Chapter 11 Oil and Gas Reserve Estimation Methods

505

Chapter I 1

OIL AND GAS RESERVE ESTIMATION METHODS F E R N A N D O SAMANIEGO V. and HEBER CINCO LEY

INTRODUCTION

Oil and gas reserve estimation is one of the keystones on which many of the important decisions of the oil and gas business rest. Proper planning and operational control is based on an estimate of the quantity of oil and/or gas available in a reservoir. It is usual in engineering applications that estimates of reserve values can have a wide range of precision from a best guess, on one hand, to near-certainty, on the other. Hydrocarbon reserve estimates fall into the latter category. Recently, important efforts toward the implementation of standards for the definition of reserves and estimation of oil and gas reserves have been made by the Society of Petroleum Engineers of AIME (1979, 1980, 1982a). These three papers discussed the basic concepts and methods for reserve evaluation. With regard to reserve information it is important to mention, first, that there are two types of reserves: (1) the volume of hydrocarbons in-place; and (2) the recoverable reserves. Reserve estimates classified in order of decreasing certainty are “proved”, “probable”, and “possible”. The “proved reserves” definition agreed upon by SPE/AAPG/API is: “Proved reserves of crude oil, natural gas, or natural gas liquids are the estimated quantities which geologic and engineering data demonstrate with reasonable certainty to be recoverable in the future from known reservoirs under existing economic conditions”. Four items usually comprise the information on reserves discussed in these reports: ( 1 ) proved reserves; (2) future rates from proved reserves; (3) future net revenue from proved reserves; and (4)the present value of such future revenue. In this chapter the term reserves will refer to proved reserves. Proved reserves are commonly divided into two subcategories: (1) Developed reserves, i.e., those that can be expected to be recovered through existing wells with proven equipment and operating methods. These reserves are, in essence, the primary developed and secondary developed reserves defined by Sheldon (1954). (2) Proved undeveloped reserves, which include those additional reserves to be recovered from (a) drilling new wells; (b) completion of existing wells in deeper or shallower reservoirs; and (c) implementation of an enhanced recovery project. These encompass the primary undeveloped and secondary undeveloped reserves of Sheldon (1954). Arps (1945, 1956) presented excellent discussions on the evaluation of reserves. Reserve estimates are needed more during the early stages of development, when

5 06

available reservoir information is scarce. Figure 11-1 presents a modified version of Arps’ (1956) fig. 1, in which three periods during the life of an imaginary oil propert y are shown. Time is shown on the horizontal axis, and the cumulative production and estimated ultimate recovery are plotted vertically. This figure shows no particular units and is not to scale. During the first period, before any wells are drilled, reserve estimates will be of a general nature, basically empirical, and are usually expressed in barrels per acre. This period is called the “barrels-per-acre” period, and estimates vary within the wide range AB (property non-producible) on the pessimistic side to CD on the optimistic side. The second period starts after one or more wells are drilled and assuming the field is productive; it is now possible to narrow the estimated recovery range within the much closer limits EF and G H . Types of data that could be collected during this period include geologic information, well logs, petrophysical data, PVT data, transient pressure analysis data, subsurface and isopach maps, and early reservoir simulation information. Interpretation of these data based on early pressure behavior may lead to the identification of the productive mechanism acting in the reservoir. Most reserve estimates during this period are of the volumetric type. The third period follows after sufficient actual performance data of reservoir behavior have become available with which to check previous volumetric estimates against other methods, such as material balance, decline curve, or reservoir simulation. Verification is of particular importance when dealing with

I

I I I I

‘I

True Ultimate Recovery Range of I Recovery Estimotes 1

I

I

I

I

I

I I I I

/

I

I A

/’

I

Abandonment at Economic L i m i t

I

B T i m e -

PERIOD

EVALUATION METHOD( S 1

I

I1

VOLUMETRIC, RESE RVOl R EMPIRICAL LIMIT TESTS AND EARLY MATERI A L BALANCE

I11

VOLU M E T R l C , D E C L I N E CURVES AND RESERVOIR SI M U LATlON

Fig. 1 1 - 1 . Schematic presentation of recovery estimates made during three periods in [lie life of a reservoir. J K = optimistic curve, LK = pessimistic curve. (Modified after Arps, 1956, fig. I , p. 183; courtesy o f SPE of A I M E . )

507 carbonate reservoirs because proper initial reservoir characterization is a difficult task, and volumetric estimates frequently fail to provide accurate reserve figures. With an increase in information, the lines HJK for optimistic and FLK for pessimistic estimates converge at point K, which corresponds to the true ultimate recovery or cumulative production at abandonment conditions. The dashed curve BK represents the cumulative recovery from the reservoir. The objective of this chapter is to present a summary of the methods currently used in the estimation of proved reserves, with specific application to carbonate reservoirs. Description is brief because of space limitation, but it is expected that readers would be able to enhance their knowledge on the subject by referring to the cited references. BASIC METHODS

The basic methods employed in reserve estimation calculations can be broadly classified into analytical and empirical categories. The analytical methods are: (1) volumetric; (2) evaluation of the performance history, which may include an analysis and extrapolation of established performance trends, such as rates, reservoir pressures, oil/water ratios, gadoil ratios, and gadliquid ratios; and (3) development of a mathematical model by means of material balance or computer simulation techniques. The empirical methods are based on the analogy of the particular reservoir to older, similar reservoirs for which, because of advanced depletion stage, the ultimate recovery can be determined with a high degree of accuracy. Reserve estimates based on an empirical or analogue approach are usually a qualitative generalization, because of the lack of accuracy, inasmuch as the characteristics of two oil reservoirs seldom are sufficiently alike for this method to provide reliable results. The Society of Petroleum Engineers of AIME (1979) recommended the use in reserve estimations of the particular methods and the number of methods that, in the professional judgment of the estimator, are most appropriate depending on (1) the geographic location, formation characteristics, and nature of the property or group of properties for which proved reserves are being estimated; and (2) the amount and quality of available data. As a general rule, it can be stated that reserves should be estimated with as many of the various methods at hand, as data, time, and available resources permit. Reserve estimations, as most of the reservoir engineering estimates, constitute a continuous problem. Improvement comes with production time as more data become available for refinement of these studies. Each year’s production - pressure performance might change visualization of the reservoir pore volume, possible cut-offs because of porosity or permeability, and the effect of the surrounding environment on the reservoir. Dickey (1980) stated that the engineer hesitates to make an estimate until he or she has what is considered a sufficient amount of data. The only time one knows with real assurance what are the recoverable reserves of a field is just after the last hydrocarbon unit volume has been produced (Arps, 1956).

508 It is early in the life of a reservoir that reserve estimates are most needed. This is particularly important in remote areas where there is no infrastructure nor pipelines. A problem dealt with in offshore exploration is that wildcat wells are drilled without any intention of completing them as producers. This may delay to some extent the decision to develop such fields, which is taken only after some idea of the field size and its productivity is gained by performing initial well tests on the wildcat well.

DATA REQUIRED

As in all reservoir engineering studies, the completeness and accuracy of the database is of prime importance in the estimation of proved reserves. It is well known that the answer to an engineering problem is no better than the accuracy of the data used. The type and the amount of the required data will vary in accordance with the methods employed in the reserve estimation study. The fact that the data available for estimating reserves are insufficiently complete or exact to render rigorous results is a source of particular concern to management. This justifies the previous comment that reserves should be calculated by all possible methods. When differences result, a cautious judgement must be made to arrive at the most probable value (van Wingen, 1972). Excellent papers regarding the interpretation, averaging, and collection of data have been presented by Havlena (1966, 1968). He stated that discretion and judgment have to be exercised with great care in working with the basic data, in order to arrive at the most probable and physically logical answer to a specific reservoir problem. Because of its idiosyncratic nature, it is impossible to describe specific rules which satisfy all conditions: each reservoir or even each well presents a unique problem to be considered separately during planning and interpretation stages using the surface and subsurface data. Havlena suggested that data-collecting programs should be formulated with the following chief objectives in mind: (1) To obtain an optimum coverage of data and testing among the reservoirs and wells at reasonable frequency, keeping the cost of obtaining such data in mind. (2) To achieve consistency in procedures, as much as practicable, so that data actually represent reservoir conditions and, thus, are directly comparable to each other. (3) To understand that quality is always better than quantity. (4) To consider at all times the purpose of collecting the data and their intended usage; unnecessary data collecting constitutes an economic waste. (5) To obtain a maximum usefulness from the available data with a minimum of assumptions. A list of the most commonly used data in the different reserve estimation methods is presented next. This is not intended to be exhaustive; instead, it is merely a good guide to start with in conducting a reserve study. (1) Variation of bottomhole pressure and rate versus time during a well test. It is a well-known practice to make a drawdown (and the subsequent buildup) test as long as wellsite conditions permit on every wildcat well.

