Applications of the Coefficient ofIsothermal Compressibility to Various
Reservoir Situations With New Correlationsfor Each Situation
John P. Spivey, SPE, Phoenix Reservoir Engineering, and Peter P. Valkó, SPE, andWilliam D. McCain, SPE, Texas A&M U.
SummaryThe coefficient of isothermal compressibility (oil compressibility)is defined as the fractional change of oil volume per unit change inpressure. Though the oil compressibility so defined frequently ap-pears in the partial-differential equations describing fluid flow inporous media, it is rarely used in this form in practical engineeringcalculations. Instead, oil compressibility is usually assumed to beconstant, allowing the defining equation to be integrated over somepressure range of interest. Thus, the oil compressibility in theresulting equations should be regarded as a weighted average valueover the pressure range of integration.
The three distinct applications for oil compressibility in reser-voir engineering are (1) instantaneous or tangent values from thedefining equation, (2) extension of fluid properties from values atthe bubblepoint pressure to higher pressures of interest, and (3)material-balance calculations that require values starting at initialreservoir pressure. Each of these three applications requires a dif-ferent approach to calculating oil compressibility from laboratorydata and in developing correlations.
The differences among the values required in these three ap-plications can be as great as 25%. Most published correlations donot indicate the particular application to which the proposed cor-relation applies.
A correlation equation for oil compressibility has been devel-oped using more than 3,500 lines of data from 369 laboratorystudies. This correlation equation gives the average compressibil-ity between the bubblepoint pressure and some higher pressure ofinterest. Equations to calculate appropriate values of compressibil-ity for the other two applications are presented.
IntroductionThe equation defining the coefficient of isothermal compressibilityat pressures above the bubblepoint pressure is rather simple:
Moving oil compressibility outside the integral sign requires theassumption that it is constant. Because it is not constant, the use ofthis equation requires a value of oil compressibility that is a pressure-weighted average across the pressure range used in the calculations.
There are three applications for oil compressibility in reser-voir engineering:
• The defining equation, for which the oil compressibilityshould be calculated as a single value at the pressure of interest,often used in pressure-transient analysis.
• The extension of fluid properties from correlations starting atthe bubblepoint pressure to pressures above the bubblepoint pres-sure. This application is also used in black-oil reservoir simulation.
• The use of oil compressibility in black-oil material-balanceequations in which the starting point is the initial reservoir pressure.
Values of oil compressibility should be calculated from labo-ratory data with these applications in mind. Most published cor-relations for oil compressibility do not indicate the particular situ-ation to which the correlation applies, although values calculatedfor these three applications can differ significantly. For example,Fig. 1 gives values of oil compressibility calculated with the con-stant-composition-expansion data from a widely available black-oil laboratory report (Reservoir Fluid Study 1988). Two things arereadily apparent. First, coefficients of isothermal compressibilityare not constant as pressure changes. Second, the three applica-tions require values that differ by up to 25%.
Using more than 3,500 data points measured in constant-composition-expansion experiments from 369 laboratory studiesof worldwide origins, we have developed a correlation equation foroil compressibility calculated as a chord from bubblepoint pressureto a pressure of interest. We propose two equations to adjust valuesfrom this correlation either to chord values starting at initial res-ervoir pressure or to instantaneous values. These equations allowestimation of oil-compressibility values for the correct application.
Applications of CompressibilityIn this section, we discuss the three applications of the coefficientof isothermal compressibility.
Extension of Fluid Properties From Correlations Starting atthe Bubblepoint Pressure. When correlations are used to deter-mine values of fluid properties, values of oil compressibility areused to extend values of some fluid properties from the bubble-point pressure of the oil to higher pressures. One example is oilformation volume factor.
Placing oil compressibility, co, outside of the integral sign in Eq.4 implies that it is constant. Oil compressibility certainly is notconstant as pressure changes (as seen in Fig. 1). Eq. 5 can be usedto calculate the oil formation volume factor at pressures abovebubblepoint pressure using the value of Bob calculated at thebubblepoint with a correlation. However, the value of co to be usedin Eq. 5 must be a pressure-weighted average of oil compressibilityfrom the bubblepoint pressure to the pressure of interest. In thiscase, the derivative in Eq. 3 is approximated by the slope of achord from the bubblepoint pressure to the pressure of interest. We
This paper (SPE 96415) was first presented at the 2005 SPE Annual Technical Conferenceand Exhibition, Dallas, 9–12 October, and revised for publication. Original manuscript re-ceived for review 1 July 2005. Revised manuscript received 5 September 2006. Paper peerapproved 10 October 2006.
will call this the coefficient of isothermal compressibility from thebubblepoint, cofb. Values of cofb can be determined easily from theconstant-composition-expansion experiment of a routine black-oilfluid property report (McCain 1990) and used to create a correla-tion for cofb as a function of various properties, including reservoirpressure, that are readily available from field data.
