1.818J/2.65J/3.564J/10.391J/11.371J/22.811J/ESD166J

SUSTAINABLE ENERGY

Prof. Michael W. Golay Nuclear Engineering Dept.

RESOURCE EVALUATION

AANND D DDEPEPLETILETION ON AANNAALYLYSES SES

1

WAYS OF ESTIMATING ENERGY RESOURCES

• Monte Carlo

• “Hubbert” Method Extrapolation

2

• Expert Opinion (Delphi)

FACTORS AFFECTING RESOURCE RECOVERY

• Nature of Deposit

• Fuel Price

• Technological Innovation

3

� Deep drilling � Sideways drilling � Oil and gas field

pressurization � Hydrofracturing � Large scale mechanization

4

URANIUM AREAS OF THE U.S.

Courtesy of U.S. Atomic Energy Commission.

,

MAJOR SOURCES OFURANIUM

Class 1 – Sandstone Deposits U3O8 Concentration

Share (Percent) Tons U3O8

New Mexico .49 0.25 Total Wyoming .36 0.20 315,000 Utah .03 0.32 Š $10/lb Colorado .03 0.28 Texas .06 0.28 Other .03 0.28

Class 2 – Vein Deposits 7 100Class 2 – Vein Deposits 7,100

Class 3 – Lignite Deposits 0.01-0.05 1,200

Class 4 – Phosphate Rock 0.015

Class 5 – Phosphate Rock Leached 0.010 54,600 Zone (Fla.)

Class 6 – Chattanooga Shale 0.006 2,557,300

Class 7 – Copper Leach Solution 0.0012 30,000 Operations

Class 8 – Conway Granite 0.0012-Uranium 1x106

0.0050-Thorium 4x106

Class 9 – Sea Water 0.33x10-6 4x109 5

ESTIMATES OF URANIUM AVAILABILITY FROM GEOLOGICAL FORMATIONS AND OCEANS IN

THE U.S.

6

Conventional Shale Shale ShaleGranite Granite Seawater

60-80 ppm 25-60 ppm 10-20 ppm 10-25 ppm 4-10 ppm 0.003 ppm

S 10

S 30

0

2

4

6

8

200

1800

4000

Mill

ions

of T

ons U

3O8

700-2100 ppm

Image by MIT OpenCourseWare.

DECLINE IN GRADE OF MINED COPPER ORES SINCE 1925

7

2.5

2.0

1.5

1.0

0.51925 1930 1935 1940 1950 1955 1960 19651945

% C

oppe

r in

OR

E

Image by MIT OpenCourseWare.

RECOVERY BY IN-SITU COMBUSTION

8

MONTE CARLO ESTIMATION

Yield from Region Y

Y = Y

(Eq. 1)

Σ jj=1

n

Y

Probability density functions are obtained subjectively, using information about deposit characteristics, fuel price, and technology used.

Yield from Zone 1, y1

Yield from Zone n, yn

Yield from Zone 2, y3

y1 y2 yn

9

i

MONTE CARLO ESTIMATION OF THE PROBABILITY DENSITY FUNCTION OF A

FUNCTION OF A SET OF RANDOM VARIABLES, AS

Yi is a random variable (i = 1,n)

G = G Z( ), where

Z = y1, y2, K , yn[ ], and

(Eq. 1)

Note that and G are also random variables. Z

10

⇒

yiyimin yimax

yiyimin yimax

dyi

Area = 1

1

MONTE CARLO ESTIMATION OF PROBABILITY DENSITY AND CUMULATIVE

DISTRIBUTION FUNCTIONS

11

Prob. yi < Yi < yi + δyi( )= fYi yi( )dyi (Eq. 2) Prob. Yi < yi( )= FYi

yi( ) = fYi y imin

y i

∫ ′yi( )d ′yi

(Eq. 3)

Consider Y to be a random variable withini yimin , y imax [ ]

MONTE CARLO ESTIMATION, Continued

Note: is uniformly distributed within [0, 1] FYi yi( )

12

3. For the k-th set of selected values of oneZ = y , y , , y

1. Utilize a random number generator to select a value of F(yi) within range [0, 1] ⇒ corresponding value of yi (Eq. 3).

2. Repeat step 1 for all values of i and utilize selected values of

to calculate a value of ,

(Eq. 1)

(note is also a random variable).

Z1 = y11 , y21

, L , yn1[ ]

MONTE CARLO ESTIMATION, Continued

Z1 Z

[ ]

13

3. For the k-th set of selected values of

one

can obtain the corresponding value of

.

