1304
Ind. E n g . Chem. Res. 1987,26, 1304-1312 Smith, J. W.; Reddy, K. V. S. Can. J . Chem. Eng. 1964, 42, 206. Tsvik, M. Z.; Nabiev, M. H.; Rizaev, N. U.; Merenkov, K. V.; Wyzgo, V. S . Uzb. Khim. Zh. 1967, 11, 50. Uemaki, 0.;Fujikama, M.; Kugo, M. Jpn. Pet. Inst. 1977,20, 410. Venuto, P. B.; Habib, E. T. Fluid Catalytic Cracking u i t h Zeolite Catalysts; Marcel Dekker: New York, 1979. Volpicelli, G. Ing. Chim. It. 1965, 1, 37. Wan-Fyong, F.; Romankov, P. G.; Rashkovskaya, N. B. Zh. Prikl. Khim. (Leningrad) 1969, 42, 609. Weng, H. S.; Chen, T. L. Chem. Eng. Sci. 1980, 35, 915.
Piccinini, N. Proceedings of the International Fluidization Conference; Grace, J. R., Matsen, J. M., Eds.; Wiley: New York, 1980; p 279. Piccinini, N.; Grace, J. R.; Mathur, K. B. Chem. Eng. Sci. 1979, 34, 1257. Piccinini, N.; Cancelli, T. Proceedings of the International Fluidization Conference: Kunii, D., Toei, R., Eds.; Wiley: Tokyo, 1983; p 533. Piccinini, N.; Rovero, G. Can. J . Chem. Eng. 1983, 61, 448. Romankov, P. G.; Rashkovskaya, N. B. Drying in Suspension State; Khimiya: Leningrad, 1979. Rovero, G.; Piccinini, N.; Grace, J. R.; Epstein. N.; Brereton. C. M. H. Chem. Eng. Sci. 1983, 38, 557.
Received for review October 28, 1985 Accepted March 20, 1987
Thermodynamics of Petroleum Mixtures Containing Heavy Hydrocarbons: An Expert Tuning System Rafiqul Gani* and Aage Fredenslund Instituttet for Kemiteknik, Danmarks Tekniske H ~ j s k o l e 2800 , Lyngby, Denmark
The problem of improving the prediction accuracy of equation of state models for petroleum mixtures containing heavy hydrocarbons is studied. Procedures for establishing the sensitivity type of the mixture and the prediction problem type are developed. On the basis of these procedures, a tuning policy is proposed. Knowing the sensitivity type of the mixture and the prediction problem type, the tuning policy first decides if tuning is possible. If so, it selects a set of candidate adjustable variables. The adjustable variables may be the pure-component hydrocarbon fraction properties, or they may be the binary interaction parameters. Depending on the number of experimental data points available, a subset of the candidate adjustable variables is tuned to satisfy the necessary requirements. The applicability of the proposed tuning policy is demonstrated for several petroleum mixtures. Accurate prediction of phase equilibria and PVT properties is an important requirement in process and reservoir calculations involving heavy gas condensates and reservoir oils. Usually, an equation of state (EOS) is used for this purpose. For some applications the accuracy of the prediction may be insufficient, specially near critical conditions. The usual approach for such situations is to “tune” the parameters in the EOS to attempt to improve the accuracy of prediction. There is, however, a lack of acceptable/reliable procedures and/or criteria for employing such measures. Questions regarding which parameter to tune, feasibility of tuning, or how much tuning is possible remain unanswered. Recently, Pedersen et al. (1985b) have pointed at the dangers of tuning equation of state parameters for prediction of properties of reservoir fluids. Coates and Smart (1982) have previously also studied the problem of tuning equation of state parameters. The objective of this work is, through a sensitivity analysis of the different terms of an equation of state, to develop a set of criteria and procedures which can be applied to improve the predictions. The Soave-RedlichKwong (SRK) equation of state (Soave, 1972) will be used for the calculations. The SRK equation of state is only used in this paper for illustrative purposes and also since it is the one most commonly used for prediction of properties of petroleum mixtures. This paper presents criteria for determining the sensitivity type of mixtures, and it characterizes the prediction problems in terms of complexity. Also, a well-defined tuning policy consisting of two parts is proposed. The first part consists of a sensitivity analysis, selection of the pa-
* To whom all correspondence should be addressed. 0888-5885/87/2626-1304$01.50/0
rameters (i.e., candidate parameters) to tune, and the definition of the problem type. The second part consists of the actual tuning of the selected parameters and evaluation of the results. Finally, results showing the applicability of the proposed policy to different problems are presented. Sensitivity Analysis The SRK equation of state may be written as a p = - -RT u - b U(U + b ) The equation can be rearranged to the cubic form in terms of the compressibility factor 2, as 2 3
-
2 2
+ AZ - B = 0
(2)
where A =Ap-Bp-Bp2
B = ApBp AP = a P / ( R T I 2
Bp = b P / ( R T ) For the mixtures, the terms a and b are obtained from mixing rules of the type
a = CCyiVjaij
(3)
b = xyibi
(4)
i
j
i
The a i j term (for i # j ) may be calculated from the pure-component values aii and aj using a binary interaction parameter, hi,,as 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1305 a11 , . = (a..a. 1, 1 1, ) 1 / 2 ( 1
- k 11. .)