The Material-Balance Equation for Undersaturated Oil Res-ervoirs. Another use of oil compressibility is in the material-balance equation for undersaturated oil reservoirs (i.e., reservoirsin which the pressure is higher than the bubblepoint pressure). Eqs.6 and 7 show this application (Craft and Hawkins 1959).
Eq. 7 demonstrates that this application starts at initial reservoirpressure, pi, and goes to a lower pressure, p, which occurs aftersome oil production. Thus, the oil-compressibility value should bea pressure-weighted average starting at the initial reservoir pres-sure. In this case, the derivative is approximated by the slope of achord from initial pressure to the pressure of interest. We will callthis the coefficient of isothermal compressibility from initial pres-sure, cofi. This definition of compressibility is analogous to thecumulative compressibility terms defined by Fetkovich et al.(1998). Values of this property can be obtained using the correla-tion for cofb with initial reservoir pressure and the lower pressure.
Tangent Compressibility for Pressure-Transient Analysis. Thepartial-differential equations describing single-phase fluid flow inporous media do not require that the equation be integrated. In-stead, oil compressibility is calculated at the pressure of interestwith the value of the derivative of Eq. 1 determined by measuringthe slope of the tangent line to the volume-vs.-pressure curve at thepressure of interest. In other words, the value of oil compressibilityis not a weighted average but an “instantaneous” value at thepressure of interest. For instance, for single-phase flow, Eqs. 8 and9 apply.
This is oil compressibility by definition; we will use the symbol co.An equation for determining values of co at any pressure of interestwith the correlation for cofb will be presented later.
For multiphase flow at pressures below the bubblepoint pressure,a term must be added to account for gas coming out of solution:
co = −1
Bo��Bo
�p �T
+1
1,000
Bg
Bo��Rso
�p �T
. . . . . . . . . . . . . . . . . . (11)
We prefer taking derivatives of the correlation equations for Bo
and Rs to calculate co directly from Eq. 11, rather than using aseparate correlation for co.
Development of New CorrelationsWe have developed new correlations for the coefficient of isother-mal compressibility and equations for calculating values for eachof the three applications for oil compressibility. Because of theway fluid-property data are reported in standard laboratory reports,it is more convenient to develop a correlation for cofb than for cofi
or co. Thus, our correlation gives cofb. Equations for calculatingvalues for cofi and co with the cofb correlation have been developed.
Correlation for Coefficients of Isothermal CompressibilityFrom the Bubblepoint to a Pressure of Interest. A correlationfor cofb has been developed using a technique to reveal the under-lying statistical relationships between variables corrupted by ran-dom error. The method of alternating conditional expectations(ACE) is intended to alleviate the main drawback of parametricregression: the mismatch of the assumed model structure and theunderlying relationship of the actual data (Breiman and Friedman1985). Thus, an a priori knowledge of the functional relationshipbetween the dependent variable and the independent variables isnot required. The program GRACE, a user-friendly implementionof the ACE algorithm, was used for this work. GRACE first cre-ates individual transformations for each variable in an optimumway. Then, these single-variable transformations may usually beapproximated by curve fitting in a commercial spreadsheet pro-gram using low-order polynomials (Xue et al. 1997).
A database was assembled comprising 3,537 lines of constant-composition-expansion data at pressures above bubblepoint pres-sures from 369 service-company reservoir-fluid studies of blackoils. The statistics of these assembled data, which cover the rangesof the independent variables to be expected in black-oil reservoirs,are given in Table 1. Extreme caution should be used in dealingwith oils with properties outside the ranges in this table; extrapo-lation of the equations presented is risky.
Applying the GRACE technique produced the following equa-tions for estimating values of cofb with the independent variableslisted in Table 2.
To apply these equations, use the natural logarithm of each vari-able and the coefficients listed in Table 2 to calculate a value of zn
for each of the six independent variables. The values of zn are added,and the sum is used in the cofb equation (Eq. 13). A discussion ofthe accuracy and precision of this equation will be given later.
Fig. 1—Coefficients of isothermal compressibility for Good OilCo., Oil Well No. 4 (Reservoir Fluid Study 1988).