509 (2) Core analysis data: routine data furnished by core analysis are porosity, permeability, and residual fluids contents, accompanied by observed lithology and texture. These data are the common denominator for reservoir description (Keelan, 1982), and comprise a portion of all information that can be obtained from cores. Specialized core data may include, among other important information, capillary pressure data and petrographic studies (e.g., thin-section analysis and scanning electron microscopy). It is common usage to perform the analysis of carbonate rocks (which are frequently heterogeneous, fractured, and/or vuggy) by means of fulldiameter cores of cylinder geometry up to 10 inches (25 cm) long and approximately 5 inches (13 cm) in diameter. (3) Pressure - volume - temperature (PVT)data: typical PVT information needed in reserve estimation studies includes the formation volume factors for the oil ( B J , gas (B& and water (B,), the solution gadoil ratio, R,, and the gas deviation factor z. As with all data, it is best to measure properties directly using a representative sample of the reservoir fluids. If this is not possible, then a number of correlations can be used, e.g., those presented by Standing (1952), Cronquist (1973), Earlougher (1977), Vazquez and Beggs (1980), G l a s ~(1980), and Lee (1982). (4) Production - pressure performance data: production (daily and cumulative) of oil, gas, and water versus time. In regard to wellbore pressure, it is necessary to collect both flowing and static pressures at fixed production times. (5) Well logging data: well logging information is of utmost importance in the determination of reserves in carbonate rocks. Typical data include porosity, water saturation, net thickness of the producing formation, and permeability.

DESCRIPTION OF METHODS OF ESTIMATION

Volumetric method The volumetric method involves the estimation of the total volume of oil or gas originally in-place, based on the knowledge of volume of the reservoir. This method requires the determination of representative values of porosity, 4, water saturation S,,, and net thickness, h , of the reservoir. The general expression for the estimation of hydrocarbons in-place, Vhc,is given by the following equation:

(1 1-1) where A is the reservoir area. Equation 1 1 - 1 can be applied to only fully developed reservoirs, assuming that extensive data are available for each well. Its use is justified for cases where rapid changes in the reservoir rock characteristics occur over short distances. The suggested method of integration is as follows:

1

h

( 1 ) Evaluate I , = 4 (1 - S,) d h for each well, using any appropriate graphical or numerical method.

510 (2) Evaluate V,,, = / . A f , d A by means of an iso-fl map and integration (traditionally with a planimeter). In carbonate rocks, application of a formula of the type of Eq. 11-1 is usually required to achieve an estimate of the volume of hydrocarbons in-place, because of the heterogeneous nature of these rocks. Carbonates typically present abrupt changes in lithology and petrophysical properties, and wide variations in porosity and permeability as a result of the random distribution and orientation of vugs and fractures. A synergistic approach (Harris, 1975; Halbouty, 1976; Stewart et al., 1981; Aminian et al., 1982; SOC.Petrol. Engrs., 1982b) is particularly useful in the study of carbonate reservoirs, where team effort can provide the best reservoir characterization possible. As an example of unique problems found in carbonate rocks one can mention the partial accumulation of hydrocarbons in low-porosity and low-permeability areas of the reservoir; it is uncertain whether or not these hydrocarbon fluids would contribute to production from the wells. Reservoir portions that do not contribute significantly to production should not be included in the reserve calculations. Essential to the evaluation of reserves by the volumetric method is the determination of the net hydrocarbon thickness, after the location of the gas - oil and/or oil-water contacts have been established. The basic methods currently in use to locate fluid contacts and net thickness are well logs (electrical, sonic, radioactivity, etc.); pressure gradient and RFT (Repeat Formation Test) measurements; core analysis; and geologic descriptions. The other two parameters that enter into Eq. 11-1 are porosity, 6, and water saturation, Swc.These three parameters, i.e., hydrocarbon thickness, porosity, and water saturation, are interrelated. In carbonate rocks the relationship depends on the degree of heterogeneity of the formation. It appears that capillary pressure is the best correlating parameter. Capillary pressure is most important because it influences the distribution of fluids in the reservoir and the mobility of the fluids. Mannon (1972) presented an excellent discussion on the influence of the last two parameters (distribution of fluids and their mobility) on reserve estimates. He concluded that meaningful interpretation of oil-in-place values, in order to arrive at an accurate reserve estimate, requires careful examination of the relative mobility of the fluids in a vertical section of the formation. Mannon (1972) discussed in great detail the paper by Aufricht and Koepf (1957), which dealt with carbonates of relatively uniform porosity, and concluded that for these cases the correlating cutoff parameter is the rock permeability. He also presented a summary of the resuits of Rockwood et al. (1957) for determination of tank oil-in-place in carbonate reservoirs using porosity and rock type as parameters for correlation. This technique does not make use of relative permeabilit y data, and, thus, is somewhat more qualitative. Capillary pressure curves are used in a field example presented by Rockwood et al. (1957) to point out once again that, for reserve estimation purposes the proportion of large and small pore spaces in an oil-producing interval is of utmost importance. As previously stated, it is essential when dealing with carbonate reservoirs to identify sections of low permeability and porosity that are questionable, and to decide whether or not these zones should be included in the calculation of reserves.

511 The basic methods available for estimation of porosity are electrical and radioactivity logs, petrophysical data, and well test analysis. It is strongly advisable to combine all possible methods because the first two would mainly reflect the intergranular (and/or intercrystalline) porosity. In carbonate formations that contain vugs, crevices, and fractures, the error involved may be large. Thin-section examination is also recommended as a means to handle this problem (van Wingen, 1972). is the sum of the In general, for fractured carbonate formations total porosity and the secondary (vugs, crevices, and fractures) primary (interparticle) porosity porosity C$2. Detailed discussions on the evaluation of porosity in carbonate rocks have been presented by van Golf-Racht (1982) and Lucia (1983). Water saturation is normally estimated by the same methods as porosity, i.e., logs, direct petrophysical measurements, and capillary pressure tests. It must be kept in mind that in non-fractured carbonate rock systems the variation of water saturation with depth should be taken into consideration when making reserve estimates, because application of the irreducible water saturation concept would result in erroneously high reserve values. In addition, the fact that many carbonate reservoirs are oil wet should be kept in mind. Lindseth (198 1) discussed the use of seismic inversion to obtain synthetic sonic logs from seismic reflection data recorded at the ground surface. Previously, Ausburn et al. (1977) elaborated on this important technique. These synthetic logs are similar to, and have most of the properties of, long-source receiver borehole sonic logs. The synthetic logs have been found particularly useful in the determination of three of the basic parameters needed in the evaluation of Eq. 11-1 : porosity, thickness, and areal extent of the reservoir. Best results are obtained when sonic logs are available for calibration of the synthetic logs, at least for the discovery well. Sonic logs are also used for an estimation of the remaining fourth parameter of Eq. 1 1 - 1 , i.e., water saturation. An example was shown on the application of the method t o carbonate formations by Ausburn et al. (1977; fig. 13). The latest efforts toward the volumetric evaluation of hydrocarbon reserves have been in the direction of an automatic determination by means of digital computers. Sol6rzano et al. (1982) described a computer code for this purpose where data are handled on a well-by-well basis. Typical output of the program includes porosity and water saturation isopach maps. Basic to this model, and in any volumetric reserve estimation, is the information on reservoir continuity which may be acquired through pressure transient analysis of preliminary well test data or reservoir continuity analysis (Delaney and Tsang, 1982). Another aspect of volumetric reserve estimation that deserves attention is the uncertainty of the parameters in Eq. 1 1 - 1 (Walstrom et al., 1967; Ford, 1968; Smith, 1968; Pritchard, 1970; Stout, 1972; Aguilera, 1978). The most widely used approach is the Monte Carlo method, which consists of placing a range and type of distribution on the parameters that enter into Eq. 11-1. For reservoir engineering purposes, the uniform and triangular distributions are typical. The uniform distribution confines the variable between an upper and lower limit. The triangular distribution (Fig. I 1-2) is used for a variable when more data are available to indicate a central tendency of distribution. This allows for postuIation of a “most likely” value for the distribution and lower and upper limits. Normally, an integrated form of Eq. 1 1 - 1 ,

