ESTIMATION of PRIMARY OIL RESERVES J . J. ARPS MEMBER AlME
ABSTRACT This paper reviews the methods currently in use for estimating primary oil reserves and discusses the principles on which these methods are based. Particular emphasis is placed on how these methods change with the type of information available during the life cycle o f an oil property. This paper contains various novel estimating methods and shortcuts heretofore unpublished. INTRODUCTION Estimating oil reserves is one of the most important phases of the work of a petroleum engineer since the solutions to the problems he deals with usually depend on a comparison of the estimated cost in terms of dollars, with the anticipated result in terms of barrels of oil. His recommendations to management regarding the best course of action are therefore normally based on the most favorable balance between these two. Specific engineering problems which require such a knowledge of Original manuveript received in Petroleum Branch office on Feb. 26, 1956. Revised manuscript received June 29. 1956. Paper presented at the Petroleum Conference-Economics and Valuation, Dallas. Tex., March 29-30. 1956. Discussion of this and all following technical papers is invited. Discussion in w r i t ~ n g( 3 copies) may be sent to the offices of the Journal of Petroleum Technology. Any discussion offered after Dee. 31. lQ56, ghould hc in the foim of a new paper. 182
BRITISH-AMERICAN O I L PRODUCING C O . DALLAS, TEX.
recoverable oil reserves and a projection of future rates are: [a.] the exploitation and development of an oil reservoir; [b.] the construction of gasoline plants, pipelines and refineries; [c.] the division of ownership in unitized projects; [d.] the price to be paid in case of a sale or purchase of an oil property, and the magnitude of the loan which it will support; [e.] the proper depreciation rate for the investment in oil properties; and [f.] evaluation of the results of an exploration program. This discussion will be confined to the various methods and tools which are currently in use for estimating oil reserves to be obtained during the primary phase of an oil-producing reservoir and for a projection of the future production rates. Reserves which may be obtained by secondary recovery methods or fluid injection programs and gas and gas condensate reserves will not be discussed in this paper. Unfortunately, reliable oil reserve figures are most urgently needed during the early stages when only a minimum amount of information is available. Management's interest in the oil recovery from a property-aside from its use for accounting purposes-usually declines when the property approaches its economic limit, just at the time when the reliability of the estimates is at its best. T o illustrate this general idea n
chart is presented (Fig. 1) showing the three periods in the life of an imaginary oil property. Time is shown on the horizontal axis, while the cumulative production and estimated ultimate recovery are plotted vertically. No particular units are used and this schematical chart is not to scale. During the first period, before any wells are drilled on a property, any estimates will of necessity be of a very general nature, based on experience from similar pools o r wells in the same area, and usually expressed in barrels per acre. This will
PETROLEUM TR i h S A L
I I0\5,
\ IM1
thcrcfore be called the "barrels per acre" period. The estimates can range all the way from AB (property non-productive) to CD, reprcsenting the most optimistic view. The second period follows after one or more wells are drilled, and, assuming that the property is productive, it is now possible to pin the estimated ultimate recovery down within the much closer limits E F and GH. During this time interval most estimates are on a volumetric basis and are based on acre-feet of pay and a unit recovery expressed in harrels per acre-foot. The type of information available during this period consists of well logs, core analysis data, bottom-hole sample information and a subsurface map. Interpretation of this data in the light of early pressure behavior may lead to conclusions regarding the type of productive mechanism to be expected. Production decline curves or other trends are not yet available during this second, or "barrels per acre-foot", period. The third period follows after sufficient actual performance data on the property have become available to make a check of previous volumetric estimates against decline curve trends possible. Also, the pressure hehavior may now make material balance work possible, thereby offering other valuable clues regarding the type of production mechanism and active oil in place. This increased amount of information causes the lines HJK and FLK, which delineate the range between optimistic and pessimistic estimates, to converge gradually in point K. which represents the true ultimate rccovery or cumulative production at abandonment time. The dashed curve BK represents the cumulative recovery from the property. It follows from the foregoing that it is desirable for the engineer making a reserve estimate to provide his client o r his management with the possible sprcad in estimated ultimate recovery, both high and low, or, if he provides them with a single figure, to point out the probable error in his estimate. In reviewing the histories of many reserve estimates over an extended period of time in many different fields, it seems to be a common experience that the really good fields such as East Texas, Oklahoma City, Yates or Redwater, etc.. have been generally under-estimated during the early barrels per acre-foot period compared to their later performance, while the poorer ones such as
West Edmond, Spraberry and the like are usually over-estimated during the early stages. It should be emphasized that, as in all estimates, the accuracy of the results generally cannot be expected to exceed the limitations imposed on it by inaccuracies in the available basic data. In other words, the better and more complete the available data, the more reliable will be the result. Another factor which is often overlooked - particularly in cases where property values are involved -is that additional investment in acquiring good basic data during the early stages oftentimes pays off later. With good basic data available the engineer making the estimate naturally feels more sure of his results, and will be less inclined to the cautious conservatism which sometimes creeps in when many of the basic parameters are based on guesswork only. VOLUMETRIC ESTIMATES
