 not accurate; ε =10-6 or 10-7 too few skips. Higher M requires even smaller ε 9

10 TTL with pre-calculated tie-lines (TTL-PRE) The tie-line table can be calculated in advance to reduce simulation time Use M = 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 1 Gas saturation Accurate solution Recovery 0.8 CSAT-PRE eps=1e-4 80 CSAT-PRE eps=1e-5 70 Gas saturation CSAT-PRE eps=1e-6 CSAT-PRE eps=1e-7 Recovery (%) Accurate solution CSAT-PRE eps=1e-4 CSAT-PRE eps=1e-5 CSAT-PRE eps=1e-6 CSAT-PRE eps=1e Cell number PVI 10

11 System 1: simulation times 11 PVI=0.5 Time Direct Time (sec) approximation in (sec) two-phase * PVI=1.2 Direct approximation in two-phase * Conventional/ Full stability TTL M=100, ε = % % M=500, ε = % % M=1000, ε = % % M=5000, ε = % % TTL-PRE (three tielines) ε = % % ε = % % ε = % % ε = % % Shadow region * Reported numbers are percentages of total flashes in two-phase region

12 Tie-line Distance Based Approximation (TDBA) an alternative and simpler Just compare one tie-line in the same cell from a previous rigorous flash using tie-line distance. Procedure Calculate e k as before (only one) If e>ε, do new flash, and update the tie-line if it is two-phase If e<10-4 ε, use the previous results as a solution without any adjustment If ε >e>10-4 ε, use the previous results with adjustment Option 1 (TDBA1): use old K values to solve Rachford-Rice Eq. Option 2 (TDBA2): use vapor split factors θ i to adjust directly θ β y i i = β yi + ( 1 β ) xi old v z θ = l = z ( θ ) i i, new i i i, new 1 i 12

13 System 1: simulation times Conventional/ Full stability TTL-PRE (three tielines) Time (sec) PVI=0.5 Approx. with adjustment in two-phase * Direct approximation in two-phase * Time (sec) PVI=1.2 Approx. with adjustment in two-phase * Direct approximation in two-phase * ε = % % ε = % % TDBA1 ε = % 11% % 12% ε = % 11% % 11% ε = % 8% % 10% ε = % 7% % 10% TDBA2 ε = % 11% % 13% ε = % 10% % 11% ε = % 9% % 11% ε = % 6% % 9% Shadow region * Reported numbers are percentages of total flashes in two-phase region

14 TDBA1 results for System Accurate solution TDBA1 eps=1e-4 80 Gas saturation Gas saturation Accurate solution TDBA1 ε=10-4 TDBA1 ε=10-5 TDBA1 ε=10-6 TDBA1 ε=10-7 TDBA1 eps=1e-5 TDBA1 eps=1e-6 TDBA1 eps=1e-7 Recovery (%) Recovery (%) Accurate solution TDBA1 ε=10-4 TDBA1 ε=10-5 TDBA1 ε=10-6 TDBA1 ε=10-7 Accurate solution TDBA1 eps=1e-4 TDBA1 eps=1e-5 TDBA1 eps=1e-6 TDBA1 eps=1e Cell number Cell# PVI PVI 14

15 TDBA s potential : speeding up complicated EoS s 6-component gas injection simulated by PC-SAFT and SRK Simulation time (sec) SRK CPA PC-SAFT CPA new PC-SAFT new SRK+TDBA CPA+TDBA PC-SAFT+TDBA Simulation time ratio 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 Number of components Number of components Speed-up 1: SPE (solid dashed ) Speed-up 2: TDBA1 (dashed dash-dot) 15

16 Summary on approximation methods CSAT/TTL saves the time for rigorous flash but managing the tieline 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. 16

17 Reduced variables methods basics Solution procedure for equilibrium calculations with a cubic EoS where the matrix of BIP s is of low rank B C = i b n i i P nrt A = V B V ( V + B ) C C = aijni n j aij = aiia jj (1 kij ) i j If all BIPs are zero, Ai = 2 aii aij n j and A ˆ ϕ = C + C A + C b ln i n a i b i C A A = = 2 n Consequently, the vector of ln ˆ ϕ i can be written as a linear combination of 3 pre-calculated vectors, with i th elements a1, ii and b i. Same applies to the lnk i. C a n i ij j i j j * ln ˆ ϕ i = C n + C a a ii + C b b i 17

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 ) 18

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 A i are today very inexpensive (SIMD) Our experience: Effort of the conventional approach grows approximately linearly with C, not proportionally with C 2 or C 3. 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! 19

20 Reduced variables by spectral decomposition Consider the matrix P with elements P ij =1-k ij. We calculate the spectral decomposition P C = k= 1 λ u u T k k k λ k 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 << C. For example: All BIP s equal to zero: M=1 Upper triangle of BIP s zero except for a few rows, i.e., k = 0 for i > m and j > i ij In this case, M = 2m + 1 and the match is exact 20

21 Cont d T We then get P λ u u and and A = 2 a n = 2 λ e e n = d e with Net results M = k= 1 k k k C M C M i ij j k ik jk j k ik k = 1 k = 1 j= 1 k = 1 ln ˆ ϕ i Vector of : Linear combination of 2m + 3 vectors Results identical to full approach Computational effort reduced from C 2 to 2CM plus overhead! Successive substitution Conventional implementation, where the reduction method is only implemented to calculate A i Acceleration as usual No effect on convergence behavior ij k ii ik jj jk k ik jk k= 1 k= 1 Used for stability analysis, as well as for phase split M a = λ a u a u = λ e e d M C = 2λ e n k k jk j j= 1 21

22 Second order Gibbs energy minimization ln K M + 2 i = cleil i, M 1 1 l= 1 e = + ei, M 2 + = b i Independent variables c 1, c 2,, and c M+2 Gradient C G G vi = or g c = Wg c v c v i j i= 1 i j where is vapor moles i and W = v / c (from Rachford-Rice equation) ij i j Hessian H c WHW T W ij looks complex to calculate, but simple algebraic expressions for the elements can be derived. 22

23 Procedure for the 2 nd order minimization Calculate the K-factors from c Solve the Rachford-Rice equation to get v i 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? 23

24 Alternative simplification: sparse k C j= 1 a a (1 k ) n = a ( S S ) ii jj ij j ii i where S C = j= 1 a n jj j and S i C a k n i m jj ij j j= 1 = m jj ij j > j= 1 a k n i m Uses approximately 2mC multiplications. 24

25 Test examples Example 1 Modified SPE3 with 9 components. Modified such that all k ij = 0 for i > 3, j > 3. Non-zero interaction coefficients for N 2, CO 2 and CH 4. Example 2 Removal of N 2 and CO 2. Only CH 4 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. 25

26 Test example 1 26

27 Test example 2 27

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. 28

29 Acknowledgment Danish Council for Technology and Production Sciences 29