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\title{Cumulative Production Forecast of An Oil Well Using Simplified "Hyperbolic-Exponential" Decline Models}
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\begin{document}

             \author[1]{Dr Makinde F A  .}

             \author[2]{Orodu O.  D.}

             \author[3]{Anawe P. A.  L}

             \affil[1]{  Covenant University}

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\date{\small \em Received: 9 December 2011 Accepted: 5 January 2012 Published: 15 January 2012}

\maketitle


\begin{abstract}
        


 Decline  Curves are important tools employed in the petroleum production industry to establish a good production performance forecast of production wells. Studies have shown that neither hyperbolic nor exponential decline could accurately produce dependable forecast results, which in turn affects the various economic decisions being made on both investment and future production processes. New simplified models for decline curve analysis are developed. The models are applicable to naturally producing wells that have not been secondarily enhanced. These models use exponential decline to extrapolate hyperbolic decline behaviour in making future production performance forecasts. Estimating different needed parameters and engaging some assumptions, the forecasted cumulative production increment using the model is .This compares favourably with the existing models.

\end{abstract}


\keywords{Production forecast, Reserves, Decline curve, Exponential, Hyperbolic.}

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\let\tabcellsep& 	 	 		 
\section[{INTRODUCTION}]{INTRODUCTION}\par
he petroleum industry is summarily driven by profit. This is the topmost factor to consider in the development of petroleum fields. Reserves have to be well estimated before the company's limited available resource is expended on a given project.\par
One of the most important tasks of a petroleum engineer is estimating, by factual prediction, the amount of oil and gas that could be recovered from a reservoir. Choosing the methodology is critical for accurate forecasts that are, in turn, vital for sound managerial planning \hyperref[b4]{5} . Risks have to be minimized in the process of making decisions based on the recoverable percentage of the original hydrocarbon in place, as well as the residual amount of this oil when the economic limit is reached before the need for any secondary recovery mechanism is involved, for the case of a naturally producing reservoir-well relationship.\par
Extrapolation of production history has long been considered the most accurate and defendable method of estimating the remaining recoverable reserve from a well and, in turn, a reservoir \hyperref[b4]{5} . Various methods have been developed so far, ranging from the most commonly used Decline Curve Analysis (DCA), to the Type Curve Matching (TCM) method based on the well or reservoir production history. While Decline Curve Analysis is independent of any reservoir characteristics, Type Curve Matching is a very subjective procedure.\par
DCA is a method used for the prediction of future hydrocarbon production by analyzing past production. A decline curve of a well is simply a plot of the well's production rates against the respective times of recording. It was recognized early in the history of petroleum engineering that calculating reserves and production forecasts were possible by studying past production trends.\par
Upon all the various works done in the past on forecasting the recoverable amount of hydrocarbons from an oilfield, the problem has been how to forecast with decline curves (production history) without overestimating or under-estimating cumulative production/reserves, most especially with Arp's Decline Curve Analysis. This study combines hyperbolic and exponential declines in developing simplified empirical decline curve models (hyperbolic-exponential models), to forecast the oil production performance of a primarily flowing well till economic limit. The production performance includes predicting future production rates, cumulative production increments, production time or period, ultimate recovery and the residual reserves \hyperref[b5]{6} but this paper presents only the models for predicting the cumulative production.\par
Arps \hyperref[b0]{1} published several excellent papers in 1944 and 1945. These papers contained several excellent equations which have essentially remained unchanged and are still referred to as the "Arps" equations.\par
Arps categorized decline curves into exponential, hyperbolic and harmonic declines. The hyperbolic equation is the universal equation upon which the exponential and harmonic declines are special cases of. The difference is just the changes in the hyperbolic constant used for the two cases. The hyperbolic constant determines the degree of change of each decline. The hyperbolic constant "b" equals 0 for exponential declines, 1 for harmonic declines, and ranges from 0 to 1 for hyperbolic declines.\par
