A Two-Stage Stochastic Model Applied to a North Sea Reservoir
Authors: Damsleth Elvind , Tjolsen Charlotte B. , Omre Henning , Haldorsen Helge H. 
Published: December 2007

Summary
We present a two-stage stochastic model that caters to the large-scalegeological heterogeneities resulting from different rock types and the inherentspatial variability of rock properties. The suggested approach combines severalelements from a variety of models, methods, and algorithms that have emergedduring the last few years. This twostage procedure can be used to generateseveral geologically sound realizations of a reservoir in an efficient manner. Stage 1 preserves the important geological architecture, while Stage 2 providessmall-scale variability in the rock properties. At both stages, the stochasticmodels are conditional on the actual values observed in wells. Hence, everyrealization honors the observations. An example from a highly heterogeneous North Sea reservoir, deposited in an upper shore-face environment illustratesapplication of the model.

Introduction
As a result of high costs in offshore areas like the North Sea, only aminimum of exploration and appraisal wells can be justified before importantfield development decisions are made. The use of oversimplified geologicalmodels based on data from a limited number of widely spaced wells is probablyone of the most important reasons probably one of the most important reasonsfor the failures in predicting field performance. Oversimplification and theuse of performance. Oversimplification and the use of unrealistic geologicalmodels partly results from the paucity of well data but also results from theinappropriate use of available data. Experience shows, for example, that linearinterpolation of petrophysical characteristics between wells some kilometersapart usually will not give a realistic image of the heterogeneity required topredict fluid flow. To give a realistic description of the point-to-pointvariation, we resort to point-to-point variation, we resort to stochasticmodels and simulation.
A reservoir is intrinsically deterministic. It exists, and its propertiesand features are potentially measurable at all scales. A potentially measurableat all scales. A reservoir is the product of many complex processes(sedimentation, erosion, burial, processes (sedimentation, erosion, burial, compaction, diagenesis, etc.) that operate over millions of years. Why, then, do we have to apply stochastic modeling? Haldorsen and Damsleth list thefollowing reasons:the incomplete information about a reservoir'sdimensions, internal architecture, and its rock-property variability at allscales;the complex spatial disposition of reservoir building blocks orfacies;the difficult-to-capture rock-property variability and variabilitystructure with spatial position and direction;the unknown position anddirection;the unknown relationships between the property value and thevolume of rock used for averaging (the scale problem);the relativeabundance of static (point values along the well for kH, Sw, and seismicdata) over dynamic (time-dependent effects, how the rock architecture affects arecovery process, etc.) reservoir data; andconvenience and speed--handdrawing reservoir architectur and point-value realizations in three-dimensionsis a very difficult and time-consuming process.
The phenomena or variables that we normally describe with stochastic modelsare those that influence the amount, position, accessibility, and flow offluids through reservoirs. Thus, stochastic modeling or simulation in thiscontext usually refers to the generation of synthetic geological architectureand/or property fields in one, two, or three dimensions. The differentrealizations are conditioned to observations and possess a number of otherdesirable reservoir/geological features that should provide an improved basisfor recovery predictions. In addition, the uncertainty and risk associated withdifferent development options can be quantified better
Dubrule gives a very good review of stochastic models for reservoirdescription, while Weber and van Geuns discuss the problem from a geologist'spoint of view, problem from a geologist's point of view, including some of thepossible pitfalls. Several authors present valuable contributions to the theoryand applications of stochastic modeling within the petroleum industry.

Two-Stage Model
To mimic reality, heterogeneity must be accounted for because it is one ofthe most important factors governing fluid flow. A number of differentapproaches exist for the stochastic modeling of heterogeneities. The choice oftechnique depends on (1) the objective and scale of the study, (2) theavailable input data, (3) the theoretical skills of the people involved, and(4) the software available. The goal is to improve the evaluation of theproduction capacity of the field by introducing small- and/or large-scaleheterogeneities into the reservoir description.
Discrete vs. Continuous Models. The distinction between two main classes ofstochastic models (discrete and continuous) is convenient. A finerclassification of the discrete models has been proposed. Discrete models weredeveloped to describe geological features of a discrete nature (e.g., locationsof sand in fluvial depositional environments or locations of shales suspendedin sands).
JPT
P. 402