Oil Production Forecasting by Decline Curve Analysis

SOCIETY OF PETROLEUM ENGINEERS OF AJJ.1E 6300 North Central Expressway Dallas 6, Texas

PAPER NUMBER SP E 1254

THIS IS A PREPRINT --- SUBJECT TO CORRECTION

OIL PRODUCTION FORECASTING

BY DECLINE CURVE ANALYSIS

By

Robert W. Mannon, * Member AJJ.1E Montana College of Mineral Science and Technology, Butte, Mont.

Publication Rights Reserved

This paper is to be presented at the 40th Annual Fall Meeting of the Society of Petroleum Engineers of AJJ.1E, to be held in Denver, Colo., Oct. 3-6, 1965, and is considered the property of the Society of Petroleum Engineers. Permission to publish is hereby restricted to an abstract of not more than 300 words, with no illustrations, unless the paper is specifically released to the press by the Editor of the Journal of Petroleum Technology or the Executive Secretary. Such abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the Journal of Petroleum Technology or the Society of Petroleum Engineers Journal is granted on request, providing proper credit is given that publication and the original presentation of the paper.

Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines.

ABSTRACT

The subject of analyzing oilwell production-decline curves is examined in detail. The problem is approached from the standpoints of raw data validity, techniques of data handling, modes of graphic representation, and final interpretation procedures. Forms of analysis based on mathematical treatments, factors of analogy and experience, and predicted reservoir behavior efforts are discussed. It is evident that all of the methods of decline curve analysis are subject to gross forecasting error in specific situations. Examples are given to illustrate the over-all problem of reliability. The accepted method for classifying decline curves is reviewed. The mathematical definition for hyperbolic decline is found to be too restrictive and a broader definition is suggested.

Gradual or abrupt changes in the producing rate of a well due to reservoir depletion, fluctuation in bottom-hole producing pressure, and changes in conditions in or immediately adjacent to the wellbore, are examined. A method is presented utilizing comparative theoretical and actual productivity index behavior that can be useful in predicting future rates of production.

*Present address: U. of Southern California, Los Angeles, Calif. References and illustrations at end of paper.

INTRODUCTION

Of the myriad tasks that face the petroleum engineer in his professional duties, the forecasting of future production from a well or group of wells is one of the most formidable. Webster's Dictionary defines the word forecast as meaning: to predict; to foretell; to prophesy. Many a petroleum engineer has felt the need to be possessed of prophetic powers when called upon to predict the future performance of his company's oil wells.

The job of making reliable production forecasts can be laborious and time-consuming. Notwithstanding the effort required, moreover, industry insists that the petroleum engineer's forecasts be trustworthy. We might ask, "In view of the rigors of preparing dependable predictions, are oil industry people justified in demanding such exactness?" The answer to this question is, of course, "Yes", and the reasons are quite evident. Oil companies and related organizations rely heavily on production forecasts as an integral part of profitability studies, financing and loan work, capital budgeting, and the trading of oil and gas properties. Much of their capacity for future growth lies in these potential areas. Why reliable forecasts are often so difficult to

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2 OIL PRODUCTION FORECASTING BY DECLINE CURVE ANALYSIS SPE-l2S4

prepare 1s not so apparent, however; which brings us to the purpose of this paper. The purpose is threefold: [l] to examine critically the problem of predicting reliably future rates of oil production, specifically through the use of oil production decline curves; [2] to review the work that has been done in this areaj and [3] to present methods of analysis that may aid in obtaining reliable forecasts under certain conditions.

The Importance of the Oil Production Forecast

Evidently, early in the history of the oil industry, oil operators realized the importance of having some idea of the amount of oil to be forthcoming from their wells and the length of time re~uired to produce this amount.

The early-day producers rapidly became aware of the fact that the net profit generated from a well is disproportionately sensitive to changes in production rate. The business of producing oil is somewhat uni~ue in this respect. The gross revenue from a well is usually directly proportional to the well's production rate since the oil is sold with little difficul ty at the prevailing posted price. On the other hand, the cost of producing this oil is very insensitive to the relative production rate, i.e. the rate at which a well produces oil normally has no significant effect on the operating expense to the producer. As a result, the net income from oil producing operations is affected by changes in production rate to a greater extent than in other businesses. It was soon apparent, also, that wells demonstratec characteristically initial rates over a wide range with subse~uent differing types of production-decline performance. The need for reliable production forecasts was evident and expert help was sought for this task. This may have been one of the first duties of the petroleum engineer.