(5)
The pure-component parameters, aii and bi, are defined as
+
aii = 0.042748R2T,~/Pci[l mi(l - T/Tci)1/2]2(6)
bi = 0.08664RTc./Pc.
(7)
where mi = 0.480 + 1 . 5 7 5 ~~ 0.176~~~
(8)
For the case where ki = 0, for all i and j , the above mixing rule simplifies to a =
(9)
cy2
where = cyicy;
cy
i cy. 1
= a.,ll2 11
The expression for the fugacity coefficient ( k i = 0 for all i and j ) can be found to be In
-(
= bi PU - 1) - In
b RT
(A(u -
b)) -
The above equation is valid for both the liquid and vapor phases. The equilibrium ratio, ki = y i / x i ,expressed in fugacity coefficients, is given by
K i= a:/@:
(11)
The objective of the sensitivity analysis is to study how each parameter ( A , B, Ap, B p , etc.) or property (2,V, @i, K,, etc.) may vary with respect to the adjustable variables (T6;,P&,w;, ki$. By use of the equations given above, the required derivatives of the terms or properties with respect to any of the adjustable variables may be obtained analytically. Variables which depend on the mixture, the state, and a particular component will be known as individual properties (or parameters). These are ai, Ki, etc. (properties), and aii, bi,etc. (parameters). Bulk-phase variables (and properties) will be known as global parameters (and properties). These are A , B, Ap, Bp,etc. (parameters), and 2, V, etc. (properties). The k i j will be known as adjustable interaction variables to distinguish from the adjustable component variables (Tci,Pci,wi). The scaled derivatives of the global variables (parameters and properties) with respect to each of the adjustable component variables will be represented as
where f) = A , B, Apt Bp, etc., and 2, V, etc., and pi = Tc;, Pci,wi. NG is the total number of terms (properties and parameters) whose derivatives are to be determined. NC is the total number of components. F represents the sensitivities of all the global terms, and each column of it will represent the sensitivity of a particular term with respect to a particular variable (for example, with respect to TCJ.There will be three (NG = 3) F matrices for the three sets of adjustable component variables, P,. In case of the individual terms (parameters and properties), the scaled derivatives with respect to each adjustable component variable will be represented as
Pi D ik. = -dfk2 -
k = 1(1),NI i = l(l), NC (13) dpi f k 2 where NI is the total number of individual terms (properties or parameters) whose derivatives are to be determined, and fk2= @,, K,, a,,, etc. Thus, a total of 3NI matrices (D) will be obtained from eq 13. Once the F and D matrices have been obtained, the components of the mixture which give rise to the largest terms in their respective columns will be identified. These information are then analyzed to identify the sensitivity type of the mixture and to select the candidate parameters (i.e,, adjustable variables) that could be used if tuning was required. Also, the values of the elements in the matrices give an estimate of how much change can be obtained from tuning a particular variable. Characterization of Sensitivity Type Gas condensates and reservoir oils are usually mixtures of hydrocarbons and non-hydrocarbons (like Nz and COP). The methane percentage can vary from 40 to nearly 90 mol %. The heaviest hydrocarbon can vary from CZ0to Cm (or even heavier). Usually, gas condensates and reservoir oils are characterized as mixtures of real and pseudocomponents (Pedersen et al., 1985a). This is necessary because these mixtures contain a very large number of