- Jef:In this short presentation, we will talk about a field called multiple-point geostatistics. Before explaining in detail what this field is, consider first the definition of geostatistics itself. You may probably know that geostatistics is a branch of statistical science where if you study spacial phenomenon, such as this digital elevation map shown here, and capitalize on spacial relationships established on sample data extracted from the single truth, to either do something that's called estimation, also, in its general from, universal kriging, or stochastic simulation, where we try to mimic the truth as much as possible and create multiple realizations. What will you notice immediately is that neither does universal kriging nor these Gaussian Simulation models, or these Gaussian realizations, actually have much in common with the truth. This is a common problem in the traditional application of geostatistics. Let's ask ourselves a question. Why does this occur? The reason for this lies in the limitations of the variogram. The variogram is essentially a 2-point statistics. Given these 3 images, which evidently are very different numerical models, all matching this 30-sample data, they essentialy show the same variogram. Here we see the variogram calculated in the east/west direction, and the variogram in the north/south direction of these 3 exhaustive images. The variogram, neither the data can discriminate between these 3 very different geological heterogeneities. Instead of relying on the variogram to capture spacial variability, multiple point geostatistics relies on another way of describing spacial variability. Consider here these same or similar 30-sample data. What typically happens in a reservoir, and consider these are wells drilled in the subsurface, is that the first thing that a geologist does is not calculate a variogram. The first thing a geologist often does is to interpret geological variability to essentially make a conceptual view of what he or she thinks is down in the subsurface. In geostatistics, we ask the geologist to convey this information in an exhaustive image, which is called the training image. In this case we have a simple 2D image, but later we see that, in reality, we have much more complexity and we will deal with 3D training images. There are several methods in multiple-point geostatistics that have been developed that essentially capture the patterns that we see in this training image and anchor them to the data to create multiple realizations, also called MPS realizations. As you can see very clearly, all realizations capture the variablity that is shown on the training image and, if you would verify it, it would also match the data. This is a very simple problem that has many applications. Essentially what we are trying to do here is to provide the geologist with a very easy means to convey his or her understanding of what is the geological heterogeneity in the subsurface. The only thing that needs to be done is not deal with a mathematical model such as a variogram, but to provide, essentially, a conceptual description of the subsurface using various techniques. We'll see later which techniques we can use, such as boolean techniques and process-based models. Then the idea of multiple-point geostatistics is to extract these very complex patterns and anchor them to subsurface data, which could be either well data, seismic data, and production information. Here we see, for example, 4 training images, and from each training image a single realization generated with the same multiple-point geostatistical algorithm. In order to understand a little bit what the nature of multiple-point geostatistics is and how it applies in reality, I will present 2 actual case studies. In this first case study, we will use object-based techniques to generate training images. Object-based techniques essentially represent subsurface geology using objects and has been very popular in creating realistic geological models. The problem, often, with object-based techniques is that they are very difficult to constrain to data, and particularly when such data is very dense, or when such data is indirect, such as production data and seismic data. Here we have a real case, it was one of the first cases used around 2000, to illustrate the use of multiple-point geostatistics. We have a case with 140 wells. Along each of these wells we have recorded lithology. Based on this information and based on other background geological information of these tidal-dominated reservoirs, geology provided an object-based description of the reservoir, which can be seen in this table. In this table we see that there is various facies types, conceptual description, basically their geometries, and also the spacial distribution, where they occur and how they occur in relationship with other facies types. Given this information, geologists can then use basic object-based techniques; however, without the need to constrain these to any specific reservoir data and generate what is called an unconditional object-based, or unconditional boolean model. This unconditional model is essentially an explicit reflection of the table that has been provided here, and can be used as a training image. Then, multiple-point geostatistical algorithm essentially come in in generating facies model realizations, which I've shown here, that are exactly constrained to this data. There is no approximation made. At the same time, these techniques allow some constraining to other information such as vertical proportion curves or aerial proportion maps, which can be derived from seismic or well data. As we notice here, the model looks quite realistic. It generates the elliptical shapes of the sandbars. It generates correctly the relationships. It is conditioned to the data and it also reflects the seismic data that says that in this corner, for example, we should have more shale and less sand. In a second case study, I would like to discuss the use of process-based models to generate training images. This is, again, a real reservoir case study. On the left, we see a horizontal section of a seismic image, which clearly shows channel variation. This is in a turbidite type reservoir. Geologists can now use process models, which are actually much more physically realistic than object-based models, to generate heterogeneity in the subsurface, conceptually, and somewhat constrained to the seismic information that we have here. The problem with these process models, often, is that they take a very long time to run, if they are physical models. The second problem is that they are not constrained to any well data, and not exactly constrained to the seismic data. What we can do with multiple-point geostatistics is to use this 3D process model as a training image to further be constrained to such data. This is done in the next slide. Here we see that 10 wells are available, vertical wells. This is a reservoir that is still pre-production with recorded lithology. We have various MPS realizations that constrain exactly to this very complex variation of facies along the wellbore and reflect, as given in the previous slide, the variability we notice in the turbidite process model. Let's have a more detailed look. Here we see the well, the lithofacies recorded, and we see that the MPS model conditions nicely to this well data. In summary, we can say that multiple-point geostatistics relies on training images for geologists to convey their understanding about geological heterogeneity and various sources of information can be used to generate these training images. We showed 2 applications, 1 with object models and 1 with process models. The idea of MPS algorithms is then to capture the patterns in these training images and anchor them to subsurface data. For such applications there has been applications conditioned to wells, seismic as well as production data.