Arps in his document, coupled with the work of Curtler showed that approximately two-third of the T Author : Petroleum Engineering Department, Covenant University, Ota, Nigeria. various cases of declines studied have a hyperbolic constant between 0.1 and 0.4, suggesting a log normal distribution.\par
Arps was able to derive and establish the various basic equations for Decline Curve Analysis, but unfortunately, he could not establish this transition point from hyperbolic to exponential decline.\par
To tackle this problem, in early 1980s, Fetkovich \hyperref[b1]{2} designed Type Curve models from different studied wells and fields based on Arps' empirical equations and known reservoir flow properties from which the decline curve derived from the production data of analysis is compared to look for a perfect match of which the well automatically assumes the Type Curve properties, whose values are then used for the forecasting of the well of interest.\par
Long and Davis  {\ref 3 (1988)} further added to the work of Arps. They tried to look at how the problem of the transition point could be surmounted. In their pursuits, they developed a log-rate-versus-time overlay to cope with this problem. They introduced the use of Type Curve Matching method, where the curve from a well production history is matched with already developed curves from different wells based on different reservoir-well properties for correlation.\par
Robertson \hyperref[b3]{4} (1988) also developed a production rate equation that is hyperbolic initially but asymptotically exponential with time. He introduced a dimensionless constant; its value ranges from 0 to 1 and is related to the abandonment pressure, and the rock and fluid properties.\par
Long and Davis, and Robertson ignored determining the precise transition point between the hyperbolic and exponential declines, they assumed the exponential decline rate was known based on experience with analogous wells or experience with particular reservoirs. This method proposed by them is limited in application because: ? Different wells situated in the same reservoir at times exhibit different reservoir-well properties such as the skin effect along their production history, thus, making them to exhibit different productivity indices, effective permeabilities etc, thereby drastically varying their production decline rates. This limits the dependability on the value. ? Also, in a situation when the well to be analyzed is the first producing well on the field or probably the other wells on the field are in a hyperbolic decline state, a dependable exponential decline rate (constant) that could have served as a reference for comparison with the decline rate value of the well-in-question is no more readily available.\par
Due to the fact that the hyperbolic decline rate at which exponential decline starts is the rate that is used to forecast the various production rates and cumulative production at any time or point after this transition point, conciseness is necessary in determining this value for improved accuracy in any estimated forecast made after this point (that is in the exponential decline state of the well productions).\par
Khaled \hyperref[b4]{5} from King Saud University, Saudi Arabia, in his publication "Predicting Production Performance using a Simplified Model", looked at combining hyperbolic and exponential decline models empirically together to predict production performance using the available well production history data. In his method, he stated that at a point in the production life of a well, the rate of decline of the decline rates, D with production time will behave as a constant (that is ). The time, at which this occurs, he described as the transition point from hyperbolic to exponential declines, from where production forecast can now be made exponentially. In that publication, Khaled assumed the value of this constant, based on the behaviour of the curve gotten from the constant-time graph he plotted. Thus, he employed a deterministic approach in estimating the transition time. 
\section[{II.}]{II.} 
\section[{METHODOLOGY}]{METHODOLOGY}\par
In this work, a set of empirical models are developed for estimating the cumulative production at the economic limit as follows: 
\section[{Model for Cumulative Production Forecast}]{Model for Cumulative Production Forecast}\par
Cumulative Production Forecast of An Oil Well Using Simplified "Hyperbolic-Exponential" Decline Models From the developed models, the required parameters for forecasting the performance of a well in a reservoir, and subsequently estimating the cumulative production and residual reserve at a set or predetermined economic limit production rate are: 
\section[{Global Journal of Researches in Engineering}]{Global Journal of Researches in Engineering}\par
-, the time at which the hyperbolic decline -, the initial decline rate at the onset of the hyperbolic decline behavior of the well's production rates, and-\par
