1
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 (DTU),
DK 2800, Kgs. Lyngby, Denmark
* Corresponding author, Tel.: +45 45252914, Fax: +45 45882258, E-mail: [email protected] Abstract
Compositional reservoir simulations are usually used for simulating enhanced oil
recovery processes involving large composition changes, such as gas injection. Phase
equilibrium calculation, which is often solved in terms of flash calculation, is the most
time consuming part in those simulations. Improving the flash calculation speed has
therefore become one of the central issues in compositional reservoir simulations, and
new algorithms for fast flash calculations are frequently proposed in the literature. Two
types of fast flash methods were recently investigated at DTU CERE, namely the
Compositional Space Adaptive Tabulation (CSAT) method and the reduced variables
methods. This paper summarizes the major results obtained in those studies.
CSAT is a table look-up method, which saves computation time by replacing rigorous
phase equilibrium calculations by the stored results in a tie-line table whenever the new
feed composition is on one of the stored tie-lines within a certain tolerance. With a
slimtube simulator, it has been investigated whether the table look-up method can
compete with an efficient implementation of phase split calculation in two-phase regions.
The number of tie-lines stored for comparison and the tolerance set for accepting the feed
composition greatly influence the simulation speed and the accuracy of simulation results.
An important observation is that the table look-up itself is not free and its cost can be
comparable to the efforts for a rigorous flash. An alternative method, the Tie-line
Distance Based Approximation (TDBA) method, was proposed to get the approximate
results without performing a table look-up. TDBA can cut the simulation time by half or
even higher for the tests with cubic EoS’s. It has a bigger potential for speeding up
simulations with more advanced and complicated EoS’s.
2
The reduced variables methods or the reduction methods reformulate the original phase
equilibrium problem with a smaller set of independent variables. Various versions of the
reduced variables have been proposed since the late 80’s while efficiency of the methods
was recently questioned by Haugen and Beckner (2011). With the recent formulations by
Nichita and Garcia (2010), it is possible to code the reduced variables methods without
extensive modifications of Michelsen’s conventional flash algorithm. A simple test using
the SPE 3 example was performed, showing that the best reduction in time was less than
20% for the extreme situation of 25 components and just one row/column with non-zero
binary interaction parameters. A better performance can be achieved by a simpler
implementation using the sparsity of the interaction parameter matrix.
1. Introduction
Equation of State (EoS) based compositional reservoir simulations are usually employed
when the composition effects can no longer be accounted for by a black oil model. In
compositional reservoir simulations, solution to the phase equilibrium equations is often
separated from solution to the transport equations. In that case, the phase equilibrium
equations are solved by the so called flash algorithm, where the phase amounts and the
equilibrium phase compositions are computed for a given feed composition at specified
temperature and pressure. The calculation algorithm for two-phase isothermal flash is
rather matured (Michelsen, 1982a and 1982b; Michelsen and Mollerup, 2007) and the
solution procedure consists of two steps, a stability analysis step to determine whether the
feed will split into two phases, and a phase split step to calculate the equilibrium
compositions using the initial estimates from the first step. The algorithm is proposed for
general situations where there is not a priori information about the possible results. It
stresses more on safety than on speed. However, speed is a crucial concern in
compositional reservoir simulations since flash calculation with a complicated EoS model
increases the computation time dramatically. Speeding up the flash calculation without
too much compromise in accuracy and reliability has always been a research topic in
compositional reservoir simulations.
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The computation time spent on flash calculations can be reduced by different strategies.
One recent effort is the shadow region method (Rasmussen et al., 2006; Michelsen and
Mollerup, 2007) which reduces the computation time mainly by skipping stability
analysis for a large portion of compositions in the single phase region. In the two-phase
region, a highly efficient Newton-Raphson algorithm can be employed with initial
estimates from the previous step. The shadow region method has been applied to
compositional transient pipeline simulations (Rasmussen et al., 2006). Another recent
effort is represented by the Compositional Space Parameterization (CSP) framework and
the Compositional Space Tabulation (CST) method (Voskov and Tchelepi, 2007, 2008a
and 2008b), which is based on CSP. In the CST method, a table of converged flash
calculations (tie-lines) is built to parameterize the compositional space. The table can be
updated adaptively and the corresponding method is known as the Compositional Space
Adaptive Tabulation (CSAT) method. During a simulation run with CSAT, it is checked
if the feed composition lies on one of the stored tie-lines within a certain tolerance. The
standard EoS-based phase equilibrium calculation will be replaced if a stored tie-line is
identified in the tie-line table look-up. This look-up strategy can be applied to replace
both stability analysis and phase split (Voskov and Tchelepi, 2007, 2008a and 2008b).
