Using Multiple Grids in Markov Mesh Facies Modeling
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Abstract
A multigrid Markov mesh model for geological facies is formulated by defining a hierarchy of nested grids and defining a Markov mesh model for each of these grids. The facies probabilities in the Markov mesh models are formulated as generalized linear models that combine functions of the grid values in a sequential neighborhood. The parameters in the generalized linear model for each grid are estimated from the training image. During simulation, the coarse patterns are first laid out, and by simulating increasingly finer grids we are able to recreate patterns at different scales. The method is applied to several tests cases and results are compared to the training image and the results of a commercially available snesim algorithm. In each test case, simulation results are compared qualitatively by visual inspection, and quantitatively by using volume fractions, and an upscaled permeability tensor. When compared to the training image, the method produces results that only have a few percent deviation from the values of the training image. When compared with the snesim algorithm the results in general have the same quality. The largest computational cost in the multigrid Markov mesh is the estimation of model parameters from the training image. This is of comparable CPU time to that of creating one snesim realization. The simulation of one realization is typically ten times faster than the estimation.
Keywords
Multiple grids Markov mesh models Sequential simulation Generalized linear models1 Introduction
The spatial distribution of facies is a crucial part of any reservoir model since it is often one of the main sources of variability in flow (Skorstad et al., 2005). Multipoint statistics is one class of methods for geological facies modeling, proposed nearly two decades ago (Guardiano and Srivastava 1993), and it has developed along two main paths: the statistical model approach (Tjelmeland and Besag 1998) and the algorithmic approach (Strebelle and Journel 2000). Common to many multipoint methods is the use of a training image that represents the geologic patterns typically found in the reservoir of study. The multipoint methods aim at reproducing the essential aspects of these patterns, but with a variability that can be adapted to the case at hand. During simulation, algorithmic multipoint methods create situations where the conditioning event is not found in the training image. This creates artifacts in the simulations (Strebelle and Remy 2005). Statistical models on the other hand, can interpolate between observed patterns to compute the probability of patterns that are not explicitly present in the training image, and hence artifacts can potentially be reduced. With the introduction of the Markov mesh model (Stien and Kolbjørnsen 2011), the statistical model based approach also overcame its original timeconsumption problems in parameter estimation and simulation. In this paper, we proceed yet another step, and formulate a multigrid Markov mesh model.
The use of multiple grids has previously been used in the algorithmic approach to multipoint methods (Strebelle 2002), and for general geostatistical approaches (Tran 1994). The strategy has proved invaluable for capturing patterns at different scales. With a multigrid Markov mesh model, we combine an advantage developed for algorithmic methods—the use of multiple grids, with the consistency and flexibility of the statistical model. Combining multiple grids with Markov random fields was recently explored by Toftaker and Tjelmeland (2013). Markov mesh models are a subclass of Markov random fields (Tjelmeland and Besag 1998) defined through a unilateral path (Daly 2005; Daly and Knudby 2007). In Stien and Kolbjørnsen (2011), the authors propose to model facies geometries through a singlegrid Markov mesh model defined using the framework of generalized linear models (McCullagh and Nelder 1989; Cressie and Davidson 1998). In this paper, a hierarchy of grids is defined, and a Markov mesh model analogous to that of Stien and Kolbjørnsen (2011) is defined for each grid, but such that it takes into account information also from coarser grids. The result is what we denote a multigrid Markov mesh model. The framework of generalized linear models and systematic grid specification enable fast parameter estimation. The estimation is done once per grid level. During simulation, the coarse patterns are first laid out, and by simulating increasingly finer grids we are able to create patterns at different scales. We present several threedimensional examples, illustrating that the multigrid Markov mesh model can be successfully applied for a range of training images. For each training image, the simulation results are quantitatively evaluated by comparing facies fractions and upscaled permeability tensors of realizations and training images.
2 Multigrid Markov Mesh Model
Multigrid Markov mesh models are defined by a hierarchy of grids, a unilateral path per grid level, and a conditional probability for each cell value given the cell values in a sequential neighborhood. Mathematically, the multigrid Markov mesh model is nothing but a singlegrid Markov mesh model where the cells are visited according to the overall path, and the sequential neighborhood for any cell consists only of cells from the past part of this path. It is nevertheless useful to explicitly discuss the model in terms of the multiple grid levels, since it is the systematic model specification in terms of these levels that makes it an efficient and useful tool for capturing patterns at different scales.
2.1 The SingleGrid Model
2.2 Capturing Large Scale Patterns
2.3 Defining the Multigrid Markov Mesh Model
2.4 Data Conditioning
3 Model Specification
The statistical model is defined by specifying the path and parameterization of the conditional probabilities in Eq. (10). The path is uniquely determined by the sequence of grid. The statistical model specification is based on generalized linear models (McCullagh and Nelder 1989). The formulation is such that the parameters are efficient to estimate and simple to interpret.
3.1 Specifying the Sequential Neighborhood