During the second, o r "barrels per acre-foot" period, reserve estimates are based on a knowledge of the oilor gas-bearing reservoir volume and the physical characteristics of the reservoir rock and the reservoir fluids contained therein. These data must be supplemented with an informed guess as to the reservoir mechanism which will govern the withdrawals during the primary production period. The petroleum engineering literature of the last 25 years has dealt extensively with the determination of physical characteristics of the reservoir rock - such as porosity, permeability, relative permeability to the different phases, wetting characteristics, capillary pressure, and interstitial water-as well as with the significant physical characteristics of the reservoir fluids such as their viscosity, compressibility, gas solubility, shrinkage, and such other parameters as surface tension and interfacial tension. Prediction of the reservoir mechanism under which recovery will be obtained during the primary phase of production also has found widespread treatment in the technical literature since the middle thirties. particularly since the introduction of the material balance concept and a better understanding of the inter-reaction between the oil reservoir and its adjacent aquifer, if such an aquifer is present. Tn this discussion it
will therefore be assumed that the reservoir mechanism can be predicted and that the necessary data on the reservoir rock and reservoir fluids are o r can he made available. Pools without an active water drive and which produce solely as the result of expansion of natural gas liberated from solution in the oil are said to produce under a depletion mechanism; also termed an internal o r solution gas drive. When a free gas cap is prescnt this mechanism may be supplemented by an external o r gas cap drive. When the reservoir permeability is sufficiently high, the oil viscosity low, and when the pay zone has sufficient dip or a high vertical permeability, the depletion mechanism may be followed o r accompanied by gravity segregation. When a depletion-type reservoir is first opened to production its pores contain :qterstitial water and oil with gas in solution under pressure. N o free gas is assumed to be present in the oil zone. The interstitial water is usually not produced, and its shrinkage upon pressure reduction is negligible, compared to some of the other factors governing the depletion-type recovery. When this reservoir reaches the end of its primary producing life, and disregarding for the time bcing the possibility of gas cap drive o r gravity segregation, it contains the same interstitial water as before, together with a certain amount of residual oil under low pressure. The void space vacated by the oil produced and by the shrinkage of the remaining oil is now filled with gas liberated from the oil. During the depletion process this gas space has increased gradually to a maximum value at abandonment time. The amount of gas space thus created is the key to the estimated ultimate recovery under a depletion mechanism. It is reached when the produced free gas-oil ratio in the reservoir, which changes according to the relative permeability ratio relationship and the viscosities of oil and gas involved, causcs exhaustion of the available supply of gas in solution. FINALFREEGASSPACE COREANALYSIS If a sufficiently large number of accurate determinations of the oil and water saturation on freshly-recovered core samples is available, an approximation of the total free gas space to be expected in a reservoir can be obtained from them. This FROM
method is based o n the assumption that the depletion process taking place within t h e core upon the rcduction in pressure by bringing it to the surface is somewhat similar to the actual depletion process in the reservoir. Possible loss of liquids from the core before analysis may cause this total saturation to be too low. On the other hand. the smaller amount of gas in solution in the residual oil has a tendency to reduce the final gas phase. Those using this method hope that these two effccts somewhat compensate for each other.
In 1945 R. C. Craze and S. E. Buckley', of the special API study committee on Well Spacing and Allocation of Production, collected a large amount of statistical data on the performance of some 103 oil reservoirs in the United States for the purpose of a statistical analysis of the well spacing problem. Twenty-seven of these fields appeared to be producing under a depletion-typc mechanism. They produce from formations widely differing in geologic age and characteristics and have been operated with different spacings and efficiencies. A study of the final gas saturation of these fields shows considerable scattering of the data, as is to be expected. Most points seem to fall. however, between 30 and 40 per cent of the pore space, with the average at 30.4 per cent. The average crude oil viscosity under initial reservoir conditions of these fields was 2.2 cp. whilc the average solution gas-oil ratio was 378 cu ft/bbl. As shown by Muskat and Taylor', higher solution gas-oil ratio or lower oil viscosity in the same reservoir rock tend to increase the final gas saturation and vice versa.
Under initial conditions the reservoir rock contains, in barrels per acre-foot : Interstitial water Reservoir o i l Stock tank oil
7758.f.S~7758.f.11-Su.1 7758.f.11 -Swl Bi
in which f stands for porosity as a fraction and S, for interstitial hater as a fraction of the pore space; B , is the single phase formation volume factor for oil undcr initial conditions. U n d e r abandonment conditions 1Refercnces r i v e n at end of paper.
the reservoir rock contains, in barrels per acre-foot: Interstitial water Free gas Reservoir oil Stock tank oil
tics of the reservoir rock and its fluid content, this table has been found very helpful in estimating the possible range of depletion recovery factors. It may be noted that in general the most important single factor governing the recovery factor is the K , / K , relationship of thc reservoir rock. Unconsolidated intergranular material seems to be the most favorable, while increased cementation or consolidation tends to affect recoveries unfavorably. Information obtained from various sources since publication of this paper:' seems to indicate that the K,/K, curve No. 23 as shown in Fig. 2 for Wasson Dolomite was very likely in error and the maximum case for limestones and dolomites is therefore too high. It is recommended that the maximum case for sands and sand-
7758.f.S~ 7758.f.5, 7758.f.11-Su--Sg) 7758.f.ll-Sw-SI) 6.