components, and the computational methods require a smaller number. The F and D matrices were found to have characteristic patterns for different petroleum mixtures. Based on these matrices for different mixtures, the following criteria for characterization of the sensitivity type of the mixture are proposed: A mixture may be characterized as (i) light sensitive, (ii) heavy sensitive, or (iii) mixed sensitive. A light sensitive mixture is one whose sensitive component in the F matrix is a light hydrocarbon and the D matrix has only significant non-zero elements in the upper triangular section. These elements will belong to the row (of the D matrix) corresponding to the sensitive component. In the case of light sensitive mixtures, adjustable component variables (T,,, P,,, a,)cannot be tuned, since these values are already known accurately. A heavy sensitive mixture is one whose sensitive component in the F matrix is a heavy pseudocomponent and the D matrix has only significant lower nondiagonal elements. In this case, adjustable component variables of the sensitive component can be tuned as explained below. Also, for both cases, the diagonal elements are the largest. When neither of the conditions above are satisfied, the mixture is of the mixed sensitive type. This type of mixture can have a light sensitive component in the F matrix, and the D matrix may have significant upper nondiagonal elements in the row corresponding to the sensitive component and also in other rows. In this case, the mixture will be called light-mixed sensitive. Another type of mixed sensitive mixture can have a heavy sensitive pseudocomponent in the F matrix and significant upper and lower nondiagonal elements. In this case, the mixture will be called heavy-mixed sensitive. Most gas condensates (if they are not light sensitive) are of the former type. Most reservoir oils (if they are not heavy sensitive) are of the latter type. At the separator conditions, most gas mixtures are light (or light-mixed) sensitive and most liquid mixtures are heavy (or heavy-mixed) sensitive. The vapor product from a flash of a gas condensate is almost always light sensitive. The liquid product is mixed or heavy sensitive. Table I shows the sensitive types of different mixtures. Details of these mixtures can be found from the references given in the table. Table I1 gives the list of components
1306 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 Table I. Mixture Types According to Sensitivity Analysis mixture 1 2 3 4
5 6 m
8
sensitivity type heavy-mixed light-mixed heavy light-mixed heavy-mixed light-mixed light heavy
ref a b b b a c a a a
identification in ref sample C sample A sample R gas 14 North seal Oil A gas 1 Table 1 (vapor) Table 1 (liquid)
“Pedersen et al., (a) 1984, (b) 1985a, and ( c ) 1985b.
Table 11. Details of the Three Types of Mixtures Characterized by the Sensitivity Analysis component
N* COz c1 c2
(23-6 (27-9 C12*
CZ1*
critical property Tc,K Pc, atm w 33.50 0.0400 126.20 72.80 0.2250 304.20 45.40 0.0080 190.60 0.0980 305.40 48.20 38.26 0.1862 408.45 31.58 0.4126 654.87 24.67 0.6847 657.14 14.47 1.1509 770.90
composition of mixture 6 7 8 0.0071 0.0075 0.0864 0.0996 0.0010 0.7085 0.7428 0.0032 0.0853 0.0862 0.0029 0.0823 0.0712 0.0286 0.0171 0.0270 0.4410 0.3284 0.0084 0.1948 0.0049
and the original values of their adjustable component variables (i.e., T,, P,, and w ) for three of the mixtures of Table I. The F and D matrices for different sensitivitytype mixtures are given in the supplementary material.
