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Geomath

Geomath. Geology 351 -. Formulating integral problems. tom.h.wilson tom.wilson@mail.wvu.edu. Dept. Geology and Geography West Virginia University. Start reviewing materials for the final! … current to-do list. Problem 9.7 is due today Hand in the gravity computation

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Geomath

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  1. Geomath Geology 351 - Formulating integral problems tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University Tom Wilson, Department of Geology and Geography

  2. Start reviewing materials for the final!… current to-do list • Problem 9.7 is due today • Hand in the gravity computation • I will give you till next Tuesday to finish up problems 9.9 and 9.10 • Start reviewing class materials. Next week is a final review week No class this Thursday Tom Wilson, Department of Geology and Geography

  3. Take advantage of this day off and … Turn in any late assignments by Friday afternoon, April 26th. Put all late assignments in my mailbox (mailroom, 3rd floor Brooks) Tom Wilson, Department of Geology and Geography

  4. In this simple example, we didn’t have to employ calculus This formula can be evaluated – as is – for points and equidimensionally shaped objects g gv  x  z r Sulfide deposit Tom Wilson, Department of Geology and Geography

  5. Sulfide minerals L. Morgan, 2010, Geophysical characteristics of volcanogenic massive sulfide deposits; USGS report 210-5070-C, 19p. Tom Wilson, Department of Geology and Geography

  6. Excess density or high horizontal density contrast produces observable changes in the g L. Morgan, 2010, Geophysical characteristics of volcanogenic massive sulfide deposits; USGS report 210-5070-C, 19p. Tom Wilson, Department of Geology and Geography

  7. In this location the gravity anomaly was less distinctive than other geophysical features L. Morgan, 2010, Geophysical characteristics of volcanogenic massive sulfide deposits; USGS report 210-5070-C, 19p. Tom Wilson, Department of Geology and Geography

  8. In this area, a distinctive gravity anomaly is associated with a sulfide deposit M. Thomas, 1997, Gravity prospecting for massive sulfide deposits in the Bathurst mining camp, New Brunswick Canada: Proceedings of Exploration 97, Fourth Decenial International Conference on Mineral Exploration, p837-840. Tom Wilson, Department of Geology and Geography

  9. However, we had to modify it algebraically since we wanted to solve for gv and express as a function of x & z (rather than r) We looked at the geometry and did this in a series of steps • Find r as a function of x and z • Rewrite g in terms of x and z • Express gv in terms of g and  • Replace cos with its spatial equivalent to get gv as a function of x & z • Simplify by factoring z out of the denominator Tom Wilson, Department of Geology and Geography

  10. In the form below it is easy to compute gv at arbitrary x along the surface. Given that G=6.6732 x 10-11nt-m2/kg2, x=1km, z=1.7km, Rdeposit=0.5km and =2gm/cm3, you would find that Tom Wilson, Department of Geology and Geography

  11. Spatial variation in the gravity anomaly over the sulphide deposit Note than anomaly is symmetrical across the sulphide accumulation Tom Wilson, Department of Geology and Geography

  12. One of the first problems you did in the class was a units conversion problem for acceleration gvat 1 km is 0.0000155 m/sec2 Remember what a Gal is? How about a milliGal? Make the units conversion from m/sec2 to milliGals Tom Wilson, Department of Geology and Geography

  13. The same governing equation but with more complex geometry could be used to calculate gcore and gmantle Another slant on the text problem Approximate the average densities 11,000 kg/m3 4,500 kg/m3 Here, the gravitational field is associated with shells of differing density and the problem is a little more complex. Tom Wilson, Department of Geology and Geography

  14. Approximate the average densities 11,000 kg/m3 4,500 kg/m3 In the book problem we are just trying to estimate the mass of the earth and simplify the problem by assuming the earth can be represented by two regions: 1) an inner core of average density, i, and 2) an outer shell (mantle and crust) represented by another average density, o. Tom Wilson, Department of Geology and Geography

  15. We can simplify the problem and still obtain a useful result. Approximate the average densities 11,000 kg/m3 4,500 kg/m3 What do we get when we integrate the surface area over r? Tom Wilson, Department of Geology and Geography

