SPE

Society of Petroleum Engineers

SPE 16343

A Regression Technique With Dynamic-Parameter Selection

for Phase Behavior Matching
by R. Agarwal, Y-K. Li, and L. Nghiem, Computer Modelling Group
SPE Members

Copyright 1987, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE California Regional Meeting held in Ventura, California, April 8-10, 1987.

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the
author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the
author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, ItS officers, or members. Papers
presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy IS
restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of
where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836, Telex, 730989 SPEDAL.

ABSTRACT The dynamic selection of the most meaningful

regression parameters from a larger set of variables
The major problem associated with phase- is described in Section 3. This feature is extremely
behavior matching with a cubic equation of state is useful in EOS fitting because it alleviates the
the selection of regression parameters. There are problem of deciding apriori the best regression
many parameters that can be selected as the best set variables, which is extremely difficult.
of parameters, and therefore a dynamic parameter-
solution scheme is desired to avoid tedious and It should be stressed that the regression
time-consuming trial-and-error regression runs. procedure will not correct the deficiencies of the
EOS used, and the EOS predictive capability depends
This paper proposes a regression technique entirely on the type and the accuracies of the data
where the most significant parameters are selected used in the regression. For predictive purposes,
from a large set of parameters during the regression attempts should be made to ensure that the "tuned"
process. This reduces the regression effort parameters are within reasonable physical limits.
considerably and alleviates the problem associated
with the apriori selection of regression parameters. All calculations in this report are performed
with the Peng-Robinson EOS (Peng and Robinson, 1976)
The success of the technique is demonstrated by al though the scheme is general and could be applied
matching the experimental data for a light oil and a to any EOS.
gas condensate.
THE REGRESSION METHOD
INTRODUCTION
Outline
It is well known that cubic equations of state
(EOS) will not generally predict accurately The implementation of the dynamic-parameter-
laboratory data of oil/gas mixtures without the selection strategy for tuning the EOS requires the
tuning of the EOS parameters (Coats and Smart, solution of a nonlinear optimization problem. In
1986). It has often been the practise to adjust the terms of least-squares, the optimization problem may
properties of the components (usually the heavy be stated as
fractions), e.g. Pc, Tc ' w etc., to fit the experi-
mental data. minimize f(x) = R(x)T R(x)
x
The objective function in the regression
involves the solution of complex nonlinear equations
such as flash and saturation-pressure calculations. (1)
A robust minimization method is therefore required
for rapid convergence to the minimum. In this
report a modification of the adaptive least-squares where x = [X1 ,x2,'" ,x n ] T is the regression-
algorithm of Dennis et al (1981) is used. The parameter vector, wi th rfr being the number of
modification involves the use of some other regression parameters and nm the number of measure-
nonlinear optimization concepts on direction and ments to be fitted. Usually nm > n r • The elements
step-size selection due to Chen and Stadtherr of R(x) are denoted by ri (x) which are nonlinear in
(1981) •

207
 A Regression Technique With
2 Dynamic-Parameter Selection for Phase-Behavior Matching

x. lfuen the equation of state is adjusted to match If the jacobian J is to be calculated by finite
a set of experimental data Y differences, the perturbation in the independent
variables x must be such that it is not masked by the
convergence accuracy €i or the truncation and round-
off errors associated with the computation. It has
Yi been found that a perturbation of 1% in the
independent variables is adequate to compute J by
numerical differentiation.
with R(x) = [e1(x), e 2 (x), ••• ,e nm (x)]T
Choice of Regression Parameters
and Y = [Yi' Y2"",Ynm]T
Given a global set of regression parameters,
where R(x) are the equation-of-state results and y
the experimental data points. In this case the xj,j=1, ••• ,np, the method selects an active subset of
nonlinear least-squares problem consists of nr parameters with which regression will be
performed. The global set of regression parameters
adjusting x so that the EOS results match the
experimental measurements. is supplied by the user and includes all or some of
the following:
The problem (1) may be solved by various
methods for nonlinear parameter estimation (Bard, Pci critical pressure of component i
1974), and for nonlinear optimization (Himmelblau, Tci critical temperature of component i
1972; Schittkowski, 1981). The general purpose vci cri tical vol ume of component i which
optimization methods however do not take advantage affects the interaction coefficients
of the special structure of the nonlinear least between hydrocarbons (see Equation 2)
squares optimization problem (1). Several accent ric factor of component i
strategies are available to exploit this structure. volume translation of component i (see
Coats and Smart (1986) used a modified linear Equation 3)
programming least squares algorithm to solve (1). interaction coefficient between components
i and j
Watson and Lee (1986) use a modification of the
Lenenberg-Marquardt algorithm (see More, 1978) to power for computing interaction coeffici-
solve a nonlinear least-squares problem. ents between hydrocarbons (see Equation 2)

