Analysis of Different Statistical Models in Probabilistic Joint Estimation of Porosity and Litho-Fluid Facies from Acoustic Impedance Values

We discuss the influence of different statistical models in the prediction of porosity and litho-fluid facies from logged and inverted acoustic impedance (Ip) values. We compare the inversion and classification results that were obtained under three different statistical a-priori assumptions: an analytical Gaussian distribution, an analytical Gaussian-mixture model, and a non-parametric mixtu re distribution. The first model assumes Gaussian distributed porosity and Ip values, thus neglecting their facies-dependent behaviour related to different lithologic and saturation conditions. Differently, the other two statistical models relate each component of the mixture to a specific litho-fluid facies, so that the facies-dependency of porosity and Ip values is taken into account. Blind well tests are used to validate the final predictions, whereas the analysis of the maximum-a-posteriori (MAP) solutions, the coverage ratio, and the contingency analysis tools are used to quantitatively compare the inversion outcomes. This work points out that the correct choice of the statistical petrophysical model could be crucial in reservoir characterization studies. Indeed, for the investigated zone, it turns out that the simple Gaussian model constitutes an oversimplified assumption, while the two mixture models provide more accurate estimates, although the non-parametric one yields slightly superior predictions with respect to the Gaussian-mixture assumption

A data-driven transdimensional approach to include lateral constraints on 2D target-oriented AVA inversion

Seismic inversion aims to infer subsurface properties from processed seismic data; since these are often ill-conditioned procedures, numerous strategies can be investigated. To date currently adopted procedures assume an a priori structural knowledge of the investigated area and impose such constraints to the recovered solution. To overcome this downside we apply a transdimensional reversible jump-Markov chain Monte Carlo (Rj-McMC) algorithm to solve the interval-oriented amplitude versus angle (AVA) inversion on 2D synthetic seismic data. This approach considers the model parametrization as an unknown, together with the elastic properties of the investigated area. The algorithm samples models discretized in Voronoi cells characterized by similar AVA responses. The elastic values associated with each Voronoi cell are obtained taking the average of the elastic properties of the CDPs falling within it. This data-driven approach does, therefore, need no external assumption over the investigated area and ensures an automatically inferred strategy to include lateral variability of data inside the inversion kernel. We compare results obtained to a standard Bayesian approach for different SNR, showing how the increase of random noise contaminating the data strongly affects the linear approach, while the Rj-McMC generates model predictions in accordance with the true model, producing more reliable results

1D elastic full-waveform inversion and uncertainty estimation by means of a hybrid genetic algorithm-Gibbs sampler approach

Stochastic optimization methods, such as genetic algorithms, search for the global minimum of the misfit function within a given parameter range and do not require any calculation of the gradients of the misfit surfaces. More importantly, these methods collect a series of models and associated likelihoods that can be used to estimate the posterior probability distribution. However, because genetic algorithms are not a Markov chain Monte Carlo method, the direct use of the genetic-algorithm-sampled models and their associated likelihoods produce a biased estimation of the posterior probability distribution. In contrast, Markov chain Monte Carlo methods, such as the Metropolis-Hastings and Gibbs sampler, provide accurate posterior probability distributions but at considerable computational cost. In this paper, we use a hybrid method that combines the speed of a genetic algorithm to find an optimal solution and the accuracy of a Gibbs sampler to obtain a reliable estimation of the posterior probability distributions. First, we test this method on an analytical function and show that the genetic algorithm method cannot recover the true probability distributions and that it tends to underestimate the true uncertainties. Conversely, combining the genetic algorithm optimization with a Gibbs sampler step enables us to recover the true posterior probability distributions. Then, we demonstrate the applicability of this hybrid method by performing one-dimensional elastic full-waveform inversions on synthetic and field data. We also discuss how an appropriate genetic algorithm implementation is essential to attenuate the "genetic drift" effect and to maximize the exploration of the model space. In fact, a wide and efficient exploration of the model space is important not only to avoid entrapment in local minima during the genetic algorithm optimization but also to ensure a reliable estimation of the posterior probability distributions in the subsequent Gibbs sampler step

Assessment of different approaches to rock-physics modeling: A case study from offshore Nile Delta