The CSAT method has been implemented to a General Purpose Research Simulator
(GPRS) for simulating multicomponent immiscible, and miscible gas injection scenarios
(Voskov and Tchelepi, 2007, 2008a and 2008b). It is reported that the CSAT strategy can
lead to significant gains in computational efficiency compared to standard EoS based
compositional simulation (Voskov and Tchelepi, 2009) The authors also showed that the
adaptive tabulation approach can be extended to systems with component that partition
among three, or more, fluid phase equilibrium (Voskov and Tchelepi, 2009). Another
way of reducing the flash calculation time is through the reduced variables methods
(Michelsen, 1986; Hendriks, 1988; Firoozabadi and Pan, 2002; Pan and Firoozabadi,
2003). The reduced variables methods take advantage of the form of cubic equations of
state and approximate the original set of equations by a smaller set of equations.
This paper is based on two recent studies carried out at Center for Energy Resources
Engineering (CERE), DTU (Belkadi et al., 2011; Michelsen, 2011). The first part in this
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paper is about the table loop-up approach (Belkadi et al., 2011) and the second part is
about the reduced variables method (Michelsen, 2011). The purpose is to compare those
methods with the existing fast flash or conventional flash methods.
2. Table look-up methods
In the study of the table look-up methods, we use the shadow region method as the basis
for a fast flash calculation strategy. The method provides a simple but reliable way to
skip stability analysis in single phase regions, and no compromise in accuracy in two-
phase regions. We also notice that the tie-line table look-up technique employed by
CSAT has the potential to further reduce the computation time in the two-phase region by
approximating the rigorous phase split results with the results of an existing tie-line. The
approximation in the two-phase region has been implemented in two different ways: the
Tie-line Table Look-up (TTL) approach, which is similar to CSAT, and the tie-line
distance based approximation (TDBA) approach, which is improved based on the
analysis of the first one. In the following subsections, a brief description of the shadow
region will be given first. Then, the TTL methods and the TDBA methods will be
introduced. Finally, several 1-D multicomponent gas injection problems will be used to
compare the TTL method and the TDBA method with the shadow region method as
reference.
2.1. Shadow region method
The shadow region method takes advantage of the previous results to speed up flash
calculations (Rasmussen et al., 2006; Michelsen and Mollerup, 2007). For a given feed
composition, the shadow region refers to the part in the single phase region where a non-
trivial positive minimum of the tangent plane distance exists. The flash results of a feed
composition in a previous time step may fall into three different regions: the two-phase
region, the shadow region (single phase), and outside the shadow region (single phase).
Depending on which region it is, the flash strategies for the new feed composition in the
same grid block is different. In the two-phase region, a second order Newton-Raphson
algorithm is directly used with initial estimates given by
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,i i new iv z , 1i i new il z (1)
with the vapor split factors defined by
1i
ii i old
y
y x
(2)
If it does not converge, the calculation procedure is reverted to the safe traditional flash
procedure (Michelsen and Mollerup, 2007). In the shadow region, the non-trivial solution
from the previous tangent plane distance minimization is stored as the shadow phase. Its
composition is used as initial estimates for the stability analysis of the new composition.
Different from the traditional flash procedure, only one-sided stability analysis starting
from a second-order minimization is needed here. For a feed composition outside the
shadow region, if it is not too close to the critical point, it is safe to assume the new flash
results will be still in the single phase region and stability analysis can therefore be
skipped. If the composition is close to the critical point, the safe flash approach with
stability analysis must be employed. Rasmussen et al. (2006) suggested using the
minimum eigenvalue, λ1 of the Hessian matrix H used in minimizing tangent plane
distance, as such a measure of the closeness to the critical point. If the changes in
composition, pressure, and temperature satisfy the following criteria
, 10.1i i oldz z , 10.1oldP P P , 110 (K)oldT T (3)
we skip stability analysis.
2.2. Tie-line Table Look-up (TTL)
The TTL methods were modified from the CSAT method (Voskov and Tchelepi, 2007,
2008a and 2008b). CSAT was inspired by the special feature in the analytical solution of
1-D gas injection processes that the 1-D analytical solution usually contains just a limited
number of tie-lines (Orr, 2007). It is argued that a table without too many representative
tie-lines can provide good approximation for the injection simulation. Although CSAT is
based on 1-D gas injection processes, it can be applied to speeding up phase equilibrium
calculations in general compositional reservoir simulations.
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CSAT relies on building up a table of tie-lines and comparing the new feed composition
with the existing tie lines (Voskov and Tchelepi, 2007, 2008a and 2008b). To check if the
composition lies on one of the stored tie-lines or its extension, the following equations are
used
kj j
k kj j
z x
y x
(4)
2( (1 ) )k k
i i ii
z y x (5)
Here, the component j can be arbitrarily chosen. β is the vapor fraction, zi denotes the
mole fractions of component i in the new feed. kjx and k
jy represent the mole fractions in
the liquid and vapor phases for tie-line k. If one of the stored tie-lines satisfies the above
criteria, it is possible to skip the EoS-based phase equilibrium calculation. Otherwise, a
standard flash calculation is used to calculate the results and generate a new tie-line. The
tie-line table is constructed in advance and updated during the simulation. For simulations
not at a constant pressure, tie-line tables at several pressures must be constructed for
pressure interpolation during the simulation.
