Applications of the Coefficient of Isothermal ...

Applications of the Coefficient of
Isothermal Compressibility to Various
Reservoir Situations With New Correlations

for Each Situation

John P. Spivey, SPE, Phoenix Reservoir Engineering, and Peter P. Valkó, SPE, and
William D. McCain, SPE, Texas A&M U.

Summary • The defining equation, for which the oil compressibility
should be calculated as a single value at the pressure of interest,
The coefficient of isothermal compressibility (oil compressibility) often used in pressure-transient analysis.
is defined as the fractional change of oil volume per unit change in
pressure. Though the oil compressibility so defined frequently ap- • The extension of fluid properties from correlations starting at
pears in the partial-differential equations describing fluid flow in the bubblepoint pressure to pressures above the bubblepoint pres-
porous media, it is rarely used in this form in practical engineering sure. This application is also used in black-oil reservoir simulation.
calculations. Instead, oil compressibility is usually assumed to be
constant, allowing the defining equation to be integrated over some • The use of oil compressibility in black-oil material-balance
pressure range of interest. Thus, the oil compressibility in the equations in which the starting point is the initial reservoir pressure.
resulting equations should be regarded as a weighted average value
over the pressure range of integration. Values of oil compressibility should be calculated from labo-
ratory data with these applications in mind. Most published cor-
The three distinct applications for oil compressibility in reser- relations for oil compressibility do not indicate the particular situ-
voir engineering are (1) instantaneous or tangent values from the ation to which the correlation applies, although values calculated
defining equation, (2) extension of fluid properties from values at for these three applications can differ significantly. For example,
the bubblepoint pressure to higher pressures of interest, and (3) Fig. 1 gives values of oil compressibility calculated with the con-
material-balance calculations that require values starting at initial stant-composition-expansion data from a widely available black-
reservoir pressure. Each of these three applications requires a dif- oil laboratory report (Reservoir Fluid Study 1988). Two things are
ferent approach to calculating oil compressibility from laboratory readily apparent. First, coefficients of isothermal compressibility
data and in developing correlations. are not constant as pressure changes. Second, the three applica-
tions require values that differ by up to 25%.
The differences among the values required in these three ap-
plications can be as great as 25%. Most published correlations do Using more than 3,500 data points measured in constant-
not indicate the particular application to which the proposed cor- composition-expansion experiments from 369 laboratory studies
relation applies. of worldwide origins, we have developed a correlation equation for
oil compressibility calculated as a chord from bubblepoint pressure
A correlation equation for oil compressibility has been devel- to a pressure of interest. We propose two equations to adjust values
oped using more than 3,500 lines of data from 369 laboratory from this correlation either to chord values starting at initial res-
studies. This correlation equation gives the average compressibil- ervoir pressure or to instantaneous values. These equations allow
ity between the bubblepoint pressure and some higher pressure of estimation of oil-compressibility values for the correct application.
interest. Equations to calculate appropriate values of compressibil-
ity for the other two applications are presented. Applications of Compressibility

Introduction In this section, we discuss the three applications of the coefficient
of isothermal compressibility.

The equation defining the coefficient of isothermal compressibility Extension of Fluid Properties From Correlations Starting at
at pressures above the bubblepoint pressure is rather simple: the Bubblepoint Pressure. When correlations are used to deter-
mine values of fluid properties, values of oil compressibility are
co = − Vͩ ͪ1 ѨV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) used to extend values of some fluid properties from the bubble-
Ѩp point pressure of the oil to higher pressures. One example is oil
T formation volume factor.

However, in application the situation becomes somewhat complex. ͩ ͪ1 ѨBo
Usually the equation is integrated by separating variables: Ѩp
co = − Bo
͐ ͐co V2 dV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)
.
p2 dp = − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2) T