In this report a modification of the adaptive The interaction coefficients between hydro-
least-squares algorithm of Dennis et al ( 1981) is carbons are estimated from the following equation
used. The algorithm departs from the method of
Dennis et al in using some other nonlinear optimiza- (Li, et aI, ':8[': v ;. voj ~j A
tion concepts on step-direction and step-size
selection due to Chen and Stadtherr ( 1981 ) • The d =ij
oi (2)
regression method is described in detail in Appendix 1/3 1/3
A. V ci + v cj _

APPLICATION OF THE REGRESSION METHOD TO ROS TUNING The volume translation technique of Peneloux et
al (1982) is used to correct the molar volume as
Outline follows

The regression method discussed in Section 2 is v t = vO - v

used to fit an EOS to experimental data involving
oil and gas systems. Methods for choosing the where v
regression parameters and details of the computation
are discussed. where Xi is the mole fraction of each component and
vti is the individual translation volume for each
Calculation of R and J component. The superscripts and t correspond
respectively to the results before and after the
°
It was found very early in the investigation of volume translation.
the regression application that the key to an
efficient algorithm would be the fast and accurate The parameters Xj are scaled by using the upper
(as far as possible) estimation of the jacobian bound. x j, max and lower bound x j, min of the corre-
matrix J. In Appendix A it is shown that the matrix spondlng parameter such that they always lie between
J also determines the second derivative hessian zero and unity.
matrix V 2 f. Consequently, a small change in the
determination of J affects the performance of the (4)
regression method quite dramatically.
The regression scheme sorts the n p parameters in
The derivatives of the residuals R are calcu- the descending order of faf/aXjl. From these np
lated by numerical differentiation, since in most parameters, the first n r parameters are chosen for
cases it is not possible to obtain exact analytical regression, Le., the n r parameters with the largest
derivatives. The calculation of R at all times laf/aXj!. nr is supplied by the user.
involves iterative processes, where the solution is
only available to some accuracy €i' The regression proceeds on these nr parameters
and i f at any time during the regression, ,a flax jl
becomes less than 'ril, i=1, ••• ,n m, the variable Xj