The TTL method is formulated in a way different from CSAT. Instead of using Eqs. (5)
and (6) to calculate the distance from the new feed composition to an existing tie-line, it
is suggested that the shortest distance between the feed and tie-line k should be calculated
(Michelsen, 2010). If we denote the shortest distance to tie-line k by dk. (dk)2 or the
minimum error ek can be obtained by the following minimization
2
min 1k k ki i i
i
e z y x (6)
The vapor fraction from the minimization is given by
Tk k k
Tk k k k
z x y x
y x y x (7)
In Eqs.(4) and (5), is arbitrarily calculated and there are Nc possible distances to a tie-
line for a Nc component system. The new formulation by Eqs. (6) and (7) gives a unique
distance to tie-line k (hereafter called tie-line distance) and the calculation is also direct.
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For a new feed composition in the two-phase region, we calculate its shortest distance to
tie-line k using Eqs. (6) and (7). If ek<ε, we accept the tie-line as flash solution; if ek>ε
for all the M tie-lines in the table, we flash the new composition, and in the case of two
phases, we include the solution in the tie-line set. The TTL method is only applied in the
two-phase region to speed up the calculation by approximating the rigorous results with
existing tie-lines. In the single phase region, the shadow region criteria are used to judge
whether stability analysis can be skipped.
2.3. Tie-line Distance Based Approximation (TDBA)
Instead of building a tie-line table for comparison, a simple way is suggested to find a
neighboring tie-line and to utilize its information to approximate flash results (Michelsen,
2010). For a new composition in a given grid block, it is assumed that the tie-line from
the previous rigorous flash calculation can be a good approximation to the new flash. No
tie-line table is constructed here. The distance between the new composition and the tie-
line from the previous flash calculation is calculated using Eqs. (6) and (7). Since there is
only one tie-line for comparison, only one distance (or error), e, needs to be calculated.
The flash is then proceeded in three different ways depending on the magnitude of e:
if e > ε, we do new flash;
if ε > e > 10-4ε, we use old results but adjustment is needed;
if e < 10-4ε, we use the previous results as a solution without any adjustment.
Since the method uses the distance to a tie-line to judge whether we can make
approximation to the rigorous flash results, we hereafter call it Tie-line Distance Based
Approximation (TDBA). It should be noted that when e is in the intermediate range (ε > e
> 10-4ε), we can either solve the Rachford-Rice equation with the old K-values or directly
adjust the material balance using Eqs. (1) and (2). These two options are denoted by
TDBA1 and TDBA2, respectively.
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2.4. Results and Discussion
A one-dimensional slimtube simulator was used here to test different methods.
Description about the slimtube simulator can be found in Michelsen (1998) and Yan et al.
(2004). The shadow region criteria were used in single-phase regions to skip stability
analysis. In the two-phase region, the TTL and the TDBA options were implemented to
study their computational performance as compared with the efficient implementation of
direct Newton-Raphson method originally employed in the slimtube simulator. The
option of using direct Netwon-Rapshon method is termed as the shadow region method
here. It serves as a reference option to compare with the other two options: TTL and
TDBA. Another reference option used in the computation speed comparison is the full
stability analysis method, where full stability analysis is performed whenever a flash
calculation is made. The full stability analysis method is much slower than the shadow
region method, whereas both methods make no approximations in phase equilibrium
solutions.
Four gas injection systems have been studied here (Table 1): System 1 is a 13-component
system, with detailed fluid description given in Table 2; Systems 2 to 4 correspond to the
systems studied by Zick (1986). The fluid description for Systems 2 to 4, listed in Table
3, is originally created by Jessen (2000). The experimental MMPs for Systems 2 to 4 are
150, 220, and 240 atm, respectively (Zick, 1986). Jessen’s description predicts similar
MMP values. Soave-Redlich-Kwong (SRK) EoS (Soave 1972) is used for System 1 and
Peng-Robinson (PR) EoS (Peng and Robinson 1976) is used for Systems 2 to 4.