Selection of Candidate Adjustable Variables Besides allowing the selection of the candidate adjustable variables for possible tuning, the sensitivity type of the mixture and the table of the largest derivative values (see Table 111)give an indication of the feasibility of tuning and how much tuning is possible. Also, as will be shown later, the table of largest derivative values can give an indication of whether tuning is necessary even without any comparison with experimental data. The selection of the candidate adjustable variables and the corresponding tuning policy will also depend on the type of prediction problem. Most prediction problems involving petroleum mixtures can be divided into three groups. (a) Simple Problems. These are prediction problems which require the prediction of bulk properties (e.g., compressibility factor) of the mixture. The component present in the greatest amount has the biggest influence on the bulk property of the mixture. Individual properties are not required for analyzing these problems. (b) Semicomplex Problems. In this class of problems, although bulk properties of the given mixture are being predicted, individual properties like fugacity coefficients also need to be calculated. Typical problems in this category are dew point and bubble point calculations. (c) Complex Problems. Problems requiring single- or multiple-flash calculations belong to this class. These problems require the prediction of bulk properties (densities) as well as individual properties ( K values). The properties of each product (Le., of the phase present) mixtures are also calculated. For all types of problems, if the mixture is light sensitive, the adjustable component variable cannot be tuned. Usually, they are accurately known. Thus, only in the case of mixed and heavy sensitive types, tuning with adjustable component variables (of the pseudocomponent) is feasible. It should be noted that simple problems for all types of mixtures require a volume correction procedure. The best alternative (before tuning) for these problems is to use the
Table 111. Largest Scaled Elements in the Columns of the F Matrix and Rows of the D Matrix term or property AP BP
Z
largest derivative component value F Matrix 3 1.380 3 0.532 3 -0.260
aarameter
T, TC Tc
D Matrix Fug,, i = 1
-0.2142 -0.5168 -12.3300 -1.0500 -1.9200 -1.8600 -2.8300 -5.3100
1 2 8 7 8 6 7 8
Tc TC TC Tc Tc Tc
Tc Tc
Peneloux method (1982). Hence, selection of the candidate adjustable variables will be mainly for semicomplex and complex problems. Also, the selection policy will be the same for the light-mixed and heavy-mixed types of mixtures. The amount of tuning necessary can be determined from PI - PI* =
VI’
- f,’*)/Fl,
(14)
where i refers to component number of the pseudocomponent only and * indicates the given (or known) value. Tuning with adjustable component variables should be done only if small changes can achieve the desired accuracy requirement. For fine tuning, which requires small improvements to meet the necessary accuracy, the adjustable component variables can be used. In general, an adjustment of more than *lo% change from the original value of the variable should not be used. Larger changes may lead to physically meaningless values for the variable in question. For example, a large negative change in the critical temperature of the heaviest pseudocomponent can take its value near to its adjacent (lighter) pseudocomponent. When greater improvements are required, the best alternative may be to tune the adjustable interaction variables ( k , , ) . Examination of the F matrices shows that the term “Ap”,which is a function of k,,, is more sensitive to T, than the other terms (i.e., parameters). Also, in the case of the D matrix, the terms that affect the fugacity coefficients most are the terms that contain k , , . It may therefore be assumed that the derivatives with respect to k , , will also be large. Ideally, all k,, values should be adjusted to get the most accurate prediction. As there is a maximup of NC(NC1)/2 candidate interaction variables, this problem is quite difficult if NC is large. However, tuning a smaller, more sensitive set of variables can lead to a feasible reduced problem. For the hydrocarbon-non-hydrocarbon interactions, optimal k,, values have already been proposed (Reid et al., 1977). The adjustment therefore involves the k,! values for some of the hydrocarbon-hydrocarbon interactions. It should ideally be between light and heavy hydrocarbons. Analysis of the F and D matrices gives an indication for the best (i.e., most sensitive) pairs of k , , values to adjust. Since, in the case of simple and semicomplex types of problems, only bulk property prediction is required, only sensitivity analysis of the feed mixture is necessary. In the case of complex problems, however, in addition to the sensitivity type of the feed mixture, analysis of the product mixtures is also necessary.
Ind. Eng. Chem. Res., Vol. 26, No. 7 , 1987 1307
-
160
-
120
-
80
-
sensitive mixtures, the largest derivatives will belong to the pair methane-heavy pseudocomponent (heavy sensitive). In the case of mixed sensitive mixtures, the largest derivatives will belong to a light and heavy hydrocarbon. For complex problems, the selection of the most sensitive pair must also take into account the F and D matrices of the product mixture. Since in most cases the vapor product is light sensitive and the liquid product is mixed or heavy sensitive, the identification of the best pairs for adjustment is quite straightforward.
-adjusted’
Q E
QI
3 VI
2
p.
100
300
200
400 Temperature ( K )
- original 1
- 30- ---- adjusted‘ E
..........
4
adjusted3
2a
f
20-
L
10-
300
400
500
600
700
Temperature ( K )
Figure 1. (a, top) Phase envelope for a light sensitive mixture: (1) component 7 present in trace amounts (0.00001mole fraction), (2) original composition plus T,of component 6 changed from 654 to 587 K, (3) mole fraction as given in Table VI11 (column 5). (b, bottom) Phase envelope of a heavy sensitive mixture: (1) mole fraction as given in Table VI1 (column 7), (2) composition of first four components altered (yl= 0, y z = 0.00001,y3 = 0.0022,y4 = 0.0306), (3) original composition plus T,of component 7 changed from 657 to 672 K.