512

Most likely value

Variable

value

Fig. 11-2. Triangular probability distribution. (After Walstrom et al., 1%7, fig. Ib, p. 1596; courtesy of SPE of AIME.)

which considers constant values of porosity and saturation, is used in the calculations:

where V, is the reservoir rock volume (Vr = A h ) . In the Monte Carlo simulation technique, Eq. 11-2 is solved many times (usually between 500 and 5000) by a digital computer on sample parameters that have been generated by a Monte Carlo random technique. At the end of these calculations, the reserve estimate is available in a statistical probability format and its mean value and variance can be computed. Pritchard (1970) presented an example of the application of this method to estimate the rPserves of the Devonian Zama Keg River CC Pool, which is a pinnacle reef overlain by the Zama Dolomite. Figure 11-3 shows the results of these calculations. Figure 11-3a is the histogram of oil-in-place values obtained from the simulation and Fig. 11-3b is the cumulative probability curve of the same data. As shown, the oil-in-place ranges from about 2.5 x lo6 to 7.0 x lo6 STB, with a modal value of 3.2 x lo6 STB. Hernandez and Berlanga (1983) have introduced a promising method for the evaluation of hydrocarbon reserves based on geostatistical kriging theory (Haas and Jousselin, 1975). The first step of this method involves structural analysis (spatial variation) of the variables that enter into Eq. 11-2, namely, thickness, porosity, and hydrocarbon saturation. Basic to the structural analysis is the evaluation of the experimental semivariograms for these variables. In a geostatistical sense, the variogram is the variance of two regionalized random variables separated by a vec-

513

U

Millions of stock tank barrels

( a )

5 M i l l i o n s of stock tonk barrels

( b )

Fig. 11-3. Histogram and cumulative probability curve of volumetric oil-in-place for the Devonian Zama Keg River cc Pool. (After Pritchard, 1970, figs. 8 and 9, p. 1362; courtesy of SPE of AIME.)

tor distance h. A semivariogram y * (h) can be calculated using the experimental information and the following expression: (1 1-3) where N' is the number of data pairs separated by a vector distance h. The experimental semivariograms for three variables of interest, i.e., thickness, porosity, and hydrocarbon saturation, are usually calculated in several directions through the reservoir, with the purpose of evaluating the anisotropic behavior of the three variables. Figure 11-4 shows the experimental semivariogram for thickness presented by Hernandez and Berlanga (1983) for the Miguel Aleman Field, for four different directions and a spherical model adjusted to the weighted averaged experimental semivariograms. In the application of this method, the projected area of the reservoir is divided into cells and Eq. 11-2 is applied to each one. The original oil-in-place is calculated adding the Ni values for the m cells, which make up the reservoir area. Usual square cell size ranges from 0.25 to 1 km per side. For each cell, values of the variables of interest are estimated through normal kriging by means of the following expressions:

(11-4)

514 where the number of terms ( k , I, or rn) in these expressions depends on the range of the corresponding experimental semivariogram. For instance, in the case of Fig. 11-4 the range is 2.4 km a n d , consequently, in the estimation of hJi the index k is equal to the number of neighbor thickness data, location of which V , falls inside a circle of radius equal t o the range. The weights A . in Eq. 11-4 are calculated through a solution of the linear system J of N + 1 equations, called the krigeage system, resulting from the krigeage application (JourneI and Huijbregts, 1978): N

-

C Aj y ( v n , v,)

j=I

+

p =

7 ( vn, A;), n

=

1, 2, . . ., N

N

c

x j = l

(11-5)

j=1

where p is the Lagrange parameter. The index N could be k, I or rn depending o n which reservoir parameter is being estimated. The evaluation of Eq. 11-5 is multiple in the sense that it has to be computed for each estimation of the three parameters. This is due t o the fact that, according to the method, the estimation depends not only o n the number of data points but also on their configuration in relation t o the main features of the regionalization as characterized by the structural function y * ( h ) in the various terms (vn, 13). The term (vn, v,) represents the value

7

0 0240

00200

I

DIRECTION

----

- 15'

-_-.

30' 750 120° SPHERICAL THEORETICAL MODEL

*-+-t

0 0160

N

E

0.0120

1

. .

c

1

--

x

z'

\

*\ ?a

4

I

I

t I t I

:', { i

I

/x

i

-f

i

0.0080

x

\ \

0 0000 00

10

20

30

40

5.0

60

70

80

\

90

10'

I h l , km

Fig. 11-4. Experimental semivariograrn 0 1 thicknes3 calculared in four directions and the spherical weighted average model for the Miguel Alernan Field. (After Hernandez and Berlanga, 1983, fig. 5 ; courtesy of Reidel Publ. Co.)

515

estimated from the theoretical semivariogram model or read from the experimental semivariograms, such as that shown in Fig. 11-4, for e ( = lhl) equal to the distance between the points v, and the point at which the parameter is being estimated, v,. The term (vn, A,) represents the mean value of the experimental semivariogram y * ( h ) when one extremity of the vector h is fixed at the data point vn and the other extremity independently describes the domain A i (“volume” of the ith cell), where the reservoir parameter is being estimated. This, expressed in an equation form, may be written as follows:

7

(1 1-6)

In this expression, a numerical approximation has been used for the integral over the domain A . This means that the domain A has been represented by a discrete number of x, points, specifically 16 (J.M. Berlanga, personal communication, 1983). The main advantage of the estimation method based on kriging over the Monte Carlo procedure is that the former is a local estimation technique which provides the best linear unbiased estimator of the unknown reservoir parameter studied, whereas the latter does not. On the other hand, programming the kriging method may require considerably more effort, especially in the three-dimensional case, as compared with the Monte Carlo method.

Material balance method From the previous discussion on the volumetric method applied to carbonate reservoirs, it is concluded that in many cases it is extremely difficult to obtain accurate reserve estimates. Van Wingen (1972) and Mannon (1972) clearly stated that the complexity of the problem is due to the high heterogeneity of these formations. Wide variations in values of rock properties, such as porosity, permeability, and pore-size distribution, occur over short distances. An alternative method for the estimation of in-place volume of hydrocarbons is the material balance equation. The method is based on the premise that the reservoir pore volume remains constant or varies in a predictable manner with the reservoir pressure when oil, gas, and/or water are produced. This allows one to equate the expansion of the reservoir fluids, caused by the pressure drop, to the reservoir voidage caused by the withdrawal of oil, gas, and water minus the water influx. Optimum application of this technique requires an accurate record of the following data: ( 1 ) average reservoir pressure; (2) oil, gas and water production; and (3) PVT data of the reservoir fluids. Usually at least 5 - 10% of the oil or gas originally inplace has to be produced before reliable results can be obtained from this method. The material balance equation can be derived by simply applying the law of conservation of mass, stated as follows: expansion of the system (rock plus fluids) is equal t o the fluid production (oil, gas, and water) minus fluid influx (water from an associated aquifer and water or/and gas injected from the surface) into the

5 16 original hydrocarbon zone. The resulting expression is given by Eq. 11-7 (Craft and Hawkins, 1959; Dake, 1978):

(1 1-7) where:

N

=

Np

=

B, B,

= =

Ap pi p R, R, G

=

= =

= = =

de

=

W, m

=

S,, cf

= = =

original oil-in-place, stock tank barrels (STB); cumulative oil production, STB; oil formation volume factor, reservoir bbl/STB (RB/STB); gas formation volume factor, reservoir bbl of gas per standard cubic feet of gas (RBISCF); pressure drop, (Ap = p i - p ) , psi; initial reservoir pressure, psi; reservoir pressure at time t, psi; solution gadoil ratio, SCF/STB; net cumulative gas/oil ratio (RP = Gp/Np), SCF/STB; cumulative gas produced, SCF; cumulative water influx from the associated aquifer into the reservoir, STB; cumulative amount of aquifer water produced, STB; ratio of the initial hydrocarbon volume of the gas cap to the initial hydrocarbon volume of oil; connate or irreducible water saturation; and formation compressibility, psi- ’.

