Uncertainty Analysis of Production Decline Data

Transcription

1 Energy Sources, 27: , 2005 Copyright Taylor & Francis Inc. ISSN: print/ online DOI: / Uncertainty Analysis of Production Decline Data ZSAY-SHING LIN CHANG-HSU CHEN Department of Resources Engineering National Cheng Kung University Tainan, Taiwan G. V. CHILINGAR Environmental Engineering Department University of Southern California Los Angeles, CA, USA H. H. RIEKE Department of Petroleum Engineering University of Louisiana at Lafayette Lafayette, LA, USA SUMIT PAL Superior Energy Services Broussard, LA, USA Decline rate-time equations, including uncertainty characteristics of production data, were developed to estimate the probabilistic reserves and production life of hydrocarbon reservoirs. The uncertainty characteristic, or residual function, is a statistical distribution of the difference between field data and the deterministic production decline curve. The reserves may be estimated directly by summing the production rates, including the residual function, which is referred to as the summation method. Alternatively, other equations for estimating the probabilistic reserves and production life are also derived, based on the integration of the decline time-rate equations that include uncertainty. This is referred to as the integration method. In both methods, Monte-Carlo simulation is used. The uncertainty analysis is applied to the production data from two gas fields located in Taiwan: the Chinshui and the Tiechenshan gas fields. In the case studies, production data for the Chinshui gas field shows an exponential decline, whereas the Tiechenshan gas field exhibits hyperbolic decline. The residual function for the Chinshui gas field is a normal distribution with a mean of zero. In the Tiechenshan gas field, the residual function approximates a normal distribution. Thus, the determined probabilistic reserves for both fields also have a normal Received 27 June 2003; accepted 21 July Address correspondence to G. V. Chilingar, 101 S. Windsor Blvd., Los Angeles, CA 90004, USA. gchiling@usc.edu. 463

2 464 Z.-S. Lin et al. distribution. The range of reserves, which is defined as the difference between proven and mean reserves, is estimated using both summation and integration methods. The Chinshui gas field is 10.6% of mean reserve value, whereas for the Tiechenshan range of probabilisitc reserves of value is 8.9%. The methods derived in this study are more accurate than those determined by the conventional probability method where the range of reserves depends on the assumed ranges of the parameters used. A set of new decline curves equations in residual form, instead of conventional equations of decline curves, are derived. In addition, the equations of cumulative production in residual form are derived. A new method based on the equations of decline curves in residual form and those of cumulative production in residual form are developed (uncertainty analysis method). Case studies with conventional decline curve analysis, probabilistic decline curve analysis, and a new method of uncertainty analysis are presented. The results of the three methods are compared and discussed. Keywords decline curve analysis, gas field, probabilistic reserves, Taiwan, uncertainty analysis Oil and gas are currently the most important energy sources worldwide and have a significant influence upon global economic development. In the 1970s and 1980s, oil and gas supply shortages resulted in economic disorder. Hence, many nations have invested heavily in exploration for oil and gas to stabilize the energy supply. Knoring et al. (1999) pointed out that an exploration strategy applied in a region is reflected in the results of exploration, and defined either as the history of the regional reserve accumulation or the oil and gas field discoveries. Prospecting for oil and gas has risks and high cost, but huge profits are assured if the exploration is successful and the price of oil is above $25 per barrel. In addition to oil and gas prospecting, petroleum companies purchase and develop producing oil and gas fields at discounted prices to maintain or increase reserves. Reserve information is critical to this kind of investment since the recoverable reserves are directly related to the sales value of oil and gas producing properties. The accumulation of exploration results over time is quantified using the sequence and tempo of economic discovery, reserve forecasting, and reserve estimation data. The reserve accrual is achieved by field appraisal. The goal of predicting the validity of the production decline curve information is a precondition of control. Control here is understood as the means by which the uncertainty in the decline curves is accounted for. A new set of decline curve equations that incorporate uncertainty analysis are used to formulate optimum reserve estimation. All reserve estimates involve some degree of uncertainty or probability distribution, depending mainly on the amount of reliable data (e.g., geological, geophysical, engineering, and production) available at the time of the estimate and on the interpretation of these data. The uncertainty of reserves can be classified into proved and unproved reserves. Unproved reserves are further sub-classified into probable and possible reserves (SPE/WPC, 1996). Proved reserves are defined as those quantities of petroleum that can be estimated with reasonable certainty to be commercially recoverable from a given date forward and under current economic and operating conditions and government regulations. In general, these quantities of oil and gas have an existence probability greater than 85 to 95%. Thus, a symbol for proved reserves can be P 90. Probable reserves are the quantities of recoverable hydrocarbons whose data are similar to those used for proved reserves, but lacking in certainty. Probable reserves, denoted by P 50, are generally considered to be those that have a probability for overall reserves to be produced greater than 50%.