208
SPE 16343 Rajeev Agarwal, Yau-Kun Li & Long Nghiem 3

is dropped from the regression set and the next Gas Condensate System (Third SPE Comparative Solution
variable on the original sorted list is added on. Project)
Indeed, since all IX j are scaled betweep z;eros and
unity, if I(Iflax j, is less than all i ri:, it is The composition of the reservoir fluid used in
likely that Xj has to go beyond its bounds to the third SPE comparative solution project (Kenyon
further reduce rio Therefore, it is logical that Xj and Behie, 1983) is given in Table 4. The fluid
should be dropped from the active parameter set. contains a large percentage of methane. The seven
Another condition where x j is dropped is when it component model fluid characterization is given in
tries to go out of bounds for more than two Table 5.
successive iterations.
The experimental data to be matched is the lean
At convergence, if the total number of gas swelling experiment which consists of a satura-
regressed variables (including those which have been tion pressure-composition data set and a swelling
dropped) is less than five, then new variables are factor-composition data set.
added to the active regression set and the original
ctive regression variables with the smallest For this example several different cases were
jdflax j I are removed from the active set such that run to illustrate the pitfalls in EOS tuning, to
nr is preserved. stress that caution must be used when selecting the
potential regression parameters, and finally to show
The flowchart of the parameter selection the importance of choosing the upper and lower bounds
procedure is given in Figure 1. of the regression parameters. The following cases
were run:
EXAMPLES
Case 1 The cri tical pressures and temperatures of
Light-Oil System all the hydrocarbon components except C6 were
treated as potential regression parameters, with
The composition of alight reservoir fluid is the lower and upper bounds being 0.7 and 1.3
given in Table 1. This fluid is modeled by seven times the initial values, respectively.
hypothetical components shown in Table 2. The
experimental data to be matched is the pressure- Case 2 The cri tical pressures and temperatures of
composition diagram when C02 is the injection fluid. all components except those of C6 were treated
as regression parameters. Also the upper and
The possible 29 regression parameters for this lower bounds were much narrower than in Case 1
system are: to ensure that, after regression, Pc and Tc
would follow the trend of decreasing Pc with
Acentric factors, Wi increasing carbon number and increasing Tc wi th
Critical pressures, Pci increasing carbon number.
Critical temperatures, Tci
C02-hydrocarbon interaction coefficients, Case 3 The critical properties of only the two
dC02-i, and heaviest fractions were chosen as regression
Power for hydrocarbon-hydrocarbon interaction parameters, again with restrictions on the upper
coefficients, A (see equation 2) and lower bounds to maintain the monotonic
increase and decrease in Tc and Pc wi th carbon
where the subscript i refers to all the hydrocarbon number.
components of the fluid.
/ For all the above cases the interaction
The properties of the original fluid and the coefficient between the first component and the
regressed fluid are shown in Table 3. Figure 2 second through seventh component were also part of
shows the experimental pressure-composition data as the potential regression parameter set.
well as the calculated values obtained by using the
tuned and untuned EOS. It can be seen that the fit The properties of the original flui.d model,
obtained by regression is excellent. along with those determined by Cases 1, 2 and 3 are
shown in Table 6.
Table 3 also shows the five variables that were
chosen by the regression routine. The properties The experimental data and the regression results
chosen are the Tc of the two lightest hydrocarbons, for the saturation pressure-composition data are
the Pc and Tc of the next-to-heaviest hydrocarbon. plotted in Figure 3, and those for swellinp; factor-
(Component 6) , and the interaction coefficient composition data in Figure 4.
between C02 and the heaviest hydrocarbon fraction.
Case 1 gives the best fit, followed by Case 2
At first glance, the choice of Pc and Tc of and then Case 3. It should be noted however that for
Component 6 seems surprizing as one would expect the Case 1, the critical pressures Tc have lost the
properties of the heaviest fraction to have more property of monotonic increase with increasing carbon
impact on the residuals. However, examination of number.
Table 2 shows that the composition of Component 6 is
much larger than that of the heaviest component Case 2 on the other hand maintains the desirable
(Component 7) (12.01 mol% versus 5.57 mol%) and trend of decreasing Pc and increasing Tc • However,
therefore have probably a bigger impact. This the fit is not as good as Case 1. It should also be
example demonstrates that the five variables noted that in Case 1 the EOS is tuned by changing a
selected by the program among the 29 candidates are few parameters by large amounts. Whereas, in Case 2,
very adequate to fit the experimental data.

209
 A Regression Technique With
4 Dynamic-Parameter Selection for Phase-Behavior Matching

the tuning is done by adjusting more parameters by ri individual components of R

smaller amounts. S approximation to the second order terms of
the Hessian of f
As expected, the match in Case 3 is not as good vector of regression variables/parameters
as in Cases 1 and 2, as the available regression individual components of x
parameters are further restricted. This case is experimental data
included to show the possible effects of including
only a small number of component properties in the Subscripts
regression parameter set. It should be noted that i index of a vector or matrix element
the results from Case 3 may still be adequate for j index of a vector or matrix element
most engineering calculations. k iteration number

