1.818J/2.65J/3.564J/10.391J/11.371J/22.811J/ESD166J
SUSTAINABLE ENERGY
Prof. Michael W. Golay
Nuclear Engineering Dept.
RESOURCE EVALUATION
AND DEP
EPLETI
LETION
ON ANALY
LYSES
SES
1
WAYS OF ESTIMATING ENERGY
RESOURCES
• Monte Carlo
• “Hubbert” Method Extrapolation
• Expert Opinion (Delphi)
2
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
URANIUM AREAS OF THE U.S.
4
Courtesy of U.S. Atomic Energy Commission.
MAJOR SOURCES OF
URANIUM
Class 1 – Sandstone Deposits
New Mexico
Wyoming
Utah
Colorado
Texas
Other
Share
.49
.36
.03
.03
.06
.03
U3O8 Concentration
(Percent)
0.25
0.20
0.32
0.28
0.28
0.28
Class 2 – Vein Deposits
Tons U3O8
Total
315,000
Š $10/lb
7,100
7
,100
Class 3 – Lignite Deposits
0.01-0.05
Class 4 – Phosphate Rock
0.015
Class 5 – Phosphate Rock Leached
Zone (Fla.)
0.010
54,600
Class 6 – Chattanooga Shale
0.006
2,557,300
Class 7 – Copper Leach Solution
Operations
0.0012
30,000
Class 8 – Conway Granite
Class 9 – Sea Water
1,200
0.0012-Uranium
0.0050-Thorium
1x106
4x106
0.33x10-6
4x109
5
ESTIMATES OF URANIUM AVAILABILITY FROM
GEOLOGICAL FORMATIONS AND OCEANS IN
THE U.S.
4000
Millions of Tons U3O8
1800
200
8
6
4
S 30
2
S 10
0
Conventional
700-2100 ppm
Shale
Shale
Granite
Shale
Granite
Seawater
60-80 ppm
25-60 ppm
10-20 ppm
10-25 ppm
4-10 ppm
0.003 ppm
Image by MIT OpenCourseWare.
6
DECLINE IN GRADE OF MINED
COPPER ORES SINCE 1925
% Copper in ORE
2.5
2.0
1.5
1.0
0.5
1925
1930
1935
1940
1945
1950
1955
1960
1965
Image by MIT OpenCourseWare.
7
RECOVERY BY IN-SITU
COMBUSTION
8
MONTE CARLO ESTIMATION
Yield from
Region Y
n
Y = ΣYj
j=1
(Eq. 1)
Y
Yield from
Zone 1, y1
Yield from
Zone 2, y3
Yield from
Zone n, yn
y1
y2
yn
Probability density functions are obtained subjectively, using information
about deposit characteristics, fuel price, and technology used.
9
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 = y1, y 2 , K , y n , and
Yi is a random variable (i = 1,n)
Note that Z and G are also random variables.
10
MONTE CARLO ESTIMATION OF
PROBABILITY DENSITY AND CUMULATIVE
DISTRIBUTION FUNCTIONS
1
⇒
Area = 1
dyi
yimin
yi
Prob. (y i < Yi < y i + δy i ) = fYi (y i )dy i
yimax
yimin
(
yimax
yi
)
( )= ∫
(Eq. 2) Prob. Y < y = F y
i
i
Y
i
[
i
yi
yi
Consider Yi to be a random variable within y i min , y i max
min
( )
fY y′i dy′i
i
(Eq. 3)
]
11
MONTE CARLO ESTIMATION,
Continued
Note: FYi (y i ) is uniformly distributed within [0, 1]
12
MONTE CARLO ESTIMATION,
Continued
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
Z1 = y1 , y 2 , L , y n to calculate a value of Z (Eq. 1)
1
1
1
1
,
(note Z is also a random variable).
3. For the k-th set of selected values of Z K = y1 , y 2 , L , y n one
[
]
[
K
(
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 Gk.
5. Their abundance distributions
will approximate those of the
.
pdfs of the variables
and
Z
G⎛⎜ Z⎞⎟ as
one
⎝ ⎠
)
K
]
13
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
improved extraction technologies
technologies
�
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 Innovation
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
U.S. Natural Gas Reserves
Trillions of cubic feet
320
300
Proved Reserves
320
300
280
280
260
260
240
240
220
220
200
200
180
180
160
40
160
(As of Dec. 31)
35
30
25
20
15
40
35
30
25
20
15
Additions
10
5
0
5
10
5
0
5
10
15
20
25
10
15
20
25
Production
1947
1950
1955
1960
1965
1970
1975
AGA committee on natural gas reserves
Image by MIT OpenCourseWare. Adapted from the American Gas Association.