3 Uncertainty Analysis of Production Decline Data 465 Possible reserves, denoted by P 15, incorporate more uncertain data that indicate possible reserves to be less likely than probable reserves to be commercially recoverable. Possible reserves usually have a probability for overall reserves to be produced over 15%. The aim of this study is to use the uncertainty characteristic of production data and decline curve analysis to estimate probabilistic reserve and production life. Production data were analyzed from two gas fields (Chinshui and Tiechenshan) in Taiwan. Both deterministic and probabilistic methods were employed and the results from these analyses were compared. Production Data Analysis with Uncertainty Forecasting future production for reserve estimates is the most important part of the economic analysis of exploration and production expenditures. Capen (2001) pointed out that treating reserves probabilistically provided necessary information for reservoir managers to plan future exploration and development. Decline curve analysis, using production data, is a powerful and convenient tool for forecasting the future production from wells, leases, or reservoirs. However, there are several assumptions that should be made before analyzing the production decline data with any degree of reliability. These assumptions include: (1) Production must have been stable over the period being analyzed; that is, a flowing well must have been producing with a constant choke size or constant wellhead pressure. This indicates that the well must have been producing at a capacity under a given set of conditions. (2) Observed production decline should truly reflect the reservoir productivity and not be the result of external causes, such as change in production conditions, well damage, production controls, or equipment failure. (3) Stable reservoir conditions must prevail in order to extrapolate the decline curves with any degree of reliability (Ikoku, 1984). Three types of production decline curves have commonly been used, including constant percentage (exponential), hyperbolic, and harmonic decline. Garb (1985) pointed out that the three types of decline curves can be generalized by the hyperbolic decline equation. In considering the uncertainty of production data, the following three types of decline curves are expressed as: q t(hyp) = q i (1 + bd i t) 1/b + r (1) q t(exp) = q i e Dt + r (2) q t(har) = q i (1 + D i t) 1 + r (3) where q t(hyp) is defined as the half year production rate at time t for hyperbolic decline, q t(exp) is defined as the half year production rate at time t for exponential decline, q t(har) is defined as the half year production rate at time t for harmonic decline, and r is a residual and referred to as the r-function, which is a function of time. It is the difference between the production data and the calculated data from conventional decline curve Eqs. (1), (2), or (3) with r = 0. In terms of probability distribution, the r-function is close to a normal distribution because it is based on regression analysis. In uncertainty analysis, the r-function is a statistical distribution. In a declining stage, a field or reservoir will produce until the production rate reaches an economic limit, which is based on the product price expressed as a production rate at that price. Based on the fact that revenue is balanced with operation cost and taxes,

