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.

Rock physics model‐based seismic inversion

2006 ◽  
Author(s):  
Kyle Spikes ◽  
Jack Dvorkin ◽  
Gary Mavko
Keyword(s):  
Rock Physics ◽  
Seismic Inversion ◽  
Model Based ◽  
Physics Model ◽  
Rock Physics Model

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>

Seismic inversion for reservoir properties combining statistical rock physics and geostatistics: A review

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. 75A165-75A176 ◽  
Author(s):  
Miguel Bosch ◽  
Tapan Mukerji ◽  
Ezequiel F. Gonzalez
Keyword(s):  
Elastic Properties ◽  
Quantitative Estimation ◽  
Rock Physics ◽  
Seismic Inversion ◽  
Elastic Parameters ◽  
Reservoir Properties ◽  
Sequential Approach ◽  
Geostatistical Methods ◽  
Bayesian Formulation ◽  
Statistical Rock Physics

There are various approaches for quantitative estimation of reservoir properties from seismic inversion. A general Bayesian formulation for the inverse problem can be implemented in two different work flows. In the sequential approach, first seismic data are inverted, deterministically or stochastically, into elastic properties; then rock-physics models transform those elastic properties to the reservoir property of interest. The joint or simultaneous work flow accounts for the elastic parameters and the reservoir properties, often in a Bayesian formulation, guaranteeing consistency between the elastic and reservoir properties. Rock physics plays the important role of linking elastic parameters such as impedances and velocities to reservoir properties of interest such as lithologies, porosity, and pore fluids. Geostatistical methods help add constraints of spatial correlation, conditioning to different kinds of data and incorporating subseismic scales of heterogeneities.

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.

Seismic Quantitative Interpretation for Uncertainty Reduction of Subsurface Modeling of the Deep Visean Reservoirs

2021 ◽  
Author(s):  
Pavlo Kuzmenko ◽  
Rustem Valiakhmetov ◽  
Francesco Gerecitano ◽  
Viktor Maliar ◽  
Grigori Kashuba ◽  
...  
Keyword(s):  
Elastic Properties ◽  
Seismic Data ◽  
Seismic Anisotropy ◽  
Rock Physics ◽  
Seismic Inversion ◽  
Well Logs ◽  
Quantitative Interpretation ◽  
Structural Interpretation ◽  
Elastic Inversion ◽  
Subsurface Modeling

Abstract The seismic data have historically been utilized to perform structural interpretation of the geological subsurface. Modern approaches of Quantitative Interpretation are intended to extract geologically valuable information from the seismic data. This work demonstrates how rock physics enables optimal prediction of reservoir properties from seismic derived attributes. Using a seismic-driven approach with incorporated prior geological knowledge into a probabilistic subsurface model allowed capturing uncertainty and quantifying the risk for targeting new wells in the unexplored areas. Elastic properties estimated from the acquired seismic data are influenced by the depositional environment, fluid content, and local geological trends. By applying the rock physics model, we were able to predict the elastic properties of a potential lithology away from the well control points in the subsurface whether or not it has been penetrated. Seismic amplitude variation with incident angle (AVO) and azimuth (AVAZ) jointly with rock-derived petrophysical interpretations were used for stochastical modeling to capture the reservoir distribution over the deep Visean formation. The seismic inversion was calibrated by available well log data and by traditional structural interpretation. Seismic elastic inversion results in a deep Lower Carboniferous target in the central part of the DDB are described. The fluid has minimal effect on the density and Vp. Well logs with cross-dipole acoustics are used together with wide-azimuth seismic data, processed with amplitude control. It is determined that seismic anisotropy increases in carbonate deposits. The result covers a set of lithoclasses and related probabilities: clay minerals, tight sandstones, porous sandstones, and carbonates. We analyzed the influence of maximum angles determination for elastic inversion that varied from 32.5 to 38.5 degrees. The greatest influence of the far angles selection is on the density. AI does not change significantly. Probably the 38,5 degrees provides a superior response above the carbonates. It does not seem to damage the overall AVA behavior, which result in a good density outcome, as higher angles of incidence are included. It gives a better tie to the wells for the high density layers over the interval of interest. Sand probability cube must always considered in the interpretation of the lithological classification that in many cases may be misleading (i.e. when sand and shale probabilities are very close to each other, because of small changes in elastic parameters). The authors provide an integrated holistic approach for quantitative interpretation, subsurface modeling, uncertainty evaluation, and characterization of reservoir distribution using pre-existing well logs and recently acquired seismic data. This paper underpins the previous efforts and encourages the work yet to be fulfilled on this subject. We will describe how quantitative interpretation was used for describing the reservoir, highlight values and uncertainties, and point a way forward for further improvement of the process for effective subsurface modeling.