in which S, stands for the final gas space as a fraction of the total pore space and B, is the single phase formation volume factor for oil at abandonment reservoir pressure. The unit recovery factor, DR, for a depletion-type reservoir is therefore:
Another approximatc approach to the problcm of estimating recoveries from depletion-type reservoirs is the use of the correlations developed for six different types of reservoir rocks, representing maximum, average and minimum conditions for sands and sandstones and for limestones, dolomites and cherts" In this study these six types of reservoir rocks were assumed to be saturated with 12 synthetic crude oil gas mixtures, representing gravities from 15" to 50" API, and gas solubilities from 60 to 2,000 cu ft/bbl, and their production performancc computed by means of the wcll-known depletion equation (see Eq. 2 on this page) in which the symbols correspond to the letter symbols recently submitted by the Petroleum Branch Executive Committee of the AIME', with the units defined as in Ref. 3. The results of this study, expressing recovery factors as a percentage of stock tank oil in place, are reproduced here as Table 1. In cases where no detailed data are available concerning the physical characteris-
n .%, --
B,
-
TABLE I-PRIMARY O i l Solution GOR ICu f t l b b l l
Oil Grovity ["APII
B.,
AP
B,
- --
/I<
k,,
AP
RECOVERY I N PER CENT O F O I L I N PLACE FOR DEPLETION-TYPE RESERVOIRS
Maximum
Sand or Sandstones Average Minimum
Limestone, Dolomite or Chert Moximum Average Minimum
stones should be used for both types of rock. Next in importance is crude oil gravity with viscosity as its corollary. Higher oil gravities and lower viscosities appear to improve the recovery. The effect of gas solubility on recovery factors is less pronounced and shows no consistent pattern. Apparently the beneficial effects of lower viscosity and more effective gas sweep with higher gasoil ratios is sometimes offset by the higher formation volume factors. I n the previous refercnce3 the results were expressed in Table 1 as recovery in barrels per acre-foot per per cent porosity rather than as recovery in per cent of oil in place. In general, these data seem to indicate a recovery range from the poorest combinations of 2 or 3 bbl/acreft for each per cent porosity to the best combinations of 17 or 18 bbl/ acre-ft/per cent. An over-all average seems to be around 10 bbl/acre-ft/ per cent. Analysis of the Craze-Buckley data indicates an over-all average of the order of 13 bbl/acre-ft for each per cent porosity. If the actual PVT data on the reservoir fluids are available, as well as all the necessary characteristics of the reservoir rock, the best way to arrive at a volumetric estimate is to actually compute the pressure-saturation relationship according to Eq. 2. The accuracy of this type of calculation falls off rapidly if the pressure increments chosen are too large, particularly during the final stages when the gas-oil ratio is increasing rapidly. In the study of Ref. 3 where computations were carried out with IBM equipment, pressure increments of 10 psi proved satisfactory. BabsonI4 and Tarner'j have advanced other computation methods which require a much smaller number of pressure increments and can therefore be handled by desk calculator. These methods are based on a trial-and-error type of solution of material balance and gas-oil ratio equations.
The permeability distribution in most reservoirs is sufficiently nonuniform in vertical and horizontal directions so that the depletion calculations as above on average material should be fairly representative. However, when distinct layers of high and low permeability are known to be present the depletion process may advance much more rapidly in a high permeability hank than in a
low permeability zone. In such a case separate calculations should be made for each permeability bank which is known to be continuous and the results converted into rate-time curves for each. The estimated ultimate should then be based on a superposition of such curves for the different zones. In that case one may find that the more permeable banks are depleted and have yielded their full unit recovery, while tighter zones are still partially saturated. The production rate from these tighter zones may, however, be so slow that the combined rate for all zones has already reached the economic limit.
Additional recovery can sometimes be realized when a high pressure reservoir is undersaturated so that it may produce by virtue of the expansion of its own reservoir oil, its interstitial water and of the reservoir rock itself, until the bubble point is reached. This additional recovery by expansion above the bubble point is sometimes quite substantial. In the case of the D-7 zone in the Ventura Avenue field, described by E. V. Wattss, 40 per cent additional recovery was indicated from this source. When conditions are favorable for gravity segregation or gravity drive, such as low oil viscosity, large pores and high vertical permeability or steep dips, the ultimate recovery can be estimated from a knowledge of the residual oil saturation remaining after such gravity segregation has taken place. Very little factual information on residual saturation after gravity segregation is available in the literature. D. L. KatzGreports that in the Oklahoma City Wilcox reservoir oil saturations were found in the gas zone varying between 1.0 and 25.9 per cent, while oil saturations of between 52.7 and 92.8 per cent were found below the gas-oil contact. These figures are in general agreement with actual recoveries computed by comparing the oil withdrawn with the volume of sand depleted. The writer found by studying the receding gas-oil contact, from detailed Oklahoma City Wilcox fluid level data, and by comparing the oil produced with the sand volume from which it was drained, that recoveries as high as 70 per cent of the pore space were not unusual.