4 466 Z.-S. Lin et al. the economic limit production rate (q a ) is calculated as follows (Ikoku, 1984): q a = opc G p (1 T x ) (4) where q a is the production rate at the economic limit under present-day existing economic condition, kscm/hy, G p is the produced gas price, $/kscm, T x is the tax fraction, and opc is the producing costs, $. If q a is known, the production rate from q i to q a, can be calculated by using Eqs. (1), (2), or (3), all of which consider uncertainty. The cumulative production from q i to q a, or the reserves, can be numerically estimated by summing the average production rate and multiplying it by the time interval. This calculation is referred to as the summation method. The time, taken from q i declining to q a, is called production life (t a ). In numerical calculations, the production life can be obtained by increasing time, t, in Eqs. (1), (2), or (3) with r = 0, until the production rate declines to q a. Alternately, the production life (t a ) can be calculated by rewriting Eqs. (1), (2), and (3) as follows: t a(hyp) = 1 bd i [ ( ) q b i 1] (q a r) t a(exp) = ln(q i/(q a r)) D t a(har) = [(q i/(q a r)) 1 (7) D i The above equations can be used to estimate the production life with uncertainty. In addition to using Eqs. (1) through (3) directly, cumulative production or reserves can be calculated by integrating Eqs. (1), (2), and (3) with respect to time, t: ta 0 ta 0 ta 0 q t(hyp) dt = q t(exp) dt = q t(har) dt = ta 0 ta 0 ta 0 q i (1 + bd i t) 1 b dt + ta 0 (5) (6) rdt (8) ta q i e Dt dt + rdt (9) 0 ta q i (1 + D i t) 1 dt + rdt (10) 0 Other forms of the cumulative production equations for the hyperbolic decline (N p(hyp) ), exponential decline (N p(exp) ), and harmonic decline (N p(har) ) can be obtained by carrying out integrations on the left sides of Eqs. (8), (9), and (10): [ ( ) ] q (1 b) i qa N p(hyp) = 1 + rt a (11) (1 b)d i q i N p(exp) = q i q a D + rt a (12) N p(har) = q i D i ln q i q a + rt a (13)

5 where Uncertainty Analysis of Production Decline Data 467 n i=1 r = n, and r is the average residual function, r i is the residual or r-function (sampled randomly during Monte Carlo calculations), and n is the number of data points in the production lifespan. If the r-function is zero, then the above equations are the same as those in the conventional decline curve equations (Garb, 1985). The above equations are for estimating the cumulative production from q i (t = 0) declining to q t (t = t). For estimating the reserves, q, (at t = t) should be replaced by q a (t = t a ). This calculation is called the integration method. In using the above uncertainty equations to analyze the production decline data, the parameters in the rate-time equations should be estimated by making the time-rate equation best fit the production data as in deterministic decline curve analysis. The production rate residues, which are functions of time, are then calculated from the difference between rate-time equation and production rate data. The probability distribution or a probability density function for the production rate residuals is obtained and used in the calculations of reserves and production life. The Monte Carlo simulation is conducted with these calculations. r i Case Study of the Chinshui Gas Field The Chinshui gas field in Taiwan has been producing since December of Production rate data was collected semi-annually (half year, hy) from December of 1961 to December of 1998 (Figure 1). The production rate for the Chinshui gas field reached a peak by December of 1973, and the declines in the production data after December of 1973 were used to perform production analysis. Figure 1. Production data for the Chinshui gas field in Taiwan.

6 468 Z.-S. Lin et al. Deterministic Decline Curve Analysis A regression analysis was used to fit three types of time-rate equations (exponential, hyperbolic and harmonic declines) using the production data of the Chinshui gas field. The objective function in the regression analysis was a summation of the residual squares between the regression curve and the production rate data. The objective function value from the harmonic decline curve was found to be (kscm/hy) 2, which is greater than the objective function value from the exponential decline curve (Figure 2) or the hyperbolic decline curve. Both the residual square values from the exponential and hyperbolic decline curves were found to be 2.72 l0 9 (kscm/hy) 2. The harmonic decline curves were thus excluded from further analysis. The hyperbolic decline curve was also excluded because the b value was negative ( ), and gives an undefined value for q. The production data from the Chinshui gas field exhibits an exponential decline. The set of parameters obtained for the exponential decline curve in regression analysis was: q i = 126,976 kscm/hy and D = /hy (Figure 2). The time rate equation can be expressed as follows: q t = e t (14) In the above equation, the initial production rate (at t = 0), which corresponds to December 1973, is about 126,977 kscm/hy. To simplify the future prediction, the initial time was set at the time of the last data point in December The decline curve equation can be rewritten as follows: q t = e t (15) Figure 2. Field data with best-fitted exponential decline curve for the Chinshui gas field.