Case 4 Finally the fluid model obtained from Case Superscripts

---3 was used to fit the liquid dropout volume g Gauss-Newton model given by equation (A.7)
percent for the constant volume depletion study s secant model given by equation (A.5)
by adjusting the volume translation factors T transpose of a matrix
(Peneloux et al 1982) of the lightest and
heaviest components. The results are shown in Other symbols
Figure 5. The values of the volume translation !S trust region radius
factors before and after the regression are !', increment in variable
also given in Table 6. This case has been V gradient operator
included to show that more than one type of :\ parameter in equations (A.9)
data from different experiments can be matched
reasonably by changing very few regression
II·' I the Euclidean norm
parameters. REFERENCES

CONCLUSIONS Bard, Y., "Nonlinear Parameter Estimation," Academic

Press Inc., 1974.
This paper proposes the use of a regression
scheme to tune the EOS that does not require the Chen, H-S., and Stadtherr, M.A., "A Modification of
apriori knowledge of regression parameters. The Powell's Dogleg Method for Sol ving Systems of
scheme also discards those active regression Nonlinear Equations," Compo and Chern. Engg., Vol. 5,
parameters that are found to have outlived their No.3, 1981, pp. 143-150.
usefulness in the regression process.
Coats, T<.H., Smart, G. T., "Application of a
It is seen that this regression scheme performs Regression-Based EOS PVT Program to Laboratory Data,"
very well and chooses regression parameters in a SPE Reservoir Eng., Vol. 1, No.3, May 1986, pp. 277-
very sensible and expected manner, thus eliminating 299.
tedious trial-and-error runs.
Dennis Jr., J.E., and Schnabel, R.B., "Numerical
A modification of the adaptive technique Methods for Unconstrained Opt imization and Nonl inear
proposed by Dennis et al (1981) has been used Equations," Prentice-Hall Series in Computation Math,
successfully in this work. It involves an automatic Cleve Moler, Advisor, 1983.
switching between a Gauss-Newton and a Secant method
to approximate the Hessian matrix. The search Dennis, Jr., J.E., Gay, D.M., and Welsch, R.E., "An
directions are taken by using a region of confidence Adaptive Nonlinear Least-Squares Algorithm," ACM
approach. Trans. Math. Software, Vol. 7, No.3, September 1981,
pp. 348-368.
It should be stressed that caution should be
used when utilizing an EOS that has been tuned to Himmelblau, D.M., "Applied Nonlinear Programming,"
some experimental data. The predictive range of a McGraw-Hill Inc., 1972.
tuned EOS should not be very different from the
experimental conditions that have been matched, Kenyon, D.E., and Behie, A., "Third SPE Comparative
otherwise unreliable calculations may result. For Solution Project: Gas Cycling of Retrograde
example, an EOS tuned using hydrocarbon gas Condensate Reservoirs," paper SPE 12278 presented at
injection data that contains small amount of C02 the Seventh SPE Symposium on Reservoir Simulation,
should not be used for predicting C02 flooding San Francisco, California, November 16-18, 1983.
processes.
Li, Y.-K., Nghiem, L.X., and Siu, A., "Phase Behavior
NOMENCLATURE Computations for Reservoir Fluids: Effect of Pseudo-
Components on Phase Diagrams and Simulation Results,
E Equation-of-state results J. Can. Petrol. Technol., Vol. 24, No.6, November-
f performance index defined by equation (1) December 1985, pp. 29-36.
H Hessian of objective function f
I Identity matrix More, J. J. , "The Levenberg-Marquardt Algori thm:
J Jacobian of R defined by equation (A.5) Implementation and Theory," In Lecture Notes in
nm number of experimental points Mathematics, No. 630, Numerical Analysis, G. Watson,
nr number of regression variables Ed., Springer-Verlag, New York, 1978, pp. 105-116.
q quadratic model of f around xk
R vector of residuals