One of the best approaches to the application of Eq. 11-7 is that presented by Havlena and Odeh (1963) and Havlena (1964). The technique is based on the interpretation of the material balance as the equation of a straight line. This algebraic arrangement attaches a dynamic meaning to the static characteristic of the material balance equation. This method was presented in the first paper, and the second paper illustrated its application to field cases. The above equation requires the definition of the following terms: Fluid production F: F

=

I

N, B, + ( R ,

-

R,)Bg]

+

WpBw

(1 1-8)

Expansion of the oil and its originally dissolved gas, E,: (11-9)

517 Expansion of the gas cap, Eg: (11-10) Expansion of the connate water and reduction in pore volume, Ef,,: (11-11) Substituting these terms into Eq. 11-7,

F

=

N (Eo + mEg

+ E f , w) +

WeBw

(1 1-12)

Experience has shown that Eq. 11-8 is a linear function for the most commonly found field situations. For undersaturated reservoirs with no water drive and for which the connate water and formation compressibility term may be neglected, Eq. 11-12 can be written as follows:

F

=

(1 1-13)

NE,

A graph of F versus Eo should result in a straight line that passes through the origin and has slope N (Fig. 11-5). As pointed out by Havlena and Odeh (1963) and Havlena (1964), the origin is an essential point that provides a guide to the straight-

t

/

LL

/’ ,/A’

Eo

-

Fig. 11-5. Material balance straight line for an undersaturated oil reservoir. (After Havlena and Odeh, 1963, fig. 1, p. 897; courtesy of SPE of AIME.)

518 line graph. If the graph is non-linear, then this fact in itself can be diagnostic in determining the drive mechanism used in the reservoir (Dake, 1978). The application of the straight-line technique to carbonate reservoirs has been successfully demonstrated in a number of papers (Platt and Lewis, 1969; DesBrisbay and Daniel, 1972; Teran et al., 1974). The use of this technique for carbonates may be especially valuable, because the accurate determination of the individual parameters in the integrated form of Eq. 11-1 may be particularly difficult. For gas reservoirs that can be assumed to produce under volumetric conditions, Eq. 11-2 can be modified to estimate the recoverable reserves measured at standard conditions of pressure, psc, and temperature, Tsc, for a fixed abandonment pressure pa as follows: (Gas reserves),,

= q5

(1

-

S,,)Vr

s‘c psc

(:

pab)

(1 1-14)

zab

where z is the gas deviation factor and the subscripts i and a b represent initial and abandonment conditions, respectively. The material balance equation for a gas reservoir may be derived based on the same principles as used in the derivation of Eq. 11-3 for oil reservoirs. For a volumetric reservoir, the material balance equation is: (1 1-15)

In the majority of the cases, Eq. 11-15 describes adequately the behavior of a volumetric (depletion) type gas reservoir. This equation indicates that there is a linear relationship between p / z and G, or the fractional recovery Gp/G, as shown in Fig. 11-6a and Fig. 11-6b, respectively.

GP

-

I

RF

( 0 )

Fig. 11-6. Graphical representation of the material balance for a volumetric gas reservoir.

519 For gas-condensate reservoirs, the formation of a liquid phase in the formation as the reservoir pressure decreases below the dew point should be considered. Usually, even in reservoirs with high liquid contents, the maximum liquid saturation will not exceed the critical liquid saturation. For such reservoirs, Eq. 11-15 can still be used provided the single-phase gas deviation factor z is replaced by the two-phase gas deviation factor z’ , determined from laboratory measurements conducted at reservoir conditions and Eq. 11-16: (1 1-16)

V G‘

GA

= = =

cell volume used in the reservoir fluid study; initial gas volume in the cell; and cumulative gas volume removed to pressure p.

Excellent discussions on the estimation of gas reserves were presented by Agarual et al. (1965), Root et al. (1965), Dranchuk (1967), Ramey (1971), Miranda and Raghavan (1975), Ford (1978), and Shehabi (1979).

Example I . Calculation of initial oil-in-place by the straight-line material balance method The example presented by Mannon (1972) is solved here using the Havlena - Odeh (1963) straight-line method. The reservoir was initially saturated with gas and had no free gas cap (m = 0). Water influx We into the reservoir is neglected based on the non-significant water production. Table 11-1 shows the pressure - production history, the flash liberation data, and reservoir data. (Also see Chapter 1.) TABLE 11-1 Data for calculation of initial oil-in-place: flash liberation data (pertaining to production through one separator at 75°F and 100 psig) .~ ~ _ _ __ ~ ~ ~

~

~

Pressure, p (psis)

(bbl/STB)

2827 2800 2600 2400 2200 2000

1.309 1.305 1.295 1.282 1.269 1.256

Bo

3, (bbl/STB)

4

~

(SCF/STB)

3bVSCF)

1.005

587 585

I .004 1.004 1.004 1.003

523 485 445

0.000996 0.000997 0.00103 0.00109 0.001 17 0.00130

1.005

Average porosity, qba 23 To Average permeability, k,, mD 200 Average interstitial water saturation, Sam, 35.5% Productive area, A , acres 269 Average net pay thickness, h , ft 92 Saturation pressure, p , psia 2827

555

~

vl

t4 0

TABLE 11-11: Pressure ~ (1)

Date

~

production history and material balance calculations ~ ~ _ _ ~~~

(2) Static resercoir pressure, p (psis)

~

~~

Initial conditions 2/14/65 4/19/66 3/31/67

~

2827 2535 2301 2207

-

(3) Cumulative production N,,(bbl) _

(4) Cumulative gas production G, (MCF) _

(5) Cumulative uater production Wp (STB)