7 Uncertainty Analysis of Production Decline Data 469 The initial time (t = 0) in Eq. (15) corresponds to December The initial production rate (q i ) is about 25,147 kscm/hy. If the economic limit production rate of 5,000 kscm/hy is used in the production life equation, Eq. (6), with r = 0, then the production life of the Chinshui gas field is years (i.e., half years (hy)), from 1999 to 2024). Using Eq. (12) with r = 0, the reserves or cumulative production (declining from q i = 25,146 kscm/hy to q a = 5,000 kscm/hy) is estimated at kscm. Similarly, if the economic limit production rate is assumed to be 3000 kscm/hy, the estimated reserve is kscm and the production life of 32.8 years. The smaller the economic limit production rate, the larger are the reserves and production life estimates. In this case study, an economic limit production rate of 5,000 kscm/hy is used. Probabilistic Decline Curve Analysis In the probabilistic analysis, the probability distribution of parameters in rate-time equation is assumed and given. From the above deterministic analysis for the Chinshui gas field, the basic parameters of exponential time-rate equation are: q i = 25, kscm/hy, D = /hy. These values were used as the most likely values in a triangle distribution. The variation in these parameters is assumed to be ±10%. The production life and reserves can be calculated using the Monte Carlo simulation from Eq. (6), with r = 0, and Eq. (12), with r = 0, respectively. The results presented in Figure 3 show that gas reserves for the Chinshui gas field are in the range of 546,978 kscm (P 90 ) and 703,800 kscm (P 10 ) (Figure 3). The mean reserve (P 50 ) is 624,168 kscm, and the estimate reserves with the highest probability are 644,036 kscm. The Chinshui gas field has the capability to produce gas for another 45.5 to 54.7 half years (or 22.7 to 27.4 years) (Figure 4). The estimated production life with highest probability is 49.9 half years (or 25 years) (Figure 4). The above calculations assume that the variation in these parameters is 10%. If both the q i and D parameters vary 5%, 15%, and 20%, then the range for the reserves (Figure 5) and production life (Figure 6) increase with the increase in the varying range of these parameters. The percent of variation is arbitrary. In the above calculations, both the q i and D parameters vary in a range between 5% and 20%. In the prediction data, the initial production rate (q i = 25, kscm/hy) is known and fixed. If the variation in the D value is in the range of ±10%, then the reserve estimate is between 575,133 kscm and 675,846 kscm, and the production life is between hy and hy. Performing the simulation with variations of 5%, 15%, and 20% for the D value, the reserves (Figure 7) and production life (Figure 8) increase with increasing range of the parameters. Again, the percent of variation is arbitrary. Uncertainty Analysis Uncertainty analysis requires a residual function that can be derived from the difference between the production rate from the rate-time equation and the actual production rate. In terms of probability distribution, the residual function (or r-function) for the Chinshui gas field is a normal distribution with a mean of zero and a standard deviation ( q std ) of 7, (Figure 9). Equation (12) can be used to estimate production life with r being zero or nonzero. Using Eq. (12), the production life is 49.6 half years with r = 0. When Eq. (6), with r being non-zero, is used, the estimated production life is in the range of 14 half

8 470 Z.-S. Lin et al. Figure 3. Reserves estimated from probability analysis for the Chinshui field. Figure 4. Production life estimated from probability analysis for the Chinshui gas field.