The Use of Decline-Curve Analysis as a Tool

Engineers, in their search for techni~ues to provide a reliable forecast of future performance, have long recognized the possibilities of using some form of graphic representation of the production data from a well as a guide. References to this techni~ue first appeared in the literature around the turn of the century. Since that time considerable work has been done attempting to formulate interpretation procedures that will produce reliable forecasts. Unfortunately the inVestigators in this field have met with only limited success. It would be helpful to determine, if possible, why more progress has not been made.

In decline-curve analysis, graphs of data from a well or group of wells are constructed in an attempt to detect trends to aid in estimating

future performance. The data usually consist of monthly oil production rates. The resulting curves are in terms of oil rate vs time or oil rate vs cumulative oil produced, and are called production-decline curves. Other types of performance curves may be prepared, however. Through the years, investigators have prepared graphs of virtually every imaginable type of data relating to the production of oil and gas in an endeavor to hit upon more useful relationships. Plots of produced water, or "water cut", pressure, depth of oil-water contact, or cumulative gas vs cumulative oil are ~uite common. Various types of graph paper are used, such as rectangular coordinate, semi-log, and log-log paper. Most of these graphic relationships are attempts to depict trends linearly, for ease in extrapolation. Most of the trends are non-linear, however; although there are some notable exceptions that will be discussed later. Noble attempts have been made to straighten out non-linear plots of oil rate vs time and oil rate vs cumulative oil by devising special semi-log graph paper2 or by suitable shifting on log-log paper; but, unfortunately, these techni~ues have had limited applicability.

Some Aspects of Well Performance

An examination of the more salient features that play a role in the ~roduction performance of a well would be helpful.

Muskatl has shown that the factors affecting the production rate of individual oil wells from the standpoint of reservoir behavior are essentially the effective permeability to oil [ko ], the net oil-zone thickness [h], the oil formation volume factor [Bo ], the oil viscosity [~o]' the wellbore radius [rw]' and the effective drainage radius of the well reo These factors define the productivity index [J] of a well under steady-state conditions:

J • f • . . . [l]

Furthermore, the production rate from a well [~] is seen to be:

Qo • J (Pe - Pv) • • • • • • [2]

where Pe = pressure in reservoir at distance re from wellbore

Pw = pressure at wellbore.

Expressed in this form, the decline in the production from a well appears to be a function of the depletion of the area being draine~ by the well. Moreover, the relationship is not a simple oneo As a reservoir is depleted, the ~uantity [p - 'Pw] normally decreases. The productivity i~dex of the well will also decrease due to the sensitivity of the oil permeability

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SPE-l254 ROBERT W. MANNON 3

to fluid saturation and the effect of reservoir pressure on oil viscosity and oil formation volume factor. Only in rare instances, in which the oil saturations and values for oil viscosity and oil formation volume factor are maintained constant, will the productivity inde approach a constant value. Notable examples are reservoirs under active water drive in which the reservoir pressure remains above the bubble pOint, and gas-drive pools 'subjected to pressure maintenance. The relationship of productivity index as a function of the degree of depretion is further complicated by transient behavior and wellbore damage effects incurred subsequent to completion. This topic will be discussed subsequently.

It is evident, therefore, that the production performance of a well or group of wells producing from the same pool cannot normally be described mathematically in a simple manner. Moreover, a great deal of work has been done in this area, but the mathematical relationships ~

derived are of very little practical use. Arp~

has reviewed the work in this field quite exhaustively and it is not necessary to repeat it here. The work in general represents an attempt to depict mathematically the decline trend of production as an aid in extrapolation.

Classification of Decline Curves

It might be well to examine the present classification of oil production-decline curves as generally accepted in the industry.

It is apparent that some wells exhibit production histories indicating that the loss in production rate [~] is proportional to the production rate or

• -& •• • • • • • • • • • • • • [3]

As Arps2 has shown, to recognize this type of decline, called exponential decline, where the decline rate is constant, production rates at equal time intervals are entered in one column with the loss in production rates per unit time entered in the second column. In the third column, the ratio of production rate to loss in production rate per unit time, known as the loss ratio, is entered. If the calculated values of the loss ratio in column three represents substantially a constant value [a], the decline is considered to be exponential.