Table 1. Overview of the four gas injection systems
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
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Table 2. Fluid description for gas injection system 1
zoil zgas Tc (K) pc (atm) ω MW (g/mol) kCO2,j
CO2 0.003699 0.8 304.2 72.8 0.225 44.01 0.00
C1 0.435813 0.2 190.6 45.4 0.008 16.04 0.12
C2 0.074085 305.4 48.2 0.098 30.07 0.12
C3 0.062088 369.8 41.9 0.152 44.09 0.12
i-C4 0.010398 408.1 36.0 0.176 58.12 0.12
C4 0.031394 425.2 37.5 0.193 58.12 0.12
i-C5 0.011498 460.4 33.4 0.227 72.15 0.12
C5 0.016497 469.6 33.3 0.251 72.15 0.12
C6 0.022396 507.4 29.3 0.296 86.17 0.12
C7 0.034293 529.5 32.3 0.4591 100.20 0.10
C8 0.040692 547.1 30.4 0.4854 114.23 0.10
C9 0.025295 568.1 27.4 0.5228 128.25 0.10
C22 0.228254 810.3 15.1 1.0315 310.60 0.10
Table 3. Fluid description for gas injection systems 2 to 4 (Zick oils and gases) zoil1
zoil2
zoil3
zgas1
zgas2
zgas3
Tc (K)
pc (atm)
ω
MW (g/mol)
kCO2,j
CO2 0.0483 0.0652 0.0673 0.2218 0.1774 0.1708 304.2 72.9 0.228 44.01 0
C1 0.2067 0.3542 0.3792 0.2349 0.3879 0.4109 190.6 45.4 0.008 16.04 0.105
C2 0.048 0.0534 0.0537 0.235 0.188 0.181 305.4 48.2 0.098 30.07 0.130
C3 0.0408 0.0377 0.037 0.2745 0.2196 0.2114 369.8 41.9 0.152 44.09 0.125
C4 0.0322 0.0266 0.0257 0.0338 0.027 0.0208 425.2 37.5 0.193 58.12 0.115
C5 0.0246 0.0192 0.0184 469.6 33.3 0.251 72.15 0.115
C6 0.0296 0.0225 0.0214 507.4 29.3 0.296 86.17 0.115
C7+(1) 0.2515 0.1859 0.1755 616.2 28.5 0.454 137.69 0.115
C7+(2) 0.128 0.0946 0.0891 698.9 19.1 0.787 243.95 0.115
C7+(3) 0.0851 0.0629 0.0593 770.4 16.4 1.048 347.69 0.115
C7+(4) 0.0629 0.0465 0.0438 853.1 15.1 1.276 481.03 0.115
C7+(5) 0.0425 0.0315 0.0296 1001.2 14.5 1.299 735.68 0.115
In all the simulations, a simple relative permeability model is used where the relative
permeability for a specific phase is set to the square of its saturation. A fixed viscosity
ratio between oil and gas (5:1) is used. All the simulations have used an implicit single-
point upstream scheme with 500 grid blocks. The time step is set to Δt/Δx =0.1 unless
otherwise mentioned. All experiments were run on an Intel Core™ Duo CPU with 3 GHz
processor and 1.94 GB of RAM.
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First, System 1 is used to analyze the performance of the TTL method. For TTL with the
tie-line table generated and updated during the simulation, its performance is influenced
by the tolerance used and the maximum number of tie-lines allowed in the tie-line table,
M. During the calculation, a new tie-line is stored whenever a new set of two-phase
results is obtained from the rigorous flash calculation. The time used in comparing with
the stored tie-lines and managing the tie-line table will increase with the number of tie-
lines in the table. In principle, sorting the tie-lines can move the most hit tie-lines to the
top of the table and thus reduces the searching time. However, sorting itself is also time-
consuming. We found that a partial sorting, i.e., simply moving a tie-line forward if its
number of hits is higher than that of the neighboring tie-line just before it, only reduces
the simulation time obviously if a larger number of tie-lines are used. But the simulation
time in that case is usually too long, which makes the method unattractive.
Table 4 - Performance of TTL method for System 1 with PVI=0.5 and different M and ε and time step.
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% Ntieline 100 100 100 100
M = 500 Time (sec) 2.0 18.1 21.2 21.5 % skips 99.9% 10% 0.3% 0.2% Ntieline 482 500 500 500
M = 1000 Time (sec) 2.0 28.8 37.3 38.3 % skips 99.9% 18% 0.9% 0.4% Ntieline 482 1000 1000 1000
M = 5000 Time (sec) 2.0 66.4 134.8 157.2 % skips 99.9% 64.7% 25.8% 7% Ntieline 482 5000 5000 5000
Table 4 shows the TTL results for System 1 with PVI=0.5. The simulation time is
strongly influenced by the maximum number of tie-lines, M. In general, the actual
number of tie-lines used during the simulation, Ntieline, increases with M. More tie-lines
will increase the time used in managing the tie-lines and the overall simulation time also
increases. For ε = 10-4, Ntieline does not increase with M, and the change in simulation time
does not follow the aforementioned trend. Besides, the simulation results are not accurate
at big tolerances. At ε = 10-4, none of the simulation results are acceptable. Figure 1
provides an example at M=1000. Compared to the accurate gas saturation profile, i.e., the
profile from the full stability analysis method or the shadow region method, where no
approximation is made in flash solutions, the TTL results are unacceptable at ε = 10-4 and
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10-5. At ε = 10-6 and 10-7 for M=1000, the CSAT results are closer to the accurate
solution, but the number of approximations (% skips) are actually very small. For higher
M, the tolerance that can give an acceptable solution becomes even smaller. Although the
number of skips will increase, the simulation time becomes too long and even longer than
doing full stability analysis for all the flashes.