210
SPE 16343 Rajeev Agarwal, Yau-Kun Li & Long Nghiem 5

Pen810ux, A., Rauzy, E., and Freze, R., "A fl T T T

qk (x) = Rk Rk + 26xk Jk Rk
Consistent Correction for Redlich-Kwong-Soave
Volumes," Fluid Phase Equil., Vol. 8, 1982, pp. 7- + b.xJ [JJ Jk + Sk] L\xk ( A.5)
23.
where the subscript k denotes values evaluated at xk'
Peng, D.Y. and Robinson, D.B., "A New Two-Constant The superscript s in qS indicates that the S term is
Equation of State," Ind. Eng. Chem. Fundam., Vol. included. The minimum of the quadratic model can be
15, 1976, pp. 59-64. located easily using the Newton's method. This
resul ts in the following itera ti ve scheme
Schittkowski, K., "The Nonlinear Programming Method
of Wilson, Han, and Powell With an Augmented [Jk T Jk + Sk] (Xk+1 - Xk) = -Jk T Rk (A.6)
Lagrangian Type Line Search Function - Parts I and
II," Numer. Math, Vol. 38, 1981, pp. 83-127. Ini tially, So is set to 0 and subsequent Sk,
k= 1,2, •• " are calculated from a recursive correla-
Watson, A.T., and Lee, W.J., "A New Algorithm for tion resulting from a secant approximation. For
Automatic History Matching Production Data," paper convergence, the quadratic model in Equation (A.5)
SPE 15228, presented at the Unconventional Gas will be referred to as the secant model in subsequent
Technology Symposium of SPE held in Louisville, discussions.
Kentucky, May 18-21, 1986.
Note that initially, So = 0, Equation (A.5)
APPENDIX A - Adaptive Nonlinear Regression reduces to the Gauss-Newton model, qg with the
following iterative scheme
Adaptive Quadratic Model
The least-square problem is restated here as
JJ Jk (Xk+1 - Xk) = -Jk T Rk (A.7)
follows: It is well known that for small k, the Gauss-Newton
model, qg, gives a better prediction to the objective
minimize f(x) = R(x)T R(x) ( A.1) functions than the secant model, qS until the second
x
derivative information, S, is sufficiently accurate
To facilitate the regression effort, a through secant approximations. An ideal scheme thus
simplified quadratic model, q, of the objective would use qg initially and then would switch to qS
function f is first established. Expanding f in once an accurate S is available. To retain
Taylor series around the k-th iterate point xk and general i ty, the proposed scheme allows the switching
ignoring the third and higher order terms, resul ts from qg to qS and vice versa. The choice of the
in quadratic models and the trust radius of the models
are intimately connected. Further description of the
qk (x) = f(Xk) + 7f(Xk)T I'Ixk SWitching criterion is given later.
+ ~ I'IXk T 7 2 f(Xk) ~xk (A.2) Quadratic Hodel Constrained Minimization
where /\Xk = x xk' The first and second Since quadratic approximations to f(x) are used,
derivatives of f are as follows: there exists a region ~, outside which the model
fails. This imposes a validity limit to the size of
/\.x in the interati ve scheme resulting from these
quadratic models. This is expressed as follows:
and
minimize qk such that II b.xk II ~ IS (A.8)
(A.4)

where J is the Jacobian matrix

where q could be either qg or qS and I! denotes
the Euclidean norm. The solution. to the minimization
I'.
(A.8) does not consist of simply truncating the step
size as calculated from Equations (A.6) or (A.7), but
may involve the calculation of a different step
and direction because of the constraint.
m
S (xk) = 1: ri (Xk) • 'iJ2 ri (Xk) The solution to the minimization problem (A.8) is
i=1
Equation (A.4) shows that the first term of the
second expansion of f is available through the
= - (Hk + AI)-1 (JJ Rk)
Jacobian J. However the second term, S, is not T
where Hk = Jk Jk + Sk
readily available and is quite difficult to
evaluate. Usually, S is neglected in practice. Note that S = 0 when the Gauss-Newton model is used.
This is valid when the residuals ri (x) are small. Equation (A.9) is subjected to the following
For large residuals, it is necessary to include S, constraint
and the latter can be approximated by using a secant
method (Dennis, et ali 1981). o (A.10a)
Using Equations (A.2), (A.3) and (A.4), the and
quadratic model becomes
I 'b.xkl! = IS when A > 0 (A.10b)