For each grid level, we specify the maximal extension in different directions, and then include all cells inside these limits provided they belong to the past path. Since each grid Open image in new window is a regular grid, assuming the onedimensional indexing is also regular, all cells Open image in new window are then assigned sequential neighborhoods of exactly the same shape. This is important for the efficiency of the model. When all cells have the same neighborhood, we only have to estimate one model at each grid level. This is faster than estimating several models and the estimates are more precise since we have more available data. For a sequential neighborhood bounded by a rectangular box, six parameters are sufficient for parameterization, the extent of the box and the location of the reference cell. If the box is assumed symmetric around the reference cell, three parameters describing the extent of the box (l _{ x },l _{ y },l _{ z }) establishes the neighborhood.
3.2 Using the Framework of Generalized Linear Models
The use of a generalized linear model (GLM) for specifying Markov mesh models for facies modeling was first suggested in Stien and Kolbjørnsen (2011). The idea in GLM is that the distribution of a response variable depends on a linear combination of explanatory variables through a nonlinear link function. We let the facies x _{ i } be the response variable, and the explanatory variables be functions of the sequential neighborhood Γ _{ i }. Consider a given grid level l and let the cell i be on this grid level that is Open image in new window . Let z _{ i } be a P _{ l }dimensional vector of explanatory variables with elements that are functions of cells from the sequential neighborhood Γ _{ i }. We propose particular functions below, but for now simply write \(z_{ij} = f_{j}(x_{\varGamma_{i}})\) for j∈{1,2,…,P _{ l }}. The same set of P _{ l } functions is used for any neighborhood Γ _{ i } if Open image in new window . The value \(f_{j}(x_{\varGamma_{i}})\) varies with i, since it depends on the facies configuration in the neighborhood Γ _{ i }. In the model, we propose there is one model parameter for every pair of neighborhood function and facies value at each grid level l. Assuming there are K different facies values, we let the K vectors \(\mathbf{\theta}_{l}^{1},\ldots,\mathbf{\theta}_{l}^{K}\) hold the parameters. Each vector is P _{ l }dimensional.
3.3 Neighborhood Functions
As noticed by Stien and Kolbjørnsen (2011), the challenge with multipoint statistics is that there generally are too many possible patterns. A finite training image does not hold information about all possibilities. To overcome this problem, we extract a subset of properties, represented by neighborhood functions that are important in order to reproduce geological structures and aim at making robust choices. The specification of the neighborhood functions is similar to the specification of Stien and Kolbjørnsen (2011) in the sense that the threedimensional model consists of combining three twodimensional models. For each of three orthogonal grid slices, intersecting at the reference cell, we add offtwodimensional extensions and consider two point interactions, multipoint interactions representing continuity and transitions of facies, and multipoint interactions representing all possible patterns for a very limited number of cells.
3.3.1 TwoDimensional Specification
3.3.2 ThreeDimensional Specification
3.4 Grid Refinement
The purpose of the multigrid formulation is to be able to reproduce patterns on large scales using small neighborhoods or templates. This is done by initially applying the neighborhood or template on a coarse grid and applying the same neighborhood or template on successively finer grids. The grids are refined by inserting grid levels to increase the grid resolution (Fig. 3). There is no unique way of refining the grid that will work equally well for all training images. Two different grid refinement sequences will represent two different models. We will proceed by selecting one refinement scheme that is adapted to the training image and estimate the parameters for this model. The alternative of using multiple refinement schemes, each with an attached probability, is abandoned in order to have a parsimonious model representation. The challenge is therefore to find a nice way of obtaining grid Open image in new window from the grid at the previous level, Open image in new window by a refinement in the x, y, and zdirections. It is possible to refine in a single direction or to combine two or three directions at the same time. The simplest is either to choose a single refinement in one direction or single refinement in all three directions. Treating all directions equal is possible, but training images are seldom isotropic so we have chosen to refine in one direction at a time. This also gives the most flexible model and the additional computational cost is acceptable since most of the CPU time is spent on the final grid resolutions. Refining in one direction means that grid Open image in new window has twice as many grid cells as the grid at the previous level, Open image in new window . We start by selecting a coarse grid and choose the direction of refinement along the direction of the weakest spatial correlation. This has the consequence that the direction of refinement will change several times so that spatial structures in all directions interact at several resolutions. This consequence is not obvious and is best explained by considering the opposite approach: Assume that all directional correlations are monotonically decreasing with distance. If the direction of maximum correlation is selected, then the direction that have the maximum correlation at the coarsest grid level will be refined until the finest grid level before any of the two other directions are refined. So, this approach will prevent structures in different directions from interacting. The second important property is that by selecting the direction of the weakest spatial correlation we keep more randomness after more steps in the refinement sequence. This will hopefully reduce the chance of getting stuck into improbable patterns at an early stage of the simulation.
 1.
Estimate the three directional correlation functions, c _{ x }(h _{ x }), c _{ y }(h _{ y }), and c _{ z }(h _{ z }), in the x, y, or zdirections from the training image. The lags, h _{ x },h _{ y }, and h _{ z }, are measured as distances in terms of the finest grid resolution in each direction.
 2.
Compute the absolute value of these functions: g _{ k }(h)=c _{ k }(h), for k∈{x,y,z}.
 3.
Compute monotonized correlation functions: r _{ k }(h)=max_{ u≥h } g _{ k }(u), for k∈{x,y,z}.
 4.For grid level (g _{ x },g _{ y },g _{ z }) to grid level (0,0,0):

