- Let's study Sequential Indicator Simulation, Steps of Sequential Indicator Simulation, and Indicator Variogram. Sequential Indicator Simulation is a procedure that uses the indicator kriging mean to generate the conditional probability distribution field. It uses input data and simulated data when computing a value at an unsimulated grid cell. It generates many equally probable realizations which can be post-processed to quantify and assess uncertainty. It preserves the variance observed in the data, instead of just the mean value, as in interpolation. It is the most commonly used form of geostatistical simulation for reservoir modeling for discrete variables like facies. The family of sequential simulation methods makes use of the same basic algorithm. There are following basic steps in the SIS process. One, Binary Indicator Transform is to transform the original continuous well data into indicator variables. Two, assign transformed data into simulation grid. Three, generate a random path through the grid nodes. Four, visit a node in the random path and use indicator kriging to estimate a mean for each indicator variable based on surrounding data, variogram parameters, search parameters, and global facies proportions. That generates a local conditional probability distribution, that is lcpd. Five, select at random a value from the lcpd based on indicator threshold and set the node value to that number. Six, include the newly simulated value as part of the conditioning data. Seven, repeat steps four to six until all grid nodes have values. SIS requires the input data are discrete indicator variables. In this example, original data are net to gross. If threshold is 0.3, a net to gross which is equal to or less than 0.3 will be transformed to shale, that is zero. Please see, 0.08 is transformed into zero. 0.12 is transformed into zero. 0.11 is transformed into zero. 0.23 is transformed into zero. A net to gross which is larger than 0.3. will be transformed to sand, that is one. Please see 0.8 is transformed into one. 0.75 is transformed into one. 0.68 is transformed into one. This is an example of two indicators. The procedures are similar for more than two indicators. Assign transformed data into simulation grid which will be used as conditioning data. Seven conditioning data are assigned to their nearest gird cell centers. When multiple data are assigned to same node, we can take the maximum occurrence one, or closest one to the cell center. In this example, we do maximum occurrence for one, one, and zero, so it is one. At the end, one cell has one datum. A random path is used to avoid the artifacts induced by visiting grid nodes in a regular fashion. The random path must go through each cell once and in random order. This is an example of a random path for nx equal to 8, ny equal to seven grid. The order is from first to second, et cetera, until 56th. Input data are conditional data, global facies proportions, indicator variogram parameters, search parameters, seed. It is a number used to initialize a pseudorandom number generator. And a number of realizations. Input data table is copied here. This is a global facies proportion table. The facies counts of the input data are computed and shown in the table. There are three zeros, two ones. The facies proportion values of the input data are computed and shown in the table. Three divided by five is 0.6. Two divided by five is 0.4. The facies proportion values are editable and are used as global facies proportions in simulation. When the variogram correlation ranges are short and the control data are scarce, the final results might reach the input global facies proportions. Before computing variogram, indicator transform applies to the input data. It is one if facies k present at location u. It is zero if not. Indicator variogram two gamma for lag h is defined as the average squared difference of values separated approximately by h. Two gamma h is variogram. Gamma h is semivariogram. Sometimes, two gamma is denoted by gamma and called variogram. It is a confused concept in geostatistics. Let's pay attention to the definition of the equation. N h is the number of pairs for lag h. I is investigated property value. I, u, k is the ith I value for facies k at u. I, u plus h, k is the ith I value for facies k at u plus h. K is for facies one, two, et cetera. Input data is copied here. First apply indicator transform, then compute the variance of the transformed data. It is 0.24 which is used as the sill. The variogram values, green dots shown here, are not real data. Input data is copied here. First apply indicator transform, then compute the variance of the transformed data. It is 0.24 which is used as the sill. The variogram values, green dots, shown here are not real data. The first node, marked as one, is not yet simulated. Let us work on that. The underlying algorithm of Sequential Indicator Simulation is indicator kriging. Indicator kriging gives us an estimate of mean for each facies which is the probability of this facies, using conditional data, indicator variogram parameters, and search parameters, and global facies proportions. The standard deviation is not in use. In this example, we get mean, that is probability, of sand is 0.18, mean of shale is 0.67. Input global facies proportions are assigned to the cell when there are no conditional data obtained in the search area. Sorting mu sand equal to 0.18 and mu shale equal to 0.67 from shale to sand. That is first is mu shale equal to 0.67, and the second is mu sand is 0.18. Rescale them into zero to one cumulative probability space. That is mu shale equal to 0.79, mu sand equal to 0.21. SIS uses mu equal to 0.79 and mu equal to 0.21 construct a cumulative probability distribution. See blue lines. Run uniform random number generator which represents the probability level. We get probability 0.71. Follow the arrowed gray lines, we get indicator value zero. Set the simulated value zero to the grid node. See blue cell. Include the simulated value x equal to 35, y equal to 45, indicator equal to zero in the set of conditioning data. That ensures that closely spaced values have the correct short-scale correlation. The second node, marked as two, is not simulated yet. The similar procedures as the first node will be conducted. One difference is the simulated value zero, shown in a purple circle, can be used as the conditional data if it is in the search region in order to preserve the proper covariance structure, spatial continuity, between simulated values. Repeat steps four to six until all grid nodes have simulated values. Let's review the steps of SIS. One, Binary Indicator Transform. Two, assign transformed data into simulation grid. Three, generate a random path through the grid nodes. Four, visit a node in the random path and use indicator kriging to estimate a mean for each indicator variable which are used to build lcpd. Five, select at random a value from the lcpd based on indicator threshold and set the node value to that number. Six, include the newly simulated value as part of the conditioning data. Seven, repeat steps four to six until all grid nodes have values.