0
0,2
0,4
0,6
0,8
1
0 100 200 300 400 500
Gas Saturation
Cell Number
Accurate solution
TTL, M=1000 eps=1E-4
TTL, M=1000 eps=1E-5
TTL, M=1000 eps=1E-6
TTL, M=1000 eps=1E-7
0
0,2
0,4
0,6
0,8
1
0 100 200 300 400 500
Gas Saturation
Cell Number
Accurate solution
TTL, M=1000 eps=1E-4
TTL, M=1000 eps=1E-5
TTL, M=1000 eps=1E-6
TTL, M=1000 eps=1E-7
(a) (b)
Figure 1. Gas saturation profile for System 1 at 0.5 PVI by TTL method a) Δt/Δx =0.1 b) Δt/Δx =0.5.
0
0,2
0,4
0,6
0,8
1
0 100 200 300 400 500
Gas Saturation
Cell Number
Accurate solution
TTL-PRE, eps=1E-4
TTL-PRE, eps=1E-5
TTL-PRE, eps=1E-6
TTL-PRE, eps=1E-7
0
0,2
0,4
0,6
0,8
1
0 100 200 300 400 500
Gas Saturation
Cell Number
Accurate solution
TTL-PRE, eps=1E-4
TTL-PRE, eps=1E-5
TTL-PRE, eps=1E-6
TTL-PRE, eps=1E-7
(a) (b) Figure 2. Gas saturation profile for System 1 at 0.5 PVI by TTL-PRE method a) Δt/Δx =0.1 b) Δt/Δx
=0.5.
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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
Recovery (%)
PVI
Accurate solution
TTL-PRE, eps=1E-4
TTL-PRE, eps=1E-5
TTL-PRE, eps=1E-6
TTL-PRE, eps=1E-7
Figure 3. Oil recoveries for System 1 by TTL-PRE method.
Using a pre-calculated tie-line table can dramatically cut the time in managing tie-lines.
We used M = 20000 and ε = 10-8 to find the most frequently used tie-lines during the
simulation. Three tie-lines were identified, which account for 27%, 36% and 25% of the
total tie-line usage. These three tie-lines were used to construct a fixed tie-line table
which is used throughout the simulation. This variation is referred to as TTL-PRE. Figure
2 shows the results for TTL-PRE, the results are generally better than TTL but still not
satisfactory. There is always some discrepancy between the TTL-PRE solution and the
accurate solution. For smaller ε, the simulation time gets closer to that for the shadow
region method where no approximation is made. Figure 3 shows the recovery curves
calculated by TTL-PRE using different ε. All the tolerances except ε = 10-4 give almost
the same recovery as the accurate solution, despite the discrepancy observed in Figure 2.
Table 5 compares the simulation times used for different methods for System 1 and
provides details on how flash calculations are approximated in TTL, TTL-PRE, TDBA1
and TDBA2. Calculation with full stability analysis is given as a reference. The shadow
region method is around 15 times faster than the full stability analysis method. Table 5
only provides the TTL with acceptable gas saturation profiles (although for M = 5000, ε =
10-7, there is still some discrepancy). Their simulation times are actually longer than that
for the shadow region method. This is because that very few percentages of
approximation have been made and a lot of time is spent on tie-line management. TTL-
PRE is much faster than TTL and for ε = 10-4 to 10-6, it is also faster than the shadow
region method. However, as Figure 2 indicates, the TTL-PRE solution is not satisfactory
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for those tolerances. At ε = 10-7, there is no significant difference in simulation time
between the TTL-PRE method and the shadow region method. Among all the methods,
TDBA1 and TDBA2 are the fastest. Their simulation times can be 1/3 to 2/3 shorter than
that for the shadow region method, depending on PVI and ε. TDBA1 and TDBA2 show
almost the same performance in terms of simulation time and percentage of skips. Their
saturation profiles are presented in Figures 4 6, respectively. At ε = 10-4, their results
show some fluctuations but still acceptable, the results at lower tolerances are almost the
same as the accurate solution. The fluctuations for TDBA1 seem to be smaller than
TDBA2, especially at ε = 10-4, whereas TDBA2 is a little faster than TDBA1. Figures 5
and 7 show the recoveries by TDBA1 and TDBA2, respectively. At all the tolerances
tested with TDBA1 and TDBA2, the recovery curves obtained are almost the same as the
accurate solution. We can observe that the recovery results are not so sensitive to the
tolerance as the gas saturation results.
Table 5. Comparison of different methods for system1 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*
Full stability 47.4 163.3 TTL
M=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 (3 tie-lines)
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% TDBA1 ε =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% TDBA2 ε =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
* reported numbers are percentages of total flashes in two-phase region
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0
0,2
0,4
0,6
0,8
1
0 100 200 300 400 500
Gas Saturation
Cell Number
Accurate solution
TDBA1 eps=1E-4
TDBA1 eps=1E-5
TDBA1 eps=1E-6
TDBA1 eps=1E-7
0
0,2
0,4
0,6
0,8
1
0 100 200 300 400 500
Gas Saturation
Cell Number
Accurate solution
TDBA1 eps=1E-4
TDBA1 eps=1E-5
TDBA1 eps=1E-6
TDBA1 eps=1E-7
(a) (b)
Figure 4. Gas saturation profile for System 1 at 0.5 PVI by TDBA1 method a) Δt/Δx =0.1 b) Δt/Δx =0.5.
