Changes

  | pdf    = http://archives.datapages.com/data/specpubs/methodo1/images/a095/a0950001/0300/03480.pdf

  | pdf    = http://archives.datapages.com/data/specpubs/methodo1/images/a095/a0950001/0300/03480.pdf

}}

}}

The Monte Carlo technique consists of generating many different joint outcomes of random processes ([[:Image:Charles-l-vavra-john-g-kaldi-robert-m-sneider_capillary-pressure_1.jpg|Figure 1c]]) and then observing the behavior of response values that are functions of these outcomes. Such behavior can be characterized by probability density functions (pdf) of the response variables, as depicted on the right of [[:Image:Charles-l-vavra-john-g-kaldi-robert-m-sneider_capillary-pressure_1.jpg|Figure 1c]]).<ref name=Journel_1989>Journel, A. G., 1989, Fundamentals of geostatistics in five lessons: Washington D.C., American Geophysical Union, Short Course in Geology, v. 8, 40 p.</ref>

The [http://energy.cr.usgs.gov/WEcont/chaps/MC.pdf Monte Carlo] technique consists of generating many different joint outcomes of random processes ([[:Image:Charles-l-vavra-john-g-kaldi-robert-m-sneider_capillary-pressure_1.jpg|Figure 1c]]) and then observing the behavior of response values that are functions of these outcomes. Such behavior can be characterized by probability density functions (pdf) of the response variables, as depicted on the right of [[:Image:Charles-l-vavra-john-g-kaldi-robert-m-sneider_capillary-pressure_1.jpg|Figure 1c]]).<ref name=Journel_1989>Journel, A. G., 1989, Fundamentals of geostatistics in five lessons: Washington D.C., American Geophysical Union, Short Course in Geology, v. 8, 40 p.</ref>

For example, the input variables might be porosity (&phi;), oil saturation (S₀), and a binary indicator (I) set equal to 1 or 0 depending on whether the sample location belongs to a certain pay formation. The unique response value is the volume of oil in place defined by a particular function of the various input variables, called a ''transfer function'' (TF). In this example, the transfer function is a summation representing the total volume V.

For example, the input variables might be porosity (&phi;), oil saturation (S₀), and a binary indicator (I) set equal to 1 or 0 depending on whether the sample location belongs to a certain pay formation. The unique response value is the volume of oil in place defined by a particular function of the various input variables, called a ''transfer function'' (TF). In this example, the transfer function is a summation representing the total volume V.

==Stochastic simulation==

==Stochastic simulation==

[[File:Charles-l-vavra-john-g-kaldi-robert-m-sneider capillary-pressure 1.jpg|thumbnail|'''Figure 1.''' The monte carlo approach for transferring input uncertainty into a distribution of response values. (a) ideal case—the input space and transfer function are perfectly known, resulting in the exact response values. (b) traditional approach—the input values are interpolated from sparse data and the actual transfer function is estimated, resulting in estimated response values with usually no assessment of their uncertainty. (c) monte carlo approach—input uncertainty is modeled by a series of equiprobable input sets which, after processing, provide a probability distribution (pdf) for the response value(s).]]

[[File:Monte-carlo-and-stochastic-simulation-methods fig1.png|400px|thumbnail|'''Figure 1.''' The monte carlo approach for transferring input uncertainty into a distribution of response values. (a) ideal case—the input space and transfer function are perfectly known, resulting in the exact response values. (b) traditional approach—the input values are interpolated from sparse data and the actual transfer function is estimated, resulting in estimated response values with usually no assessment of their uncertainty. (c) monte carlo approach—input uncertainty is modeled by a series of equiprobable input sets which, after processing, provide a probability distribution (pdf) for the response value(s).]]

