Joint Bayesian spatial inversion of lithology/fluid classes, petrophysical properties and elastic attributes – A Norwegian Sea gas discovery

2020 ◽  
Author(s):  
Torstein Fjeldstad ◽  
Henning Omre
Keyword(s):  
Markov Chain ◽  
Random Field ◽  
3D Model ◽  
Gaussian Mixture ◽  
Monte Carlo Algorithm ◽  
Petrophysical Properties ◽  
Norwegian Sea ◽  
First Order ◽  
Spatial Coupling ◽  
Likelihood Model

<p>A Bayesian model for prediction and uncertainty quantification of subsurface lithology/fluid classes, petrophysical properties and elastic material properties conditional on seismic amplitude-versus-offset measurements is defined. We demonstrate the proposed methodology  on a real Norwegian Sea gas discovery in 3D in a seismic inversion framework.</p><p>The likelihood model is assumed to be Gaussian, and it is constructed in two steps. First, the reflectivity coefficients of the elastic material properties are computed based on the linear Aki Richards approximation valid for weak contrasts. The reflectivity coefficients are then convolved in depth with a wavelet.  We assume a Markov random field prior model for the lithology/fluid classes with a first order neighborhood system to ensure spatial coupling. Conditional on the lithology/fluid classes we define a Gauss-linear petrophysical and rock physics model. The marginal prior spatial model for the petrophysical properties and elastic attributes is then a multivariate Gaussian mixture random field.</p><p>The convolution kernel in the likelihood model restricts analytic assessment of the posterior model since the neighborhood system of the lithology/fluid classes is no longer a simple first order neighborhood. We propose a recursive algorithm that translates the Gibbs formulation into a set of vertical Markov chains. The vertical posterior model is approximated by a higher order Markov chain, which is computationally tractable. Finally, the approximate posterior model is used as a proposal model in a Markov chain Monte Carlo algorithm. It can be verified that the Gaussian mixture model is a conjugate prior with respect to the Gauss-linear likelihood model; thus, the posterior density for petrophysical properties and elastic attributes is also a Gaussian mixture random field.</p><p>We compare the proposed spatially coupled 3D model to a set of independent vertical 1D inversions. We obtain an increase of the average acceptance rate of 13.6 percentage points in the Markov chain Monte Carlo algorithm compared to a simpler model without lateral spatial coupling. At a blind well location we obtain a reduction of at most 60 % in mean absolute error and root mean square error for the proposed spatially coupled 3D model.</p>

A one-step Bayesian inversion framework for three-dimensional reservoir characterization based on a Gaussian mixture model – A Norwegian Sea demonstration

Geophysics ◽  
2020 ◽  
pp. 1-68
Author(s):  
Torstein Fjeldstad ◽  
Per Avseth ◽  
Henning Omre
Keyword(s):  
Random Field ◽  
Markov Random Field ◽  
Gaussian Mixture ◽  
Three Dimensions ◽  
Monte Carlo Algorithm ◽  
Petrophysical Properties ◽  
Norwegian Sea ◽  
Spatial Coupling ◽  
Markov Random ◽  
One Step

A one-step approach for Bayesian prediction and uncertainty quantification of lithology/fluid classes, petrophysical properties and elastic attributes conditional on prestack 3D seismic amplitude-versus-offset data is presented. A 3D Markov random field prior model is assumed for the lithology/fluid classes to ensure spatially coupled lithology/fluid class predictions in both the lateral and vertical directions. Conditional on the lithology/fluid classes, we consider Gauss-linear petrophysical and rock physics models including depth trends. Then, the marginal prior models for the petrophysical properties and elastic attributes are multivariate Gaussian mixture models. The likelihood model is assumed to be Gauss-linear to allow for analytic computation. A recursive algorithm that translates the Gibbs formulation of the Markov random field into a set of vertical Markov chains is proposed. This algorithm provides a proposal density in a Markov chain Monte Carlo algorithm such that efficient simulation from the posterior model of interest in three dimensions is feasible. The model is demonstrated on real data from a Norwegian Sea gas reservoir. We evaluate the model at the location of a blind well, and we compare results from the proposed model with results from a set of 1D models where each vertical trace is inverted independently. At the blind well location, we obtain at most a 60 % reduction in the root mean square error for the proposed 3D model compared to the model without lateral spatial coupling.