p1 V1 V ͐ ͐co Bo dBo .
p
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)
dp = −
Moving oil compressibility outside the integral sign requires the Bpb Bob o
assumption that it is constant. Because it is not constant, the use of
this equation requires a value of oil compressibility that is a pressure- Bo = Bob exp͓−co͑p − pb͔͒. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)
weighted average across the pressure range used in the calculations.
Placing oil compressibility, co, outside of the integral sign in Eq.
There are three applications for oil compressibility in reser- 4 implies that it is constant. Oil compressibility certainly is not
voir engineering:
constant as pressure changes (as seen in Fig. 1). Eq. 5 can be used
Copyright © 2007 Society of Petroleum Engineers
to calculate the oil formation volume factor at pressures above
This paper (SPE 96415) was first presented at the 2005 SPE Annual Technical Conference
and Exhibition, Dallas, 9–12 October, and revised for publication. Original manuscript re- bubblepoint pressure using the value of Bob calculated at the
ceived for review 1 July 2005. Revised manuscript received 5 September 2006. Paper peer bubblepoint with a correlation. However, the value of co to be used
approved 10 October 2006. in Eq. 5 must be a pressure-weighted average of oil compressibility

from the bubblepoint pressure to the pressure of interest. In this

case, the derivative in Eq. 3 is approximated by the slope of a

chord from the bubblepoint pressure to the pressure of interest. We

February 2007 SPE Reservoir Evaluation & Engineering 43

1 Ѩ␳o = co Ѩp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)
␳o Ѩt Ѩt

Eq. 10 is the equation describing single-phase-fluid flow of a
slightly compressible liquid in porous media (Lee et al. 2003).

ͩ ͪ1 Ѩ Ѩp = ␾␮co Ѩp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)
r Ѩr k Ѩt
r Ѩr

This is oil compressibility by definition; we will use the symbol co.
An equation for determining values of co at any pressure of interest
with 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:

ͩ ͪ ͩ ͪ1 ѨBo ѨRso
Ѩp Ѩp
co = − Bo
+ 1 Bg . . . . . . . . . . . . . . . . . . (11)
T 1,000 Bo
T

Fig. 1—Coefficients of isothermal compressibility for Good Oil We prefer taking derivatives of the correlation equations for Bo
Co., Oil Well No. 4 (Reservoir Fluid Study 1988). and Rs to calculate co directly from Eq. 11, rather than using a
separate correlation for co.

will call this the coefficient of isothermal compressibility from the Development of New Correlations

bubblepoint, cofb. Values of cofb can be determined easily from the We have developed new correlations for the coefficient of isother-
constant-composition-expansion experiment of a routine black-oil mal compressibility and equations for calculating values for each
of the three applications for oil compressibility. Because of the
fluid property report (McCain 1990) and used to create a correla- way fluid-property data are reported in standard laboratory reports,
it is more convenient to develop a correlation for cofb than for cofi
tion for cofb as a function of various properties, including reservoir or co. Thus, our correlation gives cofb. Equations for calculating
pressure, that are readily available from field data. values for cofi and co with the cofb correlation have been developed.

The Material-Balance Equation for Undersaturated Oil Res- Correlation for Coefficients of Isothermal Compressibility
ervoirs. Another use of oil compressibility is in the material- From the Bubblepoint to a Pressure of Interest. A correlation
balance equation for undersaturated oil reservoirs (i.e., reservoirs for cofb has been developed using a technique to reveal the under-
in which the pressure is higher than the bubblepoint pressure). Eqs. lying statistical relationships between variables corrupted by ran-
6 and 7 show this application (Craft and Hawkins 1959). dom error. The method of alternating conditional expectations
(ACE) is intended to alleviate the main drawback of parametric
co ≡ Bo − Boi , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6a) regression: the mismatch of the assumed model structure and the
Boi͑pi − p͒ underlying relationship of the actual data (Breiman and Friedman
1985). Thus, an a priori knowledge of the functional relationship
where between the dependent variable and the independent variables is
not required. The program GRACE, a user-friendly implemention
ce ≡ Soco + Swcw + cf , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6b) of the ACE algorithm, was used for this work. GRACE first cre-
So ates individual transformations for each variable in an optimum
way. Then, these single-variable transformations may usually be
giving the material-balance equation approximated by curve fitting in a commercial spreadsheet pro-
gram using low-order polynomials (Xue et al. 1997).
NBoice͑pi − p͒ = NpBo − We + BwWp. . . . . . . . . . . . . . . . . . . . . . (7)
A database was assembled comprising 3,537 lines of constant-
Eq. 7 demonstrates that this application starts at initial reservoir composition-expansion data at pressures above bubblepoint pres-
pressure, pi, and goes to a lower pressure, p, which occurs after sures from 369 service-company reservoir-fluid studies of black
some oil production. Thus, the oil-compressibility value should be oils. The statistics of these assembled data, which cover the ranges
a pressure-weighted average starting at the initial reservoir pres- of the independent variables to be expected in black-oil reservoirs,
sure. In this case, the derivative is approximated by the slope of a are given in Table 1. Extreme caution should be used in dealing
chord from initial pressure to the pressure of interest. We will call with oils with properties outside the ranges in this table; extrapo-
this the coefficient of isothermal compressibility from initial pres- lation of the equations presented is risky.
sure, cofi. This definition of compressibility is analogous to the
cumulative compressibility terms defined by Fetkovich et al. Applying the GRACE technique produced the following equa-
(1998). Values of this property can be obtained using the correla- tions for estimating values of cofb with the independent variables
tion for cofb with initial reservoir pressure and the lower pressure. listed in Table 2.