9 Uncertainty Analysis of Production Decline Data 471 Figure 5. Range of estimated reserves by varying the initial production rate and decline rate for the Chinshui gas field. Figure 6. Range of estimated production life by varying the initial production and decline rates for the Chinshui gas field. years (P 90 ) to 81 half years (P 10 ) with a mean of 44 half years (P 50 ). The production life result from uncertainty analysis shows that the production life distribution is not a normal distribution and that the ranges of the production life are wide (Figure 10). The fact that the calculated production life has a non-normal distribution and is unreasonably wide is due to the log-function included in Eq. (6). Alternatively, the production life for the Chinshui gas field is estimated by increasing time, t, in Eq. (2) until the calculated production rate declines to the economic production limit rate. The production life is 49.6 half years. In probabilistic reserves estimation using uncertainty analysis, production rates, declining from q i (= 25,147 kscm/hy) to q a (= 5,000 kscm/hy) for each half year interval between the initial time to production life, can be calculated using Eq. (2) and/or Eq. (12)

10 472 Z.-S. Lin et al. Figure 7. Range of estimated reserves by varying the decline rate for the Chinshui gas field. Figure 8. Range of estimated production life by varying the decline rate for the Chinshui gas field. including an uncertainty or an r-function. In the summation method, using Eq. (2), the summation of the production rate per half year in half year intervals indicates probabilistic reserves in the range of 561,074 kscm (P 90 ) and 694,920 kscm (P 10 ) with a mean of 628,038 kscm (Figure 11). The ranges of estimated reserves are 10.6% of the mean reserves. In the integration method, the reserves of the Chinshui gas field can also be calculated using Eq. (12). The probabilistic reserves estimation can then be obtained by Monte Carlo

11 Uncertainty Analysis of Production Decline Data 473 Figure 9. Probability distribution of residuals for the Chinshui field. Figure 10. Probability of production life by using derived equation in uncertainty analysis for the Chinshui field. simulation. In this case, the proven gas reserves (P 90 ) are estimated to be 547,229 kscm, the probable gas reserves (P 50 ) are 613,401 kscm and the possible gas reserves (P 10 ) are 679,423 kscm (Figure 12). The range of probabilistic reserves (about 10.6% of the mean reserve) estimated from Eq. (12) is almost equal to that calculated from Eq. (2). In the above uncertainty analysis, the residual function or uncertainty characteristic is calculated from production data and used in the probabilistic reserves. In contrast, the parameter variations are assumed to be arbitrary in probabilistic decline curve analysis.

12 474 Z.-S. Lin et al. Figure 11. Probability density function for the reserves estimated by using the summation method in uncertainty analysis for the Chinshui gas field. Figure 12. Probability density function for the reserves obtained by using the integration method in uncertainty analysis for the Chinshui gas field.

13 Uncertainty Analysis of Production Decline Data 475 Figure 13. Production data for the Tiechenshan gas field in Taiwan. Case Study of the Tiechenshan Gas Field The Tiechenshan gas field in Taiwan has been producing since June of Data for the production rate was collected from June of 1965 to December of 1998 in half-year intervals (hy) (Figure 13). The Tiechenshan gas field production rate reached a peak by June of Decline in the production after June of 1977 was adopted for performing the production analysis. Deterministic Decline Curve Analysis Regression analysis was used to fit the three types of time-rate equations, in order to evaluate the Tiechenshan gas field production data. The objective function values from the harmonic decline curve [7.13 l0 10 (kscm/hy) 2 ] and exponential decline curve [ ((kscm/hy) 2 )] were greater than the residual squares from the hyperbolic decline curves [ (kscm/hy) 2 ] (Figure 14). The production data for the Tiechenshan gas field is of the hyperbolic decline type. The parameters obtained for the hyperbolic decline curve were: q i = 744,967 kscm/hy, b = and D = /hy. The time rate equation is as follows: q t = [1 + ( )( t)] 1/ (16) In the above equation, the initial production rate (at t = 0), which corresponds to June of 1977 is 744,967 kscm/hy. The production rate at the end of the production data (December 1998) is 119,480 kscm/hy. In estimating the reserves, Eq. (11) should be applied twice because of the characteristics of the equation and is explained as follows. First, Eq. (11) is used to estimate the cumulative production (N p1 ) from q i (= 744,967 kscm/hy on December 1977) declining to the economic production rate limit of 30,000 kscm/hy. Next, Eq. (11) is used again to estimate the cumulative production (N P 2 ) from