Get refinement direction \(k_{s}=\text{arg}\min_{k\in\{ x, y, z\}} r_{k}(2^{g_{k}}1) \).

Set \(g_{k_{s}}=g_{k_{s}}1\).

4 Examples
Listing of the test cases, and settings for the simulation runs. Column 1 is the simulation name of the test case. Column 2 is the dimensions of the training image and simulation grid (which are identical). Column 3 gives the maximum neighborhood extension for the Markov mesh model. Column 5 gives the number of cells in the template used by the snesim algorithm. Column 4 and 6 gives the number of grid refinements in the x y and zdirection for the Markov mesh model and the snesim algorithm. The number of grid levels where chosen to obtain similar results
Test case  Grid size  Markov mesh  snesim  

Neighborh. size (l _{ x },l _{ y },l _{ z })  Grid levels (g _{ x },g _{ y },g _{ z })  Template size  Grid levels (g _{ x },g _{ y },g _{ z })  
Turbidite  100×100×100  (3, 3, 3)  (2, 2, 2)  102  (3, 3, 2) 
Azimuth Channels  100×100×100  (2, 2, 2)  (2, 2, 2)  102  (6, 6, 4) 
Isolated Channels  125×125×75  (4, 2, 2)  (5, 5, 4)  100  (6, 6, 6) 
Channel Crevasse  100×100×100  (4, 4, 2)  (2, 2, 2)  80  (6, 5, 4) 
4.1 Conceptual Geology
4.2 Upscaled Permeability
The permeability values in milli Darcy used for each facies in the test cases when computing the upscaled permeability
Test case  Sand  Shale  Crevasse 

Turbidite  1000 mD  1 mD  – 
Azimuth Channels  1000 mD  1 mD  – 
Isolated Channels  1000 mD  1 mD  – 
Channel Crevasse  1000 mD  1 mD  700 mD 
4.3 Volume Fractions
Comparison of sand fraction in training image av realizations
Test case  Training image  Realizations  Markov mesh  Realizations snesim  

Mean  Mean  Std. dev.  Mean  Std. dev.  
Turbidite  0.1679  0.1591  0.0004  0.1764  0.0024 
Azimuth Channels  0.1470  0.1358  0.0007  0.1558  0.0017 
Isolated Channels  0.2693  0.2566  0.0022  0.2723  0.0041 
Channel Crevasse  0.3757  0.3799  0.0001  0.3812  0.0009 
4.4 Time Usage
5 Conclusions
We have presented a multigrid Markov mesh model for geological facies modeling. This combines an advantage originally developed for algorithmic multipoint methods—the use of multiple grids—with the flexibility and consistency of the statistical approach to multipoint methods. The model consists of a hierarchy of nested grids, with a singlegrid Markov mesh model for each grid, but such that information from coarser grids are taken into account. We have adopted the specification of earlier published singlegrid Markov mesh models, by using the framework of generalized linear models and a parameterization that captures continuity/discontinuity of geological structures. The result is a model that gives results that are comparable to the established snesim algorithm for several test cases.
Notes
Acknowledgements
We would like to thank Statoil and Roxar for financial support, and for making available the tool for doing permeability upscaling and the Irap RMS software. We would also like to thank the Norwegian Research Council for financial support.
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