The estimation of a reliable rock-physics model (RPM) plays a crucial role in reservoir characterization studies. We assess different methods in deriving a reliable RPM that will be used in conjunction with amplitude-versus-angle inversion for the characterization of a clastic reservoir located in offshore Nile Delta. The reservoir zone is located in gas-saturated sand channels surrounded by shale sequences within a depth interval ranging between 2.3 and 2.7 km. One theoretical and three empirical approaches to derive a RPM are analyzed: The theoretical RPM is established using the well-known rock-physics equations valid for granular materials, whereas the empirical RPMs are derived using one multilinear stepwise regression and two nonlinear regression procedures based on neural networks (NNs) and genetic algorithms (GAs). A proper calibration and validation of the derived RPMs is conducted by using the extensive log suite of four existing wells drilled over an area of 100 km2. For the investigated reservoir interval and for the encasing shales, all the analyzed methods give a final RPM that is able to reliably predict the elastic attributes (P-wave velocity, S-wave velocity, and density) from the petrophysical properties of interest (porosity, water saturation, and shaliness). Among the empirical approaches, the RPM predicted by the multilinear regression is characterized by a prediction capability very similar to the RPMs predicted by the nonlinear GA method, thus demonstrating that in the investigated zone, the relation linking the petrophysical properties to the elastic attributes can be conveniently described by a multilinear model. Differently, the NN method seems to be affected by the overfitting problem that produces a RPM with a lower prediction capability than the RPMs estimated by the other methods. The theoretical method yields predictions of elastic properties very similar to those produced by multilinear regression

Probabilistic estimation of reservoir properties by means of wide-angle AVA inversion and a petrophysical reformulation of the Zoeppritz equations

We apply a target-oriented amplitude versus angle (AVA) inversion to estimate the petrophysical properties of a gas-saturated reservoir in offshore Nile Delta. A linear empirical rock-physics model derived from well log data provides the link between the petrophysical properties (porosity, shaliness and saturation) and the P-wave, S-wave velocities and density. This rock-physics model, properly calibrated for the investigated reservoir, is used to re-parameterize the exact Zoeppritz equations. The so derived equations are the forward model engine of a linearized Bayesian AVA-petrophysical inversion that, for each data gather, inverts the AVA of the target reflections to estimate the petrophysical properties of the reservoir layer, keeping fixed the cap-rock properties. We make use of the iterative Gauss-Newton method to solve the inversion problem. For each petrophysical property of interest, we discuss the benefits introduced by wide-angle reflections in constraining the inversion and we compare the posterior probability distributions (PPDs) analytically obtained via a local linearization of the inversion with the PPDs numerically computed with a Markov Chain Monte Carlo (MCMC) method. It results that the porosity is the best resolved parameter and that wide-angle reflections effectively constrain the shaliness estimates but do not guarantee reliable saturation estimates. It also results that the local linearization returns accurate PPDs in good agreement with the MCMC estimates

Characterisation of shallow marine sediments using high-resolution velocity analysis and genetic-algorithm-driven 1D elastic full-waveform inversion

We estimate the elastic properties of marine sediments beneath the seabed by means of high-resolution velocity analysis and one-dimensional elastic full-waveform inversion performed on twodimensional broad-band seismic data of a well-site survey. A high-resolution velocity function is employed to exploit the broad frequency band of the data and to derive the P-wave velocity field with a high degree of accuracy. To derive a complete elastic characterisation in terms of P-wave and S-wave velocities (Vp, Vs) and density of the subsurface, and to increase the resolution of the Vp estimates, we apply a one-dimensional elastic full-waveform inversion in which the outcomes derived from the velocity analysis are used as a priori information to define the Vp search range. The one-dimensional inversion is done using genetic algorithm as the optimisation method. It is performed by considering two misfit functions: the first uses the entire waveform to compute the misfit between modelled and observed seismograms, and the second considers the envelope of the seismograms, thus relaxing the requirement of an exact estimation of the wavelet phase. The full-waveform inversion and the high-resolution velocity analysis yield comparable Vp profiles, but the full-waveform inversion reconstruction is much more detailed. Regarding the full-waveform inversion results, the final depth models of P- and S-wave velocities and density show a fine-layered structure with a significant increase in velocities and density at shallow depth, which may indicate the presence of a consolidated layer. The very similar velocities and density-depth trends obtained by employing the two different misfit functions increase our confidence in the reliability of the predicted subsurface models

Effect of different statistical models in probabilistic joint estimation of porosity and litho-fluid facies from acoustic impedance values