211
 A Regression Technique With
6 Dynamic-Parameter Selection for Phase-Behavior Matching SPE 16343

In nonlinear programming, A is usually referred TABLE 1 - Light-On Composition

to as the Levenberg-Marquardt weighting factor.
Since A is not known apriori, an iterative method Component Mole %
for updating A is usually required (More', 1978;
Wa tson and Lee, 1986) • However, it has been found
that a good step L}Xk can be computed in a much CO2 3.03
simpler way by using Powell's Dogleg method (Chen
and Stadtherr, 1981). N2 0.33
C1 41.33
Quadratic Model Switching
C2 8.89
As discussed earlier, the Secant model, qS, and C3 5.95
the Gauss-Newton model, qg, are accurate at
different times during the regression process. This iC4 1.34
requires a switching criterion between the two nC4 2.78
quadratic models. The traditional approach is to
iC5 1.22
start the regression with So = 0, i.e. with qg.
Since it takes a certain number of iterations to nC6 1. 44
build sufficient information in Sk for it to be
useful, the Gauss-Newton model is used for a C6+ 33.69
prescribed number of iterations and is replaced
subsequently by the Secant model. To retain flexi-
bility and generality, the dynamic switching method API Gravity of C6+ @ 15 0 C = 35.3
proposed by Dennis, et al (1981) is used in this Specific Gravity of C6+ 0.8478
work. Since both the trust radius and the choice of
the quadratic model affect the efficiencies of the Molecular Weight of C6+ = 202
regression scheme, the strategy is to adjust both
during the regression to maintain reasonable
convergence rate. This approach consists of trying TABLE 2 - Model Fluid for the Light Oil in Table 1
the alternate quadratic model with the same (\ when
the present model fails to reduce the objective
function. If the alternate model does not lead to a Component Hypothetical
more successful step, then the present model is Numbe'" Component Mole %
retained and the trust radius is reduced.
Alternately, an attempt will be made to increase '\ CO2 3.03
for valid steps.
2 C1,N2 42.93
3 C2- C3 15.30
4 C4- C5 6.99
5 C6- C12 17 .16
6 C13- C20 12.01
7 C21+ 5.57

TABLE 3 - Properties of Original and Regressed Light on

Component Component pc,atm Tc,K

Number Initial Final Initial Final

CO2 72.80 72.80 304.2 304.2

2 C1,N2 45.31 45.31 190.1 220.4*
3 C2+ C3 45.64 45.64 332.9 233.9*
4 C4+ C5 35.54 35.54 438.2 438.2
5 C6- C12 26.69 26.69 621.0 621.0
6 C13- C20 19.84 15.17* 747.3 800.1*
7 C12+ 14.65 14.65 888.2 888.2

Initial dC02-i = 0.1 i=2, 3, ••• ,7

Final dC02-1 = 0.1 i=2,3, ••• ,6
dC02-7 = 0.107*

.Parameters used during regression.

212
 SPE 1£"~43
TABLE 4 - Gas-Condensate Composition TABLE 5 - Hodel Fluid ~or Gas Condensate in Table 4

Component Mole % Component Hypothetical

Number Component Mole %
CO2 1.21
C1,N2 67.93
N2 1.94 2 C2, C02 9.90
C1 65.99 3 5.91
C3
C2 8.69 4 C4+C5 7.86
C3 5.91 5 1.81
C6
iC4 2.39 6 C7- C12 5.18
nC4 2.78 7 1. 41
C13+
iC5 1.57
nC5 1.12
C6 1. 81
C7+ 6.59

