1.818J/2.65J/3.564J/10.391J/11.371J/22.811J/ESD166J SUSTAINABLE ENERGY. Prof. Michael W. Golay Nuclear Engineering Dept.

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1 1.818J/2.65J/3.564J/10.391J/11.371J/22.811J/ESD166J SUSTAINABLE ENERGY Prof. Michael W. Golay Nuclear Engineering Dept.

2 RESOURCE EVALUATION AND DEPLETION ANALYSES 1

3 WAYS OF ESTIMATING ENERGY RESOURCES Monte Carlo Hubbert Method Extrapolation Expert Opinion (Delphi) 2

4 FACTORS AFFECTING RESOURCE RECOVERY Nature of Deposit Fuel Price Technological Innovation Deep drilling Sideways drilling Oil and gas field pressurization Hydrofracturing Large scale mechanization 3

5 URANIUM AREAS OF THE U.S. 4 Courtesy of U.S. Atomic Energy Commission.

6 MAJOR SOURCES OF URANIUM Class 1 Sandstone Deposits Share U3O8 Concentration (Percent) Tons U3O8 New Mexico Total Wyoming ,000 Utah Š $10/lb Colorado Texas Other Class 2 Vein Deposits 7,100, Class 3 Lignite Deposits ,200 Class 4 Phosphate Rock Class 5 Phosphate Rock Leached ,600 Zone (Fla.) Class 6 Chattanooga Shale ,557,300 Class 7 Copper Leach Solution ,000 Operations Class 8 Conway Granite Uranium 1x Thorium 4x106 Class 9 Sea Water 0.33x10-6 4x109 5

7 ESTIMATES OF URANIUM AVAILABILITY FROM GEOLOGICAL FORMATIONS AND OCEANS IN THE U.S Millions of Tons U 3 O S 30 S 10 Conventional Shale Shale Granite Shale Granite Seawater ppm ppm ppm ppm ppm 4-10 ppm ppm Image by MIT OpenCourseWare. 6

8 DECLINE IN GRADE OF MINED COPPER ORES SINCE % Copper in ORE Image by MIT OpenCourseWare. 7

9 RECOVERY BY IN-SITU COMBUSTION 8

10 MONTE CARLO ESTIMATION Yield from Region Y n Σ j j=1 Y = Y (Eq. 1) Y Yield from Zone 1, y 1 Yield from Zone 2, y 3 Yield from Zone n, y n y 1 y 2 y n Probability density functions are obtained subjectively, using information about deposit characteristics, fuel price, and technology used. 9

11 MONTE CARLO ESTIMATION OF THE PROBABILITY DENSITY FUNCTION OF A FUNCTION OF A SET OF RANDOM VARIABLES, AS G = G( Z), where (Eq. 1) Z = [ y 1, y 2, K, y n ], and Y i is a random variable (i = 1,n) i Note that Z and G are also random variables. 10

12 MONTE CARLO ESTIMATION OF PROBABILITY DENSITY AND CUMULATIVE DISTRIBUTION FUNCTIONS 1 Area = 1 dy i y imin y i y imax y imin y i y imax Prob. ( y i < Y i < y i + δy i )= f Yi ( y i )dy i (Eq. 2) Prob. ( Y i < y i )= F Yi y i y i ( ) = f Yi ( )d y imin y i y i (Eq. 3) Consider Y to be a random variable within i [ y imin, y ] imax 11

13 MONTE CARLO ESTIMATION, Continued Note: F Yi ( y i ) is uniformly distributed within [0, 1] 12

14 MONTE CARLO ESTIMATION, Continued 1. Utilize a random number generator to select a value of F(y i ) within range [0, 1] corresponding value of y i (Eq. 3). 2. Repeat step 1 for all values of i and utilize selected values of Z 1 = y 11, y 21, L, y n1 to calculate a value of (Eq. 1), [ ] (note Z is also a random variable). 3. For the k-th set of selected values of Z K = y 1K, y 2K,L, y n one K can obtain the corresponding value of G K = G ( K Z ) K 4. Repeat step 2 many times and obtain a set of values of vector Z, and corresponding value of G k. 5. Their abundance distributions will approximate those of the pdfs of the variables Z and. G Z as Z 1 [ ] 13

15 M. KING HUBBERT S MINERAL RESOURCE ESTIMATION METHOD ASSUMED CHARACTERISTICS OF MINERAL RESOURCE EXTRACTION As More Resource Is Extracted The Grade Of The Marginally Most Attractive Resources Decreases, Causing Need for improved extraction technologies ogi es Search for alternative deposits, minerals Price increases (actually, rarely observed) 14