, the hyperbolic decline constant or exponent.\par
b) Estimation of is an important parameter needed for forecasting a well's production and residual reserve using a combination of hyperbolic and exponential decline models.\par
There are three methods which could be used in the estimation of . Each method has its own applicability and conditions of usage. They include:\par
-Graphical method, -Comparison with already producing well in the same field, and -Company's Production Policy. The value of the hyperbolic constant b is very crucial in any hyperbolic DCA forecast. The shape of the hyperbolic curve is controlled by both the hyperbolic constant , and the initial decline . The higher the factor, the lower the decline in production rate, the longer the life of the well to the economic limit and the higher the ultimate recovery is.\par
Over the years, engineers calculate the hyperbolic constant using simple and approximate methods. Apart from the various empirical methods developed for calculating the various parameters needed in hyperbolic decline models, various DCA softwares such as the Oilfield Manager (OFM) are available in the industry for estimating them.\par
Empirically, the following methods can be employed in solving for the hyperbolic decline constant, b.\par
They are:\par
-The three-point method, and -The initial rate and decline method. 
\section[{d) Calculation Of The Initial Decline Rate,}]{d) Calculation Of The Initial Decline Rate,}\par
The initial decline rate plays an important role in a hyperbolic-exponential decline models as seen above. Apart from the widespread DCA computer softwares available presently in the oil and gas industry for estimating the various hyperbolic decline constants, this paper identifies two simplified and empirical methods for achieving the task of calculating . These are:\par
-The tangential method, and -The two-rate method. 
\section[{III. MODEL VALIDATION AND DISCUSSION}]{III. MODEL VALIDATION AND DISCUSSION}\par
OF RESULTS\par
There are various methods of estimating the forecast parameters as highlighted above but for the sake of simplicity and time, quite a number of computer softwares have been developed nowadays to ease seemingly complex calculations such as estimation of the hyperbolic decline exponent, the hyperbolic initial decline rate, and the initial decline rate.\par
In this work, the Schlumberger Oilfield Manager (OFM) was used in carrying out the various estimations.\par
format of the semi-log plot obtained from the OFM software showing both hyperbolic and exponential decline models fit for the region of best historical decline behaviour and their respective decline parameters as well as their regression coefficients.\par
It is understood that most naturally-producing wells start their production history with a hyperbolic decline behaviour, which does occur for just some periods after production commences, before finally exponential decline behaviour sets in. This understanding aided the development of the models in this paper. The result of the decline rates, Di above clearly shows that the hyperbolic initial decline rate is higher than that of the exponential decline rate, therefore permitting the hyperbolic initial decline to reduce to the exponential decline rate, from where the decline rate remains constant till the end of the forecast. This is a good pointer to establishing the intent of this work.\par
The forecast starts with a hyperbolic rate, then the time at which this rate declines to the exponential rate is calculated from one of the models (equation xxx), hence, from this time, the exponential decline is employed to forecast the remaining future production time behaviour of the well till the economic limit is reached assuming no work-over activities are being carried out on the well.\par
After estimating all these parameters (for Hyperbolic Decline):\par
Then, estimating the time at which the Cumulative Production Forecast of An Oil Well Using Simplified "Hyperbolic-Exponential" Decline Models Shown in Figure  {\ref 1}  is the most important tool in forecasting using combined models of two decline behaviours. Its value is what mostly determines how valid our forecast result is and this is not known to have been determined with any known software yet. It is best solved empirically by analyzing the historical production decline behaviour of the well. The method used here in calculating it is employed because we have a well that is showing a decline behaviour that is very close to both hyperbolic and exponential declines. The graphical method cited under the methodology would have been the only applicable method assuming there is a great difference between the hyperbolic and exponential decline fits.\par
Having estimated and the other required parameters for forecast with the developed models, the cumulative production forecast is done using the developed models.\par
The validity of the models developed and used here is authenticated with the data set analyzed. 