Tangent Compressibility for Pressure-Transient Analysis. The ln cofb = 2.434 + 0.475z + 0.048z2, . . . . . . . . . . . . . . . . . . . . (12a)
partial-differential equations describing single-phase fluid flow in
porous media do not require that the equation be integrated. In- 6
stead, oil compressibility is calculated at the pressure of interest
with the value of the derivative of Eq. 1 determined by measuring ͚z = zn , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12b)
the slope of the tangent line to the volume-vs.-pressure curve at the n=1
pressure of interest. In other words, the value of oil compressibility
is not a weighted average but an “instantaneous” value at the zn = C0n + C1n xn + C2n xn2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12c)
pressure of interest. For instance, for single-phase flow, Eqs. 8 and
9 apply. 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
ͩ ͪ1 Ѩ␳o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8) for each of the six independent variables. The values of zn are added,
Ѩp and the sum is used in the cofb equation (Eq. 13). A discussion of
co = ␳o T the accuracy and precision of this equation will be given later.

44 February 2007 SPE Reservoir Evaluation & Engineering

Correlation for Coefficients of Isothermal Compressibility Combining Eqs. 15a and 15b yields these results:

From Initial Pressure to a Pressure of Interest. Oil compress- 1Ѩ
co = − Bo Ѩp ͕Bob exp͓−cofb͑p − pb͔͖͒
ibility can be determined from initial pressure to a lower pressure
= − Bob exp͓−cofb͑p − pb͔͒ Ѩ ͓−cofb͑p − pb͔͒
of interest, cofi, by use of Eq. 14 and the cofb correlation equations Bo Ѩp
(Eq. 13). Values of cofb at the initial pressure and the pressure of
interest are calculated and then used to calculate cofi.

Eq. 14 was developed as follows:

͐͐cofb͑p͒ = − Bo dBo ln Bo ln Bob − ln Bo Ѩ
Bob Bo Bob p − pb = Ѩp ͓cofb͑p − pb͔͒

p dp = − p − pb = , . . . . . . . . (13a) Ѩcofb
Ѩp
pb = cofb + ͑p − pb͒ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15c)

͐͐cofb͑pi͒ = − Boi dBo = − ln Boi = ln Bob − ln Boi , . . . . . . . (13b) Thus, the correlation for cofb, Eqs. 12a, 12b, and 12c, can be used
Bob Bo Bob pi − pb to develop an equation for calculating co.

pi dp pi − pb Ѩcofb Ѩz
Ѩp Ѩp
pb = cofb͑0.475 + 0.096z͒ , . . . . . . . . . . . . . . . . . . . . . . (16a)

͐͐cofi = − Bo dBo = − ln Bo = ln Boi − ln Bo . . . . . . . . . . . . . (13c) where cofb is calculated with Eq. 12a, z is calculated with Eq. 12b,
Boi Bo Boi p − pi and Ѩz/Ѩp is given by

p dp p − pi p

pi Ѩz −0.608 + 0.1822 ln pb
Ѩp p
The difference between Eqs. 13a and 13b can be used to determine = . . . . . . . . . . . . . . . . . . . . . . . . . . (16b)
an equation for cofi.

