1. Tom Wilson, Department of Geology and Geography Exponentials and logarithms – additional perspectives tom.h.wilson wilson@geo.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV

2. Tom Wilson, Department of Geology and Geography In our first lecture we discussed age/thickness relationships. Initially we assumed that the length of time represented by a certain thickness of a rock unit, was a constant for all depths. However, we noted that as a layer of sediment is buried it will be compacted. Water will be squeezed out of pore spaces, the porosity will decrease, and the grains themselves may be deformed through the process of fracturing or pressure solution. We realized that the age relationship for a certain thickness of rock could not be constant, but must vary in a more complicated way as a function of the depth of burial. Porosity Depth Relationships

3. Tom Wilson, Department of Geology and Geography assumes that the initial porosity (0.6) decreases by 1/2 from one kilometer of depth to the next. Thus the porosity (  ) at 1 kilometer is 2 -1 or 1/2 that at the surface (i.e. 0.3),  (2)=1/2 of  (1)=0.15 (I.e.  =0.6 x 2 -2 or 1/4th of the initial porosity of 0.6. Equations of the type Are referred to as _________ growth laws or exponential functions. allometric

4. Tom Wilson, Department of Geology and Geography The porosity-depth relationship is often stated using the natural base e, where e equals 2.71828.. In the geologic literature you will often see the porosity depth relationship written as  0 is the initial porosity, c is a compaction factor and z - the depth. Sometimes you will see such exponential functions written as In both cases, e=exp=2.71828

5. Tom Wilson, Department of Geology and Geography Waltham writes the porosity- depth relationship as Note that since z has units of kilometers (km) that c (in the previous equation) will have units of km -1 and, units of km. Also recall that when z=, Hence, represents the depth at which the porosity drops to 1/e or 0.368 of its initial value. In the formc is the reciprocal of that depth.

6. Tom Wilson, Department of Geology and Geography Can you evaluate the natural log of

7. Tom Wilson, Department of Geology and Geography is a straight line. Power law relationships end up being straight lines when the log of the relationships is taken. In our next computer lab we’ll determine the coefficients c (or ) and ln(  0 ) that define the straight line relationship above between ln(  ) and z. We will also estimate power law and general polynomial interrelationships using Excel.

8. Tom Wilson, Department of Geology and Geography Thus far we’ve worked with four exponential functions Porosity depth Radioactive decay Reef growth rate Population growth So exponential functions relatively common

9. Tom Wilson, Department of Geology and Geography Problem 3.10 You’re given that Where  is the bulk density of the rock;  g, the grain density, V the total rock volume and V p is the volume of the pore space. Question > Can you re-write this expression in terms of the porosity  ?

10. Tom Wilson, Department of Geology and Geography How would you rewrite this equation if the pore space were filled with water of density  w ? Next, assume that the porosity varies with depth exponentially as Can you develop an exponential representation of the bulk density in terms of  &  ?

11. Tom Wilson, Department of Geology and Geography Substitution of known interrelationships to extract information about other variables Where settling velocity = lake depth/settling time, you were able to show that

12. Tom Wilson, Department of Geology and Geography This allowed us to determine that t 2 If we are given the viscosity of water then we can solve Stokes equation explicitly for the velocity With a viscosity of water equal to about 0.01 poise [gm/(cm-s)]. In the lab exercise, you are asked to do this for a range of particle sizes extending from 0.001cm to 0.1cm (0.01mm to 1 mm).

13. Tom Wilson, Department of Geology and Geography You will get a continuous plot of velocity versus particle radius. 1 mm radius

14. Tom Wilson, Department of Geology and Geography 1 mm radius According to Waltham, it took 10 days for the 1mm radius particle to settle to the bottom of the lake. How deep is it?

15. Tom Wilson, Department of Geology and Geography In the expanded lab version of problem 3.11, you are also asked to develop a plot of settling time versus particle radius. Comment on how the plot of settling time compares to the plot of settling velocity. Think about this in the context of comments in class about the relationship of Stokes’ equation for velocity compared to the expression modified to show how time varies with particle radius.

16. Tom Wilson, Department of Geology and Geography Look at the relationships side-by-side and also consider the logs of v and t? What kind of relationships do you get? What do you get when you solve for vt

17. Tom Wilson, Department of Geology and Geography Complete problems 3.10 and 3.11 Hand in Tuesday Feb 23 rd

18. Tom Wilson, Department of Geology and Geography Chapter 4 has been on your reading list. We won’t spend much time on the examples in this chapter. The chapter is short and is a good review of some of the concepts discussed to date. In particular - working with exponentials and logarithms

19. Tom Wilson, Department of Geology and Geography Chapter 4 problems to look over … In problem 4.7 we have another exponential relationship associated with a geological process – variations in the thickness of the bottom set bed.

20. Tom Wilson, Department of Geology and Geography Chapter 4 problems to look over … Also look over problem 4.10. Units are always an issue that we shouldn’t loose sight of when we are doing calculations. Using dimensional analysis consider whether the statement Age =(Depth x Rate) + Age of Top Has the correct dimensions The equation should look familiar. Remember what the slope was in A=kD+A o

21. Tom Wilson, Department of Geology and Geography Let’s return to the lab exercise, but also note Hand in the take-home isostacy problem before leaving today Start reviewing class and textbook materials for a test next Thursday We’ll have a review session next Tuesday