Stochastic Modeling (includes associated papers 21255 and 21299 )
Authors: Haldorsen Helge H. , Damsleth Elvind 
Published: December 2007

Summary.
To describe and visualize reservoirs and reservoir heterogeneities, geoscientists combine classic geoscience principles with geologic engineering, statistics, computer technology, and graphics to create 3D reservoir architecture representations (spatial arrangement of facies) with associated 3D arrays of rock properties. The resulting reservoir description honors all data from all disciplines and is easily stored, displayed, edited, and updated as additional data (static and/or dynamic) become available, While the reservoir description always contains many deterministic elements and features, some elements are most conveniently generated by stochastic models. The current emphasis on geologic quantification coupled with high-power computers and graphics will within a few years convert geoscience disciplines in general and production geology in particular to "screen-oriented disciplines" much like geophysics. The petroleum industry should benefit greatly in terms of increased effectiveness and interdisciplinary communications because totally integrated teams are responsible for generating, updating, and using (for forecasting and reservoir management) the same 3D geologic domain with all its intricate detail.

Introduction
A reservoir is intrinsically deterministic. It exists; it has potentially measurable, deterministic properties and features at all scales, and it is the end product of many complex processes (e.g., sedimentation, erosion, burial, compaction, and diagenesis) that occurred over millions of years. Reservoir description is a combination of observations (the deterministic component), educated aiming (geology, sedimentology), and formalized "guessing"(the stochastic component).
A stochastic phenomenon or variable is characterized by the property that a given set of circumstances does not always lead to the same outcome(so that there is no deterministic regularity) but to different outcomes in such a way that there is statistical regularity.
We apply stochastic techniques to describe deterministic reservoirs for six major reasons:incomplete information about dimensions, internal(geometric) architecture, and rock-property variability on all scales;complex spatial disposition of reservoir building blocks or facies;difficult-to-capture rock-property variability and variability structure with spatial position and direction;unknown relationships between property value and the volume of rock used for averaging (the scale problem);the relative abundance of static (point values along the well for kH, and seismic data) over dynamic (time-dependent effects. how the rock architecture affects a recovery process, etc.) reservoir data; andconvenience and speed.
The stochastic approach to reservoir description involves first capturing reliable data about the variability and variability structure at a certain volume scale for the reservoir in terms of distributions, correlations, and spatial disposition measures. Analog outcrops, mature fields with an abundance of wells, recent sediments, and seismic interpretations, are the main sources of information about lateral extents and horizontal changes in rock characteristics. Data from cores and logs in wells supply detailed information about the rock variability along "lines." Synthetic variability is then generated in a certain phenomenon or variable that provides the details between observations and descriptive needs. In all cases and at all volume scales it is desirable to condition the synthetic (or generated)realization to the actual observations. Geologic or sedimentologic data of any kind (even "soft" information-e.g., a geologic model) are extremely valuable in all phases.
The phenomena or variables we normally seek to describe with stochastic techniques are those that influence the amount, position, accessibility, and flow of fluids through reservoirs. The circumstance may be interpreted as a particular depositional environment (in which the reservoir was formed)and all its operational processes. The outputs (e.g., permeabilities. shale lengths, sand widths, and fault throws) vary according to a statistical model whose features can be obtained from observations and/or hypotheses.
If we are to account for geologic uncertainties in classic reservoir-engineering forecasting tools (e.g., in a numerical simulation model), we are faced with simulating a large number of plausible realizations because the description must be deterministic each time. The goal, in this context, is to optimize the development and to convert uncertainties about the geologic description into uncertainty ranges about important project profitability indicators(Fig. 1). The alternative would be to formulate the appropriate stochastic differential equations (SDE) and solve these for the mean and variance of the response (e.g., rates and recovery) resulting from the distribution of possible values in the input parameters. No fully successful SDE approach has been reported that can handle all nonlinearities in multidimensional, multiphase, and multifacies reservoir models with capillary, gravity, viscous, and dispersive driving forces.
JPT
P. 404⁁