͑p − pb͒cofb͑p͒ − ͑pi − pb͒cofb͑pi͒ = ln Boi − ln Bo. . . . . . . . . (13d) Calculating the Coefficient of Isothermal Compressibility for

Thus, Saturated Oils. For consistency, the tangent compressibility for
pressure-transient analysis should be calculated from correlation
͑p − pb͒cofb͑p͒ − ͑pi − pb͒cofb͑pi͒ = ln Boi − ln Bo = cofi. . . . (13e) equations for Bo, Rso, and Bg by differentiating the equations for Bo
p − pi p − pi and Rso:

Finally, rearrange to obtain the equation for calculating cofi with co = − 1 ѨBo + 1 Bg ѨRso . . . . . . . . . . . . . . . . . . . . . . . . . (17)
the correlation equations for cofb. Bo Ѩp 1,000 Bo Ѩp

cofi = ͑p − pb͒cofb͑p͒ − ͑pi − pb͒cofb͑pi͒ . . . . . . . . . . . . . . . . . . . (14)
p − pi

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 the
bubblepoint, cofb, can be rewritten from Eq. 5 as

Bo = Bob exp͓−cofb͑p − pb͔͒. . . . . . . . . . . . . . . . . . . . . . . . . . . (15a)

The tangent, or instantaneous, compressibility, co, is defined as:

co = − 1 ѨBo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15b)
Bo Ѩp

February 2007 SPE Reservoir Evaluation & Engineering 45

Fig. 2—Coefficients of isothermal compressibility calculated Fig. 3—Coefficients of isothermal compressibility calculated
from Eqs. 12a, 12b, and 12c compare well with measured values. from Eq. 15c compare well with measured values.

Evaluation of the Proposed Correlations higher values appears to be caused by data from a few laboratory
reports that may have some internal error.
Fig. 2 compares the results of calculations of cofb using Eqs. 12a,
12b, and 12c with the data used in developing the correlation. Fig. 3 A comparison of the calculations and the data in terms of average
compares these data with calculations of co based on Eqs. 15c, 16a, relative error (ARE) and average absolute relative error (AARE)
12a, 12b, and 16b. The bulk of these calculations fit the data very appears in Table 3. Figs. 4 through 7 show that the calculations
closely. In both figures, the scatter of the measured results at hold up well across the ranges of the independent variables.

Comparison of Published Correlations for
Coefficients of Isothermal Compressibility

Table 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 and
Farshad 1998; Labedi 1990; Whitson and Brule 2000; Almehaideb
1997; Hanafy et al. 1997; Calhoun 1953; Vazquez and Beggs
1980; Kartoatmodjo and Schmidt 1994; Elsharkawy and Alikhan
1997; Ahmed 1989; Farshad et al. 1996) for oil compressibility.
Table 4 gives statistics for the independent variables in the data set
used for these comparisons. The coefficients of isothermal com-
pressibility calculated with data within 200 psi of bubblepoint
pressure were eliminated because the data had only four significant
figures, making the round-off errors near the bubblepoint exces-
sively large.

Fig. 4—ARE as a function of temperature. Each data point rep-
resents an average of approximately 240 calculations.

46 February 2007 SPE Reservoir Evaluation & Engineering

Fig. 5—AARE as a function of temperature. Each data point Fig. 6—ARE as a function of bubblepoint solution-gas/oil
represents an average of approximately 240 calculations. ratio. Each data point represents an average of approximately
240 calculations.
Many of the publications proposing the correlations listed in
Table 3 did not specify the applications for which they were de- Eq. 6, psi−1
veloped, so we compared them with both cofb and co and reported cf ‫ ס‬formation (pore-volume) compressibility, psi−1
the lowest values of ARE and AARE. co ‫ ס‬coefficient of isothermal compressibility,