14 476 Z.-S. Lin et al. Figure 14. Field data with best-fitted hyperbolic decline curve for the Tiechenshan gas field. q i (= 744,967 kscm/hy on December 1977) declining to the end of the production (production rate = 119,408 kscm/hy on December 1998). The future economic cumulative production (reserves) is the difference between N P 1 and N P 2. The same procedure should be followed to estimate the production life. The future production life for the Tiechenshan gas field is half years from June 1999 to June Thus, the reserves were estimated, using Eq. (11) with r = 0, at kscm. Probabilistic Decline Curve Analysis From the above deterministic analysis for the Tiechenshan gas field, the basic parameters of hyperbolic time-rate equation are: q i = 744,967 kscm/hy, b = and D = /hy. These values are used as the most likely values in the triangular distribution. Variation in parameters is assumed to be ±10%. The production life and reserves are then calculated using a Monte Carlo simulation using Eq. (5) with r = 0 and Eq. (11) with r = 0, respectively. The results show that the gas reserves (Figure 15) for the Tiechenshan gas field range from 771,348 kscm (P 90 ) to 5,952,543 kscm (P 10 ). As shown in Figure 15, the mean reserves (P 50 ) are 3,365,106 kscm. The Tiechenshan gas field has the capability to produce gas for another to 34.4 years (Figure 16). The production life with highest probability is 27.3 years (Figure 16). If both the q i and D parameters vary by 5%, 15%, and 20%, then the range of reserves (Figure 17) and production life (Figure 18) increase with increasing variance range of the parameters.

15 Uncertainty Analysis of Production Decline Data 477 Figure 15. Estimated reserves using probability analysis for the Tiechenshan gas field. Figure 16. Estimated production life using probability analysis for the Tiechenshan gas field.

16 478 Z.-S. Lin et al. Figure 17. Range of reserves estimated by varying the initial production rate, exponent parameter, and decline rate for the Tiechenshan gas field. Figure 18. Range of estimated production life by varying the initial production rate, exponent parameter, and production decline rate for the Tiechenshan gas field. In the above calculations, the q i, b, and D parameters vary from 5% to 20%. The initial production rate (q i = 744,967 kscm/hy) is known (fixed). If only the D value varies and the simulation is performed with variations of 5%, 15%, and 20%, then the range of reserves (Figure 19) and production life (Figure 20) increase with increasing range of variance in the parameters. Uncertainty Analysis The residual function (r-function) for the Tiechenshan gas field is a normal distribution with a mean of zero and a standard deviation of 30,710 (Figure 21). Equation (11) can be used to estimate production life with r being zero or non-zero. The production life is 98.9 half years from Eq. (11), with r = 0. When Eq. (11) has a

17 Uncertainty Analysis of Production Decline Data 479 Figure 19. Range of estimated reserves by varying the production decline rate for the Tiechenshan gas field. Figure 20. Range of estimated production life by varying the production decline rate for the Tiechenshan gas field. non-zero value, the estimated production life ranges from 13.3 half years (P 90 ) to half years (P 10 ), with a mean of 63.0 half years (P 50 ). The distribution of production life, obtained using the uncertainty analysis, is not normal and the production life range is wide (Figure 22). The reason that the calculated production life has a non-normal distribution, which is unreasonably wide, and is due to the inclusion of log-function in Eq. (11). Alternatively, the production life for the Tiechenshan gas field is estimated by increasing the time, t, in the time-rate Eq. (1), until the calculated production rate declines to the economic limit. The production life is 98.9 half years. In estimating the probabilistic reserves using uncertainty analysis, the production rates declined from 119,408 kscm/hy at the end of production (December 1998) to q a

18 480 Z.-S. Lin et al. Figure 21. Probability distribution of residuals for the Tiechenshan gas field. Figure 22. Probability distribution of estimated production life using the derived equation in uncertainty analysis for the Tiechenshan field. (= 30,000 kscm/hy, for each half-year interval during the production life). The probabilistic reserves can be calculated using Eq. (1) and/or Eq. (11), including an uncertainty or r-function. By summation method, using Eq. (1), the summation of the production rate per half year in half-year intervals is the probabilistic reserves, which are in the range of 3,003,424 kscm (P 90 ) and 3,585,426 kscm (P 10 ) with mean of 3,291,043 kscm (Figure 23). The range of estimated reserves is ±8.9% of mean reserves.

