Geology 351 - geomathematicspages.geo.wvu.edu/~wilson/geomath/lectures/lec4.pdf · Geology 351 -...
Transcript of Geology 351 - geomathematicspages.geo.wvu.edu/~wilson/geomath/lectures/lec4.pdf · Geology 351 -...
Earthquakes log and exponential
relationships
tom.h.wilson
Department of Geology and Geography
West Virginia University
Morgantown, WV
Geology 351 - geomathematics
Objectives for the day
Tom Wilson, Department of Geology and Geography
• Learn to use the frequency-magnitude model to
estimate recurrence intervals for earthquakes of
specified magnitude and greater.
• Frequency magnitude and microseismic
• Learn how to express exponential functions in
logarithmic form (and logarithmic functions in
exponential form).
World seismicity – one week view
http://earthquake.usgs.gov/earthquakes/map/
Tom Wilson, Department of Geology and Geography
Lists of data in the area you select are also
available if you’d like to do your own analysis
Tom Wilson, Department of Geology and Geography
Magnitude distribution
Tom Wilson, Department of Geology and Geography
Magnitude
-1 0 1 2 3 4 5 6
Nu
mb
er
0
50
100
150
200
250
300Earthquake magnitudes histogram January 13-20, 2015
5 6 7 8 9 10
Richter Magnitude
0
100
200
300
400
500
600
Num
ber
of
eart
hquakes
per
year
m N/year
5.25 537.03
5.46 389.04
5.7 218.77
5.91 134.89
6.1 91.20
6.39 46.77
6.6 25.70
6.79 16.21
7.07 8.12
7.26 4.67
7.47 2.63
7.7 0.81
7.92 0.66
7.25 2.08
7.48 1.65
7.7 1.09
8.11 0.39
8.38 0.23
8.59 0.15
8.79 0.12
9.07 0.08
9.27 0.04
9.47 0.03
Observational data for earthquake magnitude (m)
and frequency (N, number of earthquakes per year
(worldwide) with magnitude m and greater)
What would this plot look like if we plotted the log
of N versus m?
Nu
mb
er
of
ea
rth
qu
ak
es
pe
r ye
ar
of
Ma
gn
itu
de
m a
nd
gre
ate
r
Some worldwide data
5 6 7 8 9 10
Richter Magnitude
0.01
0.1
1
10
100
1000
Num
ber
of
eart
hquakes
per
year
cbmN log
The Gutenberg-Richter Relationship
or frequency-magnitude relationship
-b is the slope
and c is the
intercept.
log(
N)
log 0.935 5.21
log 0.935(7.2) 5.21
log 1.52
N m
N
N
Magnitude
2 3 4 5 6 7
Lo
g10 N
-2
-1
0
1
2
3Frequency (log10N) Magnitude Plot (Haitian Region)
logN=-0.935 m + 5.21
Let’s determine N for a magnitude 7.2 quake.
N=10logN=10-1.52
This is number of earthquakes of magnitude m and greater per year.
Magnitude
2 3 4 5 6 7
Lo
g10 N
-2
-1
0
1
2
3Frequency (log10N) Magnitude Plot (Haitian Region)
logN=-0.935 m + 5.21
The recurrence time
To estimate the recurrence interval, simply compute 1/N. This result has units of years and provides an estimate of the number of years between magnitude 7.2 and greater (or m and greater in general) earthquakes in the region.
Earthquakes on a different scale - microseismicity
associated with hydraulic fracture treatment
Tom Wilson, Department of Geology and Geography
Downie, R., Kronenberger, Carizo, Maxwell, 2912, SPE 134772
Shear along old dead fractures in the area near
the well bore
Tom Wilson, Department of Geology and Geography
Hydraulic fracture stimulation produces a lot of
microseismic activity
Tom Wilson, Department of Geology and Geography
“out-of-zone” events
Tom Wilson, Department of Geology and Geography
Critically stressed ready-to-break area.
From Kanamori (1977) & also Boroumond and Eaton (2012)
Tom Wilson, Department of Geology and
Geography
10log ( ) 1.5 4.8s oE M
Mo is moment magnitude.
The constant 4.8 gives E in Joules. As an independent
exercise determine this constant for E(ergs)
Another area where logarithms and their manipulation become useful
To calculate E we have to take the exponential inverse of the log. Can you do it? See slides near the end of todays set.
Injection pressure compared to lithostatic
Tom Wilson, Department of Geology and Geography
0
( )
z
vS z gdz
In some cases microseismic activity continues
after pumping is completed
Tom Wilson, Department of Geology and Geography
We can also undertake frequency-magnitude
analysis of microseismic data
Tom Wilson, Department of Geology and Geography
Downie, R., Kronenberger, Carizo, Maxwell, 2912, SPE 134772
Some authors suggest that b~1 implies reactivation of pre-existing faults
And that stimulation of smaller natural fractures in the
reservoir results in higher b-value (slope)
Tom Wilson, Department of Geology and Geography
Downie, R., Kronenberger, Carizo, Maxwell, 2912, SPE 134772
A Marcellus frac. Treatments proceeds
from toe to heel
Tom Wilson, Department of Geology and Geography
Toe
Heel
Another application … See http://www.cspg.org/documents/Conventions/Archives/Annual/201
2/313_GC2012_Comparing_Energy_Calculations.pdf
Tom Wilson, Department of Geology and Geography
For applications to microseismic events produced during frac’ing.
Missing data or – how many events didn’t you hear?
