CONDITIONAL SIMULATION AND UNCERTAINTY ESTIMATION INTRODUCTION
Stochastic modeling, also known as conditional simulation, is a variation of conventional kriging or cokriging. An important advantage of the geostatistical approach to mapping is the ability to model the spatial covariance before interpolation. The covariance models make the final estimates sensitive to the directional anisotropies present in the data. If the mapping objective is reserve estimation, then the smoothing properties of kriging in the presence of a large nugget may be the best approach. However, if the objective is to map directional reservoir heterogeneity (continuity) and assess model uncertainty, then a method other than interpolation is required (Hohn, 1988).
Once thought of as stochastic “artwork”, useful only for decorating the walls of research centers (Srivastava, 1994a), conditional simulation models are
becoming more accepted into our day-to-day reservoir characterization-modeling efforts because the results contain higher frequency content, and lend a more realistic appearance to our maps when compared to kriging.
Srivastava (1994a) notes that, in an industry that has become too familiar with layer-cake stratigraphy, with lithologic units either connected from well-to-well or that conveniently pinch out halfway, and contour maps that show gracefully curving undulations, it is often difficult to get people to understand that there is much more inter-well heterogeneity than depicted by traditional reservoir models.
Because stochastic modeling produces many, equi-probable reservoir images, the thought of needing to analyze more than one result, let alone flow simulate all of them, changes the paradigm of the traditional reservoir characterization approach. Some of the realizations may even challenge the prevailing geological wisdom, and will almost certainly provide a range of predictions from optimistic to pessimistic (Yarus, 1994).
Most of us are willing to admit that there is uncertainty in our reservoir models, but it is often difficult to assess the amount of uncertainty. One of the biggest benefits of geostatistical stochastic modeling is the assessment of risk or uncertainty in our model. To paraphrase Professor Andre Journel “… it is better to have a model of uncertainty, than an illusion of reality.”
Before reviewing various conditional simulation methods, it is useful to ask what is it that we want from a stochastic modeling effort. We really need to consider the goal of the reservoir modeling exercise itself, because the simulation method we choose depends, in large part, on the goal of the study and the types of data available. Not all conditional simulation studies need the Cadillac approach, when a Volkswagen technique will do fine (Srivastava, 1994a).
WHAT DO WE WANT FROM ACONDITIONAL SIMULATION METHOD?
Srivastava (1994a), in an excellent review of stochastic methods for reservoir characterization, identifies five major types of stochastic simulation model approaches:
Assessing the impact of uncertainty.
Monte Carlo risk analysis.
Honoring heterogeneity.
Facies or rock properties (or both)
Honoring complex information.
The interested reader should refer to the original article for details, which is only summarized in this presentation.
Assessing the Impact of Uncertainty
Anyone who forecasts reservoir performance understands that there is always uncertainty in the reservoir model. Performance forecasts or volumetric predictions are often based on a “best” case model. However, the reservoir engineer is also interested in other models, such as, the “pessimistic” and
“optimistic” case. These models allow the engineer to assess whether the field development plan, based on the “best” case scenario, is flexible enough to handle the uncertainty. When used for this kind of study, stochastic models offer
many models consistent with the input data. We could then sort through the many realizations, select one that looks like a downside scenario, and find another that looks like an up-side model.
Monte Carlo Risk Analysis
A critical aspect for the use of stochastic modeling is the belief in some “space of uncertainty” and that the stochastic simulations are outcomes which sample this space fairly and adequately. We believe that we can generate a fair
representation of the whole spectrum of possibilities and hope that they do not have any systematic tendencies to show pessimistic or optimistic scenarios. This type of study involves the idea of a probability distribution, rather than simply sorting through a large set of outcomes and selecting two that seem plausible. In Monte Carlo risk analysis, we depend on the notation of a complete probability distribution of possible outcomes, and that the simulation realizations fairly represent the entire population.
Honoring Heterogeneity
Although stochastic techniques are capable of producing many plausible outcomes, many studies only use a single outcome as the basis of performance prediction. Over the past decade, it has become increasing apparent that
reservoir performance predictions are more accurate when based on models that reflect possible reservoir heterogeneity. We are painfully aware of the countless examples of failed predictions due to the use of overly simplistic models. The thought of using only a single outcome from a stochastic modeling effort is often viewed with disdain by those who like to generate hundreds of realizations.