If, on the other hand, the ratios of production rate to loss in production rate per unit time are not a constant value, but the first differences are, the decline is classified as hyperbolic where

d (d~) b ........... [4] Cit dt 8-'

In other words, in hyperbolic decline, the first derivative of the loss ratio is a constant. It can be shown that the decline rate in this case is proportional to the production rate raised to a power. The reader is referred to Arps' paper for tables illustrating examples of expoential and hyperbolic decline. 2

The tendency, currently, is to place any decline curve, regardless of the production characteristics represented, into one of the two above described categories. It appears that the nature of the decline in production from most wells is far too complicated to allow their decline curves to be adequately described mathematically in such a simple fashion. There fore, it does not appear wise to attempt to utilize the somewhat simple mathematical relationships represented by Eqs. 3 and 4 as a guide in forecasting production except in special cases. Experience indicates that such attempts may yield inadequate results. Moreover, with the advent of the electronic computer, more rigorous methods are feasible, which will be discussed later in the paper. This classification of decline curves should be retained, however, since it can be useful when used in the proper manner.

In the production performance of a well, we are dealing with a regression curve of a special type. The decline rate is either constant or decreases with time in some manner. There are no other possibilities. In cases where the decline rate is constant, true exponential is the result and Eq. 3 can be used in extrapolation. In most cases, the rate of decline decreases with time, so that Eq. 4 does not apply rigorously. We should continue to call this type of decline hyperbolic, however. It describes the general form - that of a hyperbole. On the other hand, no rigid limits dictated by Eq. 4 should be imposed on hyperbolic decline curves. This broader definition of hyperbolic decline curves should help clarify the problem and perhaps pave the way for greater use of computers in decline-curve analysis.

Mathematical Methods

Interpretation of production performance by mathematical methods should be attempted only on wells exhibiting curves that appear to be representative of future trends. For these select wells, and they are in the minority, it is convenient to represent the past production history by a power series of the form

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4 OIL PRODUCTION FORECASTING BY DECLINE CURVE ANALYSIS SPE-1254

Y' • 80 + ~X + y2 + Lj'l + ••••• BuXD • [5]

or

Y'.8o+~+~+~+ x· 2 3 x x

...... • [6]

In the infinite series, a ten degree polynomial will ordinarily suffice.

The number of pairs or [x,yJ values should be substantially more; ideally in excess of 60, or five years of monthly production data. The problem of fitting polynomial curves is best handled by the method of least squares. Computer programs of this type are readily available that use a modified Gauss method to solve the simultaneous equations. A program currently being tested has a different method of solution employing orthogonal polynomials and recursion formulas. This technique shows promise of eliminating possible problems of ill-conditioned systems.

Important Considerations in the Analysis of Decline Curves

Consider the status of the art of decline curve analysis as it is known today. To plot oil rate vs time, for example, monthly production figures of the well may be obtained from company records or from the files of the state oil and gas agency involved. Three questions arise at this point pertaining to the validity of the production data in the interpretation of decline curves.

[1] Do the production figures represent capacity production?

[2] Are the figures accurate? [3] To what extent has the well been off

prodUction during the period represented?

The first question is vital because in decline curve analysis the production decline characteristics of a well must reflect reservoir depletion. It is apparent, from Eq. 2, that the production rate [~] is a function of drainage-area depletion alone, only when the bottomhole producing pressure of the well is constant. Therefore, from a practical standpoint, the well must be producing at its maximum rate, i.e., with the bottomhole producing pressure approaching a minimum value.

Accuracy of production data is also of prime importance. Frequently, wells on the same lease producing into a common tank battery are not subjected to individual tests as often as necessary. The reported individual well production rates are then only approximated and may be grossly in error.

The Problem of Down Time

To interpret the decline curve of a well,

it is imperative that the down time be determined. In this connection, many investigators make allowances f"or any time during which the wells are off production by calculating the oil rate in bbl per producing day instead of bbl per calendar day or bbl per month. The theory is, apparently, that plotting oil rate on a calendar day basis does not give a fair representation of a well's ability to prodUce and extrapolation would be difficult and probably unreliable. The usual procedure is to enter the data on the graph as bbl per producing day vs time to extend the decline curve in the future, to convert the projection to a 365-day yearly forecast, and either to ignore the down time of the well or wells or hope that the cause for the down time can be eliminated. We have found the practice of plotting oil rate as bbl per calendar day or even in bbl per month to be a far better method In many cases, down time on a well or a lease is a fact that cannot be ignored. A decline curve of bbl-per-producing day vs time usually yields a forecast optimistic to an extent dependent on the amount of down time of the wel or lease and on the reservoir fluid and rock charac~eristics. In any event, a plot of bblper-calendar-day vs time provides a means for any significant down time to be represented graphically and interpreted in the light of circumstances. Usually, only in the cases of excessive down time is the decline trend of the well obscured. The problem of interpreting flush production of a well after being off production can also be handled more intelligent ly in this manner.

