Facies Modeling Using a Markov Mesh Model Specification

Stien, Marita; Kolbjørnsen, Odd

doi:10.1007/s11004-011-9350-9

Cited by 38 publications

(22 citation statements)

References 9 publications

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“…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 Fig.…”

Section: Using the Framework Of Generalized Linear Modelsmentioning

confidence: 99%

“…This is the same approach which is used in Stien and Kolbjørnsen (2011). The way our model is formulated makes it possible to estimate the model independently at each grid level.…”

Section: Using the Framework Of Generalized Linear Modelsmentioning

confidence: 99%

“…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 time-consumption problems in parameter estimation and simulation. In this paper, we proceed yet another step, and formulate a multigrid Markov mesh model.…”

Section: Introductionmentioning

confidence: 99%

“…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 single-grid 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.…”

Section: Introductionmentioning

confidence: 99%

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Using Multiple Grids in Markov Mesh Facies Modeling

et al. 2013

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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.

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“…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 Fig.…”

Section: Using the Framework Of Generalized Linear Modelsmentioning

confidence: 99%

“…This is the same approach which is used in Stien and Kolbjørnsen (2011). The way our model is formulated makes it possible to estimate the model independently at each grid level.…”

Section: Using the Framework Of Generalized Linear Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Using Multiple Grids in Markov Mesh Facies Modeling

et al. 2013

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“…A disadvantage is therefore, despite having an expression for the normalization constant, that it can be computationally expensive to compute. Subclasses of the MRF methods such as Markov mesh models (Stien and Kolbjørnsen 2011) and partially ordered Markov models (Cressie and Davidson 1998) avoid the computation of the normalization constant, and this advantage over the MRF methods is shared by the FM method. Moreover, in contrast to methods such as PMM and MRF, the FM method is fully non-parametric, as it does not require probability distributions to be written in a closed form.…”

Section: Introductionmentioning

confidence: 99%

A Frequency Matching Method: Solving Inverse Problems by Use of Geologically Realistic Prior Information

et al. 2012

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The frequency matching method defines a closed form expression for a complex prior that quantifies the higher order statistics of a proposed solution model to an inverse problem. While existing solution methods to inverse problems are capable of sampling the solution space while taking into account arbitrarily complex a priori information defined by sample algorithms, it is not possible to directly compute the maximum a posteriori model, as the prior probability of a solution model cannot be expressed. We demonstrate how the frequency matching method enables us to compute the maximum a posteriori solution model to an inverse problem by using a priori information based on multiple point statistics learned from training images. We demonstrate the applicability of the suggested method on a synthetic tomographic crosshole inverse problem.

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Theoretical analysis of non‐Gaussian heterogeneity effects on subsurface flow and transport

Riva

Guadagnini

Neuman

2017

Water Resources Research

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Much of the stochastic groundwater literature is devoted to the analysis of flow and transport in Gaussian or multi‐Gaussian log hydraulic conductivity (or transmissivity) fields, Ytrue(boldxtrue)=ln⁡Ktrue(boldxtrue) (x being a position vector), characterized by one or (less frequently) a multiplicity of spatial correlation scales. Yet Y and many other variables and their (spatial or temporal) increments, ΔY, are known to be generally non‐Gaussian. One common manifestation of non‐Gaussianity is that whereas frequency distributions of Y often exhibit mild peaks and light tails, those of increments ΔY are generally symmetric with peaks that grow sharper, and tails that become heavier, as separation scale or lag between pairs of Y values decreases. A statistical model that captures these disparate, scale‐dependent distributions of Y and ΔY in a unified and consistent manner has been recently proposed by us. This new “generalized sub‐Gaussian (GSG)” model has the form Ytrue(boldxtrue)=Utrue(boldxtrue)Gtrue(boldxtrue) where Gtrue(boldxtrue) is (generally, but not necessarily) a multiscale Gaussian random field and Utrue(boldxtrue) is a nonnegative subordinator independent of G. The purpose of this paper is to explore analytically, in an elementary manner, lead‐order effects that non‐Gaussian heterogeneity described by the GSG model have on the stochastic description of flow and transport. Recognizing that perturbation expansion of hydraulic conductivity K=eY diverges when Y is sub‐Gaussian, we render the expansion convergent by truncating Y's domain of definition. We then demonstrate theoretically and illustrate by way of numerical examples that, as the domain of truncation expands, (a) the variance of truncated Y (denoted by Yt) approaches that of Y and (b) the pdf (and thereby moments) of Yt increments approach those of Y increments and, as a consequence, the variogram of Yt approaches that of Y. This in turn guarantees that perturbing Kt=eYt to second order in σYt (the standard deviation of Yt) yields results which approach those we obtain upon perturbing K=eY to second order in σY even as the corresponding series diverges. Our analysis is rendered mathematically tractable by considering mean‐uniform steady state flow in an unbounded, two‐dimensional domain of mildly heterogeneous Y with a single‐scale function G having an isotropic exponential covariance. Results consist of expressions for (a) lead‐order autocovariance and cross‐covariance functions of hydraulic head, velocity, and advective particle displacement and (b) analogues of preasymptotic as well as asymptotic Fickian dispersion coefficients. We compare these theoretically and graphically with corresponding expressions developed in the literature for Gaussian Y. We find the former to differ from the latter by a factor k = true〈U2true〉/〈U〉2 ( true〈true〉 denoting ensemble expectation) and the GSG covariance of longitudinal velocity to contain an additional nugget term depending on this same factor. In the limit as Y becomes Gaussian,...

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Facies Modeling Using a Markov Mesh Model Specification

Cited by 38 publications

References 9 publications

Using Multiple Grids in Markov Mesh Facies Modeling

Using Multiple Grids in Markov Mesh Facies Modeling

A Frequency Matching Method: Solving Inverse Problems by Use of Geologically Realistic Prior Information

Theoretical analysis of non‐Gaussian heterogeneity effects on subsurface flow and transport

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