geology 351 - geomathematicspages.geo.wvu.edu/~wilson/geomath/lectures/lec4.pdf · geology 351 -...

Earthquakes log and exponential

relationships

tom.h.wilson

[email protected]

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/


IRIS Seismic Monitor

http://www.iris.edu/seismon/


Lists of data in the area you select are also

available if you’d like to do your own analysis


Magnitude distribution


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


Downie, R., Kronenberger, Carizo, Maxwell, 2912, SPE 134772

Shear along old dead fractures in the area near

the well bore


Hydraulic fracture stimulation produces a lot of

microseismic activity


Microseismicity from the top well


“out-of-zone” events


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.

Energy equivalents from an IHS webinar


Record of pump pressure & microseismicity


Injection pressure compared to lithostatic


0

( )

z

vS z gdz

In some cases microseismic activity continues

after pumping is completed


We can also undertake frequency-magnitude

analysis of microseismic data



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)



A Marcellus frac. Treatments proceeds

from toe to heel


Toe

Heel

b-values along well #1

Toe Heel

Another application … See http://www.cspg.org/documents/Conventions/Archives/Annual/201

2/313_GC2012_Comparing_Energy_Calculations.pdf


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


Zoback, 2014, online geomechanics class

Earthquakes associated with brine disposal

have much larger magnitude


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?


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


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


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


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


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


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?


8.45.1)(log10 MEs

10log ( )10 sE

Hint …

Where …

Basic notation reminders


• 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


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


We’ll save these for next time …



• 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.


• 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

geology 351 - geomathematicspages.geo.wvu.edu/~wilson/geomath/lectures/lec4.pdf · geology 351 -...

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