The Nature of the Decline Curve Under Examination

With the production rate data represented graphically in a meaningful way, the engineer is now ready to proceed with the analysis. He may be working with one or more separate graphic representations on the same well. In examining the decline data as represented, he must first decide what role he thinks the past production history shOuld play in the forecast of future production, i.e., a dominent or passive role. Can the past production data be taken at face value or is the available production history not entirely indicative of the probably future performance? If it appears that the future performance will follow the trend to date, the mathematical treatment previously discussed may be applicable.

In the alternative case, the past production history does not completely reflect the probable future performance. A common illustration would be in instances where only meagre production data is available which is not sufficient to characterize the true form of the decline cu~ve. The engineer must then attempt to devise a means apart from the produc-

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SPE-1254 ROBERT W. MANNON 5

tion history itself as a guide in extrapolating the decline curve. Some methods have been developed to accomplish this, based on reservoir behavior. A technique of this type is presented in Example B.

Frequently, the petroleum engineer is faced with the production history on a well or group of wells that seems to be adequate with an apparent unmistakable trend established Subsequently, however, the well or wells perform in a manner which may be totally unexpected. Example A illustrates this problem. Figs. 2 and 3 show decline curves on two well groups with two wells in each group. The wells are all completed on very close spacing in the same zone with an extremely high permeability to upwards of 8,000 md. Fig. 1 is a map of a portion of the pool. The close spacing is evident. The distance between wells No. 1 and No. 4 is about 500 ft. The oil gravity averages about 200 API. The two well groups indicate apparent abrupt changes in performance during the first part of the 5th year. If an investigator extrapolating the curves at this time did not anticipate these changes, his forecasts would be quite wrong. In this case well interference and gravitydrainage effects predominate because of the closeness of the wells and high formation permeability.

Let us suppose the engineer is dealing with a situation in which he sees no possibilit. for a formal type of analysis apart from using experience with wells having similar characteristics, and exercising intuition and judgment. The practicing engineer very often finds himself in this predicament and it might be of help to examine certain implications of this situation in some detail.

Analogy to Monte Carlo Methods

Consider the so-called Monte Carlo methods Any procedure which involves the use of statistical sampling techniques to approximate the solution of a mathematical or physical problem can be classified as a Monte Carlo method. These methods have an ancient and honorable history and are distinguished by their experimental nature. In recent years Monte Carlo methods are best known in connectior with problems that do no yield conveniently to classical analysis such as in areas of statistical mechanics and particle diffusion. In these cases the numerical computation of approximate solutions can sometimes be approached through random sampling.

An illustration of the Monte Carlo method would be the pastime (rather dubious to many], of playing poker. The experienced poker player knows at least subconsciously what his chances are of filling an inside straight. He has

learned it the hard way. The actual probability could be calculated, but most poker players learn from experience rather than by direct calculation. In this case the player has profited by a Monte Carlo approach.

The Problem of Uncertainties

It is apparent that the petroleum engineer must resort to measures comparable to the Monte Carlo method in many situations in predicting future production rates of an oil well. Unlike the poker player, however, he faces a problem that does not lend itself to rigorous analysis. He is confronted with a decline curve of a well of the type under discussion. The decline history to date does not adequately describe the probable future performance. He is aware of the possibility of his forecast deviating substantially from the actual ultimate performance. What course should he follow in this situation? He will ultimately have to think in terms of probabilities and will choose the forecast which he thinks will most likely be correct. His choice will be a judicious one, of course. It will be based on the total fund of engineering and geological information available on the well and similar wells. Frequently, however, engineering and geological data at hand is inadequate. In making the final judgment, then, the estimator must lean heavily on his experience with wells of similar characteristics. Because he has also learned the hard way from predictions that may have turned out badly, he is, although perhaps only subconsciously, using his previous experience as an estimator to solve the present problem.

To the uninitiated it may seem that the petroleum engineer is adopting a haphazard procedure in his evaluation work. Actually, because important variables often cannot be evaluated rigorously the petroleum engineer is simply dealing with this situation by drawing on every resource at his disposal.