44 February 2007 SPE Reservoir Evaluation & Engineering
Correlation for Coefficients of Isothermal CompressibilityFrom Initial Pressure to a Pressure of Interest. Oil compress-ibility can be determined from initial pressure to a lower pressureof interest, cofi, by use of Eq. 14 and the cofb correlation equations(Eq. 13). Values of cofb at the initial pressure and the pressure ofinterest are calculated and then used to calculate cofi.
Eq. 14 was developed as follows:
cofb�p� = −�
Bob
Bo dBo
Bo
�pb
pdp
= −ln
Bo
Bob
p − pb=
ln Bob − ln Bo
p − pb, . . . . . . . . (13a)
cofb�pi� = −�
Bob
Boi dBo
Bo
�pb
pidp
= −ln
Boi
Bob
pi − pb=
ln Bob − ln Boi
pi − pb, . . . . . . . (13b)
cofi = −�
Boi
Bo dBo
Bo
�pi
pdp
= −ln
Bo
Boi
p − pi=
ln Boi − ln Bo
p − pi. . . . . . . . . . . . . (13c)
The difference between Eqs. 13a and 13b can be used to determinean equation for cofi.
Finally, rearrange to obtain the equation for calculating cofi withthe correlation equations for cofb.
cofi =�p − pb�cofb�p� − �pi − pb�cofb�pi�
p − pi. . . . . . . . . . . . . . . . . . . (14)
Correlation for Coefficient of Isothermal Compressibility Tan-gent at Some Pressure of Interest. The chord-slope compress-ibility from the bubblepoint pressure to any pressure above thebubblepoint, cofb, can be rewritten from Eq. 5 as
Calculating the Coefficient of Isothermal Compressibility forSaturated Oils. For consistency, the tangent compressibility forpressure-transient analysis should be calculated from correlationequations for Bo, Rso, and Bg by differentiating the equations for Bo
Evaluation of the Proposed CorrelationsFig. 2 compares the results of calculations of cofb using Eqs. 12a,12b, and 12c with the data used in developing the correlation. Fig. 3compares these data with calculations of co based on Eqs. 15c, 16a,12a, 12b, and 16b. The bulk of these calculations fit the data veryclosely. In both figures, the scatter of the measured results at
higher values appears to be caused by data from a few laboratoryreports that may have some internal error.
A comparison of the calculations and the data in terms of averagerelative error (ARE) and average absolute relative error (AARE)appears in Table 3. Figs. 4 through 7 show that the calculationshold up well across the ranges of the independent variables.
Comparison of Published Correlations forCoefficients of Isothermal CompressibilityTable 3 shows ARE and AARE, both in percentages, of compari-sons of various published correlations (De Ghetto and Villa 1994;Al-Marhoun 2003; Dindoruk and Christman 2001; Petrosky andFarshad 1998; Labedi 1990; Whitson and Brule 2000; Almehaideb1997; Hanafy et al. 1997; Calhoun 1953; Vazquez and Beggs1980; Kartoatmodjo and Schmidt 1994; Elsharkawy and Alikhan1997; Ahmed 1989; Farshad et al. 1996) for oil compressibility.Table 4 gives statistics for the independent variables in the data setused for these comparisons. The coefficients of isothermal com-pressibility calculated with data within 200 psi of bubblepointpressure were eliminated because the data had only four significantfigures, making the round-off errors near the bubblepoint exces-sively large.
Fig. 2—Coefficients of isothermal compressibility calculatedfrom Eqs. 12a, 12b, and 12c compare well with measured values.
Fig. 4—ARE as a function of temperature. Each data point rep-resents an average of approximately 240 calculations.
Fig. 3—Coefficients of isothermal compressibility calculatedfrom Eq. 15c compare well with measured values.
46 February 2007 SPE Reservoir Evaluation & Engineering
Many of the publications proposing the correlations listed inTable 3 did not specify the applications for which they were de-veloped, so we compared them with both cofb and co and reportedthe lowest values of ARE and AARE.
Figs. 4 through 7 compare the results of this work with threeother published correlations. The Vazquez and Beggs (1980) cor-relation was selected for this comparison for its apparent popular-ity in the petroleum industry, and the other two (Al-Marhoun 2003;Dindoruk and Christman 2001) were chosen as recently publishedcorrelations that have low values of ARE and AARE. The resultsof this work hold up well across the full range of each of theindependent variables and generally have values of ARE andAARE that are closer to zero than the other three correlations.Sorting on stock-tank-oil gravity, °API, and reservoir pressure (notshown here), yielded similar results.