4. Repeat step 2 many times and obtain a set of values of vector , and

corresponding value of Gk. 5. Their abundance distributions

will approximate those of the pdfs of the variables and as

Z

ZK = y1K , y2K

,L , ynK[ ] GK = G K ZK( )

Z G Z⎛

⎝⎜

⎞

⎠⎟

Need for im oved extraction technol es

M. KING HUBBERT’S MINERAL RESOURCE ESTIMATION METHOD

• As More Resource Is Extracted The Grade Of The Marginally Most Attractive Resources Decreases, Causing

� Need for improved extraction technologies

ASSUMED CHARACTERISTICS OF MINERAL RESOURCE EXTRACTION

pr ogi

� Search for alternative deposits, minerals

� Price increases (actually, rarely observed)

14

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 InnovationGrowing 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

16

1975197019651960195519501947

Proved Reserves

Additions

Production25

253035

40

160

180

200

220

240

260

280

300

320

20

20

15

15

10

10

5

50

25

253035

40

160

180

200

220

240

260

280

300320

20

20

15

15

10

10

5

50

Trillions of cubic feet

U.S. Natural Gas Reserves

AGA committee on natural gas reserves

(As of Dec. 31)

Image by MIT OpenCourseWare. Adapted from the American Gas Association.

U.S. NATURAL GAS PRODUCTION

Comparison of estimated (Hubbert) production curve and actual production (solid line). 17

Courtesy of U.S. DOE.

U.S. CRUDE OIL PRODUCTION

Comparison of estimated (Hubbert) production curve and actual production (solid line). 18

Courtesy of U.S. DOE.

COMPLETE CYCLE OF WORLDCRUDE-OIL PRODUCTION

19

(ca. 10% of total resources)

Image by MIT OpenCourseWare.

RESOURCE BEHAVIOR UNDER “HUBBERT” ASSUMPTIONS

Timing: td, tr, tp are respective Qd, Qr, Qp.

times of maxima of

20

KQ X( )

LOGISTIC FUNCTION

r 4

ÝQP X( ) KÝQP =

d QP X( )[ ] dt

= rQP X( )1− QP X( )

K

⎡

⎣⎢

⎤

⎦⎥,

Hubbert’s assumed “logistical’ relationships

Cumulative Production

Rate of Production

1 2

QP X( )= KQPo

X( )

QPo X( )− QPo X( )− K( )e−rX

QPo ≡ QP X = 0( )

QP(X) = cumulative production at time, (t - to) = X

where r = relative rate of change K = carrying capacity or

ultimate production value

QP X( ) K

21

=

LOGISTIC FUNCTION, continued

QP −∞( )= 0 QP 0( )= K 2

= QPO P ∞( ) = K

ÝQ x( )≡ dQP X( )

dt max

= dQP X( )

d X X= 0

= r K 4

.

22

max X 0

QP X( ) ≈ K 2πσ

e −1 2

′Xσ

⎛

⎝⎜

⎞

⎠⎟2

d ′X , where −∞

X

∫ σ = 8 π

1 r .

Let : t - to ≡ X.

EQUATIONS

Conservation of Resource: Qd t( ) = Qr t( ) + Qp t( ) (Eq. 4)

Rate Conservation: ÝQd t( ) = ÝQr t( ) + ÝQp t( ) (Eq. 5)

Approximate Results: Approximate Results: t ÝQd = 0( )− t ÝQr = 0( )= 2τ (Eq. 6)

τ ≈ t r − t p( ) td − t r( )

⎧ ⎨ ⎪

⎩⎪ (Eq. 7)

or

t r ≈ 1 2

td + tp( ) (Eq. 8)

Qp ultimate ≈ 2Qd td( ) (Eq. 9)

23

t = exp − ⎜ ⎟ q. d 1 t − t⎛ ⎞

EQUATIONS, Continued

If we assume Gaussian distributions for

,

with each having the same standard deviation, σ, obtain Qr t( ), ÝQd t( ) and ÝQp t( )

Qr t( )= Qro

2πσ exp −

1 2

t − t r σ

⎛

⎝⎜

⎞

⎠⎟

2⎡

⎣⎢ ⎢

⎤

⎦⎥ ⎥

(Eq. 10)

ÝQ ( ) Qdo 1 t − td⎛ ⎞

2⎡ ⎢

⎤ ⎥ (E 11)ÝQd t( )= o

2πσ exp −

2 d

σ⎝⎜

⎠⎟

⎣⎢ ⎢ ⎦

⎥ ⎥

(Eq. 11)

ÝQp t( )= Qpo

2πσ exp −

1 2

t − t p

σ

⎛

⎝⎜

⎞

⎠⎟

2⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

(Eq. 12)

Then, when Qr is at a maximum t = tr and

, or

ÝQr = 0

ÝÝQr t r( )= −1xQro

σ2 ⇒ σ2 = −1xQr t r( )

ÝÝQr t r( ) (Eq. 13)

24

EQUATIONS, Continued

For the normal distribution

f z( )= 1

2π e

− 1 2

z2

f 1( )= 0.67, z ≡ t − t o

σ = 1

⇒ σ 25 yr. for U.S. petroleum and natural gas ≅

⇒ Time of exploitation 6

σ = 150 yrs ≅

⇒ End date of major U.S. oil, natural gas production

F z( )= f ′z( )d ′z − ∞

z∫ , Cumulative distribution function

F(3) = 0.99, Approximately the state of full depletion

= 1900 + 150 = 2050 25

SUBJECTIVE PROBABILITY STUDY – STATE OF NEW MEXICO

26

Courtesy of U.S. Atomic Energy Commission.

NEW MEXICO SUBJECTIVE PROBABILITY STUDY (AFTER DELPHI)

27

Courtesy of U.S. Atomic Energy Commission.

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