Stuck between a rock and a reflection: A tutorial on low-frequency models for seismic inversion

2017 ◽  
Vol 5 (2) ◽  
pp. B17-B27 ◽  
Author(s):  
Mark Sams ◽  
David Carter
Keyword(s):  
Elastic Properties ◽  
Model Building ◽  
Seismic Velocity ◽  
Low Frequency ◽  
Rock Physics ◽  
Frequency Component ◽  
Seismic Inversion ◽  
Rock Properties ◽  
Seismic Velocities ◽  
Frequency Model

Predicting the low-frequency component to be used for seismic inversion to absolute elastic rock properties is often problematic. The most common technique is to interpolate well data within a structural framework. This workflow is very often not appropriate because it is too dependent on the number and distribution of wells and the interpolation algorithm chosen. The inclusion of seismic velocity information can reduce prediction error, but it more often introduces additional uncertainties because seismic velocities are often unreliable and require conditioning, calibration to wells, and conversion to S-velocity and density. Alternative techniques exist that rely on the information from within the seismic bandwidth to predict the variations below the seismic bandwidth; for example, using an interpretation of relative properties to update the low-frequency model. Such methods can provide improved predictions, especially when constrained by a conceptual geologic model and known rock-physics relationships, but they clearly have limitations. On the other hand, interpretation of relative elastic properties can be equally challenging and therefore interpreters may find themselves stuck — unsure how to interpret relative properties and seemingly unable to construct a useful low-frequency model. There is no immediate solution to this dilemma; however, it is clear that low-frequency models should not be a fixed input to seismic inversion, but low-frequency model building should be considered as a means to interpret relative elastic properties from inversion.

Seismic petrophysics workflow applied to Delaware Basin

2020 ◽  
Vol 8 (2) ◽  
pp. T349-T363
Author(s):  
Yoryenys Del Moro ◽  
Venkatesh Anantharamu ◽  
Lev Vernik ◽  
Alfonso Quaglia ◽  
Eduardo Carrillo
Keyword(s):  
Elastic Properties ◽  
Reservoir Characterization ◽  
Weighted Average ◽  
Rock Physics ◽  
Seismic Inversion ◽  
Source Rocks ◽  
Amplitude Variation With Offset ◽  
Geomechanical Properties ◽  
Delaware Basin ◽  
Seismic Amplitudes

Petrophysical analysis of unconventional plays that are comprised of organic mudrock needs detailed data QC and preparation to optimize the results of quantitative interpretation. This includes accurate computation of mineral volumes, total organic carbon (TOC), porosity, and saturations. We used TOC estimation to aid the process of determining the best pay zones for development of such reservoirs. TOC was calculated as a weighted average of Passey’s (empirical) and the bulk density-based (theoretical) methods. In organic mudrock reservoirs, the computed TOC log was used as an input to compute porosity and calibrate rock-physics models (RPMs), which are needed for understanding the potential of source rocks or finding sweet spots and their contribution to the amplitude variation with offset (AVO) changes in the seismic data. Using calibrated RPM templates, we found that TOC is driving the elastic property variations in the Avalon Formation. We determined the layering and rock fabric anisotropy using empirical relationships or modeled in the rock property characterization process because reflectivity effects are often seen in the observed seismic used for well tie and wavelet estimation. A Class IV AVO response was seen at the top of the Avalon Formation, which is typical of an unconventional reservoir. We then performed solid organic matter (TOC) substitution to account for variability of elastic properties and their contrasts as expressed in seismic amplitudes. To complete the characterization of the intervals of interest, we used conventional seismic petrophysical methods in the workflow and found that the main driver modifying the elastic properties for the Avalon shales was TOC; this conclusion serves as a foundation in integrated seismic inversion that may target lithofacies, TOC, and geomechanical properties. Seismic reservoir characterization results are critical in constraining landing zones and trajectories of the horizontal wells. The final interpretation may be used to rank targets, optimize drilling campaigns, and ultimately improve production.