Some investigators have proposed that, where significant uncertainties do exist, the engineer work around the problem of forecasting error by simply defining a range within which he believes the actual performance will fall. The project is thus evaluated within certain limits of reasonableness. This approach has merit and can be valuable at times, but conditions usually dictate that a single estimate be given.

Depicting Trends in the Curves

Most decline curves are interpolated satisfactorily by drawing a smooth free-hand curve to represent a trend through the data points. In some instances it may be feasible to develop a mathematical curve to describe the trend of data using the method of least squares as

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6 OIL PRODUCTION FORECASTING BY DECLINE CURVE ANALYSIS SPE-1254

discussed elsewhere in the paper. The curve, which may be linear or curvilinear, is drawn free-hand so that it comes as close as possible to each point. The tendency is to draw the curve in such a way that the areas above and below the line are equal. Normally, this is a satisfactory method. When using semi-log or log-log paper, however, the fact that the scale is non-linear should be compensated for as much as possible.

The Causes for Decline in Production Rate

In decline-curve analysis, the estimator is, of course, anxious to detect "trends" in production attrition that will continue in the future. The causes of decline in production rate of a well fall into three main categories: [1] reservoir depletion, [2] fluctuation in bottom-hole producing pressure, and [3] changes in conditions in or immediately adjacent to the wellbore.

As for the first category, referring again to Eq. 1 and 2, the decrease in productivity index, J, reflects depletion in the reservoir or reservoirs from which the well is producing. Under primary production operations, depletion is directly related to the reservoir withdrawals. Production decline performance data which is mainly the result of reservoir depletion is ordinarily clearly defined and usually can be interpreted reliably. Factors to consider regarding reservoir depletion are gravity drainage and well interference. These may complicate the extrapolation of the curve, as illustrated in Example A.

We have discussed the fact that a fluctuation in bottom-hole producing pressure will obscure the decline characteristics of a well. Wells, both on natural flow and artifici~ lift, that are not producing at capacity may demonstrate this. Paying particular attention to the efficiency of artifical lift equipment may help alleviate this conditiao

In category [3] in which changes in conditions in or immediat~ly adjacent to the wellbore can cause decreases in well production a number of factors should be considered. Most of these factors serve to decrease the effectivE permeability to oil and come under the heading of cumulative well-bore damage. Some of the more common of these factors involve plugging of the well-bore by silt, sand, salt, asphaltene, and yax deposition and accummulation in and near the well-bore. Reduction in oil permeability because of increased free-gas saturation, water production, or clay swelling can also occur. Some of these changes in wellbore conditions can be remedied by well treatment and others will not ordinarily respond to treatment. Transient effects, such as flush production after shut-in might also be included

in this category.

Decline-Curve Extrapolation

Although various graphic representations of production performance are used from time to time to meet specific situations, the ultimate requirement of a forecast of future production by years never changes. Ideally, a means will be available for estimating the remaining reserves apart from the analysis of production decline curves. It may be possible to make a trustworth estimate of reserves by running a reservoir material balance, by employing the volumetric method, or by interpreting such meaningful trends as produced water cut vs cumulative oil produced. When the data for an independent estimate is lacking, the engineer will have to look to production-decline data solely.

Obviously, there is no sure-fire method for predicting, in all situations, the future rates of production from a well. It would be unwise to set forth a specific procedure to follow to arrive at meaningful answers. The problem of decline-curve analysis is much too complex to be solved in this manner.

A Numerical Method for Forecasting Production

A technique for forecasting production rates by numerical methods is illustrated by Example B [Tables I and II]. Also presented is a form of graphic representation which is valuable at times. Fig. 4 is a plot of the pool production data.

The pool was discovered in the early 1940's and produces 320 API oil from a depth of approximately 7,000 ft. It contains seven wells and there is no appreciable water influx into the reservoir. Production from the pool was curtailed until 1952. The producing history of the pool is seen in Fig. 1 which illustrates a convenient mode of data representation and extrapolation. Curves I and II are both plots of the same seven wells in the pool. The lower curve, labeled Curve II represents oil rate vs time, and the upper curve, or Curve I, depicts oil rate vs cumulative oil produced.

The pool is probably at least 75 per cent depleted. It has a relatively long history. with no recent development. In spite of these facts, the decline curve can only be extrapolated with a great deal of uncertainty. It is reasonable to expect that the decline rate will gradually decrease with time. Most pools, in the absince of an active water drive eventually will evidence a gradual flattening in decline rate. While there is an indication of a tendency toward hyperbolic decline, it is somewhat obscure due to reperforating of the zone in the late 1950's which caused an abrupt increase in oil

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SPE-1254 ROBERI' W. MANNON 7

rate. Since there is little reason, based on the decline curve, to make a hyperbolic projection, the tendency would be to extend the curve exponentially, which would probably be a conservative estimate. In this case, however, additional data were available to provide a basis for a reliable hyperbolic extrapolation of the curve.