WATER DRIVE RESERVOIRS Natural water influx into oil reservoirs is usually from the edge inward parallel to the bedding planes (edge water drive) or upward from below (bottom water drive). Bottom water drive is only possible when the reservoir thickness exceeds the thickness of the oil column, so that the oilwater interface underlies the entire oil reservoir. It is further only possible when vertical permeabilities are high and there is little or no horizontal stratification with impervious shale laminations. In either case, water as the displacing medium moves into the oil-bearing section and replaces part of the oil originally prescnt. The key to a volumetric estimate of the unit recovery by water drive is in the amount of oil which is not removed by the displacing medium. This residual oil saturation after water drive plays a role similar to the final gas saturation in the depletion-typc reservoir discussed in the previous section. In order to determine the unit recovery, it is necessary to again compare the amount of interstitial water and oil with dissolved gas initially present with the condition at abandonment time, when the same interstitial water is still present but only the residual o r non-floodable oil is left. The remainder of the original oil has at that time been removed by water displacement.
Calculation of the residual oil percentage from laboratory data on the relative permeability relationship for reservoir oil and water combined with the necessary viscosity data is possible by means of the Buckley and Leverett frontal drive method', which has been modified and simplified by Pirsonbnd Welge" The difficulty of obtaining a reliable relative permeability relationship for the reservoir rock being studied has, however, somewhat restricted the practical use of this ingenious mcthod. RESIDUAL OIL SATURATION COREANALYSIS The method most commonly used is to consider the oil saturation as found by ordinary core analysis after correction for shrinkage as the residual oil saturation to be expected from flooding with water. This is based on the assumption that water from the drilling mud invades the pay section just ahead of the core bit in a manner FROM
similar to the water displacement process in the reservoir itself.
tor and the various significant parameters. Their equation reads: Recovery fraction = .I 1403 2719 logK 25569 S, .I355 10gp<~- 1.5380 f .00035 H . . . . . . (3) in which K represents permeability in millidarcies, S, the interstitial water saturation, p, the oil viscosity in centipoises, f the porosity fraction and H the pay thickness in feet. Since Craze and Buckley's data was arrived at by comparing indicated recoveries per acre-foot with porosity data from the reservoir rock, the rcsidual oil calculated by this method includes a recovery efficiency factor which is not present when basing the estimate on residual oil as found in the cores. This recovery factor reflects the effect of spacing, if any, and of by-passing some of the oil in less permeable strata by more rapid water influx through permeable zones. Also, it includes the effect of having to abandon a property before the waterflooding action in all zones is complete, because of the water-oil ratio reaching the economic limit. When using this data, therefore, one will not arrive at the unit recovery under 100 per cent efficient water flooding, as would be obtained from the core analysis data, but rather find an over-all average recovery per acre-foot.
+
+
The analysis by R. C. Craze and S. E. Buckley' referred to in the previous section is also a valuable source for statistical data on residual oil saturation after water drive. Some 70 of the 103 fields analyzed produced wholly or partially under water drive conditions. Like the depletion-type fields, they were widely distributed geographically and produced from formations which differed in their geologic ages, physical properties, and structural characteristics, contained oils of different properties and produced under varying degrees of operating efficiency. A study of the calculated residual oil saturations for these fields shows a wide range; from 17.9 to 60.9 per cent of the pore space. However, the data seem to relate themselves distinctly according to the oil viscosity and the reservoir permeability. The average correlation indicated by the authors between oil viscosity and residual saturation, both under reservoir conditions, is shown by the following tabulation: Reservoir Oil Viscosity (in CP) 0.2
Residual O i l Saturation (per cent of pore space1 30
The deviation of the individual data from this average showed the following trend against average formation permeability: Average Rerervoir Permeability (in md)
Deviation of Residual O i l Saturation from Viscosity Trend (per cent of pare space)
50
112
According t o these statistical trends, the residual oil saturation under reservoir conditions for a formation containing 1 cp oil and having an average permeability of 500 md can 2, or 36.5 per be estimated at 34.5 cent of the pore space.
OF TIIE UNIT CALCULATION UNDER A WATER RECOVERY DRIVEMECHANISM Under initial conditions the reservoir rock contains, in barrels per acrefoot: Interstitial water Reservoir oil Stock tank oil
7758.f.S~ 7758.f.(l-SW) 7758.f.(l -Swl
BO
Under abandonment conditions the reservoir rock contains, in barrels per acre-foot : Interrtifial rmter 7758.1.5, Residual oil (Stock tonk conditions) 7758.f.So
in which So stands for the residual oil saturation under surface conditions as J fraction of the total pore space and B,, is the single phase formation volume factor for oil. The unit recovery factor for a water drive reservoir W R is, therefore:
+
I n another more recent statistical study of Craze and Buckley's water drive field recovery data by Guthrie and Greenberger'kultiple correlation analysis methods were uscd to determine the best-fitting correlation between the water drive recovery fac-
Eq. 4 shows the unit recovery factor for a 100 per cent effective water drive. In many reservoirs the permeability distribution is sufficiently nonuniform in both vertical and horizontal directions so that a flood front or