SeReMpy: Seismic Reservoir Modeling Python library

Geophysics ◽  
2021 ◽  
pp. 1-43
Author(s):  
Dario Grana ◽  
Leandro de Figueiredo
Keyword(s):  
Elastic Properties ◽  
Seismic Data ◽  
Rock Physics ◽  
Seismic Inversion ◽  
Geostatistical Simulation ◽  
Inverse Theory ◽  
Reservoir Modeling ◽  
Mcmc Methods ◽  
Two Dimensional ◽  
Seismic Reservoir

Seismic reservoir characterization is a subfield of geophysics that combines seismic and rock physics modeling with mathematical inverse theory to predict the reservoir variables from the measured seismic data. An open-source comprehensive modeling library that includes the main concepts and tools is still missing. We present a Python library named SeReMpy with the state of the art of seismic reservoir modeling for reservoir properties characterization using seismic and rock physics models and Bayesian inverse theory. The most innovative component of the library is the Bayesian seismic and rock physics inversion to predict the spatial distribution of petrophysical and elastic properties from seismic data. The inversion algorithms include Bayesian analytical solutions of the linear-Gaussian inverse problem and Markov chain Monte Carlo (McMC) numerical methods for non-linear problems. The library includes four modules: geostatistics, rock physics, facies, and inversion, as well as several scripts with illustrative examples and applications. We present a detailed description of the scripts that illustrate the use of the functions of module and describe how to apply the codes to practical inversion problems using synthetic and real data. The applications include a rock physics model for the prediction of elastic properties and facies using well log data, a geostatistical simulation of continuous and discrete properties using well logs, a geostatistical interpolation and simulation of two-dimensional maps of temperature, an elastic inversion of partial stacked seismograms with Bayesian linearized AVO inversion, a rock physics inversion of partial stacked seismograms with McMC methods, and a two-dimensional seismic inversion.

Geostatistical inversion of prestack seismic data for the joint estimation of facies and impedances using stochastic sampling from Gaussian mixture posterior distributions

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. M55-M65 ◽  
Author(s):  
Xiaozheng Lang ◽  
Dario Grana
Keyword(s):  
Elastic Properties ◽  
Posterior Distribution ◽  
Seismic Data ◽  
Gaussian Mixture ◽  
Seismic Inversion ◽  
Joint Estimation ◽  
Inversion Method ◽  
Inversion Algorithm ◽  
Data Set ◽  
Geostatistical Approach

We have developed a seismic inversion method for the joint estimation of facies and elastic properties from prestack seismic data based on a geostatistical approach. The objectives of our inversion methodology are to sample from the posterior distribution of seismic properties and to simultaneously classify the lithology conditioned by seismic data. The inversion algorithm is a sequential Gaussian mixture inversion based on Bayesian linearized amplitude variation with offset inverse theory and sequential geostatistical simulations. The stochastic approach to the inversion allows generating multiple elastic models that match the seismic data. To mathematically represent the multimodal behavior of elastic properties due to their variations within different lithologies, we adopt a Gaussian mixture distribution for the prior model of the elastic properties and we use the prior probability of the facies as weights of the Gaussian components of the mixture. The solution of the inverse problem is achieved by deriving the explicit analytical expression of the posterior distribution of the elastic properties and facies and by sampling from this distribution according to a spatial correlation model. The inversion methodology has been validated using well logs and synthetic seismic data with different noise levels, and it is then applied to a real 3D seismic data set in North Sea.

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