Stochastic simulation is a tool that allows Monte Carlo analysis of spatially distributed input variables. It aims at providing joint outcomes of any set of dependent random variables. These random variables can be

Stochastic simulation is a tool that allows Monte Carlo analysis of spatially distributed input variables. It aims at providing joint outcomes of any set of dependent random variables. These random variables can be

In a spatial context where all the random variables relate to the same attribute at different locations, condition 4 amounts to honoring the sample values at their locations. This condition is also known as the ''exactitude condition,'' and the corresponding realizations are referred as being ''conditional'' to the data values.

In a spatial context where all the random variables relate to the same attribute at different locations, condition 4 amounts to honoring the sample values at their locations. This condition is also known as the ''exactitude condition,'' and the corresponding realizations are referred as being ''conditional'' to the data values.

There are as many algorithms for generating joint realizations of a large number of dependent random variables as there are different models for the joint distribution of these random variables, with an equally large number of implementation variants. With the advent of extremely fast computers with vast memory, the field is exploding with new algorithms being proposed regularly. The book by Ripley<ref name=Ripley_1987>Ripley, B. D., 1987, Stochastic Simulation: New York, John Wiley, 237 p.</ref> gives an excellent summary and an attempt at classification of the algorithms, yet as of 1990, it can no longer be considered complete. A good generic discussion of simulation topics is given in Hohn<ref name=Hohn_1988>Hohn, M. E., 1988, Geostatistics and petroleum geology: New York, Van Nostrand Reinhold, 264 p.</ref>.

There are as many algorithms for generating joint realizations of a large number of dependent random variables as there are different models for the joint distribution of these random variables, with an equally large number of implementation variants. With the advent of extremely fast computers with vast memory, the field is exploding with new algorithms being proposed regularly. The book by Ripley<ref name=Ripley_1987>Ripley, B. D., 1987, Stochastic Simulation: New York, John Wiley, 237 p.</ref> gives an excellent summary and an attempt at classification of the algorithms, yet as of 1990, it can no longer be considered complete. A good generic discussion of simulation topics is given in Hohn.<ref name=Hohn_1988>Hohn, M. E., 1988, Geostatistics and petroleum geology: New York, Van Nostrand Reinhold, 264 p.</ref>

Particular mention should be given to stochastic simulations based on self repetitive fractal models.<ref name=Hewett_1986>Hewett, T. A., 1986, Fractal distribution of reservoir heterogeneity and their influence on fluid transport: Society of Petroleum Engineers, SPE Paper 15386.</ref> Such models correspond to patterns of spatial variability that repeat themselves whatever the distance scale used.

Particular mention should be given to stochastic simulations based on self repetitive fractal models.<ref name=Hewett_1986>Hewett, T. A., 1986, [https://www.onepetro.org/conference-paper/SPE-15386-MS Fractal distribution of reservoir heterogeneity and their influence on fluid transport]: Society of Petroleum Engineers, SPE Paper 15386.</ref> Such models correspond to patterns of spatial variability that repeat themselves whatever the distance scale used.

The present ability to generate a large number of very large stochastic simulations very quickly far outstrips the capability to look at the corresponding (stochastic) images and the capability to process them with realistic flow simulators. The bottleneck for systematic utilization of the Monte Carlo approach is no longer stochastic simulation but rather computer graphics and flow simulators that are presently much too slow.

The present ability to generate a large number of very large stochastic simulations very quickly far outstrips the capability to look at the corresponding (stochastic) images and the capability to process them with realistic flow simulators. The bottleneck for systematic utilization of the Monte Carlo approach is no longer stochastic simulation but rather computer graphics and flow simulators that are presently much too slow.

==See also==

==See also==

==References==

==References==

* [http://archives.datapages.com/data/specpubs/methodo1/images/a095/a0950001/0300/03480.pdf PDF file in Datapages]

* [http://archives.datapages.com/data/specpubs/methodo1/images/a095/a0950001/0300/03480.pdf PDF file in Datapages]

[[Category:Geological methods]] [[Category:Test content]]

[[Category:Geological methods]]

[[Category:Methods in Exploration 10]]