an oil-water contact will advance more or less in the same manner as if the formation were entirely uniform. In such a case the unit recovery factor should be representative except possibly for a correction to allow for efficiency of the drainage pattern. In other reservoirs there may be distinct layers of higher and lower permeabilities which appear to be more or less continuous across the reservoir. In such a case water may advance much more rapidly through the high permeability streaks than through the tighter zones. Allowance must then be made for this permeability distribution. In case the nature of the continuity of different permeability streaks is unknown, the estimate which takes the permeability distribution into account should be considered as a conservative one, while the one based on a uniform reservoir throughout should result in an optimistic figure. W. E. Stiles l1 in 1949 showed how the recovery from a waterflood property may be computed, while taking into account the permeability distribution. A simplified version of his method in tabular form is shown on Table 2. The data on this tabulation are adopted from a Tensleep sand reservoir in Wyoming where good statistical averages on more than 3,000 core analyses were available. Part of these cores were taken with water-base mud, which yielded the residual oil figures on Line 6. Another portion was taken with oil-base mud and yielded the interstitial water figures of Line 8. The cores were divided into the five permeability groups shown on Line 1 and appeared to have a fractional distribution of the samples as shown on Line 2. Line 3 carries the average permeability for each group, while Line 4 shows the capacity of each group in terms of darcy feet. For this purpose it was assumed that the calculation applied to a representative 100-ft total section. The relative water permeability K, behind the flood front is shown on Line 7, while the relative oil permeability K , ahead of the water in the clean section is listed on Line 9. These relative permeabilities were based on laboratory measurements. The unit recovery factor for each group was then computed by means of Eq. 4. The formation volume factor B, used was 1.07. On Line 11 is shown the cumulative +vet capacity in darcy feet under the assumption that the five groups are watering-out progrcssively in the order from high to low permeability. The figures shown are the cumulatives of those on Line 4. On Line 12 the
the produced stream vs cumulative oil recovery per acre-foot will be found on Fig. 2. Based on economic limit considerations, a final oil percentage of 2 per cent is indicated for this field and the corresponding recovery at that time can be found by interpolation as 292 hbl/acre-ft. As stated before, unless the continuity of the various permeability zones across the structure is definitely established, this recovery figure should be treated as a conservative one.
cumulative remaining or clearz oil capacity is carried, which is equal to the rota1 capacity of 3.243 darcy ft minus the wet capacity on Line 11. The water-oil ratio of the produced stream from such a linear water flood is shown on Line 13 based on:
or, with an oil viscosity of 5.5 cp and a water viscosity of 0.46 cp: (7) (11) W O R = 12 X -- X --- . . ( 6 ) (0) (12) It will be noted that the WOR at the time all permeabilities down to 100 md are wet has reached 15.5. When the watering-out process has advanced down to 50 md the W O R is 36.0. When groups 3 and 4 successively watcr out the W O R has increased to 76.5 and 307.9, respectively. On Line 14 is shown the cumulative oil production from each group at the time group 1 is watered-out (Min K,., = 100 md; W O R = 15.5).In group 1 the recovery of 61.6 listed is the product of Line 2 and Line 10 (full unit recovery). For each of the other groups the full recovery is reduced in the proportion of its average permeability to 100 md, based on Stiles' assumption that the advance of a linear flood will be approximately proportional to its average permeability. The total recovery for all groups corresponding to WOR = 15.5 is shown as 175.4 bbl/ acre-ft in the last column. In a similar manner the cumulative oil production for Min K,,.,., 50 md; WOR = 36.0, or 260.4 bbl/acre-ft. is computed on Line 15, and for Min K,c,., 25, 10 and 0 md, respectively, 344.5, 418.3 and 55!),6 bb]/acre-ft, on Lines 16, 17 and 18. A graphical presentation of the results of this computation in terms of 011 percentage in
In limestone pools producing under a bottom water drive, such as certain of the vugular D-3 Reef reservoirs in Alberta, one finds an extreme range in the permeabilities, often running from microdarcies on up to the darcy range. Under those conditions t h e modified "Stiles" method heretofore described yields results which are decidedly too low. The reason is that in pools like the Redwater D-3 there is a substantial density difference between the rising saltwater and the oil. While the water rises and advances through the highly permeable vugular material, it may at first bypass the low permeabilitv matrix material leaving oil trapped therein. However, the moment such bypassing occurs, a buoyancy gradient is set up across this tight material which tends to drive the oil out vertically into the vugular material and fractures. In the case of Redwater D-3, where the density difference between salt water and oil is 0.26. while the vertical permeabilities for matrix material are only a fraction of the horizontal permeabilities, a simple calculation based on Darcy's law applied to a vertical tube with a crosssection of 1 Sq cm shows that during the anticipated lifetime of the field very substantial additional oil recov-
-
~
-
TABLE 2-COMPUTATION OF THE WATER DRIVE RECOVERY FACTOR FOR A TENSLEEP SAND RESERVOIR I N WYOMVNG B Y T H E "PERMEABILITY-BLOCK" OR MODIFIED STILES METHOD GROUP
13. 14.
- Permeability Range ( m d ) Fraction of S a m p l e r Average Permeability (mdl Capacity i n dorcy feet Avg Porosity Fraction (I] Avg Resid. Oil Fraction IS,,) Avg 1nte:stitial Woter Froc. ISl,-) Relative Woter Perm. (KT,.) R e l a t i v e Oil Permeability ( K O ) E r t . U n i t Recovery Factor (bbl/acre-fl) C u m . "Wet" Cap. Z (41 C u m . "Clean Oil" Cap.=-3.243-111) Water-Oil Ratio Cum. Rec. WOR - 15.5
15.
C u m . Rec.
~
1.
2. 3. 4. 5. 6. 8.
7. 9.
10. I I. 12.
Min Ka.,,r
WOR
Min K w c t 16. 17. 18.