ConclusionsThe three different applications for the values of the coefficient ofisothermal compressibility require different calculations from ex-perimental data, and the calculated values among the three aresignificantly different results.
We have presented a correlation for the fluid-property applica-tion that gives results approximately as accurate as the experimen-tal data. This correlation produces a weighted average of oil com-pressibility between the bubblepoint pressure and the reservoirpressure of interest, cofb.
We have presented an equation, based on the cofb correlation, thatcan be used to obtain accurate estimates of cofi, the weighted averageoil compressibility from initial reservoir pressure to some lowerreservoir pressure, for use in the oil material-balance application.
Finally, we have presented an equation that can be used toobtain accurate estimates of the tangent compressibility co at aparticular reservoir pressure.
NomenclatureAARE � average absolute relative error, %
co � coefficient of isothermal compressibility,measured with the slope of the tangent line atthe pressure of interest, psi−1
cofb � coefficient of isothermal compressibility,measured with the slope of the chord frombubblepoint pressure to pressure of interest,psi−1
cofb(p) � coefficient of isothermal compressibilitymeasured with the slope of the chord frombubblepoint pressure to pressure p, psi−1
cofb(pi) � coefficient of isothermal compressibilitymeasured with the slope of the chord frombubblepoint pressure to initial reservoir pressurepi, psi−1
cofi � coefficient of isothermal compressibilitymeasured with the slope of the chord frominitial reservoir pressure pi to pressure ofinterest, psi−1
C0n,C1n,C2n � coefficients for use in Eq. 12c for the nthindependent variable
k � absolute permeabilityN � original oil in place, STB
Publishing.Al-Marhoun, M.A. 2003. The Coefficient of Isothermal Compressibility of
Black Oils. Paper SPE 81432 presented at the SPE Middle East OilShow, Bahrain, 9–12 June. DOI: 10.2118/81432-MS.
Almehaideb, R.A. 1997. Improved PVT Correlations for UAE Oils. PaperSPE 37691 presented at the SPE Middle East Oil Show and Confer-ence, Bahrain, 15–18 March. DOI: 10.2118/37691-MS.
Breiman, L. and Friedman, J.H. 1985. Estimating Optimal Transformationsfor Multiple Regression and Correlation. J. Am. Stat. Assoc. 80 (391):580–619. DOI: http://dx.doi.org/10.2307/2288473.
Calhoun, J.C. Jr. 1953. Fundamentals of Reservoir Engineering, 35. Nor-man, Oklahoma: U. of Oklahoma Press.
Craft, B.C. and Hawkins, M.F. 1959. Applied Petroleum Reservoir Engi-neering, 136. Englewood Cliffs, New Jersey: Prentice-Hall.
De Ghetto, G. and Villa, M. 1994. Reliability Analysis on PVT Correla-tions. Paper SPE 28904 presented at the 1994 European PetroleumConference, London, 25–27 October. DOI: 10.2118/28904-MS.
Dindoruk, B. and Christman, P.G. 2001. PVT Properties and ViscosityCorrelations for Gulf of Mexico Oils. Paper SPE 71633 presented at theSPE Annual Technical Conference and Exhibition, New Orleans, 30September–3 October. DOI: 10.2118/71633-MS.
Elsharkawy, A.M. and Alikhan, A.A. 1997. Correlations for predictingsolution gas/oil ratio, oil formation volume factor, and undersaturatedoil compressibility. J. Pet. Sci. & Eng. 17 (3–4): 291–302. DOI: http://dx.doi.org/10.1016/S0920-4105(96)00075-7.
Fetkovich, M.J., Reese, D.E., and Whitson, C.H. 1998. Application of aGeneral Material Balance for High-Pressure Gas Reservoirs. SPEJ 3(1): 3–13. SPE-22921-PA. DOI: 10.2118/22921-PA.
Farshad, F., LeBlanc, J.L., Garber, J.D., and Osorio, J.G. 1996. EmpiricalPVT Correlations for Colombian Crude Oils. Paper SPE 36105 pre-sented at the SPE Latin American and Caribbean Petroleum Engineer-ing Conference, Port-of-Spain, Trinidad, 23–26 April. DOI: 10.2118/36105-MS.
Hanafy, H.H., Macary, S.M., ElNady, Y.M., Bayomi, A.A., and El Ba-tanony, M.H. 1997. Empirical PVT Correlations Applied to EgyptianCrude Oils Exemplify Significance of Using Regional Correlations.Paper SPE 37295 presented at the SPE International Symposium onOilfield Chemistry, Houston, 18–21 February. DOI: 10.2118/37295-MS.
