SOUTHERN METHODIST UNIV Modeling of Oil Production Operations MODELING OF OIL PRODUCTION OPERATIONS SENIOR DESIGN PROJECT SPRING 1986

Transcription

1 Spring 1986 SOUTHERN METHODST UNV Modeling of Oil Production Operations William A. Carter MODELNG OF OL PRODUCTON OPERATONS SENOR DESGN PROJECT SPRNG 1986 DEPARTMENT OF OPERATONS RESEARCH AND ENGNEERNG MANAGEMENT SCHOOL OF ENGNEERNG AND APPLED SCENCE DALLAS, TEXAS 75275

2 MODELNG OF OL PRODUCTON OPERATONS SENOR DESGN PROJECT SPRNG 1986 BY WLLAM A. CARTER present Abstract An nteger Programming Model was developed to determine the value of oil field production over the life of the field using GAMS.

3 t TABLE OF CONTENTS PAGE SUBJECT... 1 PURPOSE... 1 SCOPE... 1 PLAN OF DEVELOPMENT... 2 THE PROBLEM... 2 FRSTCONSTRANT... 3 SECOND CONSTRANT... 4 THRD CONSTRANT... 4 FOURTH CONSTRANT... 5 GAMS... 5 NTERPRETATONOF RESULTS... 5 CONCLUSONS... 7 BBLOGRAPHY... 9

4 1 MODELNG OF OL PRODUCTON OPERATONS SUBJECT The subject of this report is modeling of oil production operations. The model have developed will enable the oil driller to maximize profit over a given period of time. t will also list the wells to be turned on at the most profitable time. PURPOSE The purpose of this report is to determine the maximum present value of oil field production output given a finite number of wells and a finite amount of time. This report also demonstrates how an integer programming model is applied to an oil reservoir to get this maximum profit. SCOPE This report is based on information provided to me by Dr. Julius S. Aronof sky of Gaffney, dine and Associates, nc. and by Dr. Richard Barr. Data used in the model was gathered from the article Optimization Methods in Production Operations by Dr. Julius Aronof sky and from consulting with Dr. Aronof sky. The model was written in GAMS (General Algebraic Modeling System) and was run on the BM Main Frame at the Bradfield Computing Center.

5 OA PLAN OF DEVELOPMENT The objective of this report is not only to illustrate how to maximize present value of oil field production output, but also to demonstrate how the model was developed. With this understanding, have developed the report as follows: 1. The first section will examine the problem and define the objective function along with its constraints. 2. The next section will explain why used GAMS to compute my model. 3. The third section will discuss the interpretation of the results. 4. The last section will discuss conclusions have drawn from this study. THE PROBLEM A major problem facing drillers today is when and how m oil wells should be used to maximize profit. With the somewhat new, relatively inexpensive, and easily accessible computer, the oil industry is no longer confined to guessing when to drill. The problem lam presenting is one which chose from a number of problems that Dr. Aronof sky has come up with in his article Optimization Methods in Production Operations. As have explained, the problem is how to maximize one's profit (present value) given a number of wells and a certain time period. This is the objective function. have defined the

6 3 objective functi-on 4s: - / Max: Z ( X t) - OC (X1 ) - F(1+.R.)_t(Yjt) Figures A and A-i provide a summation of the model. Z is the selling price per barrel of oil. t is multiplied by Xit which is a variable for (i), the oil well represented during (t) the time period represented. This is the oil revenues. CC is the operating costs for well (i) during time (t), and is subtracted from the oil revenues. Fixed costs (pumping costs) are also subtracted from the oil revenues as represented by the future fixed costs equations, which brings the future fixed costs back to the present time period. This part of the objective function is multiplied by it This is the integer programming part of the model. This variable will tell the computer to open or don't open well (i) in time (t) as represented by a 0 or a 1. At this time will say that due to the GAMS language not being able to run an integer program, this part of the objective function, and also part of the constraints, were not completely correct. was in hopes of having access to an MPSX crossover that would run my computer model as a true integer program. Unfortunately this was not the case. will discuss this issue further in a later part of this report. FRST CONSTRANT The first and most binding constraint is to not pump more