Figs. 4 through 7 compare the results of this work with three measured with the slope of the tangent line at
other published correlations. The Vazquez and Beggs (1980) cor- the pressure of interest, psi−1
relation was selected for this comparison for its apparent popular- cofb ‫ ס‬coefficient of isothermal compressibility,
ity in the petroleum industry, and the other two (Al-Marhoun 2003; measured with the slope of the chord from
Dindoruk and Christman 2001) were chosen as recently published bubblepoint pressure to pressure of interest,
correlations that have low values of ARE and AARE. The results psi−1
of this work hold up well across the full range of each of the cofb(p) ‫ ס‬coefficient of isothermal compressibility
independent variables and generally have values of ARE and measured with the slope of the chord from
AARE that are closer to zero than the other three correlations. bubblepoint pressure to pressure p, psi−1
Sorting on stock-tank-oil gravity, °API, and reservoir pressure (not cofb(pi) ‫ ס‬coefficient of isothermal compressibility
shown here), yielded similar results. measured with the slope of the chord from
bubblepoint pressure to initial reservoir pressure
Conclusions pi, psi−1
cofi ‫ ס‬coefficient of isothermal compressibility
The three different applications for the values of the coefficient of measured with the slope of the chord from
isothermal compressibility require different calculations from ex- initial reservoir pressure pi to pressure of
perimental data, and the calculated values among the three are interest, psi−1
significantly different results. C0n,C1n,C2n ‫ ס‬coefficients for use in Eq. 12c for the nth
independent variable
We have presented a correlation for the fluid-property applica- k ‫ ס‬absolute permeability
tion that gives results approximately as accurate as the experimen- N ‫ ס‬original oil in place, STB
tal data. This correlation produces a weighted average of oil com- Np ‫ ס‬cumulative oil production, STB
pressibility between the bubblepoint pressure and the reservoir p ‫ ס‬pressure, psia
pressure of interest, cofb.
Fig. 7—AARE as a function of bubblepoint solution-gas/oil
We have presented an equation, based on the cofb correlation, that ratio. Each data point represents an average of approximately
can be used to obtain accurate estimates of cofi, the weighted average 240 calculations.
oil compressibility from initial reservoir pressure to some lower
reservoir pressure, for use in the oil material-balance application.

Finally, we have presented an equation that can be used to
obtain accurate estimates of the tangent compressibility co at a
particular reservoir pressure.

Nomenclature

AARE ‫ ס‬average absolute relative error, %
͚ͯ ͯ1

100 n
calc − meas
meas

ARE ‫ ס‬average relative error, %

100 n ͚1 calc − meas
meas

API ‫ ס‬stock-tank-oil gravity, oAPI

Bg ‫ ס‬gas formation volume factor, RB/Mscf
Bo ‫ ס‬oil formation volume factor, RB/STB
Bob ‫ ס‬oil formation volume factor at bubblepoint

pressure, RB/STB

Boi ‫ ס‬oil formation volume factor at initial reservoir
pressure, RB/STB

Bw ‫ ס‬water formation volume factor, RB/STB
ce ‫ ס‬effective fluid compressibility, defined in

February 2007 SPE Reservoir Evaluation & Engineering 47

pb ‫ ס‬bubblepoint pressure at reservoir temperature, psia Fetkovich, M.J., Reese, D.E., and Whitson, C.H. 1998. Application of a
pi ‫ ס‬initial reservoir pressure, psia General Material Balance for High-Pressure Gas Reservoirs. SPEJ 3
r ‫ ס‬radial distance (1): 3–13. SPE-22921-PA. DOI: 10.2118/22921-PA.
Rsb ‫ ס‬solution-gas/oil ratio at bubblepoint, scf/STB
Rso ‫ ס‬solution-gas/oil ratio, scf/STB Farshad, F., LeBlanc, J.L., Garber, J.D., and Osorio, J.G. 1996. Empirical
So ‫ ס‬oil saturation, fraction PVT Correlations for Colombian Crude Oils. Paper SPE 36105 pre-
Sw ‫ ס‬water (brine) saturation, fraction sented at the SPE Latin American and Caribbean Petroleum Engineer-
t ‫ ס‬time ing Conference, Port-of-Spain, Trinidad, 23–26 April. DOI: 10.2118/
TR ‫ ס‬reservoir temperature, °F 36105-MS.
V ‫ ס‬volume
We ‫ ס‬water (brine) encroached into reservoir from Hanafy, H.H., Macary, S.M., ElNady, Y.M., Bayomi, A.A., and El Ba-
tanony, M.H. 1997. Empirical PVT Correlations Applied to Egyptian
aquifer, res bbl Crude Oils Exemplify Significance of Using Regional Correlations.
Wp ‫ ס‬cumulative water (brine) produced, STB Paper SPE 37295 presented at the SPE International Symposium on
Oilfield Chemistry, Houston, 18–21 February. DOI: 10.2118/37295-
z ‫ ס‬sum of transforms defined by Eq. 12b MS.
zn ‫ ס‬transform for independent variable n defined by
Kartoatmodjo, T. and Schmidt, Z. 1994. Large data bank improves crude
Eq. 12c physical property correlations. Oil & Gas J. 92 (27): 51–55.
␥gSP ‫ ס‬separator-gas specific gravity
Labedi, R. 1990. Use of production data to estimate volume factor, density
␮ ‫ ס‬viscosity and compressibility of reservoir fluids. J. Pet. Sci. & Eng. 4 (4): 375–
␳o ‫ ס‬oil density, lbm/ft3 390. DOI: http://dx.doi.org/10.1016/0920-4105(90)90034-Z.
␾ ‫ ס‬porosity, fraction
Lee, W.J., Rollins, J.B., and Spivey, J.P. 2003. Pressure Transient Testing,
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Elsharkawy, A.M. and Alikhan, A.A. 1997. Correlations for predicting John P. Spivey has more than 20 years of experience in the
solution gas/oil ratio, oil formation volume factor, and undersaturated petroleum industry, with interests in pressure-transient analysis,
oil compressibility. J. Pet. Sci. & Eng. 17 (3–4): 291–302. DOI: http:// production-data analysis, reservoir engineering, continuing
dx.doi.org/10.1016/S0920-4105(96)00075-7. education, and software development. From 1984 to 1990, he

