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Computing Volumes After the area of each contour is computed, the volume is computed by various techniques. Basic volume calculations can be viewed as dividing the structure into layers of a common thickness. The different techniques define the layer geometry in different ways.
The Trapezoid Rule computes the layer volume by computing the average area of the layer from the top and bottom layer and then multiplying the average area by the layer thickness. (Mathematically, this is also equivalent to computing the area underneath a thickness vs. area graph). The Trapezoid Method has historically been commonly used for computing map volumes because of its ease in computing, understandability, and accuracy. Vtrap = Sum { 1/2 (Ai + Ai-1) * (hi - hi-1) } Because isopach maps always have smaller contour areas as the thickness increases, the Trapezoid Method though accurate always slightly overestimates the reservoir volume. Another common volume method is the Pyramid Rule. This volume technique computes the areas of the frustum of a pyramid or cone (the areas within a pyramid cut by two halves with areas Ai and Ai+1). Vpyra = Sum { 1/3 (Ai-1 + Ai + Ai+1) * (hi - hi-1) } Easy to compute with a calculator or computer, the Pyramid Method and mathematically resembles the maps we are trying to compute. However, because the methods is assuming a straight line between contours for concave reservoir, most commonly encountered in the oil industry, the volume is being underestimated. Sometimes analysts use both of the volume calculations described above based upon the area ratio of adjacent contours. This method, known as the Combination Method, used the Trapezoid Rule when Ai/AI-1 >=0.5 and used the Pyramid Rule when Ai/Ai-1<0.5. Two higher order numerical integration techniques can also be used to compute volume, Simpson’s Rule and 3/8 Rule. Simpson’s Rule is a second order Newton-Cotes Numerical Integration technique for n=2 that uses three terms. The Method is derived for an even number of contours. Vsimp = Sum { 1/6 (Ai-2 + 4Ai-1 + Ai) * (hi-1 - hi) } The 3/8 Rule is also a second order Newton-Cotes Numerical Integration of n=3 that uses four terms. Derived for the number of contours as a multiple of three, this integration method is twice as accurate as Simpson’s Rule. V3/8 = Sum { 1/8) (Ai-3 + 3Ai-2 + 3Ai-1 + Ai) * (hi-1 - hi) } Though mathematically these last two methods should be more accurate that the Trapezoid Method, they both require numerous evenly-spaced contours. The Step method computes volumes assuming a flat horizontal surface between contours similar to steps on a staircase. This method will always significantly underestimate the volume. The Ratio method attempts to improve the Step method by adding volume above each step that is proportional to the ration of the the contour areas. The quadratic method curve fits a quadratic line through a graph of contour area vs thickness and computes the area under the curve and volume of map. In PLAN 2002, the area of all contours of equal thickness are added to from a single hill and then the volume is computed. For simple structures the 3/8 Rule will compute the lowest value of volume with Simpson’s Rule the next lowest and fairly close to the 3/8 Rule value. The Trapezoid Rule will give the largest value. |
Further Information & Examples
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