Srivastava (1994a) argues that “even a single outcome from a stochastic
approach is a better basis for performance prediction than a single outcome from a traditional technique that does not honor reservoir heterogeneity.”
Granted, many people will argue with this statement, because that one simulation may be the pessimistic (or optimistic) realization just by the “luck of the draw,”
probabilistically speaking.
Facies or Rock Properties (or both)
Reservoir modelers recognize two fundamentally different aspects of stochastic reservoir models. The reservoir architecture is usually the first priority, consisting of the overall structural elements (e.g. faults, top and base of reservoir, etc.), then defining the geobodies based on the depositional environment (e.g., eolian, deep-water fan, channels, etc.). Once the spatial arrangement of the different flow units are modeled, we must then decide how to populate them with rock and fluid properties. The important difference between modeling facies versus modeling rock properties is that the former is a categorical variable, whereas the latter are continuous variables. Articles by Tyler, et al. (1994), MacDonald and Aasen (1994), and Hatloy (1994) provide excellent overviews of these methods.
Though it is conventionally assumed that a lithofacies model is an appropriate model of reservoir architecture, we should ask ourselves whether this is a good assumption. Just because the original depositional facies are easily recognized and described, they may not be the most important control on fluid flow. For example, permeability variations might be due to later diagenesis or tectonic events (Srivastava, 1994 ).
Honoring Complex Information
Stochastic methods allow us to incorporate a broad range of information that most conventional methods can not accommodate. Many individuals are not so much interested in the stochastic simulation because it generates a range of plausible outcomes, but because they want to integrate seismic data with petrophysical data while obtaining some measure of reliability.
Properties of Conditional Simulation
Conditional simulation is a Monte Carlo technique designed to:
honor measured data values
approximately, reproduce the data histogram
honor the spatial covariance model
be consistent with secondary data
assess uncertainty in the reservoir model Conditional Simulation Methods
The following section is but a very brief review of stochastic simulation methods in common use, followed by a discussion of important practical advantages and limitations of each method.
The terms stochastic and conditional are sometimes used interchangeably.
Technically, they each mean something different. Stochastic typically connotes randomness to most people. In geostatistics, we define stochastic simulation as the process of drawing equally probable, joint realizations of the component Random Variables from a Random Function model. These are usually gridded realizations, and represent a subset of all possible outcomes of the spatial distribution of the attribute values. Each realization is as called a stochastic image (Deutsch and Journel, 1992). If the image represents a random drawing from a population of mean = 0 and variance = 1, based on some spatial model, we would call this type of realization a non-conditional simulation. However, a simulation is said to be conditional when it honors the measured values of a regionalized variable (Hohn, 1988). For the remainder of this discussion, stochastic and conditional will be used as equivalent processes.
Non-conditional simulations are often used to assess the influence of the spatial model parameters, such as the nugget and sill values, in the absence of control data. Each of these parameters has a direct affect on the amount of variability in the final simulation. Increasing either the sill or nugget increases the amount of variability in a simulated realization.
Srivastava (1994a) lists the following types of stochastic simulation methods:
Turning Bands
Sequential Simulation -
Gaussian
Indicator
Bayesian
Simulated Annealing
Boolean, Marked-Point Process and Object Based
Probability Field
Matrix Decomposition Methods
We will describe each of these methods in turn.
Turning Bands
This is one of the earliest simulation methods, tackling the simulation problem by first creating a smooth model by kriging, then adding an appropriate level of noise. The noise is added through a non-conditional simulation step using the same histogram and spatial model as in the kriging step, but does not use the actual data values at the well locations. The final model still honors the original data and the spatial model, but now also has an appropriate level of spatial heterogeneity (Srivastava, 1994a; Deutsch and Journel, 1992).
Sequential Simulation
Three sequential simulation (Gaussian, Indicator, and Bayesian) procedures make use of the same basic algorithm for different data types. The general process is
1. Select at random grid node GNi, a point not yet simulated in the grid.
2. Use kriging to estimate the mean, mi,andvariance, i2
at location GNi from the local Gaussian conditional probability distribution (lGcpd), with zero mean and unit variance.