7 MODEL F = FUTURE FXED COSTS. OC = OPERATNG COSTS PER BARREL. Z = SELLNG PRCE PER BARREL. t = TME PEROD R = BARRELS N RESERVOR. it = YLD = YELD PONT PER MONTH. i = WELL NUMBER Q = NTAL BARRELS PUMPED t = 1 (DO) PER MONTH. Yit = R = NTEREST RATE. i = 0 (DON'T) MAX: MAXMZE PROFT (PRESENT VALUE) OF A SPECFED OL FELD AND TS OL WELLS ***** Z ( X t) - OC ( X1t) - F(1 + R)_t(Yjt) st: DON'T PUMP MORE THAN THE YELD PONT FROM YOUR FELD 36 (1) XXil E. YLD i=1 36 X12 C. YLD i=1 ** LMT ON AMOUNT PUMPED FROM EACH WELL DURNG EACH TME PEROD ** (2) X11 Q X21 Q = t=1 X101 Q X12 Q - Q/R(X11) X22 ± Q - Q/R(X21) = t=2 X102 Q - Q/R(x101) X13 4 Q - Q/R(X11 $ X) ± Q - Q/R(X 21 + X22) = t=3 X103 Q - Q/R(x 101 $ x2) FGURE A

8 CAN'T PUMP FROM AN UNOPENED WELL, BUT ONCE WELL S OPENED YOU CAN PUMP ***** (3) X 4 YLD(Y11 ) t = 1 X12 YLD(Y 11 +Y12 ) t=2 X13 YLD(Y 11 +Y12 +Y13 ) t=3 YLD(Y 11 + Y12 + Y13 + Y14 + Y Y136 ) t = 36 OPEN EACH WELL ONLY ONCE () Yll + ' Y 136 ( Y236i + Y Y136 1 FGURE A-i

9 4 than, but as much as possible, the oil yielding point for well (i) during time (t). Figure B represents this. The straight line represents the field output at its maximum, this is also the most favorable amount pumped. The line will be straight until the oil reserves start to dwindle, at which time the model will abandon due to economic reasons. SECOND CONSTRANT The second constraint limits the amount pumped from each well (i) during each time period (t). During the first time period each of the wells can pump up to an initial amount (Q). During the second time period the wells can pump out the initial amount (Q) minus what has already been pumped out during time one (Q/R(X11)). This continues in a progressive manor over time as represented in Figure A. Figure C represents this graphically. As can be seen, the amount pumped during time 2 is added to the amount already pumped during time 1. Thus, the amount of oil being pumped is increasing. THRD CONSTRANT The third constraint prohibits an unopened well from being pumped, but once the well is opened it can be active. This constraint involves part of the integer programming as discussed earlier. (Y) is multiplied by the yield point so as to either open up the well or not open up the well (i) in time (t) (Fig. A-i). This is also progressive through time.

10 FELD OUTPUT t 2 t4 FGURE B abandon....t36

11 FELD CAPACTY PER MONTH TME FGURE C

12 5 FOURTH CONSTRANT The fourth constraint opens up each well only once as represented in Figure A-i by the integer variable Y. The sum of the entire time period for each individual well will be less than or equal to 1 (open once). This constraint is needed because it is not economical to open and close an oil well that is already pumping oil. This would require added costs of manpower. f the well runs out of oil it will close. GAMS GAMS is an algebraic computer language which enables the user to make summations of constraints for a modeling or other mathematical problem. Through my use of GAMS, saved myself from writing out many thousands of constraints as one would have had to have done with this problem had it been written with LNDO. had only four constraint 1GAMS is an incredible help because it lets the computer do all the work. Figure D is a copy of my model in GAMS. NTERPRETATON OF RESULTS The results of this report came out to be fairly reasonable considering that GAMS did not assign a 0 or 1 as a true integer programming language would have done. Before give the maximum profit, will interpret the data shown in Figure A and Figure D, that have assigned to my model. These values were given to me by Dr. Aronofsky. t is very