--
:-
100 36.0 50
WOR - 76.5 Min K n . , , t 25 C u m . Rec. WOR 307.9 Min K,,.,.t : 10 Cum. Rec. WOR =. '/i
C u m . Rec.
Min
-I=
0
I
2
3
4
5
Total
ery may be obtained due to this socalled buoyancy effect. In addition to these buoyancy phenomena there is also the effect of capillarity and preferential wetting of the reservoir rock by water. Imbibition of water from fractures and vugular material into the low permeability matrix as the water advances may materially aid the above buoyancy mechanism, but are much more difficult to evaluate quantitatively. To make a calculation of the recovery under a buoyancy mechanism it is necessary to first determine by statistical analysis of a large number of cores the average interval between high permeability zones. A separate computation is then made for each of the permeability ranges to determine what pcrcentage of the matrix oil contained in a theoretical tube of such average length may be driven out during the producing life of the reservoir under the effect of this buoyancy phenomenon. Surprisingly improved recoveries are generally indicated by this method over what one would expect from a "Stiles" type calculation and the results from recent studies of the rise in water table seem to confirm the validity of this concept. ESTIMATES BASED ON PERFORMANCE DATA Estimates of ultimate recovery by extrapolation of a performance trend fundamentally all follow the same pattern. The two quantities one wishes to determine are usually either remaining oil reserves or remaining productive life. Cumulative production and time are therefore normally selected as independent variables, and plotted as abscissae. A varying characteristic of the well performance which can be easily measured and recorded is then selected as dependent variable to produce a trend curve. For extrapolation purposes this variable has to meet two qualifications: (1) its value must be a continuous function of the independent variable and change in a uniform manner; and (2) it must have a known end point. By making a graph of the values of this continuously changing dependent variable as ordinates against the value of the independent variable (cumulative production or time) as abscissae and graphically extrapolating the apparent trend until the known end point is reached, an estimate of the remaining reserves or remaining life can be obtained. The basic assumption in this procedure is that whatever causes governed the trend of a curve in the past will continue to
govern its trend in the future in a uniform manner. This extrapolation procedure is therefore strictly of an empirical nature, and a mathematical expression of the curve based on physical considerations of the reservoir can only he set up for a few simple cases. The dependent variables most commonly selected, and their extrapolation, are as follows. Rate of production is by far the most popular dependent variable used when production is not restricted. In that case one commonly refers t o production decline curves. The two main types are rate-time and rate-cumulative curves for each of the two independent variables. Rate of production has the advantage of always being readily available and accurately recorded. When no major changes in operating procedure are made, and no stimulation treatments are applied, the curves normally show a fairly smooth declining trcnd over extended periods. This trend usually lends itself well to extrapolation. T h e second requirement is also easily met, since known o r estimated operating costs make it possible to determine the economic limit rate, o r the end point of the curve. This economic limit rate is the production rate which will just meet the direct operating expenses of a well. In determining this economic limit it is often advisable not t o use the operating expenditures charged t o a well but to analyze these costs closely and determine how much would actually be saved if the well were abandoned. This saving yields a more reliablc yardstick of the true economic limit of production, since certain expenses may have to be continued if other wells on the lease are kept in operation. T h e following is a sample calculation of the economic limit for a well: Crude Price per bbl Gar Revenue per bbl
$2.80 $0.20
Total $3.00 $0.375 Royalty (12.5 per cent) ' bbl) $0.125 Local Taxer (per h Leaves N e t Income per Gross Barrel $2.50 Ertimatsd direct operating cost ot Economic Limit $250 per month Estimated Economic Limit Rote 100 gross bbl/month
T h e mathematical background of these rate-time and rate-cumulative curves in connection with the socalled loss ratio method of extrapolation has been discussed in detail in a previous publication". Three types of decline curves are commonly recognized: constant percentage decline, hyperbolic decline and harmonic decline. With constant percentage decline the drop in production per unit of
