Geology geomathematics. Earthquakes log and exponential relationships

Transcription

1 Geology geomathematics Earthquakes log and exponential relationships tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV

2 Objectives for the day 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).

3 World seismicity one week view

4 IRIS Seismic Monitor

5 Lists of data in the area you select are also available if you d like to do your own analysis

6 Magnitude distribution 300 Earthquake magnitudes histogram January 13-20, Number Magnitude

7 Number of earthquakes per year Number of earthquakes per year of Magnitude m and greater Some worldwide data m N/year Observational data for earthquake magnitude (m) and frequency (N, number of earthquakes per year (worldwide) with magnitude m and greater) Richter Magnitude What would this plot look like if we plotted the log of N versus m?

8 log(n) The Gutenberg-Richter Relationship or frequency-magnitude relationship Number of earthquakes per year log N bm -b is the slope and c is the intercept. c Richter Magnitude

9 Let s determine N for a magnitude 7.2 quake. 3 Frequency (log10n) Magnitude Plot (Haitian Region) Log 10 N 2 1 logn= m log N 0.935m 5.21 log N 0.935(7.2) 5.21 log N Magnitude N=10 logn = This is number of earthquakes of magnitude m and greater per year.

10 The recurrence time Log 10 N Frequency (log10n) Magnitude Plot (Haitian Region) 3 2 logn= m Magnitude 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.

11 Earthquakes on a different scale - microseismicity associated with hydraulic fracture treatment Downie, R., Kronenberger, Carizo, Maxwell, 2912, SPE

12 Shear along old dead fractures in the area near the well bore

13 Hydraulic fracture stimulation produces a lot of microseismic activity

14 Microseismicity from the top well

15 out-of-zone events Critically stressed ready-to-break area.

16 From Kanamori (1977) & also Boroumond and Eaton (2012) Another area where logarithms and their manipulation become useful log ( E ) 1.5M s M o is moment magnitude. The constant 4.8 gives E in Joules. As an independent exercise determine this constant for E(ergs) 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. o Tom Wilson, Department of Geology and Geography

17 Energy equivalents from an IHS webinar

18 Record of pump pressure & microseismicity

19 Injection pressure compared to lithostatic Sv z ( z) gdz 0

20 In some cases microseismic activity continues after pumping is completed

21 We can also undertake frequency-magnitude analysis of microseismic data Some authors suggest that b~1 implies reactivation of pre-existing faults Downie, R., Kronenberger, Carizo, Maxwell, 2912, SPE

22 And that stimulation of smaller natural fractures in the reservoir results in higher b-value (slope) Downie, R., Kronenberger, Carizo, Maxwell, 2912, SPE

23 A Marcellus frac. Treatments proceeds from toe to heel Heel Toe

24 b-values along well #1 Toe Heel

25 Another application See 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?

26 Rupture area associated with microseismic events is very small Zoback, 2014, online geomechanics class

27 Earthquakes associated with brine disposal have much larger magnitude Zoback, 2014, online geomechanics class

28 Back to class example, you know b from analysis of the data. How do you solve for N 7.2? What is N 7.2? log N 0.935m 5.21 log N 0.935(7.2) log N Let s discuss logarithms for a few minutes and come back to this later.

29 Any questions about logarithms? Logarithms are based (initially) on powers of 10. We know for example that 10 0 =1, 10 1 = = =1000 And negative powers give us 10-1 = = =0.001, etc.

30 Remember the general definition of a log The logarithm of y - i.e. log(y) =x solves the equation 10 x or 10 log(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 10 3 = 1000 & log (10 y ) = y since Check your understanding on these slides else got to slide 36

31 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 x n =n log x

32 Remember the exponential functions have the independent variable in the exponent So you are dealing with equations like the following: cx y ab y a10 or cx 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.

33 For any number y, we can write x y 10 By definition, we also say that x is the log of y, and can write log y x log 10 x So the powers of the base are the logs, and when asked what is log y, wherey 45 We assume that we are asking for x such that x 10 45

34 Many suggest that the base always be specified log 10 y, wherey 45 log 10 y 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 log 10 y to say - What is the power that 10 must be raised to to get y? How about this operator? - pow y 10

35 pow y 10 The power of base 10 that yields ( ) y pow10 45 = What power do we have to raise the base 10 to, to get 45 log 10 y pow10 45 = 1.653

36 We ve already worked with three bases: 2, 10 and e. Whatever the base, the logging operation is the same. log5 10 asks what is the power that 5 must be raised to, to get 10. x log 10 x where How do we find these powers? 10 log 5 log 510 thus log log

37 In general, log base (some number) log b a or log log ( a) b Try the following on your own log 10( number ) log base 10 log10(7) log 3 7 log (3) 10? log 8 8 log 7 21 log 4 7

38 Helpful way to remember how to determine the power for an arbitrary base say n, where log (y) b Put this in exponential form Take the log base 10 of this expression and solve for x Take the log 10 of both sides of this equation to get the general rule that Otherwise stated as x x x b log 10( y) log (b) log (the number) log (base) 10 x y

39 log10 is often written as log, with no subscript log 10 is referred to as the common logarithm log is often written as ln. e log e 8 thus ln log e or ln is referred to as the natural logarithm. All other bases are usually specified by a subscript on the log, e.g. log5 or log 2, etc.

40 Return to the problem developed earlier log N 0.935m 5.21 log N 0.935(7.2) 5.21 log N 1.52 Where N, in this case, is the number of earthquakes of magnitude 7.2 and greater per year that occur in this area. How do you calculate N and what does it mean?

41 Solution review Since log N 1.52 N is the power you have to raise 10 to to get N. 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!) log N =.

42 Seismic energy-magnitude relationships more logs ( log 10 E s ) 1.5M 4.8 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!

43 How would you solve for E? Where ( log 10 E s ) 1.5M 4.8 Hint 10 Es 10 log ( )

44 Basic notation reminders log(x) implies log 10 ln(x) implies log e When in doubt ask. Also, if different bases are in use, specify: i.e. log 10 (x), log 2 (x)

45 A question to think about Where z e o How would you solve for?

46 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!

47 Spend the remainder of the class working on Discussion group problems. The one below is all that will be due today

48 We ll save these for next time

49 Don t forget to hand in the group problems (set 2) from last time

50 In the next class, we will spend some time working with Excel.

51 Next Time Hand in group problems from last Thursday before leaving today. If completed, I ll pick up today s inclass 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.