19 Uncertainty Analysis of Production Decline Data 481 Figure 23. Probability density function for the reserves estimated using the summation method in uncertainty analysis for the Tiechenshan gas field. Using the integration method, the reserves of the Tiechenshan gas field can also be calculated using Eq. (11). The probabilistic reserves can be estimated by using Monte Carlo simulation. In this case, the proven gas reserves (P 90 ) are estimated to be 3,060,223 kscm, the probable gas reserves (P 50 ) are 3,349,927 kscm, and the possible gas reserves (P 10 ) are 3,640,635 kscm (Figure 24). The range of probabilistic reserves (about 8.9% of the mean reserves), estimated from Eq. (11), is almost equal to the range calculated from Eq. (1). Summary and Conclusions The decline rate-time equations, including the uncertainty characteristics of the production data, in uncertainty analysis have been derived. The uncertainty analysis was made using two case studies of gas fields in Taiwan. The findings can be summarized as follows: (1) Equations for the summation and integration methods in the uncertainty analysis were derived and used to estimate the probabilistic reserves and probabilistic production life. (2) Uncertainty characteristics, or residual function, were derived from the production data and used in estimating the probabilistic reserves and probabilistic production life. There is no need for an arbitrary guess for the parameters to be varied as in the conventional probabilistic reserves estimations. (3) Uncertainty analysis was made using two case studies of gas fields in Taiwan (Chinshui and Tiechenshan). The production data from the Chinshui gas field shows an exponential decline, whereas the Tiechenshan gas field exhibits a hyperbolic decline.

20 482 Z.-S. Lin et al. Figure 24. Probability density function for the reserves estimated using the integration method in uncertainty analysis for the Tiechenshan gas field. (4) The range in reserves (difference between proven reserves and mean reserves), estimated using both summation and integration methods for the Chinshui gas field, was 10.6% of mean reserves value. In the Tiechenshan gas field, this range was 8.9%. (5) The methods presented in this study seem to be better than the conventional probability method in which the range of reserves depends on ranges assumed for the parameters used. Nomenclature b exponent, dimensionless D decline rate, 1/half year (or 1/hy) D i initial decline rate, 1/half year G P gas price, $/kscm N p(exp) cumulative production for exponential decline, kscm N p(har) cumulative production for harmonic decline, kscm N p(hyp) cumulative production for hyperbolic decline, kscm opc operation cost, $ q a economic limit production rate, kscm/hy q i initial production rate, kscm/ half year q t(exp) production rate (kscm/half year) at time t for exponential decline q t(har) production rate at time t for harmonic decline production rate at time t for hyperbolic decline, kscm/half year q t(hyp)

21 Uncertainty Analysis of Production Decline Data 483 r t t a(exp) t (har) t (hyp) T x residual function or r-function time, half year production life of exponential decline, hy production life of harmonic decline, hy production life of hyperbolic decline, hy tax, fraction References Capen, E. C Probabilistic reserves! Here at last? SPE Reserv. Eval. & Engr. 4(5): Garb, F. A Oil and gas reserves classification, estimation, and evaluation. J. Petrol. Techn. 37(4): Ikoku, C. U Natural Gas Reservoir Engineering, New York: John Wiley & Sons, 503 pp. Knoring, L. D., Chilingar, G. V., and Gorkunkel, M. V Strategies for Optimizing Petroleum Exploration, Houston, TX: Gulf Publ. Co., 323 pp. SPE/WPC Draft reserves definitions. J. Petrol. Techn. 48(8):