The production decline trend in Fig. 4 has been extrapolated in three different ways: Forecast A, perhaps the most optimistic forecast, Forecast B, a very pessimitic projection, and Forecast M, somewhere between these limits. Forecast A, the optimistic estimate, assumes that the character of the decline in terms of oil rate vs cumulative oil is linear. In this type of decline, representing a special form of hyperbolic decline and called harmonic decline by some,2 the rate of decline is proportional to the production rate. By means of a desk calculator this linear projection can be converted to an oil rate vs time relationship and is shown as the uppermost projection of Curve II, also labeled Forecast A.

A pessimistic forecast assuming exponentia decline, on the other hand, is Forecast B which appears as a straight line on the lower curve [Curve II]. This forecast, when converted to oil rate [bbl/day] vs cumulative oil production takes the form of the projection labeled Forecast B on Curve I. The actual performance, therefore, should be within the area delineated by the Forecasts A and B which represent the upper and lower limits of the probable range of future production from the wells.

In the case of the subject pool, sufficient additional data were available to provide the forecast labeled Forecast M in Fig. 1 that lies between Forecasts A and B. In the procedure followed, adequate static shut-in pressure data were available so that an analytical prediction of ultimate oil recovery by Tamer's method could be made. The prediction indicated an ultimate recovery of 4,751,000 bbl to a reservoir pressure of 300 psi. The question immediately arises as to what the produc- . tion rate might be at 300 psi reservoir pressurE and how it compares with the estimated economic limit of production. In most numerical predictions prepared to indicate reservoir performancE little attention is given to this problem other than to choose an abandonment pressure that seems to be reasonable. In this case, however, the problem involved further rates of oil production and it was important to determine the oil rate at one or more pressures at or near abandonment.

The procedure consisted of projecting well productivity index data to provide a basis for an intelligent forecast of future oil production.

During the period of curtailment from inception to 1952, approximately ten years, numerous bottom-hole pressure surveys, both flowing and shut-in, were made on all the wells. The data were averaged and all presented in Table I and in Fig. 5, which is a plot of productivity index vs cumulative oil production from the pool. Also shown in Fig. 5 are theoretical productivity index values calculated from Eq. 1.

Because the values for net oil zone thickness, h, and the natural logarithm of the ratio, r /r do not change, they were not included in tfie ~alculation. The relative permeability of oil, kro' was substituted for the effective permeability to oil, ko • The calculations and related data pertaining to Fig. 5 are shown in Tables I and II. Laboratory relative permeability data determined from cores on wells in the subject pool, plus PVT data on the reservoir oil were used in the calculations. The calculated values from Eq. 1 are dimensionless numbers and only COincidentally are in close agreement with the actual productivity index values are also in close agreement, particularly in recent history. This is to be expected. Using the calculations from Eq. 1 as a guide, it was then a simple matter to project the actual productivity index behavior to a reservoir pressure of 300 psi.

Next, the production rate at 300 psi was calculated from Eq. 2. Using a value for bottom-hole producing pressure [p f] of 30 psi, the total oil rate at a reservoirwpressure of 300 psi was found to be 34 B/n assuming three wells were producing at this time. The 34 B/n rate corresponded to a cumulative oil recovery of 4,751,000 bbl and this point was plotted on Curve I and is designated point X. The curve labeled Forecast M was then constructed using point X and the past production history as a guide. Again using a desk calculator, Forecast M on Curve I was converted to Forecast M on Curve II and the analysis was completed. Where justified, a series of values for total oil rate at various cumulative oil recoveries could be calculated to yield a more nearly accurate forecast.

CONCLUSIONS

The petroleum engineer is often required by management to prepare production forecasts as an integral part of valuation, budget, and profitability stUdies. The petroleum engineer's forecasts must be trustworthy. The eventual success or failure of a project frequently reflects the quality of the petroleum engineer's estimates and calculations. The petroleum engineer must exercise ingenuity and imagination. He must be skilled in a wide range of techniques for interpreting production data and be aware of the possible effect of related geological and

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8 OIL PRODUCTION FORECASTING BY DECLINE CURVE ANALYSIS SPE-l254

engineering factors on future production performance. In analyzing a situation, he must also be sensitive to conditions that would dictate a modification in method in order to prepare a trustworthy prediction.