  16. We can simplify the problem and still obtain a useful result. Approximate the average densities 11,000 kg/m3 4,500 kg/m3 Actually a pretty good approximation The result – 6.02 x 1024kg is close to the generally accepted value of 5.97 x 1024kg. Tom Wilson, Department of Geology and Geography

  17. We could then pose the question: what is the acceleration of gravity due to the core at the Earth’s surface? Considering only the core, we find it’s mass is 1.94186x1024 kg (about 1/3rd the total mass of the earth. Mass of core ~ 1.94 x 1024kg. Tom Wilson, Department of Geology and Geography

  18. With an outer radius of ~6371km (6,371,000m) The core is about 2900km beneath your feet, We have to keep units consistent and use G=6.6732x10-11 m3/(kg-sec2) M=1.94186x1024 kg And r=6,371,000 m The contribution to the total acceleration of ~9.8 m/s2 due to the core is 3.29m/s2. Tom Wilson, Department of Geology and Geography

  19. In general we express the acceleration of gravity produced by an object of arbitrary shape as We usually look for some symmetry to help simplify our problem. Tom Wilson, Department of Geology and Geography

  20. Let’s take a look at the acceleration produced by a very long horizontal cylinder This could be a cave passage or tunnel. Point of observation m r r+dr dx Tom Wilson, Department of Geology and Geography

  21. In this example, we can let the cross sectional area = dydz Point of observation Again, we are interested in the vertical component of g, so  m r r+dr dx Tom Wilson, Department of Geology and Geography

  22. Zoom in on the little element dx r+dr r  Area = R2 R Tom Wilson, Department of Geology and Geography

  23. Substitute for dx, simplify and also note that r=m/cos Point of observation  m r r+dr dx Tom Wilson, Department of Geology and Geography

  24. Note that the only variable left is  and the limits of integration would be from -/2 to /2 Point of observation  m r=m/cos r+dr dx=rd/cos Tom Wilson, Department of Geology and Geography

  25. This is an integral you should be able to evaluate What do you get? Tom Wilson, Department of Geology and Geography

  26. Assume that you run a gravity survey across a roughly cylindrically shaped cave passage Hint: replace m with r to develop this relationship g gv  x  z r Cave Passage Cylinder goes in and out of the slide Tom Wilson, Department of Geology and Geography

  27. Developing g as a function of x and z Hint2: Once again – take the vertical component! g gv  x  z r Cave Passage Tom Wilson, Department of Geology and Geography

  28. Li Strain Lf The total natural strain, , is the sum of an infinite number of infinitely small extensions In our example, this gives us the definite integral Where S is the Stretch Tom Wilson, Department of Geology and Geography

  29. Strain Strain (or elongation) (e), stretch (S) and total natural strain () Elongation Total natural strain  expressed as a series expansion of ln(1+e) The six term approximation is accurate out to 5 decimal places! Tom Wilson, Department of Geology and Geography

  30. Comparison of finite elongation vs. total natural strain Tom Wilson, Department of Geology and Geography

  31. Problems 9.9 and 9.10 Volume of the earth – an oblate spheroid In this equation r varies from re, at the equator, to r=0 at the poles. z represents distance along the earth’s rotation axis and varies from –rp to rp. The equatorial radius is given as 6378km and the polar radius, as 6457km. Tom Wilson, Department of Geology and Geography

  32. Problem 9.10 In this problem, we return to the thickness/distance relationship for the bottomset bed. Problems 9.9 and 9.10 will be due next Tuesday Tom Wilson, Department of Geology and Geography

  33. Don’t forget to hand in the answer to the gravity problem! What was the gravitational acceleration produced by the sulfide deposit? Tom Wilson, Department of Geology and Geography

  34. Start reviewing materials for the final!… current to-do list • Problem 9.7 is due today • Hand in the gravity computation • I will give you till next Tuesday to finish up problems 9.9 and 9.10 • Start reviewing class materials. Next week is a final review week No class this Thursday Tom Wilson, Department of Geology and Geography

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