A two-step inversion approach for seismic-reservoir characterization and a comparison with a single-loop Markov-chain Monte Carlo algorithm

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. R227-R244 ◽  
Author(s):  
Mattia Aleardi ◽  
Fabio Ciabarri ◽  
Timur Gukov
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Reservoir Characterization ◽  
Gaussian Mixture ◽  
Petrophysical Properties ◽  
Mcmc Algorithm ◽  
Prior Model ◽  
Single Loop ◽  
Seismic Reservoir

We have evaluated a two-step Bayesian algorithm for seismic-reservoir characterization, which, thanks to some simplifying assumptions, is computationally very efficient. The applicability and reliability of this method are assessed by comparison with a more sophisticated and computer-intensive Markov-chain Monte Carlo (MCMC) algorithm, which in a single loop directly estimates petrophysical properties and lithofluid facies from prestack data. The two-step method first combines a linear rock-physics model (RPM) with the analytical solution of a linearized amplitude versus angle (AVA) inversion, to directly estimate the petrophysical properties, and related uncertainties, from prestack data under the assumptions of a Gaussian prior model and weak elastic contrasts at the reflecting interface. In particular, we use an empirical, linear RPM, properly calibrated for the investigated area, to reparameterize the linear time-continuous P-wave reflectivity equation in terms of petrophysical contrasts instead of elastic constants. In the second step, a downward 1D Markov-chain prior model is used to infer the lithofluid classes from the outcomes of the first step. The single-loop (SL) MCMC algorithm uses a convolutional forward modeling based on the exact Zoeppritz equations, and it adopts a nonlinear RPM. Moreover, it assumes a more realistic Gaussian mixture distribution for the petrophysical properties. Both approaches are applied on an onshore 3D seismic data set for the characterization of a gas-bearing, clastic reservoir. Notwithstanding the differences in the forward-model parameterization, in the considered RPM, and in the assumed a priori probability density functions, the two methods yield maximum a posteriori solutions that are consistent with well-log data, although the Gaussian mixture assumption adopted by the SL method slightly improves the description of the multimodal behavior of the petrophysical parameters. However, in the considered reservoir, the main difference between the two approaches remains the very different computational times, the SL method being much more computationally intensive than the two-step approach.

Joint probabilistic petrophysics-seismic inversion based on Gaussian mixture and Markov chain prior models

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. R31-R42 ◽  
Author(s):  
Torstein Fjeldstad ◽  
Dario Grana
Keyword(s):  
Inverse Problem ◽  
Markov Chain ◽  
Elastic Properties ◽  
Rock Physics ◽  
Gaussian Mixture ◽  
Seismic Inversion ◽  
Monte Carlo Algorithm ◽  
Seismic Model ◽  
Prior Model ◽  
Model Based

Seismic reservoir characterization focuses on the prediction of reservoir properties based on the available geophysical and petrophysical data. The inverse problem generally includes continuous properties, such as petrophysical and elastic attributes, and discrete properties, such as lithology/fluid classes. We have developed a joint probabilistic inversion methodology for the prediction of petrophysical and elastic properties and lithology/fluid classes that combined statistical rock physics and Bayesian seismic inversion. The elastic attributes depend on continuous petrophysical variables, such as porosity and clay content, and discrete lithology/fluid classes, through a nonlinear rock-physics relationship together. The seismic model relates the elastic attributes, such as velocities and density, to their seismic response (reflectivity, traveltime, and amplitudes). The advantage of our integrated approach is that the inversion method accounts for the uncertainty associated to each step of the modeling workflow. The lithology/fluid classes are assigned by a Markov random field prior model to capture vertical continuity and vertical sorting of the lithology/fluid classes. Because rock and fluid properties are in general not Gaussian, a spatially coupled Gaussian mixture prior model based on the lithology/fluid classes is constructed. The forward geophysical operator includes a lithology-/fluid-dependent rock physics model and a linearized seismic model based on the convolution of the seismic wavelet with the reflectivity coefficient series. The solution of the inverse problem consists of the posterior distributions of petrophysical and elastic properties and lithology/fluid classes. We proposed an efficient Markov chain Monte Carlo algorithm to sample from the posterior models and assess the uncertainty. Our methodology is demonstrated on a seismic cross section from a survey in the Norwegian Sea, and it shows promising results consistent with well-log data measured at the well location as well as reliable prediction uncertainties.

Nappes d'huile: prediction de leurs deplacements comme agent efficace d'un plan de mesure d'urgence

1979 ◽  
Vol 14 (1) ◽  
pp. 89-109
Author(s):  
B. Coupal ◽  
M. de Broissia
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Finite Elements ◽  
Mass Balance ◽  
Stochastic Method ◽  
Stochastic Methods ◽  
Deterministic Approach ◽  
First Order ◽  
The Third ◽  
Comme Agent

Abstract The movement of oil slicks on open waters has been predicted, using both deterministic and stochastic methods. The first method, named slick rose, consists in locating an area specifying the position of the slick during the first hours after the spill. The second method combines a deterministic approach for the simulation of current parameters to a stochastic method simulating the wind parameters. A Markov chain of the first order followed by a Monte Carlo approach enables the simulation of both phenomena. The third method presented in this paper describes a mass balance on the spilt oil, solved by the method of finite elements. The three methods are complementary to each other and constitute an important point for a contingency plan.

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