timc is a constant fraction of the production rate. With hyperbolic decline the drop in production per unit of time as a fraction of the production rate is proportional to a fractional power of the production rate, this power being between 0 and 1. With harmonic decline the drop in production per unit of time as a fraction of the production rate is directly proportional to the production rate. Integration of the basic differential equations leads to rate-time rclationships, as shown o n the classification of production decline curves of Fig. 3. After integrating these rate-time equations a second time the ratecumulative relationships shown in this same table are obtained. On the chart of Fig. 4 arc shown the trends of these three types of rate-time and rate-cumulative curves on regular coordinate paper, semi-log paper, and log-log paper. Inspection of this chart shows that in the case of constant percentage decline thc rate-time curve becomes a straight line on semi-log paper, while the rate-cumulative curve straightens out on regular coordinate paper. I n either case the tangent of the angle of slope is equal t o the decline fraction. I n the case of hyperbolic-type dccline curves the rate-time relationship as well as the rate-cumulative relationship can be straightened out after shifting to become straight lines on log-log paper. T h e shifted ratecumulative curve in this case assumes a reverse slope. Besides the extra work involved in shifting, this type of paper also has the disadvantage that the horizontal scale on which the unknown variable is plotted usually becomes rather crowded at the point where the answer is desired. F o r this reason, special graph paper for hyperbolic decline was designed", making it possible t o plot either time or cumulative on a linear scale and still obtain the advantage of straight line extrapolation. In the case of harmonic decline it may be noted that the rate-time relationship can also be straightened out o n log-log paper after shifting, and assumcs a slope of 45". It may be of interest that in this specific case a plot of the inverse of the production rate versus time on a linear scale should also yield a straight line. T h e ratecumulative relationship for harmonic decline becomes a straight line on semi-log paper. T h e decline fraction in this case is equal to the rate times the tangent of the slope angle. It should be stressed that in this mathematical treatment of different
types of decline curves the decline fraction is instantaneous and theoretically applies only to a very short time interval. An analysis of a large number of actual production decline curves assembled by W. W. Cutler, Jr.'3 indicates that most decline curves normally encountered are of the hyperbolic type, with values for the exponent n between 0 and .7, while the majority fall between 0 and .4. The occurrence of harmonic decline ( n = 1) is apparently rare. As a matter of convenience the semi-log paper is most often used for rate-time extrapolations, while regular coordinate paper is favored for ratecumulative extrapolations. Since straight line extrapolation in these cases assumes a constant percentage decline, it will be obvious that such extrapolations therefore generally provide results which are too conservative. Experienced engineers usually allow for this by graphically flattening the declinc slope in the later stages. Another popular variable which is oftentimes substituted for the production rate in water drive fields is the oil percentage of the total fluid produced. Since projections of this oil percentage vs time are not often required, one usually finds this oil percentage variable o n l y plotted against cumulative. An example of this o n semi-log paper is shown for a T a r Springs reservoir in the Calvin field of Illinois in Fig. 5. The end point in this case is the lowest oil percentage which, combined with the fluid-producing capacity of the lease, will just cover operating expenses. CUMULATIVE GAS VERSUS CUMULATIVE OIL It is a characteristic of most oil reservoirs that only a fraction of the oil in place is recoverable by primary production methods. Gas, on the other hand, moves much more freely through the reservoir and it can generally be assumed that at abandonment time only the solution gas in the remaining oil at the then-prevailing pressure plus the free gas at that same pressure are left in the reservoir. In other words, even though it is not known exactly how much oil may be recovered, a much firmcr idea is generally available of the amount of gas that will be produced during the primary production period. This provides us with the possibility of an end point to ;I performance curve. The cumulative gas-cumulative oil method is illustrated with the chart on Fig. 6.
II
I
DECLINE
HARMONIC DECLINE
BASIC CHARACTERISTIC
I
D E C L I N E I S PROPORTIONAL TO A F R A C T I O N A L POWER (R 1 OF T H E P R O D U C T I O N R A T E
DECLINE I S CONSTANT a-o
I
I D:,.qOz
-
S u b s t ~ t u t e From
To
RELATIONSHIP
D E C L I N E IS PROPORTIONAL. TO PRODUCTION R A T E a = 8
o < n
dkr 1
I
m.
HYPERBOLIC DECLINE
Rote-itme
FOR INITILL CONOITIONS
Subslrtuie From Rote
Equotron
(
-
I
ttme Equolkon:
FOR WITIK CONOITIONS:
Substttute
-
From Rote
tune
Equotbon:
I
Find
a,
1
=
LD
I
I
1
I
D - D e c l ~ n e as o f r o c t l o n o f productLon r o t e
ql = Productron
(
D, = I n t t ~ o l decline
0, = Cumulotcve o t l productron ol t i m e
,
q , = l n l t l o l praduction
1
t = Time
rote o t
tbme
t I
K = Constont
role
n = Exponent
FIG.3-CLASSIFICATION OF PRODUCTION DECLINE CURVES. ---
-
R A T E - TIME
CURVES
COORDINATE
TIME I l l
LOG
TIME
(11
R A T E - CUMULATIVE CURVES COORMNkTE
SEMI-LOG
I
100
100.000
CUMULLTIVE 101
I-
---
ID --------
o
04000
CUYULLTIVE (01
CONSTANT PERCENTAGE DECLINE
n = 0
D= 03
HYPERBOLIC
n = %
0,:lO
n = I
D,:30
HARMONIC
DECLINE DECLINE
'
.,
110 ---HYPERBOLIC
ma - - - - - - - H A R M O N I C
DECLINE DECLINE
tpoo IOWO LOG CUMULATNE 101
100
(SHIFTED
(SHIFTED ON
100.000
ON LOG - LOG I LOG
- LOG)
FIG.&-THREE TYPESOF PRODUCTION DECLINE CURVESON COORDINATE, SEMI-LOGAND LOG-LOG GRAPHPAPER.