P (ZSi ZS1 , . . ., ZSi-1) exp [(ZSi -mi)2 / 2i2
] where:
mi is estimated by any of the kriging methods, including kriging with external drift (KED)
i2
is the error variance of mi,
3. Draw at random a single value, zi from the lGcpd, whose maximum spread is 2 around mi
4. Create a newly simulated value ZSi*
= mi + zi. 5. Include the newly simulated value ZSi*
in the set of conditioning data. This ensures that closely spaced values have the correct short scale correlation.
6. Repeat the process until all grid nodes have a simulated value.
Selection of the Simulated Grid Node
The first step in sequential simulation is the random selection of a location GNi, then GNi +1, until all grid nodes contain a simulated value. The order in which grid nodes are randomly simulated influences the cumulative feedback effect on the outcome. The selection process is random, but repeatable:
For each simulation, shuffle the grid nodes into an order defined by a random seed value.
Each random seed corresponds to a unique grid order.
Different random seed values produce a different path through the grid.
Although the total possible number of orderings is very large, each random path is uniquely identified and repeatable.
Sequential Gaussian Simulation (SGS) is a method for the simulation of continuous variables, such as petrophysical properties. In SGS, the procedure is essentially the same as (co) kriging, with the addition of a bias.
Sequential Indicator Simulation (SIS) is a method used to simulate discrete variables. By creating a grid of 0s and 1s, it uses the same methodology as SGS, which represent “lithofacies” (pay/non-pay, or sand/shale).
SIS requires the following input parameters:
The a priori probabilities (proportions) of two data classes (Indicators -denoted as I) coded as 0 or 1, for example:
I(zx) = 1 if zx is shale
I(zx) = 0 if zx is sand
Indicator histogram
The Indicator spatial correlation model
Bayesian Sequential Indicator Simulation is a later form of SIS (Doyen, ET al., 1994). This technique allows direct integration of seismic attributes with well data using a combination of classification and indicator methods.
Bayesian SIS input parameter requirements:
Code well data as 0 or 1, as in SIS.
Classify the seismic attribute into two classes (0, 1):
Assuming the two data classes are Normal Distributions, we need the:
Mean and standard deviation of the seismic attribute that is 0.
Mean and standard deviation of the seismic attribute that is 1.
The a priori probabilities of the two classes of seismic data.
The Indicator spatial correlation models.
Simulated Annealing
Annealing is the process where a metallic alloy is heated so that the molecules move around and reorder themselves into a low-energy grain structure. The probability that any two molecules will follow each other is known as the Boltzmann probability distribution. Simulated annealing is the application of the annealing mechanism of swapping the attributes assigned to two different grid node locations, using the Boltzmann probability distribution for accepting the perturbations (Deutsch, 1994). The process continues until the desired model conditions are satisfied.
Simulated annealing constructs the reservoir model via an iterative trial and error process, and does not use an explicit random function model. Rather, the simulated image is formulated as an optimization process. The first requirement is an objective (or energy) function, which is some measure of difference
between the desired spatial characteristics and those of the candidate realization (Deutsch, 1994). For example, we might want to produce an image of a
sand/shale model with a 70% net-to-gross ratio, an average shale length of 60 m, with average shale thickness of 10 m.
The image starts with pixels arranged randomly, having sand and shale in the correct global proportion. The net-to-gross is incorrect because of the random
assignment of the sand and shale. The average shale length and width are too short also. Next, the annealing mechanism swaps attributes at different grid node locations, applies the Boltzmann probability distribution for accepting the
perturbations, and continues until the model conditions are satisfied.
At first glance, this approach seems terribly inefficient, because millions of perturbations may be required to arrive at the desired image. However, these methods are more efficient than they appear in theory (Deutsch, 1994).
Boolean, Marked-Point Process and Object Based
Theses methods constitute a family of techniques that create reservoir models based on objects of some genetic significance, rather than being built up from one elementary node or pixel at a time. To use such methods, you need to select a basic shape for each lithofacies that describes its geometry. For example, you might want to model sand channels that look like half ellipses in cross section, or deltas as triangular wedges in map view. You must also specify the proportions of the shapes in the final model and choose a distribution for the parameters that describe the shapes. There are algorithms that describe how the geobodies are positioned relative to each other (that is, can they overlap, and how, or must there be a minimum distance between the shapes).