Reliable forecasts of future rates using oilwell decline curves are difficult to prepare even though engineers have been at it for over half a century. Various types of graphic representations of data are used in an attempt to depict trends that are linear, to facilitate extrapolation. Only limited success has been achieve.

Muskat'sl work on productivity indices indicates that gradual or abrupt changes in the producing rate of a well are due to [l] reservoir depletion, [2] fluctuation in bottomhole producing pressure, and [3] changes in conditions in or immediately adjacent to the well-bore. The production performance of a well is quite complicated and cannot normally be described mathematically in a simple manner. Much work has been done in this area, but the mathematical relationships derived have not had wide application.

Under the present classification, all wellf are considered to decline either exponentially or hyperbolically. These classifications are convenient as they describe the general forms of the decline trends. The wells evidencing exponential decline are consistent with true exponential decline. Few of the wells declinine hyperbolically, however, conform to true hyperbolic decline as defined by Eq. 4. Hyperbolic decline should, therefore, be considered in the broad sense as representing all wells that evidence a decrease in decline rate with tine •

The validity of production data in decline ourve analysis should be examine from the standpoint of accuracy, extent of down time, and whether the figures represent capacity production. Plotting production data on the basis of bbl per calendar day rather than bbl per producing day ordinarily will yield better results.

A decision must be made on each well concerning the extent to which the past production history of the well will represent the future performance. Every effort should be made to develop a separate estimate for remaining reserves and future rates of production, particularly on wells with decline histories that are difficult to interpret. A method is available utilizing comparative theoretical and actual productivity index behavior that can be useful in predicting future rates of production. The form of graphic representation used in Example B can be of value in delineating a working range between the extremes of conser-

vative and optimistic projections within which the actual performance should lie.

The key to analyzing decline curves with reliability is a careful, thoughtful approach, a thorough understanding of the physical and chemical factors involved, adequate experience in decline-curve analysis and in other aspects of oil production technology, and an ability to exercise ingenuity, imagination, and good judgment.

ACKNOWIEOOMENTS

The author is grateful for the assistance of Gustav Stolz, Jr. and Dr. A. J. Smith of the Montana College of Mineral Science and Technology for their valuable criticisms of the manuscript. He also wishes to extend his appreciation to H. T. Olsen, Los Angeles, and R. J. Fernandes, U. S. Natural Gas Corp., Los Angeles, for their many helpful suggestions on the topic of decline-curve analysis through the years.

REFERENCES

l. Muskat, M.: Physical Principles of Oil Production, McGraw-Hill Book Co., Inc., N. Y. [ 1949 ].

2. Arps, J. J.: "Analysis of Decline Curves", Trans., AIME [l944] l60, 228.

3. Tamer, Jack: "How Different Size Gas Caps and Pressure Maintenance Affect Amount of Recoverable Oil", Oil Weekly [June l2, 1944] ll4, No.2

4. Vance, Harold: "Evaluation", API History of Petroleum Engineering [l96l] lOOl.

5. Brown, G. W.: "Monte Carlo Methods," Modern Mathematics for the Engineer, McGraw-Hill Book Co., Inc., N. Y. [l956].

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TABIE I -- EXAMPLE B ESTIMATED PRODUCTIVITY INDICES OF INDIVIDUAL WELlS

FROM BOTTCM-HOLE PRESSURE SURVEYS

Nc Average

Well Well Well Well Well P.I. M bbl No. 2 No.4 No.5 No·7 No.8 Total BLDL12.si

90.l 248.3 435.8 0.838 0.64l l.070 0.l20 0.795 3.464 .693 663.2 0.742 0.55l .893 O.llO 0.720 3.0l6 .603 808.8 0.682 0.489 .732 O.lOO 0.660 2.662 .532

l,l38.3 0.632 0.435 .590 0·902 0.609 2.358 .472 l,425.3 0·593 0.388 .475 0.085 0.5'60 2.l0l .420 l,679.2 0.557 0.342 .394 0.080 0.5l7 l.890 .378 l,928.4 0.522 0.298 .338 0.073 0.475 l.706 .34l 2,l62.l 0.49l 0.258 .292 0.069 0.434 l.544 ·309 2,368.l 0.46l 0.220 .255 0.063 0.395 l.394 .279 2,562.l 0.43l 0.l85 .222 0.062 0.355 l.255 .25l 2,745.l 0.402 0.l53 0.l97 0.058 0.3l8 l.l28 .226 2,903.8 .l96 3,099·7 .l67 3,268.3 .l44 3,442.2 .l32 3,582.9 .ll2 3,842.4 .090 4,094.0 .073 4,399.0 .056 4,75l.0 .042