0 462,966 732,686 933,796

0 363,717 688,288 944,585

0 2017 247 1 3150

oil

_

~

(6) Oil formation volume factor, B, (bbl/STB) _ ~

1309 1291 1276 1269

~

-

(7) Solution gas/oil ratio, R , (SCF/ STB) ~

(8) Net cumulative gas/oil ratio, R ,

587 544 6 504 2 486 3

0 785 939 4 1012

~~~~~

~ ~

(9) Gas formation colume factor, B (bbl/SCF) ~ ~

0 000996 0 00 105 0 00113 000116

~ (10) F (fluid production) (bbl)

-

~ (11) E, (eupancion of oil and dissolbed gas) (bbl/STB) ~ ~

0 0 7.166 x 16 2.652 x l o - ' 1.297 x lo6 6.053 x lo-' 1.761 x 10' 7.796 x 1W2

~

521 Pressure -production history and material balance calculations are presented in Table 11-11. For the conditions of this reservoir and neglecting the expansion of the connate water and reduction in pore volume, Eq. 11-12 is reduced to Eq. 11-13. Figure 11-7 presents a Cartesian graph of F versus Eo in accordance with this equation: the straight-line relationship is reasonable. The slope of the straight line gives a value of 22.59 x lo6 STB for the original oil-in-place, N. This figure compares favorably with the average of 22.1 x lo6 STB estimated by Mannon (1972) on evaluating N at two different pressures using the volumetric method.

Numerical simulation Numerical models that solve the fundamental equations which describe the flow of fluids through porous media are in general use now. It is believed that, under suitable conditions, they provide the best means to estimate reservoir performance. Examples of the applications of this method to a carbonate reservoir can be found in the papers by McCulloch et al. (1969), Beveridge et al. (1974), Lee et al. (1974), and Harpole and Hearn (1982). These authors used a wide range of multi-phase models in their studies of recovery efficiency, including vertical one-dimensional, two-dimensional (radial and cross-sectional), and three-dimensional models. They clearly showed, based upon comparison of simulated results with field data, that conventional simulators can be used under suitable conditions to forecast the behavior of carbonate reservoirs and, in particular, to estimate reserves. For a broad

E,

,bbl/STB

x lo2

Fig. 11-7. Cartesian graph of F versus E, for Example I

522 discussion on the mathematical models that describe the flow of fluids through carbonates, Chapters 10 and 12 should be consulted. Regarding the field performance match in a reservoir simulation study, an adjustment variable is formation porosity and, consequently, through its inherent dependence, original hydrocarbons in-place. Another important variable in the match is the formation permeability, because low-permeability reservoir portions may not contribute to the well’s total production and should not be included in the reserve calculations. This procedure, under suitable conditions of reservoir characterization, provides an excellent way to check original reserves estimated by volumetric and material balance methods. The work involved in the adjustment is a direct function of the characterization of a particular reservoir. Reservoir simulation models are, in many cases, useful for complete understanding of the producing characteristics of carbonate reservoirs. As previously mentioned, it is of utmost importance to solve the problem in complex carbonate reservoirs through a synergistic approach. As an additional example of the work necessary for the description of carbonate reservoirs, the study of Jardine and Wishart (1982) may be consulted. The foregoing comments of this section assumed that the carbonate reservoir does not exhibit secondary porosity and permeability, which are common in brittle rocks such as carbonates. In these rocks, fractures contribute to the secondary porosity, resulting in a complex pore structure. The presence of the fractures often changes the formation permeability from millidarcies to darcies. Examination of fractured reservoir case histories shows that, most probably, fractured reservoirs are expected to occur in carbonate rocks of low porosity (van Golf-Racht, 1982). I t has to be kept in mind that attempts to match field behavior of these reservoirs by means of a conventional reservoir simulator without properly taking the natural fractures into consideration will lead to erroneous conclusions regarding the production characteristics of the reservoir. Several papers have appeared that deal with the simulation of naturally-fractured reservoirs (Yamamoto et al., 1971; Saidi, 1975; Kazemi et al., 1976; Kazemi and Merrill, 1979; Thomas et al., 1980; Bossie-Codreanu et al., 1982; Evans, 1982; Hill and Thomas, 1985; Leung, 1985; Waldren and Corrigan, 1985; Peng et al., 1987; Gilman and Kazemi, 1988; Bech et al., 1989; Coats, 1989; Firoozabadi and Thomas, 1989). These simulators range in complexity from single block - one phase to multiple block - compositional simulators. Reported field applications of these simulators are scarce (Saidi, 1975; Bosie-Codreanu et al., 1982). Saidi (1975) reported on the application of a compositional reservoir simulator to the naturally fractured Haft Gel Iranian oil field. He used the “sector of a reservoir” concept to obtain a manageable simulation problem. Briefly, a sector is made of a number of characteristic horizontal blocks stacked vertically. Each block is divided into several vertical and horizontal grids. The pressure and level change in the fracture system is detected by the nodes located at the top and bottom of each grid. In matching the past history of a reservoir, the fracture pressures and water-oil and gas -oil levels are fed into the model. The simulator computes the oil and gas production, which is compared with the actual production. To arrive at a good match, the most accurate and reliable information, such as pressure, fluid levels, capillary pressures, and relative permeabilities, is fixed. Oil-in-place and its distribution,

523 block size, fracture volume with depth, and permeability of the blocks are changed within acceptable limits.

Reservoir limit tests A reservoir limit test is a long-time drawdown test used to estimate the volume being drained by a well. All reservoir limit testing methods are based on the fact that pressure behavior eventually reaches pseudo-steady state for constant rate in a closed drainage reservoir (Jones and McGhee, 1956; Matthews and Russell, 1967; Earlougher, 1972, 1977). These tests find their most important use in the exploratory well that discovers the field, when it has to be decided whether the reservoir is large enough to warrant the drilling of a second well. Frequently, the test is made to determine whether or not casing should be set in the discovery well itself. At pseudo-steady state for constant-rate production, bottomhole flowing pressure p w f follows a linear variation with time given by the following expression: (11-17) where: (1 1-18) and

pint = pi -

-

~~

[In

2 kh

(:>

+

> In,-(2.2458

+

2s]

(11-19)

where a. and E are unit conversion constants (see Table 11-111). Equation 11-17 indicates that a Cartesian graph of p W fversus time should be a straight line during pseudo-steady-state flow, with slope given by Eq. 11-18 and intercept pintgiven by Eq. 11-19. From the value of the slope, the contributory reservoir drainage volume may be estimated:

6 h A = .._._~ EqB Ct mo

(11-20)

If an average value for 4h is available, an approximation for the drainage area can be obtained. Compared with the other available methods, this technique to analyze pseudo-steady-state data appears to be the simplest and least prone to error. For gas reservoirs, an expression similar to Eq. 11-17 has been presented in terms of the real gas potential m(p) (Al-Hussainy et al., 1966; Energy Resources Conservation Board, 1975):

m

(PWF)

=

mg*t + m (Pi,)

(1 1-21)

524 where: ( 1 1-22)

and

m @ill,)

=

m

@j)

-

~

2

45cT [In kh

) + 0.86859 s, ] (2.2458 6 A

( 1 1-23)

CA

The technique of analysis is similar t o that described for Eq. 11-17. A Cartesian graph of the flowing real gas potential m(pWf)versus time should be a straight line during pseudo-steady-state flow, with slope given by Eq. 11-22 and intercept by Eq. 11-23. From the value of the slope, the contributory reservoir drainage volume can be estimated. Another promising test that has found application is a n alternative to the reservoir limit test. If the intended limit test is continued a n d n o deviation from the infinite-acting reservoir behavior (semilog straight line) is observed, then the reservoir limit cannot be defined. It is possible, however, to define a minimum in-place hydrocarbon volume Vpm (Energy Resources Conservation Board, 1975). Reservoir limit tests can cause problems regarding the handling of the fluids produced, because these tests are conducted at the discovery well, for which there are n o surface facilities. In such a situation, it is possible to design a limit test t o confirm the presence of a minimum in-place hydrocarbon volume that would be necessary for economic exploitation of a reservoir. The theory involved in the derivation of the equation needed to estimate the duration of the flow period required t o perform a n economic limit test, f e l t , is straightforward. It is based on the finding that pseudo-steady-state begins at a dimensionless time based o n the drainage radius or radius of investigation of 0.25. Thus, the resulting equation would be: ( 1 1-24)