After the distribution of parameters and position rules are chosen, follow the remaining steps in the procedure (Srivastava, 1994a):
1. Fill the reservoir model background with some lithofacies (e.g., shale).
2. Randomly select a starting point in the model.
3. Randomly select one of the lithofacies shapes, and draw an appropriate size, anisotropy and orientation.
4. Check to see if the shape conflicts with any conditioning data (e.g., well data) or with other previously simulated shapes. If not, keep the shape, otherwise reject it and go back to the previous step.
5. Check to see if the global proportions are correct, if not, return to step 2.
6. Simulate petrophysical properties within the geobodies using the more classical geostatistical methods. If control data must be honored, this step is typically completed first, and then the inter-well region is simulated. Be sure that there are no conflicts with known stratigraphic and lithologic sequences in the wells.
Boolean or object-based techniques are of current interest in the petroleum industry with a number of research, academic, and commercial vendors working on new implementation algorithms. In the past, Boolean-type algorithms could not always honor all of the conditioning data, because the algorithms were not strict simulators of shape. The number of input parameters made this almost a deterministic method, requiring much upfront knowledge of the depositional system you wanted to model. Articles by Tyler, et al. (1994) and Hatloy (1994) provide excellent case studies using Boolean-type methods to simulate fluvial systems.
Probability Field Simulation
This method is an enhancement of the sequential simulation methods described earlier. In sequential simulation, the value drawn from the local cumulative probability distribution at a particular grid node is treated as if it was hard data,
and is included as local conditioning data. This ensures that closely spaced values have the correct short scale correlation. Otherwise, the simulated image would contain too much short scale (high frequency noise) variability.
The idea behind probability field, or P-field, simulation is to increase the efficiency of computing the local conditional probability distribution (lcpd) on the original well data only. P-field simulation gets around the problem of too much short scale variability by controlling the sampling of the distributions rather than controlling the distributions as in sequential simulation (Srivastava, 1994a).
Srivastava (1994b) shows how P-field simulation improves the ability to visualize uncertainty and the article by Bashore, et al, (1994) illustrates a P-field
application for establishing an appropriate degree of correlation between porosity and permeability.
Matrix Decomposition Methods
Some simulation techniques involve matrix decomposition; L-U decomposition is one such example, using a matrix represented as the product of a lower
triangular matrix, L, and an upper triangular matrix U. This decomposition can be made unique either by stipulating that the diagonal elements of L be unity, or that the diagonal elements of L and U be correspondingly identical. In this approach, different outcomes are created by multiplying vectors of random numbers by a precalculated matrix created from spatial continuity information supplied by the user, typically as a variogram or correlogram. Matrix methods can be viewed as a form of sequential simulation because the multiplication across the rows of the precalculated matrix and down the column vector of the random numbers can be construed as a sequential process in which the value of the successive node depends upon the value of the previously simulated nodes (Srivastava, 1994a;
Deutsch and Journel, 1992).
Uncertainty Estimation
Once all of these simulated images have been generated, how do you determine which one is correct? Technically speaking, any one of the simulated images is a possible realization of the reservoir, because each image is equally likely, based on the data and the spatial model. However, just because the image is
statistically equally probable does not mean it is geologically acceptable. You must look at each simulated image to determine if it is a reasonable
representation of what you know about the reservoir -if not, discard it, and run more simulations if necessary.
Some of the possible maps generated from a suite of simulated images include:
Mean: This map is the average of n conditional simulations. At each cell, the program computes the average value, based on the values from all simulations at the same location. When the number of input simulations is large, the resultant map converges to the kriged solution.
Minimum: Each cell displays the smallest value from all input simulations.
Maximum: Each cell displays the largest value from all input simulations.
Standard Deviation: A map of the standard deviation at each grid cell, computed from all input maps. This map is used as a measure of the standard error and is used to analyze uncertainty.
Uncertainty or Risk: This map displays the probability of meeting or
Uncertainty or Risk: This map displays the probability of meeting or