TABLE II -- EXAMPLE B TABULATION OF THEORETICAL AND ACTUAL PRODUCTIVITIES

VS CUMULATIVE OIL PRODUCTION kro Reservoir

Pressure S * kr ** Bou psig Bo Ua t 0 0 M bbl Original

Conditions 2,820 l.376 0.42l l.000 l.000 l.727

7/l / 45 2,765 l.367 .430 .992 .960 l.633 "90.l 7/l/46 2,660 lo353 .446 .980 .840 lo393 248.3 7/l / 47 2,540 lo337 .468 .965 .720 lol50 435.8 7/l / 48 2,370 lo3l4 .505 .946 .630 0.949 663.2 7/l/49 2,230 lo297 .540 .933 .550 0.786 808.8 7/l /50 2,llO lo283 .572 .9l4 .460 0.627 l,l38.3 7/l/5l 2,003 lo270 .630 .898 .400 0.522 l,425.3 7/l /52 l,925 lo262 .626 .886 .380 O.48l l,679.2 7/l /53 l,8l0 l.25° .662 .872 .340 0.4ll l,928.4 7/l/54 l,720 lo239 .690 .859 .290 0.339 2,l62.l 7/l /55 l,630 lo230 .720 .848 .260 0.293 2,368.l 7/l /56 l,540 lo220 ·752 .837 .250 0.273 2,562.l 7/l /57 l,440 lo209 .778 .826 .230 0.244 2,745.l 7/l /58 l,350 lol99 .82l .8l7 .220 0.224 2,903.8 7/l/59 l,260 lol9° .857 .807 .l90 0.l86 3,099·7 7/l / 6O l,l95 lol84 .882 .799 .l65 0.l58 3,268.3 7/l/6l l,l70 l.l8l .892 ·792 .l50 0.l42 3,442.2

l,065 lol70 .938 .783 .l40 0.l28 3,582.9 900 lol53 l.Ol5 .770 .l20 0.l03 3,842.4 700 lol33 loll8 .754 .llO 0.087 4,094.0 500 l.lll l.238 .739 .080 0.058 4,399. 0 300 lo086 lo385 ·72l .070 0.047 4,75l. 0

* From Tamer method for predicting reservoir performance **From relative permeability data

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10000

5000 r .... z o ~

ffilOOO a.. (/)

...J 500 UJ 0:: 0:: <t CD

100

, ~ It ...... &

I'" .,.,

'-1\

I 2 3 4

Fig.l- SDbsurface Contour Mop. Portion of Pool in Example A

5000~~--+--+--~-1---r--r--+--T--+~

Bbls Mos. vs Time

~ JZ o ~

r\ ~ • -,\ ~ V1

5 6 7 8 9 Years

10 II

~IOOO'~-+--~~~-+--~--r--+--~--r--+~ a..

~ 500~-+--~--~~'-~-~r-~-~~--r-~ UJ 0:: 0:: <t CD

100' I

'-_1..2

I I

I I I I 1

I I ~ I I i I I I : J ' i

.i.........:3"----.J'---4,.L_~ J_E) __ -'--- 1.--1, JL ~ _ J~LLJLj Years

Figure 2- Decline curve - Wells I a 2 Figure 3 - Decline curve-Wells 3 a 4

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Cumulative oil produced (thousands of barrels) 1800 3000 4200 5400 1000

o ----+-----~~~~~~----+_--~IOO'

10001----

10

IOL---~5----~I~O----~15~--~2~O----~25~--~3~0--~35 Years

Fig 4 - Production Decline History and Forecasts for Pool in Example B

CD

.8

.7

.6

.5

.4

.3

. 2

.1

A

,=

o-Theoreticol Productivity Index B~~O A-Actual Productivity Index B/D/psi

o

o

A~O 0

~....... Projection of Actual

~~ Productivity Index Data_

~o ....

~- I I 2.- 3 4 -5

Cumulative Oil Produced - Million Barrels

Figure 5 -Theoretical PI. vs. actual P.1. performance

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Oil Production Forecasting by Decline Curve Analysis

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