13 1 *THS PROGRAM WLL MAXMZE THE PROFT OF A SPECFED OL 2 *FELD AND TS OL WELLS. 3 4 SETS 5 WELL T TME T 9 /T_1*T_ ALAS (T, TPAST) SCALAR 14 F FUTURE FXED COSTS / OC OPERATNG COSTS PER BARREL / Z SELLNG PRCE PER BARREL / R BARRELS N RESERVOR / 25, YLD YELD PONT / 6, Q NTAL BARRELS PUMPED / 3, R NTEREST RATE / VARABLES 23 TOTALPROF TOTAL PROFT 24 X(,T) LMT ON AMOUNT PUMPED 25 Y(,T) OPEN EACH WELL ONLY ONCE 26 POSTVE VARABLES X, Y; EQUATONS 29 PROFT OBJECTVE FUNCTON 30 YELD(T) YELD CONSTRANT 31 SUPPLY(,T) AVALABLTY CONSTRANT 32 OPENWELL(,T) ONCE WELL S OPEN T CAN PUMP 33 OPENONCE() OPEN EACH WELL ONLY ONCE; PROFT. TOTALPROF =E= SUM( (,T), Z*X(,T)-X(,T) 36 _F*Y(,T)*(1+R)**(_ORD(T))); YELD(T).. SUM(, X(,T)) =L= YLD; SUPPLY(,T).. X(,T,) =L= 41 Q_Q/R*SUM(TPAST$(ORD(TPAST) LT ORD(T)), X(,TPAST)); OPENWELL(,T) =L= 44 YLD*SUM(TPAST $ (ORD(TPAST) LE ORD(T)), Y(,TPAST)); OPENONCE ().. SUM( T, Y (, T)) L 1; Y.UP(,T) = 1; MODEL OL /ALL/; SOLVE OL USNG LP MAXMZNG TOTALPROF; DSPLAY X.L, Y.L, TOTALPROF.L; FGURE D

14 6 important to understand that these values represent a time period when oil was very cheap by today's standards. They are true values. This problem of oil production operations was written several years ago, and because of this have decided that these values should parallel with the time period which Dr. Aronofsky wrote Optimization Methods in Production Operations. Some simple calculations can bring these values up to the present time. The number of barrels in the reservoir, the yield point, and the initial barrels pumped will remain constant. They do not need adjusting. VALUES: Future Fixed Costs (F) = $60,000 - this is a rough estimate of all the fixed costs. Operating Costs per Barrel (OC) = $ this is an average cost per barrel. Selling Price Per Barrel (Z) = $ this is the operating cost per barrel plus the profit per barrel of 30 cents. Barrels in Reservoir (R) = 25,000 - this number was derived by multiplying 50 acres (the area for 10 wells) by 500 barrels per acre. Yield Point per Month (YLD) = 6,000 - this is the maximum barrels pumped per month by each well.

15 7 nitial Barrels Pumped per Month (Q) = 3,000 - this is the initial barrels a well can pump for a month. nterest Rate (R) =.10 - this is a typical interest rate. The total profit was calculated to be $9, This is the profit of ten oil wells over a period of three years (36 months in my model). This profit is not large by any means, but as stated earlier, this is during. a time period when a barrel of oil cost $2.20 and there was only a profit of 30 cents. The model did not open up any wells during the first time period but opened up the appropriate wells after the first period in order to maximize the profit. The values assigned to the variable Y it were generated systematically by the computer program instead of as a 0 or 1 as would be the case for a true integer program. - CONCLUSONS chose a time period of 36 months because of limitations on the BM computer. The computer does not have enough storage space, at this time, to handle a model that is much greater than 36 time periods. attempted to use 48 time periods to represent four years, but, even with ten cylinders of temporary storage, could not complete the execution of this large array. chose ten oil wells because this number fits

16 8 readily with the rest of the data. Ten oil wells is sufficient to get a good understanding of this report. The results of this oil production operations model were not as accurate as they should have been due to the limitation of GAMS not being able to interpret integers as needed it to. f the MPSX crossover had been available, the results would have been more accurate because the integer values would be assigned correctly. As it stands, the GAMS language did a reasonably good job, even though the oil wells may not have opened at the precise time. The major advantage of GAMS, though, is that it saved me from writing out many constraints. CAMS is a relatively new language, but it will certainly be of great use to the oil industry because of the many areas within the industry that it can be applied to.

17 9 BBLOGRAPHY Aronof sky, Julius S. odtimization Methods in Production Operations, Gaffney, Cline and Associates, nc. Kendrick, David; Meeraus, Alexander. GAMS An ntroduction, Development Research Department - The World Bank; February Schrage, Linus. Linear, nteger, and Quadratic Programming with LNDO, The Scientific Press; 1984.