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
Recovery (%)
PVI
Accurate solution
TDBA1 eps=1E-4
TDBA1 eps=1E-5
TDBA1 eps=1E-6
TDBA1 eps=1E-7
Figure 5. Oil recoveries for System 1 by TDBA1 method.
0
0,2
0,4
0,6
0,8
1
0 100 200 300 400 500
Gas Saturation
Cell Number
Accurate solution
TDBA2 eps=1E-4
TDBA2 eps=1E-5
TDBA2 eps=1E-6
TDBA2 eps=1E-7
0
0,2
0,4
0,6
0,8
1
0 100 200 300 400 500
Gas Saturation
Cell Number
Accurate solution
TDBA2 eps=1E-4
TDBA2 eps=1E-5
TDBA2 eps=1E-6
TDBA2 eps=1E-7
(a) (b) Figure 6. Gas saturation profile for System 1 at 0.5 PVI by TDBA2 method a) Δt/Δx =0.1 b) Δt/Δx
=0.5.
15
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
Recovery (%)
PVI
Accurate solution
TDBA2 eps=1E-4
TDBA2 eps=1E-5
TDBA2 eps=1E-6
TDBA2 eps=1E-7
Figure 7. Oil recoveries for System 1 by TDBA2 method.
Since the results at tolerance ε = 10-6 seems to be accurate enough for both TDBA1 and
TDBA2, we use this tolerance for the tests of the other three systems. Table 6
summarizes the calculation results for three Zick oils (Systems 2, 3 and 4). The shadow
region method, TDBA1, TDBA2, and the full stability analysis method have been used.
The table gives detailed statistics on the number of flashes performed. The simulation
was performed for 1.2 PVI in 6000 time steps. For the full stability analysis, three million
flashes are performed with the traditional flash algorithm. For the shadow region method,
around one million flashes are in the single-phase and outside the shadow region, and
those flashes were skipped; around two million flashes are in the two-phase region, which
are calculated by the direct Newton-Raphson algorithm. For TDBA1 and TDBA2, there
are also around two million flashes in the two-phase region. Around 90% of them were
approximated: 5-10% of them by direct approximation using the old results, 80-85% of
them by approximation with adjustment. The high percentage of approximation makes
TDBA1 and TDBA2 much faster than the shadow region method. It should also be noted
that most approximations fall in the intermediate range. The approximation with
adjustment in this range is crucial to both speed and stability. The above observations on
simulation time and statistics of approximations are consistent with those in Table 5 for
System 1. Figure 8 shows the recovery curve for System 2. Although not shown here,
Systems 3 and 4 give essentially the same behavior. Again, TDBA1 and TDBA2 present
essentially the same recovery as the accurate solution.
16
Table 6. Simulation time and number of flashes (millions) for Systems 2 to 4 with PVI=1.2 using different methods
Time Number of flashes (sec)
Skipped in single-phase
Direct N-R
in two-phase
Approximation with adjustment
in two-phase
Direct approximation in two-phase
System 2 Full stability 148.5
Shadow Region 10.6 1.03 1.97 TDBA1 (ε =10-6) 4.3 0.20 1.56 0.19 TDBA2 (ε =10-6) 3.9 0.20 1.67 0.11
System 3 Full stability 172.1
Shadow Region 11.3 0.90 2.10 TDBA1 (ε =10-6) 4.0 0.16 1.63 0.31 TDBA2 (ε =10-6) 3.7 0.16 1.73 0.21
System 4 Full stability 168.5
Shadow Region 11.0 0.98 2.02 TDBA1 (ε =10-6) 4.0 0.16 1.59 0.28 TDBA2 (ε =10-6) 3.7 0.14 1.72 0.17
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
Recovery (%)
PVI
Accurate solution
TDBA1 eps=1E-6
TDBA2 eps=1E-6
Figure 8. Oil recoveries for System 2 by TBDA1 and TDBA2
The above comparison was made with Δt/Δx =0.1. Increasing the time step will increase
variation of composition in a grid block from one time step to another. We also tested the
above examples at Δt/Δx =0.5 and obtained similar results. One small change is that the
fluctuation for TDBA2 at ε = 10-4 becomes larger (Figure 6b) but it does not affect the
recovery curve. No apparent increase in fluctuation was observed either for TDBA2 at all
the other ε values or for TDBA1 (Figure 4b). Another change is that with the larger time
step, the percentage of direct approximation reduces and most approximations were in the
intermediate tolerance range. Although the total percentage of approximation is reduced
slightly, TDBA1 and TDBA2 are still two times faster than the shadow region method.