where the basic unit of Vpm is the same as the reservoir thickness h (Table 11-111). Figure 11-8 shows a general semilog graph of the flowing bottomhole pressure data recorded during the limit drawdown test. T w o values of felt are shown for explanation purposes: corresponds t o the case where for the value of telr estimated from Eq. (1) ( 1 1-24, transient flow conditions still prevail. This indicates that the reservoir limit has not yet been detected a n d , consequently, the contributory reservoir volume being drained by this first well is large enough t o warrant the drilling of a second weI1. corresponds t o the case where transient flow conditions have ended at (2) (

525 an earlier time, indicating that the reservoir dimension is not sufficient to warrant the drilling of a second well.

Example 2. Reservoir limit test in an oil reservoir (Matthews and Russell, 1967) The discovery well of this example was completed as a Muddy Sandstone oil well in the Denver Basin, Colorado. Before drilling additional wells, the engineers decided to perform a reservoir limit test. After being hydraulically fractured upon completion, the well was flowed to clean up and recover load oil, and was then shut in until stabilization of pressure. The test was started by measuring the bottomhole pressure continuously at a production rate of 800 STB/day for a period of 50 hours. Figure 11-9 is a semilogarithmic graph of bottomhole pressure versus time, and Fig. 11-10 is a linear coordinate graph of pressure versus flow time. The transient flow conditions lasted about 2 hours. From the slope of the pseudo-steady-state straight line of Fig. 11-9 and Eq. 11-20: 4hA

=

-

~p

0 23395 (800 STB/day) (1.25) 1 . ~~

=

-

~~

(15.8 psi/hour) (17.7 x

psi

0.836 x l o p 6 ft3

-I)

0.149 x lo6 bbl

This reservoir volume amounts to an equivalent drainage radius of 488 ft (17 acres). On the basis of the foregoing results, further drilling was not attempted. As time went on, the production performance of the well confirmed the earlier results of the reservoir limit test.

a-b b-c

FRONT END EFECTS SEMILOG STRAIGHT LINE

I

I

I I I

I I I

1

( t e l t )d

L o g t

Fig. 11-8. Reservoir limit test in a discovery well.

I I I

I ( t e l t )dd

Pressure decline methods The analysis of past trends in production performance, with the purpose of estimating oil and gas reserves and forecasting of production, is a frequently used estimation method. This analysis can be performed for reservoirs with reliable production trends. I t consists merely of an empirical fit of two production-related variables, for example, production rate and (1) time or (2) cumulative production. An empirical equation is a fit of existing data without formally including all the factors that affect past, present, or future performance. The basic assumption in this procedure is that the factors which controlled the trend of the curve in the past (except those shown on the graph) will continue to govern its trend in the future in a uniform manner. This empirical approach has been justified mathematically for only a few simple reservoir conditions (Frick, 1962; Campbell, 1973). Among the many dependent variables that can be used in reserve estimation based on performance trends, production rate is the most commonly used, provided production is at capacity over a sufficient period of time to adequately define the performance trend. Even in oil reservoirs where production is not controlled, capacity production has seldom been attained until the field reaches a relatively advanced depletion stage. These curves are commonly referred to as production decline

T i m e , m i n u t e s Fig. 11-9. Flowing pressure versus logarithm of flowing time for a reservoir limit test, Denver Basin Muddy Sandstone well. (After Matthews and Russell, 1967, fig. 5.4, p . 52: courtesy of SPE of AIME.)

527 curves. Production rate used as a dependent variable has the advantage of being readily available and accurately recorded. Figure 11-11 is a modification of fig. 9.1 of Campbell (1973), where eight of the most widely used curves to characterize oil production are presented. A brief comment for each part of the figure follows: Fig. 11-1l a - Production rate versus time, where production rate declines. Fig. 11-llb - Production rate versus cumulative production. This is an alternative graph to Fig. 11-lla. Fig. 11-1Ic - Logarithm of production rate versus time; frequently it is used as one of the first decline curve analysis graphs. Fig. 11-1Id - Logarithm of per cent oil in produced fluids versus cumulative oil production; used where ultimate production is limited by the percentage of water produced with the oil rather than oil production decline itself. The economic limit is fixed by the water handling cost. Fig. 11-1le - Logarithm of production rate versus cumulative production; it is used as an alternative to Fig. 11-llb, when producing conditions are suitable for such a linear relationship to exist. Fig. 11-1If - Subsea elevation versus cumulative oil production; it is used where bottomwater drive is the main producing mechanism. The broken line indicates the maximum water level at which economic production ceases.

Ti m e , minutes Fig. 11-10. Flowing pressure versus time for a reservoir limit test, Denver Basin Muddy Sandstone well. (After Matthews and Russell, 1967, fig. 5.5, p. 53; courtesy of SPE of AIME.)

528 Fig. 1 1 - 1 Ig - Logarithm of cumulative gas produced versus logarithm of cumulative oil production; it is frequently used to estimate the total primary oil produced in gas-drive reservoirs based on an estimate of the total gas released down to an abandonment pressure. Fig. 11-1 lh - A double logarithmic graph of production rate versus time; i t is used for type curve matching purposes against Fetkovich's composite type curves shown in his fig. 4 (Fig. 11-12) or the modification suggested by Ehlig-Economides

Time

(a

Cumulative Production

( b )

1

Time

(C)

Cumulative Production

(el

Logarithm of Cumulative Oil Production

( g ) Fig. 1 1 - 1 I . Basic decline ctirvc graphs (see t e \ t )

Cumulative Production

(d

1

Cumulative Production

(f)

Logarithm of Time

( h )

529 and Ramey (1981) in their fig. 6 (Fig.ll-l3), further generalized later by Harrison (1982). Because the obvious way to plot production rate is against time, this was the first method used (Nind, 1964). Field data showed that after a period during which production rate was approximately constant (at or near the well’s allowable rate, or the market demand), the well could no longer produce its allowable rate and production

lo-’

10-2

10

1

10

10

t Dd Fig. 11-12. Type curves for Arps empirical rate-time equations, unit solutions (D, Fetkovich, 1980, fig. 1, p. 1066; courtesy of S P E of AIME.)

10

-

i

I

=

1). (After

1

1

4* I

0 0) I

c

10’

Y

0 U

lo-: 10-4

10-

10-2

10-

1

~ D A

( I n ‘ e D - q3)

Fig. 11-13. Dimensionless flow rate functions for a well produced at a constant pressure from the center of a closed bounded circular reservoir. (After Ehlig-Economides and Ramey, 1981, fig. 6, p. 101; courtesy of SPE of AIME.)

530 rate decreased steadily with time (Fig. 11-1la). The extrapolation of the well’s performance into the future can be made by means of a fit (empirical or mathematical) of the trend. Extensive discussions of the theory involved in decline curve analysis have been published by Nind (1964), Mannon (1972), van Wingen (1972), Campbell (1973), Fetkovich (1980), and Harrison (1982). Production rate - time curves are generally classified into three types: exponential, hyperbolic, and harmonic. In the exponential decline type curve the change in production per unit time is a constant percentage of the production rate (Fig. 11-1lc). Other cases are better represented by the so-called hyperbolic decline, where decline rate is proportional to production rate raised to a power between zero and one. There is no single way to plot the rate data to yield a straight line for hyperbolic decline where decline rate is unknown. The use of least-squares or regression analysis techniques has been suggested (Seba, 1972). Harmonic decline is a particular case of hyperbolic decline in which the fractional power equals one (Fig. 11-1le). A suggested decline curve analysis method would be similar to the type curve matching procedure in current use in well test analysis. A log-log field data curve is drawn on tracing paper using the scale of the Arps’ type curve of Fig. 11-12. This field graph is overlaid on the Arps’ type curve until the best match is attained, which provides information on the type of decline shown by the well and a prediction of the future well performance. Once the match is completed, a convenient “match point” on the field graph is picked, such as an intersection of major grid lines. Values at that point on the field graph [(q)M and (t)M] and the corresponding values lying beneath that point on the type-curve grid [(qDd)M and (fDd)M] are recorded. The match-point data are used to estimate well production rate decline characteristics. The definitions of the decline curve dimensionless parameters are as follows: Decline curve dimensionless time tDd: tDd

Dit

=

-

I

2

(reD2 -

td

(In

‘eD

Decline curve dimensionless rate

-

3 4)

(1 1-25)

qDd:

(1 1-26) where tD and qD are the conventional dimensionless time and dimensionless production rate used in well test analysis, given by the following expressions: (11-27)