Bubble points and dew points are associated with the semicomplex type of problems. In the case of the light sensitive mixtures, usually, the dew point calculations have errors and the heaviest pseudocomponent present has a strong influence on the calculated dew point. Thus, the best pair (for kij ) is the light sensitive component and the heaviest pseudocomponent. If more than one ki value is required, then the next best pair would be the light sensitive component and the next lighter pseudocomponent. In all cases, one member of the pair should be the light sensitive component. Thus, the number of possible pairs is limited to the number of pseudocomponents present. In the case of heavy sensitive mixtures, both bubble point and dew point predictions can be erroneous. Usually, however, only bubble point predictions are required. For these predictions, the light hydrocarbons have a similar effect as the heavy pseudocomponents in the case s f the light sensitive mixtures. Figure 1shows how the phase envelope is affected by changes in composition and adjustable variable value for light and heavy sensitive mixtures. It can be seen very clearly that only dew points are affected for light sensitive mixtures and bubble points for heavy sensitive mixtures. The best pair for adjustment would be the lightest hydrocarbon and the heavy sensitive compound. In the case of mixed sensitive mixtures, the best pair would be the two most sensitive compounds. An examination of the D matrix will reveal the best pairs. In all cases, the most sensitive pair can be identified by the largest derivative value. In the case of light and heavy
Proposed Tuning Policy The problem to be solved by a tuning policy can be defined, thus, as follows. Given the following information, (a) mixture details, (b) prediction problem details, (c) accuracy required, and (d) experimental data available, determine (i) whether tuning is required, (ii) if yes, is it feasible, (iii) if yes, which variables should be changed and by how much, and (iv) how reliable the results will be. The following step by step procedure is proposed for solution of the above problem. Step 1. Solve the prediction problem (with no tuning). Compare the available experimental data with the predicted results and verify if the accuracy requirement is satisfied. If yes, no tuning is necessary. Otherwise, continue. Step 2. Establish how much improvement is required. If the necessary improvement is less than kTf%,only fine tuning is required. Otherwise, tuning of ki values will be necessary for semicomplex and complex problems. Tf denotes the value which defines if the problem involves fine tuning or not. Tfmay vary between &O% and &5%. Step 3. Establish the prediction problem type and the sensitivity type of the mixture (or mixtures). Identify the location of the largest derivatives in F and D matrices. Arrange in ascending order the ND largest derivatives and the corresponding adjustable variables. ND denotes the total number of data points available for comparison. Step 4. If the prediction problem is of type (a) simple, then (i) apply volume correction through the Peneloux method (1982) (if the desired accuracy is not achieved, continue; otherwise, stop) or (ii) select variables from the results of step 3. If the prediction is of type (b) semicomplex or complex, then select a maximum of ND adjustable variables from the results of step 3. Step 5. Update values of the selected variables from step 4. For a first trial, ND - J 2 1variables may be used. J denotes the number updated in this step. Step 6. Solve the required prediction problem and verify if desired accuracy is achieved. If yes, stop. Otherwise, continue. Step 7. Repeat from step 5 until convergence is achieved in step 6. If less than ND variables were used, add one more variable before returning to step 5 . The proposed tuning policy can be dividied into two parts. The first part consists of steps 1-4 and the second, steps 5-7. The first part identifies the problem type and the sensitivity type of the mixture, checks for feasibility of tuning, and selects the candidate adjustable variables. The second part is involved with the actual tuning of the selected variables. For this second part, a numerical minimization solution technique may be applied. A schematic diagram of the proposed tuning policy is given in Charts I and 11. The F and D matrices for different sensitivity types are available as supplementary material to this paper. In general, variables tuned by solving complex problems may be valid for semicomplex and simple problems but not the other way around. Thus, if variables are tuned to
1308 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 Chart I. Sensitivity Analysis and Selection of the Candidate Adjustable Variables" INPUT
characterization routine
y & no tuning
section
A2
*
Peneloux2 Method
semi-complex
I
I
& sensitivity
semi-complex
light
1
analysis of products
light
derivatives
perform preliminary flash calculation
"0
1
V
binary , select interaction parameters
50 'i
'Note
% : 2 : 3 : 4 :
:
6
identify the properties to be adjusted from the row column indices
mixed
composition
tion V
select binary interaction parameters
v
to adlust the selected oarametersl
select binary interaction parameters
check
compsi-
from R 2
7
1
4
identify the largest elements (off-diagonal1 of the D matrices (there will be maximum three ma trices)
Y
.