17
A potential use of the TDBA methods is to speed up simulation with advanced and
complicated EoS’s. A recent study by Yan et al. (2011) shows that by modifying the
algorithm, the simulation speed for non-cubic EoS’s, like the PC-SAFT EoS, can be
greatly improved. It is shown here that by integrating TDBA1 to the modified algorithm,
the simulation time required for a 1-D slimtube simulation with PC-SAFT can be
substantially reduced (Figure 9). The system tested in Figure 9 is a 6-component gas
injection system, the component list has been duplicated, triplicated, and quadruplicated
to study the influence of number of components on simulation time.
0
20
40
60
80
100
120
140
160
180
0 5 10 15 20 25 30
Simulation time (sec)
Number of components
SRK
CPA
PC‐SAFT
CPA new
PC‐SAFT new
SRK+TDBA
CPA+TDBA
PC‐SAFT+TDBA
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
Simulation time ratio
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
(a) (b)
Figure 9. Simulation time results for a 6-component system (Yan et al., 2011) by using the reference method, the new method (Yan et al., 2011), and the new method + TDBA1: (a) simulation time; (b)
simulation time ratio of PC-SAFT and CPA to SRK.
3. Reduced variables methods
The reduced variables methods simplify the original phase equilibrium problem with a
smaller set of independent variables. The methods are designed for a cubic EoS where the
matrix of binary interaction parameters (BIPs) is of low rank. For the SRK EoS
( )
nRT AP
V B V V B
(8)
with
C
i ii
B b n C C
ij i ji j
A a n n (1 )ij ii jj ija a a k (9)
the fugacity coefficients can be expressed as
ˆln i n a i b iC C A C b (10)
18
If all the BIPs are zero, 2C
i ii ij jj
A a a n and
*ˆln i n a i b iC C A C b (11)
To our knowledge, using reduced variables for phase equilibrium calculation can be
traced back to 30 years ago when Michelsen and Heideman (1981) applied it to critical
point calculation. The method was first suggested for flash calculations by Michelsen in
1986. Michelsen (1986) showed that only three equations are needed to solve the flash
problem when all the BIPs are zero. Later, Jensen and Fredenslund (1987) extended the
method to calculations with single nonzero BIP row/column. Hendriks et al. (1992)
proposed a generalized reduced variables method that can deal with general non-zero BIP
matrices. The generalized method has been extensively used for the last 20 years by
different researchers (Firoozabadi and Pan, 2002; Pan and Firoozabadi, 2003; Nichita and
Minescu, 2004; Li and Johns, 2006; Nichita and Graciaa, 2010). Recently, the advantages
of the reduced variables method were questioned by Haugen and Beckner (2011).
There are some arguments against using reduction methods: they are essentially restricted
to the cubic EoS; they cannot so easily be formulated as unconstrained minimization
problems— consequently, they are less safe; the composition derivatives required by
second order methods are more cumbersome to set up; the simple algebraic operations
required to evaluate iA are very inexpensive today, as compared with what can be
achieved with the computing equipments in the 1980s; finally, although it is often
claimed that the effort of the conventional approach grows approximately proportionally
with C2 or even C3, our experience does show that the dependence of simulation time on
C is almost linear. The above arguments suggest that the potential for speeding up flash
calculation with reduced variables methods may be modest, which leads to a low
incentive to compare a reduced variables method with an efficient implementation of the
conventional flash method, such as Michelsen’s code, especially if extensive coding is
needed to develop a reduced variables program with Gibbs energy minimization. A recent
development by Nichita and Graciaa (2010) enabled an adaption to Michelsen’s existing
code with moderate modifications. In the following subsections, it will be first presented
how to make such an adaptation based on Nichita and Graciaa’s formulation. Then, an
19
alternative algorithm which directly utilizes the sparsity of the BIP matrix will be
introduced. Finally, a comparison of the reduced variables method, the “sparse” BIP
matrix method, and the conventional flash method will be given. It should be noted that
the conventional flash method is for “blind” flash calculation with both stability analysis
and phase split.
3.2. Reduced variables method
For the matrix P with elements 1ij ijP k , we calculate the spectral decomposition
1
CT
k k kk
P u u (12)
where k is the k’th eigenvalue of P and u is 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, when all the BIPs are equal to zero, M=1.
A more general situation is that the upper triangle of the BIP matrix is zero except for a
few rows, i.e.
0ijk for i m and j i (13)
In this case 2 1M m and the match is exact. We then get
1
MT
k k kk
P u u (14)
and
1 1
M M
ij k ii ik jj jk k ik jkk k
a a u a u e e
(15)
and
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 e
(16)
with
1
2C
k k jk jj
d e n
(17)
The above formulation results in an expression of the vector of ˆln i exactly in linear
combination of 2 3m vectors. There is no approximation in the results as compared
20
with the full approach, while the computational effort is reduced from 2C to 2CM plus
overhead.
