derivativeexcelexercise - Geol 351 Geomathematics...

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115 Geol 351 Geomathematics Derivative Concepts & Basic Calculations The derivative is essentially the slope of a function at a given point. We’ve talked about slopes on numerous occasions throughout the semester. We estimated age-depth relationships and sedimentation rates for different periods of time. In most of the cases we’ve worked with up to this point, those slopes have been constant over a certain time period. Sedimentation rates in the North Sea for example were roughly linear during the 0 to 10, 000 period before present and the 10,000 to 15,000 year before present periods. We noted that in many instances, we would expect the age-depth, depth-porosity relationships to vary with depth because of the influence of compaction. As time passes, the increased weight of sediment bearing down on a layer deposited at an earlier time will push grains closer together and reduce the open space (or pore space) within the layer. The layer will get thinner and the porosity will decrease with increased depth of burial. The porosity depth relationship in unconsolidated sediments is often written as 0 cz e φφ = . In this equation, φ 0 is the initial porosity at a depth z = 0, and c is a constant (a compaction factor).The porosity depth relationship is shown below over the 1 to 5 km range of depths (z) for an initial porosity of 0.5 and compaction factor of 0.5 km -1 . The rate of change of porosity between 1 and 2 kilometers could be estimated as z where is the difference in porosities measured at 1 and 2 kilometers, 21 , and z is the difference in the corresponding depths z 2 z 1 .
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derivativeexcelexercise - Geol 351 Geomathematics...

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