16
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 WORLD
CRUDE-OIL PRODUCTION
(ca. 10% of total resources)
Image by MIT OpenCourseWare.
19
RESOURCE BEHAVIOR UNDER
“HUBBERT” ASSUMPTIONS
Timing:
td, tr, tp are times of
respective maxima of
Qd, Qr, Qp.
20
LOGISTIC FUNCTION
Hubbert’s assumed “logistical’ relationships
Rate of Production
Ý
⎡ Q P (X)⎤
d [Q P (X)]
ÝP =
Q
= rQ P (X)⎢1−
⎥,
dt
K
⎦
⎣
Cumulative Production
KQ Po (X)
Q P (X) =
Q P o (X) − (Q Po (X) − K )e−rX
r
4
Q P (X)
K
Q P (X)
1
K
2
where r = relative rate of change
K = carrying capacity or
ultimate production value
Q Po ≡ Q P (X = 0)
QP(X) = cumulative production at time, (t - to) = X
21
LOGISTIC FUNCTION,
continued
Q P (−∞) = 0
Q P (0) =
dQ P (X)
Ý
Q(x) ≡
dt
X
K
Q P (X) ≈ ∫
2πσ
−∞
max
K
= Q PO
2
dQ P (X)
=
dX
P (∞) = K
=
X=
X
=0
⎛ X ′ ⎞2
−1 2⎜ ⎟
⎝ σ ⎠ d X′, where
e
rK
.
4
81
.
σ=
πr
Let : t - t o ≡ X.
22
EQUATIONS
Conservation of Resource:
Qd (t ) = Q r (t ) + Q p (t )
(Eq. 4)
Rate Conservation:
Ý (t ) = Q
Ý (t ) + Q
Ý (t )
Q
d
r
p
(Eq. 5)
Approximate Results:
Ý = 0)− t(Q
Ý = 0) = 2 τ
t (Q
d
r
⎧⎪ t − t
τ ≈⎨ r p
⎩⎪ t d − t r
or
1
t r ≈ td + t p
2
Qp
≈ 2Qd t d
(
)
(
)
(
ultimate
)
()
(Eq. 6)
(Eq. 7)
(Eq. 8)
(Eq. 9)
23
EQUATIONS, Continued
Ýd (t ) and Q
Ýp (t)
If we assume Gaussian distributions for Qr (t ), Q
with each having the same standard deviation, σ, obtain
Qr
⎡ 1 ⎛t − t ⎞ 2⎤
o
r
Q r (t ) =
exp ⎢− ⎜
⎟ ⎥
(Eq. 10)
2
σ
⎝
⎠ ⎥
2πσ
⎣⎢
⎦
Qd
⎡ 1 ⎛t − t ⎞ 2⎤
o
d
Ý (t ) =
(Eq. 11)
(Eq.
Q
exp
−
⎢
⎜ σ ⎟ ⎥
d
2
⎝
⎠ ⎥
2πσ
⎣⎢
⎦
,
⎡ 1 ⎛t − t ⎞ 2⎤
Qp
p
o
Ý (t ) =
⎥
⎢
(Eq. 12)
Q
exp
−
⎜
⎟
p
2πσ
⎢ 2⎝ σ ⎠ ⎥
⎣
⎦
Ý =0
Then, when Qr is at a maximum t = tr and Q
r
−1xQ r
−1xQ r t r
o
2
Ý
Ý
Qr t r =
⇒σ = Ý
(Eq. 13)
2
Ý
,
or
σ
Qr t r
()
()
()
24
EQUATIONS, Continued
For the normal distribution
1 − 1 z2
e 2
2π
f (z) =
t − to
f (1) = 0.67, z ≡
=1
σ
⇒ σ ≅ 25 yr. for U.S. petroleum and natural gas
F(z) = ∫
z
−∞
f (z′ )dz′, 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
= 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.
MIT OpenCourseWare
http://ocw.mit.edu
22.081J / 2.650J / 10.291J / 1.818J / 2.65J / 10.391J / 11.371J / 22.811J / ESD.166J
Introduction to Sustainable Energy
Fall 2010
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