The Use of Semicontinuous Description To Model the C7+ Fraction in Equation of State Calculations

Abstract

Summary. A semicontinuous thermodynamic description has been used to modelthe C7, fraction for equation of state (EOS) calculations. Thissemicontinuous description may be used in existing discrete-component EOSprograms by picking pseudocomponents in a rigorous a priori fashion dependent only on the properties of the C7+ fraction. Excellentrepresentation of experimental PVT data considered here is obtained withonly tWO C7+ pseudocomponents, although the theory and procedure extend tothree or more components. Introduction EOS calculations are frequently burdened by the large number ofcomponents necessary to describe an oil for accurate phase-behaviormodeling. Often the problem is either lumping together the manyexperimentally determined fractions or the converse, modeling theoil properly when the only experimental data available for the C 7 +fraction are the molecular weight (MW) and density. Generally, the PVT description is Unproved by use of a largernumber of components. For compositional reservoir simulations, the upper limit to the number of components is usually controlledby simulator time considerations. Consequently, when extendedcompositional data are available, a limited number ofpseudocomponents are obtained from one of the numerous lumping methodsthat have been proposed. Only a few lumping methods discuss how to model the portions of the composition distribution not measuredexperimentally. The lumping schemes are either trial-and-errormethods, arbitrary rules, or trial-and-error methods withinspecific guidelines. The trial-and-error methods necessarily give good results if one iterates long enough. Examples of studies where the C7+ fractionis lumped by trial and error include those by Hong I and Grigg and Lingane. A single C7+ fraction was found unsuitable. Several methods have been proposed that follow specific arbitrary guidelines. Lee et al. proposed plotting all properties availablefor each fraction with the boiling point as the independent variable. Those fractions with similar slopes were then grouped together. The method attempts to group fractions by experimental data thatare often unavailable. Whitson determined the number offractions needed by one empirical rule and then assigned fractions according to another rule. Pedersen et al. grouped individualfractions together by requiring each pseudocomponent to have anequal weight fraction of the fluid. The properties were thendetermined by a weight fraction average. Li et al. assigned a givennumber of pseudocomponents for every decade range of K values. The individual fractions were then lumped together accordingly. These lumping methods do not require multiple calculations, butthey do not have a rigorous basis. The methods that incorporate multiple calculations withinspecific guidelines are not simple. Mehra et al. proposed a complexstatistical method for lumping individual fractions that minimizesthe error in phase-saturation calculations. Calculations are requiredfor each pair of consecutive fractions to determine whether theyare to be lumped into the same pseudocomponent. Colonomos etal. presented a scheme that uses linear programming in which thelumping of the fractions into pseudocomponents is done by tryingall contiguous combinations and minimizing the difference betweenthe properties of the lumped distribution and the originaldistribution. These are not trial-and-error methods, but they involvecon-siderable calculation to define the pseadocomponents. One lumping schemethat attempts to incorporate some physics was presented by Montel and Gouel. They lumped the 150 identifiable isomers of C1 to C10. The C11 +fraction was not treated. It is rare to get such detailed data on acondensate and impossible to get them over the full range of components ofan oil. There is no standard lumping scheme to date because all proposedmethods require data that are generally not available, require too muchtime to implement, or offer no clear advantage over other availableschemes. Theory Distribution. A semicontinuous fluid mixture consists ofidentifiable discrete components and a continuous distribution torepresent all other components. The discrete components include lighthydrocarbons and inorganic gases, such as CO2. The continuousdistribution F(I) describes the remainder of the fluid according tothe index 1 that is chosen to be a property, such as boiling point, carbon number, or MW. The continuous distribution is used todescribe such unidentifiable components as the heavy hydrocarbonsand identifiable components that are too numerous to beconsidered individually. Early work on semicontinuous thermodynamics was done byCotterman et al. on polymer and petroleum systems, comparingcontinuous distribution results with discrete component results. Kehlen et al. provided a mathematical analysis of continuousthermodynamics, noting that the approach facilitates treatment ofmixtures containing so many components that they are described better by distribution functions than by mole fractions for discretecomponents. Application to multistage absorption and distillationwas demonstrated by Chou and Prausnitz and Kehlen and Ratzsch. Willman and Teja recently used a bivariatecontinuous distribution to describe certain types of fluids better. An analytic distribution function is needed for a continuousdescription. Any function may be used if it describes the oil overthe desired domain. Whitson used a three-parameter gammafunction to describe the C7, fraction, with the three parameters fit toexperimental molar- and weight-distribution data. Alternatively, theparameters may be estimated by empirical relations. Many oils exhibit an exponential distribution, which is a special case of the gamma distribution, as seen from the fact that asemilog plot of composition vs. carbon number often yields a straightline beyond some initial carbon number (usually C7 or slightlyhigher). Of course, the distribution cannot continue to infinitybecause some finite upper carbon number must exist in reservoir fluids. A truncated distribution was used in this work: ..........................................(1) SPERE P. 1041