The estimation of petrophysical reservoir properties (i.e. porosity, shale content, fluid saturation) and litho-fluid facies around the target area is a common, highly ill-conditioned problem that is often casted into a Bayesian framework. Independently from the inversion approach adopted (analytical or numerical), the correct choice of the underlying statistical model always plays a crucial role in any geophysical Bayesian inversion. For what concerns the reservoir characterization problem, many authors have demonstrated that such statistical model should be able to correctly capture the facies-dependency of petrophysical and/or elastic properties related to the different lithologic and fluid-saturation conditions. In particular, it has been demonstrated that the accounting for such facies-dependency often provides more accurate descriptions of the uncertainties affecting the sought parameters. However, as the author is aware an in-depth discussion and a comparison of the results provided by different statistical models is still lacking for reservoir characterization studies. Focusing on this peculiar aspect, I use an inversion approach for the joint estimation of porosity and litho-fluid facies from logged and post-stack inverted acoustic impedance (Ip) values. The inversion approach I employ is a modification of the method proposed by Grana (2018) that is adapted to consider Gaussian-mixture and Gaussian distributions, and to jointly invert porosity and logged or inverted Ip values. This work is mainly aimed at analysing and comparing the results provided by three different statistical assumptions about the underlying joint distribution of the petrophysical model relating porosity and Ip values. To this end, I consider a simple Gaussian assumption that neglects the facies dependency of porosity and acoustic impedance values, whereas an analytical Gaussian-mixture distribution and a non-parametric mixture distribution relate each component of the mixture to a specific litho-fluid facies. In particular, the Gaussian or Gaussian-mixture models are often employed in seismic inversions because they allow for an analytical computation of the posterior model and make it possible to easily include additional constraints (i.e. geostatistical constraints) into the inversion kernel. Differently, a non-parametric distribution is not restricted by any statistical assumption about the underlying statistical model, but it impedes an analytical derivation of the posterior model and also complicates the inclusion of additional regularization operators or geostatistical constraints into the inversion framework. This work focuses the attention on well log data pertaining to a clastic gas-saturated reservoir. All the three considered statistical models are directly estimated from 5 out of 7 available wells drilled trough the reservoir zone. The kernel density technique is used to derive the non-parametric distribution. One of the two remaining wells is here used as blind test to validate the inversion results, whereas the analysis of the maximum-a-posteriori (MAP) solutions, and the coverage ratio are used to more quantitatively assess the final predictions

Some strategies to make global methods suitable to solve high-dimensional and ill-conditioned geophysical optimization problems

Geophysical optimisation problems are often non-linear, multi-dimensional, and characterised by objective functions with complex topologies (i.e. multiple local minima). Global methods are often used to solve these problems, but they are affected by the curse of dimensionality problem, that is their ability to explore the model space exponentially decreases as the dimensions of the model space increase. In addition, their limited exploitation capabilities make global search algorithms unable to converge in ill-conditioned optimization problems or in cases with highly correlated model parameters. In this work, I test three different strategies that could be used to partially attenuate the previous issues. The first strategy uses Legendre polynomials to reparametrize the subsurface model. More in detail, the subsurface model is expanded into series of Legendre polynomials that are used as basis functions. In this framework the unknown parameters become the series of expansion coefficients associated to each polynomial. The aim of this peculiar parameterization is three-fold: Efficiently decreasing the number of unknowns, inherently imposing a 1D spatial correlation to the recovered subsurface model, and finally searching for maximally decoupled parameters. This approach is applied to 1D seismic-petrophysical inversion in which the objective function to minimize is a weighted sum of data misfit and a-priori model information. The second strategy combines the global algorithm with a 1D edge-preserving smoothing (EPS) filter to solve the non-linear amplitude versus angle (AVA) inversion. In this case the simple L2 norm misfit between observed and predicted seismic gather is used as the objective function to minimize. The third example concerns a 2D cross-hole tomography. Due to the severe ill-conditioning and the non-linearity of this inverse problem, the 2D EPS filter is used in conjunction with model constraints in the objective function. In particular, following Zhang and Zhang (2012) the objective function is a weighted sum of L2 norm data misfit and an edge-preserving regularization that impose sparseness constraints into the first order derivatives of model parameters. Note that EPS filters are extensively applied to reduce the noise of geophysical subsurface images while preserving structural and stratigraphic discontinuities and/or edges (i.e. for sharpening seismic stack images for interpretation; AlBinHassan et al. 2006). In the context of global search methods, EPS filters drive the optimization in a suitably preconditioned model domain instead of relying completely on the random perturbation. This will decrease the ill-conditioning of the inversion because the modified model space is designed to be smaller than the complete suite of solutions. In all the following tests the firefly algorithm (FA) is used as the optimization tool. This is a quite new global search method inspired by the swarm intelligence that was proposed by Yang (2008). Over the last decade, this optimization strategy has been extensively applied in engineering applications but found very limited applications to geophysical optimization problems. In all cases the predictions yielded by the proposed strategies are compared with those provided by the more standard approach in which only the L2 norm misfit function and where no preconditioning strategies are applied in the optimization framework. In all the following tests, I focus the attention to synthetic data optimizations with the aim to maintain the discussion at a simple level and to draw general conclusions