53 1 and (11-28) By substituting rate scale match-point values and rearranging Eq. 11-26, the initial flow rate qi is estimated from: (11-29) Similarly, using the definition of tDd with the time-scale match-point data, an estimate of the initial decline rate D ican be obtained: (11-30) An additional piece of information that can be acquired through this matching technique is the expected life of the well. This is accomplished by entering on the field graph the economic limit rate, displacing horizontally to the traced matched curve (through the field data and its extrapolation) and then moving downward vertically to the time axis, on which the corresponding life estimate can be read. It can be mentioned that data taken prior to tDd of 0.3 will appear to be of exponential type regardless of the true value of the reciprocal of decline curve exponent b, and thus will give a straight line on semilog paper (Fetkovich, 1980) (see Fig. 11-llc). If exponentiaI decline is shown by the field data, then the type curve of Ehligh-Economides and Ramey (1981), shown in Fig. 11-13, should also be used. Based upon the above findings, the results can be checked by means of specific graphs in accordance with the particular decline type shown by the well (exponential, hyperbolic, or harmonic). As in reservoir simulation studies, the prediction period of well performance should not exceed the time used in the matching analysis. Example 3. Hyperbolic decline example (Arps, 1945; Fetkovich, 1980) Arps (1945) presented production decline data for a Kansas lease producing from the Arbuckle Limestone. Later, Fetkovich (1980) re-analyzed Arps’ example by means of the type curve matching method, and reached the previous conclusion of Arps that the lease showed hyperbolic decline with a reciprocal of decline curve exponent b being equal to 0.5. Figure 11-14 shows the results of Fetkovich shown in his fig. 10, where production data were matched to the type curve of Fig. 11-12. This match was found to be unique in that the data would not fit any other value of 6. Future producing rates can be read directly from the real time axis of the field graph. By means of Eqs. 11-29 and 11-30, the following estimates for the initial flow rate and the initial decline rate are obtained:

532 4.1

=

D.

(10' STB/month) = 30303 STB/month 0.33

0.12 ( I month)

=

'

0.12 m o n t h s -

=

I

Assuming the economic limit rate of 400 STB/month, given by Arps (1945), a n d entering this value on the field graph, gives a life estimate for the lease of approximately 130 months. As discussed by Fetkovich (1980), the fact that this example was for a lease, a group of wells rather than a n individual well, raises a key question. In general, it is expected that different results will be obtained by analyzing each well individually a n d summing the results, and by adding the production of all wells and analyzing the total production rate. Based o n comments of Fetkovich (1980) and in the light of results obtained by Ehlig-Economides and Ansari (1983), for this lease of fairly uniform reservoir properties the decline curve exponent b is similar for each well. Inasmuch as wells have been producing under conditions of capacity production, since the completion of drilling at similar wellbore pressures, identical results will be obtained for each well. Consequently, the sum of the results from all wells gives the same answer as analysis of the total lease o r field production rate.

.-

. -.

- - .- .

-.

I

!

-. -. - L TRACING

- -

PAPER

--1

I

I

!L. -. - -.-.-. 10-3

10-2

10

-.

time, months - -. -. -.-. 1

10.'

-.

I

, 10

1

J ,

.- lo'.- .

102

1(

t Dd l,ia. 11-14, Arp\ and Fetkovich hyperbolic decline example. (After Arps, 1945, fig. 4 , p. 244, and Fetko\ich, 19x0, fis. 10, p. 1071; courtesy of S P t of A I M € . )

533

Empirical methods of estimating reserves When performance trends have not been properly established with respect to oil and gas production, estimates of future production rate and proved reserves may be made by analogy to older reservoirs in the same geographic area that have similar characteristics and already established performance trends. As mentioned in relation to the discussion of Fig. 11-1, before any wells are drilled this is the only method that can be used to estimate the reserves of a potential reservoir. After the field is confirmed to be productive, the importance of this method commonly decreases in a gradual manner as more information regarding the reservoir production characteristics becomes available, which makes possible the application of the other, usually more accurate, methods described in this chapter. The use of these methods dates back to the early industry days when no other methods were available. In carbonate reservoirs, the empirical technique has been used extensively (Mannon, 1972). As the petroleum engineering technology developed, the analytical methods discussed above gained a dominant role in the estimation of reserves. One of the most well-known studies regarding the statistical determination of oil reserves is that of Arps (1967). The study was based on 226 sandstone and 86 carbonate reservoirs, classified by their predominant producing mechanism, mainly water drive and solution gas drive with and without supplementary drive. The possible application of the results of this work to carbonate reservoirs has been thoroughly discussed by Mannon (1972). The use of these results has been recommended only where the data required for more detailed studies are lacking. The APl Subcommittee on Recovery Efficiency (1980) undertook a follow-up study of the API Bulletin D14: Statistical Analysis of Crude Oil Recovery, by Arps (1967). The aim of this second effort was to assess and, if possible, to extend the previous correlations. The amount of data gathered was larger than that used in the first study: data were obtained from a total of 620 reservoirs (473 sandstones and 147 carbonates) in the U.S.A. and Canada, which had a dominant producing mechanism of solution gas (376 cases) or water drive (244 cases). The goal of this work was to study oil recovery based on actual field performance rather than on theory or laboratory data, and to develop an empirical correlation for the prediction of oil recovery. An extensive effort was made by the authors of that study to regroup, delete, and add parameters, in the search for a regression equation that would have greater significance than that obtained from the previous work. Unfortunately, this could not be achieved. It was concluded that an adequate predictive correlation cannot be achieved for the prediction of recovery and/or recovery efficiency for a reservoir based on readily definable oil and reservoir parameters. Perhaps the most important factor, reservoir heterogeneity, cannot be easily defined (van Everdingen and Kriss, 1978). These authors stated that because it is difficult to include heterogeneity and possibly other factors such as reservoir continuity as independent parameters, this precludes the development of valid general statistical correlations. They cautioned against the continued use of the correlations from API Bulletin D14 to predict recovery or recovery efficiency for a particular reservoir.

534 RECOVERY FACTOR

The recovery factor is of prime importance, and is defined as the fraction of the hydrocarbons in-place that can be recovered or an equivalent measure of the ultimate hydrocarbon recovery (Muskat, 1949). There are two main types of recovery factor. The first is a purely technical recovery factor, which depends on the nature of the production mechanism and the actual operating history. A prediction of this figure before the reservoir is fully developed and its production mechanism properly identified would be no more than a gross estimate. The second recovery factor is that governed by current economic circumstances and by environmental and ecologic considerations. Only the former recovery factor is discussed here. The recovery factor can be estimated by (1) laboratory experimentation; (2) analogy to older reservoirs with similar characteristics; and (3) theoretical considerations. The first method results in only an order-of-magnitude estimate, because it is practically impossible to properly scale the prototype with respect to the reservoir. The second method also does not give an accurate estimate because, as clearly pointed out by Muskat (1949), each reservoir has unique properties and characteristics. Only seldom does this approach give reliable results. The third method is the most commonly used, because under suitable conditions it provides the most accurate estimate of reserves. This method evolved with the advancement of the petroleum technology. It requires thorough analyses of the rock, interstitial fluids, structural conditions, and the production characteristics of the specific reservoir under consideration, which are used as input to a suitable mathematical model of the reservoir performance. With regard to the second empirical method of analogy, as discussed in the previous section, an adequate predictive correlation cannot be achieved for the prediction of recovery efficiency for a reservoir based on reservoir rock and fluids parameters. This essentially precludes, from a practical standpoint, the use of the correlations of the API Bulletin D14 or those of the continuation study. The failure to attain a statistically acceptable correlation is due to the fact that hydrocarbon recovery is a function of many factors, some of which are beyond engineering description and control. As an example, besides heterogeneity and continuity, factors such as rock wettability and pore-size distribution can be mentioned, which are difficult t o characterize and are not always measured. The second study of the API Subcommittee on Recovery Efficiency, entitled Statisrical Analysis of Crude Oil Recovery and Recovery Efficiency (1980), presented correlations of recovered oil versus original oil-in-place, both expressed in barrels per net acre-foot. A leastsquares line fit for the primary recovery of the carbonate reservoirs is shown in Fig. 11-15. The data points are not shown because they were not presented in the original study. It is expected that there would be a significant error in predicting the recovery of a particular reservoir. Thus, the recovery estimate provided by this correlation should be taken as a temporary rough approximation, until the data required for more accurate study can be acquired. The methods based on theoretical considerations are currently in common usage, because of their inherent accuracy when good reservoir information is provided.