Cun~~llallvc oil production is plottcd on the horizontal scalc while the cun~ulativc gas production is plotted on the vertical scale. As is normal in depletion-type fields, the trend of the curve appears to steepen with increasing gas-oil ratios. From a volumetric calculation an estimate is made of the total gas to be released from this reservoir down to an assumed abandonment pressure. This figurc, which in this case was 1.42 billion cu ft, is marked on the chart as a horizontal line and represents the ceiling of the cumulative gascumulativc oil curve. B~ extrapolating the current trend until it intersects the estimate for the total gas available, an estimate can be obtained for the total oil rccovery during the primary period. OIL-WATERCONTACT OR ABANDOXMENT CON.I'OUR vs SANDTOP Another method which is somctimes practiced in the larger water drive fields, such as East Texas, is to choose the depth of the oil-water contact or abandonment contour as the dependent variable to be plotted against the cumulative oil recovery as the independent variable. The end point of this type of performance curve is the average depth of the top of the sand for a given lease. The method of extrapolation in this case is based on the simple assumption that whenever the abandonment con-
tour progresses to the top of the sand the lease is ready for abandonment. An example of this is shown on the chart of Fig. 7. By plotting this type of chart for many leases in the East Texas field it has been noted that the rise in the water table appears to be more o r less proportional to the cumulative production. KELATIONSHIP BLTWEEN R'SERvES
AND
From the rate-cumulativc equation for constant Percentage decline shown on Fig. 3 it may he noted that the rem"ning reserves are equal to the difference between the present production rate and the production rate at the economic limit, divided by the decline as a fraction. The same time units should be used for dctermining both decline and production rates. This leads to thc following shortcut: When the decline is 1 per cent per month the remaining reserves are 100 times the difference in monthly production rates; for 2 per cent decline per month this ratio equals 50; for 3 per cent it is 33 1/3; for 4 per cent it is 25, etc. When production rates are on a daily or annual basis the same formula holds, provided the decline is calculated on the same time basis. CONCLUSION A very important use of rcscrvc
estimates has often been overlooked. Reference is made to the comparison between the recovery analysis based on vol~lmetricdata with the one obtained from a projection of individual well performance. Oftentimes it is rlifiicult to fit the projected performance to the volumetric estimate. If both types of estimates are based on good, reliable information but cannot be reconcilcd, some important conclusions may be drawn from this discrepancy. If the performance indicates a substantially lower ultimate recovery than the volumetric calculation would indicate, it might mean that there is something fundamentally wrong in the production practices used. Possi'ily morc drainage points are nccded or the wells need stimulation treatments o r cleanout jobs. On the other hand, if the well performance projection indicates an ultimate recovery well in excess of the volumetric estimate, it could mean that the subsurface interpretation used may be in error and that there may be a larger oil reservoir on hand than current subsurface interpretation indicates. In that case it might be highly desirable to look for a possible extension to such an oil reservoir. REFERENCES I. Craze, R. C., and Buckley, S. E.: "A Factual Analysis of the Effect of Well Spacing on Oil Recovery," Drill.and Prod. Prac. API (1945) 144. 2. Muskat, M., and Taylor, M. 0.: "Effect of Reservoir Fluid and Rock Characteristics on Production Histories of Gas-Drive Reservoirs," Tmns. AIME (1946)
165. 78.
CUMULATIVE
FIG.
PRODUCTION
I N THOUSANDS
OF
BARRELS
5-OIL PERCENTAGE-CUMULATIVE RELATIONSHIP ON SEMI-LOG PAPER-TAR SPRINGSSANDPRODUCTION, CALVINFIELD.ILL. PETROLEUM TRANSACTIONS. A I M E
C W L A T I V E OIL
PROWCTION
PER WELL
IN
THOUSANDS
Of BARRELS
FIG.7-ABANDONMENTLINE (SUBSEA)VS CUMULATIVE OIL RECOVERYSANGEAST TEXASFIELD. WOODBINE 3. Arps, J. J., and Roberts, T. C;.:
4. 5.
6.
7.
"The Effect of the Relative Permeability Ratio, the Oil Gravity, and the Solution Gas-Oil Ratio on the Primary Recovery from a Depletion Type Reservoir," Trans. AlME (1955) 204, 120. "Letter Symbols for Reservoir Engineering," Jour. of Pet. Tech. (Jan., 1955) 38. Watts, E. V. : "Some Aspects of High Pressures in the D-7 Zone of the Ventura Avenue Field," Trans. AIME ( 1948) 174, 19 1. Katz, D. L.: "Possibilities of Secondary Recovery for the Oklahoma City Wilcox Sand," Trans. AIME (1942) 146,28. Buckley, S. E., and Leverett, M.
8.
9.
10.
11.
C.: "Mechanism of Fluid Displacement in Sands," T r a n s . AIME (1942) 146, 107. Pirson, S. J.: Elements of Oil Reservoir Engineering, McGrawHi11 (1950) 285. Welge, H. J.: "A Simplified Method for Computing Oil Recovery by Gas or Water Drive," Trans. AIME (1952) 195, 91. Guthrie, R. K., and Greenberger, M. H.: " I h e Use of Multiple Correlation Analyses for Interpreting Petroleum Engineering Data," API Paper 901-31-G, New Orleans, La. ( M a r c h , 1955). Stiles, W. E.: "Use of Permeability Distribution in Water
12. 13.
14.
15.
Flood Calculations," Trans. AIME ( 1949) 186, 9. Arps, J. J.: "Analysis of Decline Curves," Trans. AIME (1945) 160, 228. Cutlcr, W. W., Jr.: "Estimation of Underground Oil Reserves by Well Production Curves," USBM Bull. 228 (1924). Babson, E. C.: "Prediction of Reservoir Behavior from Laboratory Data," Trans. AIME ( 1944) 155, 120. Tarner, 5 . : "How Different Size Gas Caps and Pressurc Maintenance Programs Affect Amount of Recoverable Oil," Oil Weekly (June 12, 1944) 32.
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