16 M. KING HUBBERT S MINERAL RESOURCE ESTIMATION METHOD, Continued POSTULATED PHASES OF MINERAL RESOURCE EXTRACTION Early: Low Demand, Low Production Costs, Low Innovation Growing: Increasing Demand And Discovering Rate, Production Growing With Demand, Start of Innovation Mature: Decreasing Demand And Discovery Rate, Production Struggling To Meet Demand, Shift To Alternatives Late: Low Demand, Production Difficulties, Strong Shift To Alternatives (rarely observed) 15

17 U.S. Natural Gas Reserves Proved Reserves Trillions of cubic feet (As of Dec. 31) Additions Production AGA committee on natural gas reserves Image by MIT OpenCourseWare. Adapted from the American Gas Association. 16

18 U.S. NATURAL GAS PRODUCTION Comparison of estimated (Hubbert) production curve and actual production (solid line). Courtesy of U.S. DOE. 17

19 U.S. CRUDE OIL PRODUCTION Comparison of estimated (Hubbert) production curve and actual production (solid line). Courtesy of U.S. DOE. 18

20 COMPLETE CYCLE OF WORLD CRUDE-OIL PRODUCTION ( ca. 10% of total resources) Image by MIT OpenCourseWare. 19

21 RESOURCE BEHAVIOR UNDER HUBBERT ASSUMPTIONS Timing: t d, t r, t p are times of respective maxima of Q d, Q r, Q p. 20

22 LOGISTIC FUNCTION Hubbert s assumed logistical relationships Rate of Production ÝQ P = d [ Q P ( X )] = rq dt P ( X)1 Q P X K Cumulative Production ( ) Q P ( X) = ( ) KQ Po X Q P o( X ) ( Q Po ( X) K)e rx where r = relative rate of change K = carrying capacity or ultimate production value Q Po Q P ( X = 0), ÝQ P ( X) K Q P ( X) K Q P (X) = cumulative production at time, (t - t o ) = X r

23 LOGISTIC FUNCTION, continued Q P ( ) = 0 Q P ( 0) = K 2 = Q P O P ( ) = K ( ) dq ( P X) ÝQ x dt = dq ( P X) d X max X=0 = = r K 4. Q P ( X) X X K 2πσ e 1 2 σ d X, where σ = π r. Let : t - t o X. 22

24 EQUATIONS Conservation of Resource: Q d ( t) = Q r ( t) + Q p ( t) (Eq. 4) Rate Conservation: ÝQ d ( t) = Q Ý r ( t) + Q Ý p ( t) (Eq. 5) Approximate Results: t ( Q Ý d = 0) t ( Q Ý r = 0)= 2τ (Eq. 6) ( t τ r t ) p (Eq. 7) ( t d t ) r or t r 1 2 t d + t p ( ) (Eq. 8) Q p 2Q ( ultimate d t ) d (Eq. 9) 23

25 EQUATIONS, Continued If we assume Gaussian distributions for Q r ( t), ÝQ d ( t) and ÝQ p ( t) with each having the same standard deviation, σ, obtain 2πσ exp 1 2 t t r 2 σ (Eq. 10) Q d 1 t t 2 o ÝQ d ( t) = 2πσ exp d 2 σ (Eq. 11) Q, po ÝQ p ( t) = 2πσ exp 1 2 t t p 2 σ (Eq. 12) Then, when Q r is at a maximum t = t r and ÝQ r = 0 1xQ ÝQ ( ro r t r )= σ 2 σ 2 = 1xQ ( r t ) r ÝQ (, or r t ) r (Eq. 13) Q r ( t) = Q ro 24

26 EQUATIONS, Continued For the normal distribution 1 f ( z) = 2π e 1 2 z2 σ f ( 1) = 0.67, z t t o σ F z = 1 25 yr. for U.S. petroleum and natural gas ( ) = f z z ( )d z, Cumulative distribution function F(3) = 0.99, Approximately the state of full depletion Time of exploitation σ = 150 yrs 6 End date of major U.S. oil, natural gas production = =

27 SUBJECTIVE PROBABILITY STUDY STATE OF NEW MEXICO 26 Courtesy of U.S. Atomic Energy Commission.

28 NEW MEXICO SUBJECTIVE PROBABILITY STUDY (AFTER DELPHI) 27 Courtesy of U.S. Atomic Energy Commission.

29 MIT OpenCourseWare J / 2.650J / J / 1.818J / 2.65J / J / J / J / ESD.166J Introduction to Sustainable Energy Fall 2010 For information about citing these materials or our Terms of Use, visit: