A one-step Bayesian inversion framework for 3D reservoir characterization based on a Gaussian mixture model — A Norwegian Sea demonstration

Abstract

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.

Joint inversion for facies and petrophysical properties based on a bi‐level optimization model

“…In the prior distribution, if the petrophysical properties given the facies 𝑓 obey a Gaussian distribution 𝑁(𝜇 𝑓 , 𝚺 𝑓 ), the label parameters jointly obey a Gaussian mixture distribution (Fjeldstad et al, 2021):…”

“…In the prior distribution, if the petrophysical properties given the facies f obey a Gaussian distribution $$N({\mu} _f,{\bf \Sigma} _f)$$, the label parameters jointly obey a Gaussian mixture distribution (Fjeldstad et al., 2021): $$$$\begin{equation} p(f,{\bf l}) = P_fC_f\exp {\left(-\frac{1}{2}({\bf l}-{\mu} _f)^\top {\bf W}_f({\bf l}-{\mu} _f)\right)}, \end{equation}$$$$where $$N({\mu} _f,{\bf \Sigma} _f)$$ is the Gaussian distribution with expectation $${\mu} _f$$ and covariance matrix $${\bf \Sigma} _f$$, $$\bf W_f = {\bf \Sigma} _f^{-1}$$ denotes the weight matrix of the prior distribution given f , $$P_f$$ is the prior probability of the facies and $$C_f$$ is the normalization constant of the Gaussian distribution $$N({\mu} _f,{\bf \Sigma} _f)$$ with the following equation: …”