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==Lognormality and log probability paper==

==Lognormality and log probability paper==

Most geological and production parameters are not distributed according to a symmetrical or ''normal'' distribution, that is, they do not form a ''bell-shaped'' frequency curve. Instead, they tend to produce a frequency distribution skewed to the right, so that there are many small values and only a few large ones. Such patterns approximate a ''lognormal distribution,'' and they arise from multiplication of several factors to produce one geological parameter.<ref name=Kaufman_1963>Kaufman, G., 1963, Statistical decision and related techniques in oil and gas exploration: Englewood Cliffs, NJ, Prentice-Hall, 307 p.</ref> <ref name=Capen_1984>Capen, E. C., 1984, Why lognormal? in E. C.Capen, R. E. Megill, and P. R. Rose, ed., Prospect Evaluation: AAPG Course Notes: Tulsa, OK, AAPG, 8 p.</ref> <ref name=Megill_1984>Megill, R. E., 1984, An introduction to risk analysis, 2nd ed.: Tulsa, OK, PennWell Books, 274 p.</ref> Good examples include field sizes, production rates of wells in a field, [[porosity]]-feet (&phi;h) of reservoirs, and effective well drainage area.

Most geological and production parameters are not distributed according to a symmetrical or ''normal'' distribution, that is, they do not form a ''bell-shaped'' frequency curve. Instead, they tend to produce a frequency distribution skewed to the right, so that there are many small values and only a few large ones. Such patterns approximate a ''lognormal distribution,'' and they arise from multiplication of several factors to produce one geological parameter.<ref name=Kaufman_1963>Kaufman, G., 1963, Statistical decision and related techniques in oil and gas exploration: Englewood Cliffs, NJ, Prentice-Hall, 307 p.</ref> <ref name=Capen_1984>Capen, E. C., 1984, Why lognormal? in E. C.Capen, R. E. Megill, and P. R. Rose, ed., Prospect Evaluation: AAPG Course Notes: Tulsa, OK, AAPG, 8 p.</ref> <ref name=Megill_1984>Megill, R. E., 1984, An introduction to risk analysis, 2nd ed.: Tulsa, OK, PennWell Books, 274 p.</ref> Good examples include field sizes, production rates of wells in a field, [[porosity]]-feet (&phi;h) of reservoirs, and effective well drainage area.

Here it is important to remind the reader that in a lognormal frequency distribution, the ''mode'' (or most likely point) is positioned to the left, at the peak of the curve. The ''median'' (or 50% point) lies in the middle, separating the area under the curve into two equal parts, whereas the ''mean'' (or average) lies to the right of the median ([[:Image:Charles-l-vavra-john-g-kaldi-robert-m-sneider_capillary-pressure_1.jpg|Figure 1]]). We shall be concerned mostly with the median and the mean in our estimates and calculations, generally discouraging use of the mode, as will be explained later.

Here it is important to remind the reader that in a lognormal frequency distribution, the ''mode'' (or most likely point) is positioned to the left, at the peak of the curve. The ''median'' (or 50% point) lies in the middle, separating the area under the curve into two equal parts, whereas the ''mean'' (or average) lies to the right of the median ([[:Image:Charles-l-vavra-john-g-kaldi-robert-m-sneider_capillary-pressure_1.jpg|Figure 1]]). We shall be concerned mostly with the median and the mean in our estimates and calculations, generally discouraging use of the mode, as will be explained later.

[[File:Charles-l-vavra-john-g-kaldi-robert-m-sneider capillary-pressure 2.jpg|thumbnail|'''Figure 2.''' Worksheet showing graphical method of combining distributions to derive the mean reserves on three-cycle log probability paper.]]

[[File:Uncertainties-impacting-reserves-revenue-and-costs_fig2.png|thumbnail|'''Figure 2.''' Worksheet showing graphical method of combining distributions to derive the mean reserves on three-cycle log probability paper.]]

In combination with the cumulative probability curve, lognormality provides us with a very useful and powerful predictive tool. Accordingly, it is important to utilize (and understand) log probability paper. Although several forms are commercially available, the three-cycle type in which the probabilities extend from 0.01% to 99.99% is recommended ([[:Image:Charles-l-vavra-john-g-kaldi-robert-m-sneider_capillary-pressure_2.jpg|Figure 2]]).

In combination with the cumulative probability curve, lognormality provides us with a very useful and powerful predictive tool. Accordingly, it is important to utilize (and understand) log probability paper. Although several forms are commercially available, the three-cycle type in which the probabilities extend from 0.01% to 99.99% is recommended ([[:Image:Charles-l-vavra-john-g-kaldi-robert-m-sneider_capillary-pressure_2.jpg|Figure 2]]).