Kartoatmodjo, T. and Schmidt, Z. 1994. Large data bank improves crudephysical property correlations. Oil & Gas J. 92 (27): 51–55.
Labedi, R. 1990. Use of production data to estimate volume factor, densityand compressibility of reservoir fluids. J. Pet. Sci. & Eng. 4 (4): 375–390. DOI: http://dx.doi.org/10.1016/0920-4105(90)90034-Z.
McCain, W.D. Jr. 1990. The Properties of Petroleum Fluids, second edi-tion, 289. Tulsa: PennWell Books.
Petrosky, G.E. Jr. and Farshad, F. 1998. Pressure-Volume-TemperatureCorrelations for Gulf of Mexico Crude Oils. SPEREE 1 (5): 416–420.SPE-51395-PA. DOI: 10.2118/51395-PA.
Reservoir Fluid Study for Good Oil Company Black Oil Well Number 4.1988. Houston: Core Laboratories.
Vazquez, M. and Beggs, H.D. 1980. Correlations for Fluid Physical Prop-erty Prediction. JPT 32 (6): 968–970. SPE-6719-PA. DOI: 10.2118/6719-PA.
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SI Metric Conversion Factors°API 141.5/(131.5+°API) � g/cm3
bbl × 1.589 873 E−01 � m3
ft3 × 2.831 685 E−02 � m3
°F (°F−32)/1.8 � °Cpsi × 6.894 757 E+00 � kPa
John P. Spivey has more than 20 years of experience in thepetroleum industry, with interests in pressure-transient analysis,production-data analysis, reservoir engineering, continuingeducation, and software development. From 1984 to 1990, he
48 February 2007 SPE Reservoir Evaluation & Engineering
worked for SoftSearch Inc. (later Dwights EnergyData) devel-oping petroleum economics and engineering software. In1990, Spivey joined S.A. Holditch & Assocs. (which was laterpurchased by Schlumberger); while there, he conducted res-ervoir-simulation, gas-storage, and tight-gas-application stud-ies and taught industry short courses in well testing and pro-duction-data analysis. In 2004, he started his own reservoir-engineering consult ing company, Phoenix ReservoirEngineering, and software development company, PhoenixReservoir Software LLC. Since 1992, Spivey has served as a vis-iting assistant professor or adjunct assistant professor at TexasA&M U., teaching undergraduate and graduate classes ingas-reservoir engineering and pressure-transient analysis. Heholds a BS degree in physics from Abilene Christian U., an MSdegree in physics from the U. of Washington, and a PhD de-gree in petroleum engineering from Texas A&M U. Spivey is theeditor of SPE Reprint Series Vol. 52, Gas Reservoir Engineering,and Vol. 57, Pressure Transient Testing, and coauthor of the SPEtextbook Pressure Transient Testing. He has published numerouspapers and articles in industry journals and trade publications.Peter P. Valkó is a professor of petroleum engineering at TexasA&M U. Previously, he taught at academic institutions in Austriaand Hungary and worked for the Hungarian Oil Co. Valkóholds BS and MS degrees from Veszprém U., Hungary, and aPhD degree from the Inst. of Catalysis, Novosibirsk, Russia.
William D. McCain Jr. has been a visiting professor in the Dept.of Petroleum Engineering at Texas A&M U. since 1991. McCainstarted his engineering career with Esso (now Exxon) ResearchLaboratories in 1956, where he assisted in research on surfaceprocessing of petroleum fluids. He was Professor and Head ofthe Petroleum Engineering Dept. at Mississippi State U. from1965 to 1976 and taught at Texas A&M U. from 1984 through1987. McCain was a consulting petroleum engineer with Caw-ley, Gillespie & Assocs. from 1987 until 1991. He was with thepetroleum engineering consulting firm S.A. Holditch & Assocs.from 1991 until 2000, retiring as Executive Vice President, ChiefEngineer, and member of the Board of Directors. McCain’sengineering specialties within the consulting company wereproperties of petroleum fluids, surface processing of petro-leum, and reservoir engineering, especially for gas conden-sate and volatile oil fields. He has written two editions of thewidely used textbook, The Properties of Petroleum Fluids, holdsthree U.S. patents, and has more than 40 publications in thepetroleum engineering literature. Involved in short-courseteaching for more than 20 years, McCain has taught for sev-eral major oil companies, independents, professional societies,and educational consulting companies throughout the world.He holds a BS degree from Mississippi State U. and MS andPhD degrees from the Georgia Inst. of Technology, all in chemi-cal engineering.