\section[{a) Cumulative Production Forecast}]{a) Cumulative Production Forecast}\par
Setting the last historical oil production rate (Table \hyperref[tab_0]{1}) of the well as the initial production rate for the forecast:\par
Assuming that no work-over activities like perforation interval plugging, pressure build-up and drawdown tests etc, as well as no secondary or enhanced recovery mechanisms are to be employed on the field till the bench-marked economic limit rate is reached, then the developed models for cumulative forecast could be used in forecasting the cumulative production increments through the well from the field, from the last historical date/rate to any forecasted production rate, at various production times. For this forecast, the minimum forecasted production rate is set at:\par
. For the cumulative production forecast, is used to estimate cumulative production increments at different forecasted production times ranging from the last historical date where t=0 to when , and is used to estimate the cumulative production hyperbolic behaviour changes to the exponential behaviour: Also, the constant\par
Calculating the various cumulative production increments at the various forecasted times using the above models at designated regions would generate Table  {\ref 2} in Appendix A.\par
A normal plot of these forecasted cumulative production increments against the respective production times is shown in Figure  {\ref 2}.\par
The forecast in Figure  {\ref 2} is very useful in analyzing at what production time a particular cumulative production increment will be achieved in the future production of the well. It is also applicable when determining the total recovery from a well at any forecasted production time in future.\par
On the other hand, the relationship between the forecasted rates and the forecasted cumulative production increments can be studied. This is very important in analyzing how the decrease in production rate with time is affecting the cumulative increments in production. It could also be used in predicting the forecasted production rate at which no economically viable increase in cumulative production will be experienced. This helps in using the forecast to decide when there will be need for a production-enhancement programme or a post-primary recovery mechanism (secondary or enhanced) to be adopted for further oil recovery from the field. This is based on the estimated reserves remaining in the field at this forecasted production rate.\par
The forecasted rate-cumulative production increment data are graphically shown in Figure \hyperref[fig_5]{3}: 
\section[{ANOTHER POSSIBLE FORECAST Forecasting Total Cumulative Production Increment Till Economic Limit}]{ANOTHER POSSIBLE FORECAST Forecasting Total Cumulative Production Increment Till Economic Limit}\par
To forecast the total possible cumulative incremental production till economic production rate, i.e till is reached, is used. In this forecast, setting the economic limit production rate at , and already estimated above, the forecasted cumulative production increment, is increments from to where . 
\section[{b) Model Validation}]{b) Model Validation}\par
Here comparison is done between the use of a hyperbolic-exponential combined model in predicting or forecasting the performance of a naturally-depleted well, and that of using either a full hyperbolic decline model In doing this, the essential forecast done so far with the developed models is carried out again using a full hyperbolic decline model as well as a full exponential model. Now, forecasting the cumulative production increment using the three models and creating a comparison between them shows how valid the developed models in this work in predicting a well's performance, as well as the lapses involved in using either a hyperbolic or exponential decline type for the full forecast.\par
A comparison using a normal plot of each model's forecasted cumulative production increment results against time is presented in Figure  {\ref 4}.\par
In addition, results obtained from the other forecasts (like cumulative production) when compared with the ones that hyperbolic and exponential declines would have generated if they were to be separately used to forecast these parameters also show that neither hyperbolic nor exponential decline alone could adequately forecast the performance of a naturallyflowing well, hence the need for a combined model as the one developed, analyzed and used in this work. It is either one decline model over-estimates the reserves and the other under-estimates the reserves or vice versa. But in most cases, the hyperbolic over-estimates while the exponential model under-estimates. This is as a result of the hyperbolic decline having its decline rate reducing with time, thereby experiencing a decreasing rate of production decline with time. Exponential decline in its own case has a constant decline rate with time, thereby making the production decline with time to be constant throughout the forecast.\par