48 February 2007 SPE Reservoir Evaluation & Engineering

worked for SoftSearch Inc. (later Dwights EnergyData) devel- William D. McCain Jr. has been a visiting professor in the Dept.
oping petroleum economics and engineering software. In of Petroleum Engineering at Texas A&M U. since 1991. McCain
1990, Spivey joined S.A. Holditch & Assocs. (which was later started his engineering career with Esso (now Exxon) Research
purchased by Schlumberger); while there, he conducted res- Laboratories in 1956, where he assisted in research on surface
ervoir-simulation, gas-storage, and tight-gas-application stud- processing of petroleum fluids. He was Professor and Head of
ies and taught industry short courses in well testing and pro- the Petroleum Engineering Dept. at Mississippi State U. from
duction-data analysis. In 2004, he started his own reservoir- 1965 to 1976 and taught at Texas A&M U. from 1984 through
engineering consulting company, Phoenix Reservoir 1987. McCain was a consulting petroleum engineer with Caw-
Engineering, and software development company, Phoenix ley, Gillespie & Assocs. from 1987 until 1991. He was with the
Reservoir Software LLC. Since 1992, Spivey has served as a vis- petroleum engineering consulting firm S.A. Holditch & Assocs.
iting assistant professor or adjunct assistant professor at Texas from 1991 until 2000, retiring as Executive Vice President, Chief
A&M U., teaching undergraduate and graduate classes in Engineer, and member of the Board of Directors. McCain’s
gas-reservoir engineering and pressure-transient analysis. He engineering specialties within the consulting company were
holds a BS degree in physics from Abilene Christian U., an MS properties of petroleum fluids, surface processing of petro-
degree in physics from the U. of Washington, and a PhD de- leum, and reservoir engineering, especially for gas conden-
gree in petroleum engineering from Texas A&M U. Spivey is the sate and volatile oil fields. He has written two editions of the
editor of SPE Reprint Series Vol. 52, Gas Reservoir Engineering, widely used textbook, The Properties of Petroleum Fluids, holds
and Vol. 57, Pressure Transient Testing, and coauthor of the SPE three U.S. patents, and has more than 40 publications in the
textbook Pressure Transient Testing. He has published numerous petroleum engineering literature. Involved in short-course
papers and articles in industry journals and trade publications. teaching for more than 20 years, McCain has taught for sev-
Peter P. Valkó is a professor of petroleum engineering at Texas eral major oil companies, independents, professional societies,
A&M U. Previously, he taught at academic institutions in Austria and educational consulting companies throughout the world.
and Hungary and worked for the Hungarian Oil Co. Valkó He holds a BS degree from Mississippi State U. and MS and
holds BS and MS degrees from Veszprém U., Hungary, and a PhD degrees from the Georgia Inst. of Technology, all in chemi-
PhD degree from the Inst. of Catalysis, Novosibirsk, Russia. cal engineering.

February 2007 SPE Reservoir Evaluation & Engineering 49