For the successive substitution step in both stability analysis and phase split, we follow
the conventional implementation except for calculation of Ai, which is calculated using
Eqs. (16) and (17). The acceleration step is used as usual. And there is no effect on
convergence behavior. The only difference is that the calculation of Ai is simplified.
For the second order minimization step, the expression of ln iK suggested by Nichita and
Graciaa (2010) is used. Nichita and Graciaa express ln iK as
2
1
lnM
i l ill
K c e
, , 1 1i Me , , 2i M ie b (18)
where c1, c2, …, and cM+2 are treated as independent variables. We express the gradients
required for minimization as
1
Ci
ij i j
vG G
c v c
, c g Wg (19)
where iv is vapor moles i and /ji i jW v c , which can be obtained from the Rachford-
Rice equation. The required Hessian matrix can be further derived as
c TH WHW (20)
Although Wji looks complex to calculate, simple algebraic expressions for the elements
can be derived. The final minimization procedure is
1. Calculate the K-factors from c
2. Solve the Rachford-Rice equation to get vi
3. Calculate “conventional” gradient and Hessian
4. Calculate transformation matrix W
5. Calculate c-based gradient and Hessian
6. Calculate corrected c using trust-region approach
A similar procedure can be developed for stability analysis.
21
3.3. An alternative simplification: sparse k
An alternative formulation to take advantage of the sparsity of the BIP matrix is proposed
here:
1
(1 ) ( )C
ii jj ij j ii ij
a a k n a S S
(21)
where
1
C
jj jj
S a n
(22)
and
1
1
C
jj ij jj
i m
jj ij jj
a k n i m
S
a k n i m
(23)
This formulation uses just approximately 2mC multiplications.
3.4. Results
Two tested mixtures used here are derived from the 9-component SPE3 mixture. The first
mixture is modified such that all kij = 0 for i > 3 and j > 3. Only the interaction parameters
with nitrogen, carbon dioxide, and methane are non-zero. For the second mixture,
nitrogen and carbon dioxide are removed, leaving just one row/column non-zero BIPs
and only five reduced variables. For both mixtures, the last species in the component list
is subdivided to increase C so that the influence of C on the simulation speed can be
tested. For all the three methods, including the reduced method, the “sparse” BIP matrix
method, and the conventional method, one million flash calculations are performed in an
equidistant 1000 by 1000 grid in T and P. All the calculations are “blind” with about 60%
in the two-phase region. The simulation time results are given in Figures 10 and 11. First,
for both mixtures and all three methods, the simulation times increase almost linearly
with C with a slight upward curvature only for very large C. This is also consistent with
our tests with more complicated EoS’s (Yan et al., 2011). If there are three rows/columns
of non-zero BIPs, the reduced variables method becomes slightly faster than the
22
conventional method only if C is larger than 20 (Figure 10). For the even simpler
situation with just one row/column non-zero BIPs, the speeding-up of the reduced
variables method is still very modest, with less than 20% reduction in time for C = 24
(Figure 11). A better reduction in simulation time can actually be obtained by simplifying
the Ai calculation using the sparsity of the BIP matrix.
Figure 10. Simulation times for the first mixture (three rows/columns non-zero BIPs)
Figure 11. Simulation times for the second mixture (just one row/column non-zero BIPs)
23
6. Conclusions
This paper summarizes two recent studies on fast flash methods. We have found
1. TTL, a variation of CSAT, approximates the flash results when the new feed is close
enough to an existing tie-line in the tie-line table. Although the time for rigorous flash is
saved, building and updating the tie-line table during the simulation leads to a non-
negligible overhead. The simulation time increases dramatically with the number of tie-
lines used. Big tolerances can lead to inaccurate results whereas small tolerances can lead
to very few uses of the tie-lines. TTL-PRE with a few pre-calculated tie-lines can
improve the speed as compared with performing rigorous flash calculations. However, at
small tolerances like 10-6 and 10-7, its speeding-up is very limited.
2. TDBA compares only the tie-line in the same grid block from a previous calculation so
that the task in managing the tie-line table is minimized. It speeds up the flash in the two-
phase region a lot, even at a tolerance of 10-7. At a tolerance of 10-5, the saturation profile
is already quite close to the rigorous solution. The recoveries are essentially the same
even for a tolerance of 10-4 for the systems studied here. TDBA also shows its potential to
speed up simulation with complicated EoS’s.
3. Compared with the conventional flash method, the reduced variable method is faster
only for large C and the speed-up is very modest. A better reduction in simulation time
can be obtained by a much simpler implementation, which directly takes advantage of the
sparsity of the BIP matrix.
Acknowledgements
The study on the table look-up methods was carried out under the CompSim project
sponsored by ENI S.p.A. The study on the reduced variables method was carried out
under the project “ADORE—Advanced Oil Recovery Methods” funded by the Danish
Council for Technology and Production Sciences, Maersk Oil, and DONG Energy.
24
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