Sequential Gaussian Simulation

Two of the most commonly used forms of geostatistical simulation for reservoir modeling are Sequential Gaussian Simulation (SGS) for continuous variables like porosity and Sequential Indicator Simulation (SIS) for categorical variables like facies. The family of sequential simulation methods makes use of the same basic algorithm.

Stochastic simulation is a means for generating multiple equiprobable realizations of a property, rather than simply estimating the mean, for example when using the kriging algorithm. Essentially, we are adding back in some variability to undo the smoothing effect of kriging. This result possibly gives a better representation of the natural variability of the property and provides a means for quantifying uncertainty.

Sequential Gaussian Simulation

The underlying algorithm of Sequential Gaussian Simulation (SGS) is simple kriging. Recall that kriging gives us an estimate of both the mean of the property value and its standard deviation at each grid node, meaning that the variable at each grid node is represented as a random variable following a normal (Gaussian) distribution.  Rather than choosing the weighted mean as the estimate at each node, SGS chooses a random deviate from this normal distribution, selected according to a uniform random number representing the probability level.

There are five basic steps in the SGS process:

  1. Generate a random path though the grid nodes.

  2. Visit the first node in that path and use kriging to estimate a mean and standard deviation at that node based on surrounding data (i.e., local conditional probability distribution, or lcpd).

  3. Select at random a value from the lcpd and set the node value to that number.

  4. Include the newly simulated value as part of the conditioning data.

  5. Repeat steps 1 to 4 until all grid nodes have a simulated value.

A random path is used to avoid the artifacts induced by visiting grid nodes in a regular fashion. Sequential methods also use previously simulated values as “data” in the lcpd in order to preserve the proper covariance structure (spatial continuity) between simulated values.

For SGS it is important that the data actually follow a Gaussian distribution, so when SGS is used, the data are first transformed using the normal score transform.

In addition to the conditioning data that we want to honor, the input to a sequential simulation depends on the procedure used to estimate the cumulative density function (CDF). SGS requires us to describe the spatial continuity in terms of a variogram. These methods are very flexible and can also accommodate secondary data, but require a variogram of the secondary data and its relationship to the primary data. In the Earth Modeling implementation the strength of the relationship is controlled by the correlation coefficient between the primary and secondary data.