535 The estimates of hydrocarbon recovery are carried out by matching a reservoir model to the reservoir’s production history and then using the matched model to predict the recovery or recovery factor. The models used range in complexity from zero-dimensional or material balance models to the multi-dimensional multicomponent models. For oil production, the recovery factor is the ratio of oil production N p to the initial oil-in-place N . As an example, for the case of an undersaturated reservoir, the recovery factor obtained from Eq. 11-7, adapted for these conditions, is as follows:

RF (fraction)

=

Np/N

=

Boi ~

BO

coS0

(

+ c,S,

+

Cf

s,

1 -

(11-31)

Expressions similar to Eq. 11-31 can be written for other reservoir conditions. For volumetric gas reservoirs, the recovery factor Gp/G can be estimated through the use of Eq. 11-14 and Fig. 11-6b. As shown in this figure, the recoverable reserves can be calculated by means of the p/z ratio evaluated at abandonment pressure conditions, P a b /zab. This information, besides the pi/zi ratio, would give the recovery factor for the particular reservoir:

(1 1-32) As mentioned in the section on the “Material balance method”, substitution of the single-phase t factor by the two-phase deviation factor permits the use of Eq.

+ 0

0 * 0)

L

0 0 c

c

0,

k CI

6oo 400

m

E L

t

1

-

0

-

200

z

0

n . . 2

I

1

I

Original oil in place, barrel per net acre-foot Fig. 11-15. Least-squares fit for primary recovery of carbonate reservoirs. (After API Subcommittee on Recovery Efficiency, 1980, fig. c-6, p. 67; courtesy of API.)

536 11-32 for the determination of the recovery factor in a gas-condensate reservoir. Additional discussion regarding this and other related matters can be found in the paper of Stoian and Telford (1966). Their study has shown that the average gas recovery for a wide sample of volumetric reservoirs is 85%. I t has been discussed in the section on “Numerical simulation” that numerical solutions for the fundamental flow equations, which describe the flow of fluids through porous media, are currently in general use. The writers believe that these, under suitable conditions, provide the best means to estimate the recovery factor.

Example 4 . Comparison of reservoir simulation calculated recovery factor and empirical recovery factor Two examples of comparison of the recovery factor calculated for carbonate reservoirs by ( 1 ) reservoir simulation techniques and (2) the API Subcommittee on Recovery Efficiency results (Fig. 11-15) are presented here. The first example is taken from the paper of McCulloch et al. (1969). They used conventional black-oil reservoir simulators in one and three dimensions to forecast the primary depletion performance of the reef pools of the Devonian Rainbow Field in Alberta, Canada. The three-phase simulator was run for the Rainbow Keg River “A” Pool in a primary depletion prediction for a period of 27 years using a 5 x 7 x 20 grid three-dimensional configuration. The calculated primary oil recovery via reservoir simulation was 57.6% of the original oil-in-place. In general, because of the particular characteristics of these reefs, the three-dimensional results were similar to those obtained using a one-dimensional simulator. McCulloch et al. (1969) stated that the primary recovery mechanism is gravity drainage. From the average reservoir properties of porosity and water saturation in the oil zone of 0.1 16 and 0.076, respectively, the calculated value of original oil-in-place is 834.4 bbl/net acre-ft. Using this value in Fig. 11-15 yields a recovery of 118 bbl/net acre-ft, which is equivalent to a recovery factor of 14%. This value is substantially different from the 57.6% estimated via reservoir simulation studies. This difference confirms previous comments related to the failure to attain an adequate predictive correlation for the prediction of recovery efficiency. The second example is taken from the paper of DesBrisbay and Daniel (1972). They used a conventional one-dimensional black-oil reservoir simulator to predict the primary depletion performance of the Libyan Intisar “D” reef field. The primary performance indicated that natural depletion from the initial pressure of 4257 psi down to 2000 psi would result in a less than 8% primary oil recovery. It can be mentioned that most of this recovery occurred above the bubble point conditions, because the oil saturation pressure was 2224 psi. From the average reservoir properties of porosity and water saturation in the oil zone of 0.22 and 0.18, respectively, a value of the original oil-in-place of 1400 bbl/net acre-ft is calculated. Using this value in Fig. 11-15 yields a recovery of 155 bbl/net acre-ft, which is equivalent to a recovery factor of 11Yo. This example shows an excellent agreement between the prediction of the recovery factor by reservoir simulation studies and that by the empirical correlation of the API Subcommittee on Recovery Efficiency.

537 SUMMARY

This chapter provides a brief review of currently available methods for the estimation of oil and gas reserves. It can be stated that accurate reserve estimates can be made for carbonate reservoirs with the available methods. It is essential for carbonate rock systems to use all available methods in light of the amount and quality of the information at hand, the formation characteristics, and time and resources available. Reserve estimations, as most of the reservoir engineering estimates, are envisioned as a continuous problem, where improvement comes with production time as more data become available. Because of the heterogeneous nature of carbonate reservoirs, the synergistic approach is particularly useful in their study, where the team effort provides the best reservoir characterization. The empirical methods still lack accuracy, mainly due to the fact that each carbonate reservoir has unique properties and characteristics. Thus, seldom would this approach render reliable results. It has been concluded that an adequate predictive correlation cannot be achieved for the prediction of recovery and/or recovery efficiency for a reservoir based on readily definable oil and reservoir parameters. The impossibility of including heterogeneity and, possibly, other factors, such as wettability and reservoir continuity, as independent parameters precludes the development of valid general statistical correlations. Regarding the analytical methods, they should all be used whenever possible. The volumetric method has been improved by the use of the geostatistics kriging theory, which has the advantage of providing the best linear, unbiased estimator of the unknown reservoir parameters. The use of the material balance equation for carbonate systems may be especially valuable because the accurate determination of the reservoir properties involved in the volumetric method could be particularly difficult. Today, numerical models that solve the fundamental flow equations are in general use. These, under suitable conditions, provide the best means to estimate the reservoir reserves.

NOMENCLATURE*

A b Ai

=

B,

=

Bo Bw

c

CA Di k h N

= =

= = = = = = = =

area reciprocal of decline curve exponent area of ith cell, Eq. 11-6. gas formation volume factor oil formation volume factor water formation volume factor compressibility shape factor initial decline rate formation permeability reservoir thickness, vector distance initial oil-in-place

= = = = = = = =

cumulative oil production ratio of the initial hydrocarbon volume of the gas cap to the initial hydrocarbon volume of oil slope of the straight line on a linear graph of pressure (real gas potential) versus time real gas potential pressure flow rate dimensionless production rate, Eq. 11-28 decline curve dimensionless rate, Eq. 11-26 cumulative gas/oil ratio solution gadoil ratio apparent skin factor ( = s + Dqs,), Eq. 11-23 van Everdingen and Hurst skin factor time dimensionless time, Eq. 11-27 dimensionless time based on drainage area, p k t / $ p c 4 connate or irreducible water saturation initial hydrocarbons in place minimum in-place hydrocarbon volume reservoir rock volume cumulative water influx cumulative amount of water produced value of a reservoir parameter at a position x experimental semivariogram, Eq. 11-3 mean value of the experimental sernivariogram, Eq. 11-6 estimated value of a reservoir parameter weights of the krigeage system, Eq. 11-5 viscosity porosity

= = = = = = = = =

abandonment initial interception formation, flowing oil produced total standard conditions well

= = = = = = = =

= = = = = = = = = = = =

* Units are defined in Table

11-111.

539 TABLE 11-Ill

SI preferred units and customary units for various parameters Parameter or variable

SI preferred units

Customary units

A B c

m2 m3/m3 Pa- I

k

w2

ft2 RB/STB psi I mD ft STB STB psi STB/day MSCF/day SCF/STB S C F / ST B bbl (ft3) bbl bbl bbl Fraction CP 141.2 1424 2.637 x 0.23395

h N NP

P 4 qsc RP

m m3 m3

kPa m3/day m3/day m3/m3

m3/m3 m3 m3 m3

€

m3 Fraction Pa s 1842 1293 3.6 x 10-9 4.1665 x

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