The forecast results have satisfactorily validated the reason for the development of the models in this paper, which is showing that in most well performance prediction or forecast involving the use of Decline Curve Analysis (DCA), only one model cannot just be employed for the forecast because most naturally producing wells exhibit mostly the combination of two models. Despite that the hyperbolic decline behaviour that they display at the onset of their production history does not really last for long before exponential decline sets in, still if this period of hyperbolic decline is not considered (or probably taken as exponential), or the remaining exponential decline period is taken to be hyperbolic, it seriously affects the forecast results as shown above with feasible space or shift between the different models' results.\par
Table  {\ref 3} in Appendix A shows in summary the effectiveness of the developed model on cumulative production increment forecast. This is done by applying the developed model in making a cumulative production forecast using an already-established set of production data employed by Long and Davis 3 and Khaled 5 in their works. This is just to show the level of deviation of the developed model's result from the results estimated by Long and Davis 3 , as well as Khaled \hyperref[b4]{5} . It significantly shows that the result generated by the developed model or an exponential model.\par
(for total cumulative production forecast) in this paper correlates well with the results from the works of Long and Davis 3 and Khaled \hyperref[b4]{5} . 
\section[{IV. a) Conclusions}]{IV. a) Conclusions}\par
The use of either hyperbolic or exponential decline model has been discovered to produce unrealistic results in predicting or forecasting the production performance of a naturally-flowing well. This serves as the platform on which this work was based.\par
New models are developed to forecast the production performance of a well in terms of cumulative production increment. These new models combined the conventional hyperbolic and exponential decline models. The transition time from hyperbolic to exponential decline along the production life of such well was estimated by a deterministic approach.\par
It indeed showed in the results generated from the analysis of the data used in this study that neither the hyperbolic nor the exponential decline model could predict or forecast the performance of a naturallyflowing well accurately. The results obtained vividly showed that the two models, (hyperbolic and exponential) either over-estimate the forecasted performance (rate, time and cumulative production increment) or under-estimate it. 
\section[{b) Acknowledgments}]{b) Acknowledgments}\par
The authors appreciate and acknowledge the faculty and staff of the Department of Petroleum Engineering, Covenant University for the conducive atmosphere provided for this study.   \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Volume}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{fig_3}Cumulative}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{fig_4}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-7.png}
\caption{\label{fig_5}CumulativeTable 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-8.png}
\caption{\label{figure8}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-9.png}
\caption{\label{figure9}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-10.png}
\caption{\label{figure10}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.3540045766590389\textwidth}P{0.07196796338672769\textwidth}P{0.2217391304347826\textwidth}P{0.2022883295194508\textwidth}}
7/15/1975\tabcellsep 0\tabcellsep 5.09\tabcellsep 164.19\\
8/15/1975\tabcellsep 1\tabcellsep 24.09\tabcellsep 612.9\\
9/15/1975\tabcellsep 2\tabcellsep 47.6\tabcellsep 783.67\\
9/15/1976\tabcellsep 14\tabcellsep 314.54\tabcellsep 620.33\\
10/15/1976\tabcellsep 15\tabcellsep 333.84\tabcellsep 622.58\\
11/15/1976\tabcellsep 16\tabcellsep 350.83\tabcellsep 566.33\\
7/15/1977\tabcellsep 24\tabcellsep 449.82\tabcellsep 310.97\\
8/15/1977\tabcellsep 25\tabcellsep 459.04\tabcellsep 297.42\\
2/14/1978\tabcellsep 31\tabcellsep 509.88\tabcellsep 257.5\\
3/15/1978\tabcellsep 32\tabcellsep 517.69\tabcellsep 251.94\\
4/15/1979\tabcellsep 45\tabcellsep 616.02\tabcellsep 232\\
6/15/1979\tabcellsep 47\tabcellsep 631.67\tabcellsep 249.33\\
7/15/1980\tabcellsep 60\tabcellsep 728.55\tabcellsep 174.84\\
8/15/1980\tabcellsep 61\tabcellsep 729.88\tabcellsep 42.9\\
3/15/1981\tabcellsep 68\tabcellsep 736.4\tabcellsep 210.32\\
4/15/1981\tabcellsep 69\tabcellsep 746.34\tabcellsep 331.33\\
6/15/1982\tabcellsep 83\tabcellsep 777.74\tabcellsep 10\\
7/15/1982\tabcellsep 84\tabcellsep 778.03\tabcellsep 9.35\\
1/15/1985\tabcellsep 90\tabcellsep 778.19\tabcellsep 5.16\\
8/15/1985\tabcellsep 97\tabcellsep 820.13\tabcellsep 122.9\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
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\end{document}