API Gravity of C7+ @ 600F = 51.4

Specific Gravity of C7+ 0.7737
Molecular Weight of Cy+ = 140

TABLE 6 - Properties o~ Original Fluid and

Regressed Fluids ~or Gas Condensate System

Component
Number Component

C1+ N2 45.08 67.62* 47.59* 45.08 188.7 245.7* 189.9* 188.7

2 C2+ C02 50.36 50.36 50.36 50.36 305.3 203.6* 336.0* 305.3
3 C3 41.90 41.90 41.90 41.90 369.8 369.8 400.3* 369.8
4 C4+ C5 35.61 35.61 35.61 35.61 433.9 433.9 468.9* 433.9
5 C6 32.46 32.46 32.46 32.46 507.5 507.5 507.5 507.5
6 C7+ C12 26.96 20.36* 23.33* 23.33* 586.7 715.6* 549.2* 549.2*
7 C13+ 19.30 9.65* 13.51* 13.51* 729.3 1094.0* 716.9* 706.6*

Initial dij = 0 for all i and j

Case d7,1 * = 0.073 others = 0
Case 2 d2,1 = 0.43; d7,1 * = 0.25
* others = 0
Case 3 d2,1 * = 0.25; d7,1 * = 0.25 others = 0
Case 4 Initial Volume Translation Factors v t = 0
Final Values: V1 t * = - 0.022; v7 t * = 0.025; Vjt = 0,i;ll1,7

213
 SPE16343
45.000 - - - - , INITIAL MODEL

START - - - - - - - - FINAL MODEL

o
o EXPERIMENTAL DATA I

A
LOAD THE INITIAL VALUES OF THE np REGRESSION PARAMETERS 40,000
0
~
CALCULATE 8f18xJ J= 1,2, ,n p AND SORT w
Xj IN THE DESCENDING ORDER OF MAGNITUDE OF afl dX
J ""
:::>
</) 35,000
~ , I
CHOOSE_T~:~nr xJ's FROM THE SO~~_ SET
""
~~/
"-
.---- z
Q 30,000
!;:(
c,,;;x ................
'S
"":::> __ x o .... 9........
!;:(
af/8xj(r, FOR All i ,.....,...x ....
</)
25,000
x .... v ....
~JRm," 1
0
eJ',fr
'J' :0R TWO ITERATIONS
NO
Xj: XJmox J 20,000
0 '0 40 60 80 '00
YES
MOLE 'l', CO,

Fig. 2-Light-oiI/C02 mixtures.

HAS
NO
CONVERGENCE BEEN
ACHIEVED? '0

YES

HAVE WE
REGRESSED ON AT LEAST FIVE
YES
STOP
o EXPERIMENTAL DATA

VARIABLES? INITIAL MODEl

CASE 1

CASE 2

- - - - CASE 3
Fig. 1-Flow chart for selecting the active regression parameters. o
z
3
3
</)
2,0

4900 o EXPERIMENTAL DATA

" 1.0 .,-=.:,...--...,.----,---,.---,.---.,._----,
'0 20 30 '0 50 60 70

4700 MOLE 'Y. OF INJECTED FLUID

Fig. 4-Gas condensate·injected fluid; swelling factor diagram injected fluid composition
(moIDAl): 94.68%, Ct. 5.27% C2 , 0.05°/(1 C3 •
4500
C
~
c. 26 a EXPERIMENTAL POINTS
4300
w INITIAL MODEL
ex
:::> FINAL MODEL
if>
~
go: 4100

Z
Q
>-- 3900 ~
.... .,.,.---- ..... ,
<t
ex 22 /
:::> w
>-- ::;; / \
<t / \
</)
3700 3 / \
0
> // \
>-- 20 / \
3500
2
:::>
/// \
0 / 0 \
/ \
""0 /
/ \
\
3300 ..j---...,...----,------.,.-----,~--,._--.,._--.,
'0 20 30 '0 50 60 70
0
'5
18 / \

MOLE % OF INJECTED FLUID

a:::; /
/ \
\
/ \
/ \
/ \
16 \
Fig. 3-Gas condensate-injected fluid; pressure-composition diagram injected fluid compo- \
sition (mol%): 94.68% Ct> 5,27% C2 , 0.05% C3 •
o

14..j----....,------,-----r------,-----,
500 1000 1500 2000 2500 3000

PRESSURE (psig)

Fig. 5-Gas condensate constant volume depletion stUdy.

214