Rupture area associated with microseismic
events is very small
Tom Wilson, Department of Geology and Geography
Zoback, 2014, online geomechanics class
Earthquakes associated with brine disposal
have much larger magnitude
Tom Wilson, Department of Geology and Geography
Zoback, 2014, online geomechanics class
7.2
7.2
log 0.935 5.21
log 0.935(7.2) 5.21
log 1.52
N m
N
N
Back to class example, you know b from
analysis of the data. How do you solve for N7.2?
What is N7.2?
Let’s discuss logarithms for a few minutes and
come back to this later.
Any questions about logarithms?
Tom Wilson, Department of Geology and Geography
Logarithms are based (initially) on powers of 10.
We know for example that 100=1,
101=10
102=100
103=1000
And negative powers give us
10-1=0.1
10-2=0.01
10-3=0.001, etc.
Remember the general definition of a log
Tom Wilson, Department of Geology and Geography
The logarithm of y - i.e. log(y) =x solves the
equation 10x or 10log(y) = y
The logarithm of y is the exponent (x) we
have to raise 10 to - to get y.
So log (y=1000) = 3 since 103 = 1000 &
log (10y) = y since …
Check your understanding on these slides else got to slide 36
Questions? - more review examples
Tom Wilson, Department of Geology and Geography
What is log 10?
We rewrite this as log (10)1/2.
Since we have to raise 10 to the
power ½ to get 10, the log is just ½.
Some other general rules to keep in mind are that
log (xy)=log x + log y
log (x/y)= log x – log y
log xn =n log x
cxaby orcxay 10
Where b and 10 are the bases. These are
constants and we can define any other
number in terms of these constants and
base raised to a certain power.
Remember the exponential functions have the
independent variable in the exponent
So you are dealing with equations like the following:
xy 10
By definition, we also say that x is the log of y, and can write
xy x 10loglog
So the powers of the base are the logs, and when asked what is
45y where,log y
We assume that we are asking for x such that
4510 x
For any number y, we can write
45y where,log10 yy10log leaves no room for doubt that we are
specifically interested in the log for a base of 10.
One of the confusing things about logarithms is the word
itself. What does it mean? You might read log10 y to say -
”What is the power that 10 must be raised to to get y?”
How about this operator? -
ypow 10
Many suggest that the base always be specified
Tom Wilson, Department of Geology and Geography
ypow 10
The power of base 10 that yields () y
653.1log10 y 1.65310 45
10 45 = pow
10 45 = 1.653pow
What power do we have to raise the base 10 to, to get 45
We’ve already worked with three bases: 2, 10 and e.
Whatever the base, the logging operation is the same.
5log 10 asks what is the power that 5 must be raised to, to get 10.
5log 10 where 5 10xx
How do we find these powers?
5log
10log 10log
10
105
431.1699.0
1 10log5
105 431.1 thus
In general, base
numberbase
10
10
log
)(log number) some(log
or
b
ab
10
10
log
)(log alog
Try the following on your own
?)3(log
)7(log 7log
10
103
8log8
21log7
7log4
Helpful way to remember how to determine the
power for an arbitrary base – say n, where
Tom Wilson, Department of Geology and Geography
log (y)b x
Take the log10 of both sides of this equation to get the general rule that
10
10
log ( )
log (b)
yx
Otherwise stated as
10
10
log (the number)
log (base)x
Put this in exponential formxb y
Take the log base 10 of this expression and solve for x
10log is often written as log, with no subscript
log10 is referred to as the common logarithm
log is often written as ln. e
2.079 ln8 8log e
thus
loge or ln is referred to as the natural
logarithm. All other bases are usually
specified by a subscript on the log, e.g.
etc. ,logor og 25l
log 0.935 5.21
log 0.935(7.2) 5.21
log 1.52
N m
N
N
Return to the problem developed earlier
How do you calculate N and what does it mean?
Where N, in this case, is the number of earthquakes of
magnitude 7.2 and greater per year that occur in this area.
Solution review
Tom Wilson, Department of Geology and Geography
log 1.52N Since
1.5210N
Take another example: given b = 1.25 and c=7, how often
can a magnitude 8 and greater earthquake be expected?
(don’t forget to put the minus sign in front of b!)
-1.52 is the power you have to raise 10 to to get N.
log N = ….
8.45.1)(log10 MEs
What energy is released by a magnitude 4 earthquake?
A magnitude 5?
Can you prove that the energy increases 31.6 times?
More logs and exponents!
Seismic energy-magnitude relationships
more logs
How would you solve for E?
Tom Wilson, Department of Geology and Geography
8.45.1)(log10 MEs
10log ( )10 sE
Hint …
Where …
Basic notation reminders
Tom Wilson, Department of Geology and Geography
• log(x) implies log10
• ln(x) implies loge
• When in doubt – ask.• Also, if different bases are in use,
specify: i.e. log10(x), log2(x) …
A question to think about
Tom Wilson, Department of Geology and Geography
z
oe
Where
How would you solve for ?
Have a look at the basics.xlsx file.
See youtube video for brief overview of file contents
Some of the worksheets are interactive allowing you to get
answers to specific questions. Plots are automatically adjusted
to display the effect of changing variables and constants
Just be sure
you can do it
on your own!
Spend the remainder of the class working on Discussion group
problems. The one below is all that will be due today
Tom Wilson, Department of Geology and Geography
Tom Wilson, Department of Geology and Geography
• Don’t forget to hand in the group problems (set 2) from last time
In the next class, we will spend some time
working with Excel.
Tom Wilson, Department of Geology and Geography
• Hand in group problems from last Thursday before
leaving today. If completed, I’ll pick up today’s in-
class work – need more time?
• Look over problems 2.11 through 2.13 for
discussion next time
• Continue reading text (everyone get a text?)
• We will examine a comprehensive approach to
solving problems 2.11 and 2.13 using Excel next time.
Next Time