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

Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus

We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) $3$-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the $3$-connected case) to this setting. Precisely, for any cylindric essentially internally $3$-connected map $G$ with $n$ vertices, we can obtain in linear time a periodic (in $x$) straight-line drawing of $G$ that is crossing-free and internally (weakly) convex, on a regular grid $\mathbb{Z}/w\mathbb{Z}\times[0..h]$, with $w\leq 2n$ and $h\leq n(2d+1)$, where $d$ is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially $3$-connected map $G$ on the torus (i.e., $3$-connected in the periodic representation) with $n$ vertices, we can compute in linear time a periodic straight-line drawing of $G$ that is crossing-free and (weakly) convex, on a periodic regular grid $\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}$, with $w\leq 2n$ and $h\leq 1+2n(c+1)$, where $c$ is the face-width of $G$. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$.Comment: 37 page

Drawing bobbin lace graphs, or, Fundamental cycles for a subclass of periodic graphs

In this paper, we study a class of graph drawings that arise from bobbin lace patterns. The drawings are periodic and require a combinatorial embedding with specific properties which we outline and demonstrate can be verified in linear time. In addition, a lace graph drawing has a topological requirement: it contains a set of non-contractible directed cycles which must be homotopic to $(1,0)$, that is, when drawn on a torus, each cycle wraps once around the minor meridian axis and zero times around the major longitude axis. We provide an algorithm for finding the two fundamental cycles of a canonical rectangular schema in a supergraph that enforces this topological constraint. The polygonal schema is then used to produce a straight-line drawing of the lace graph inside a rectangular frame. We argue that such a polygonal schema always exists for combinatorial embeddings satisfying the conditions of bobbin lace patterns, and that we can therefore create a pattern, given a graph with a fixed combinatorial embedding of genus one.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Schnyder woods for higher genus triangulated surfaces, with applications to encoding

Schnyder woods are a well-known combinatorial structure for plane triangulations, which yields a decomposition into 3 spanning trees. We extend here definitions and algorithms for Schnyder woods to closed orientable surfaces of arbitrary genus. In particular, we describe a method to traverse a triangulation of genus $g$ and compute a so-called $g$-Schnyder wood on the way. As an application, we give a procedure to encode a triangulation of genus $g$ and $n$ vertices in $4n+O(g \log(n))$ bits. This matches the worst-case encoding rate of Edgebreaker in positive genus. All the algorithms presented here have execution time $O((n+g)g)$, hence are linear when the genus is fixed.Comment: 27 pages, to appear in a special issue of Discrete and Computational Geometr

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

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

Application of different classification methods for litho-fluid facies prediction: A case study from the offshore Nile Delta

In this work we test four classification methods for litho-fluid facies identification in a clastic reservoir located in offshore Nile Delta. The ultimate goal of this study is to find an optimal classification method for the area under examination. The geologic context of the investigated area allows us to consider three different facies in the classification: shales, brine sands and gas sands. The depth at which the reservoir zone is located (2300-2700 m) produces a significant overlap of the P- and S-wave impedances of brine sands and gas sands that makes the discrimination between these two litho-fluid classes particularly problematic. The classification is performed on the feature space defined by the elastic properties that are derived from recorded reflection seismic data by means of Amplitude Versus Angle (AVA) Bayesian inversion. As classification methods we test both deterministic and probabilistic approaches: the quadratic discriminant analysis and the neural network methods belong to the first group, whereas the standard Bayesian approach and the Bayesian approach that includes a 1D Markov chain prior model to constrain the vertical continuity of litho-fluid facies, belong to the second group. The capability of each method to discriminate the different facies is evaluated both on synthetic seismic data (computed on the basis of available borehole information) and on field seismic data. The outcomes of each classification method are compared with the known facies profile derived from well log data and the goodness of the results is quantitatively evaluated using the so called confusion matrix. It results that all methods return vertical facies profiles in which the main reservoir zone is correctly identified. However, the consideration of as much prior information as possible in the classification process is the winning choice to derive a reliable and a physically plausible predicted facies profile

A feasibility study on the expected seismic AVA signatures of deep fractured geothermal reservoirs in an intrusive basement

The deep geothermal reservoirs in the Larderello-Travale field (southern Tuscany) are found in intensively fractured portions of intrusive/metamorphic rocks. Therefore, the geothermal exploration has been in search of possible fracture signatures that could be retrieved from the analysis of geophysical data. In the present work we assess the feasibility of finding seismic markers in the pre-stack domain which may pinpoint fractured levels. Thanks to the availability of data from boreholes that ENEL GreenPower drilled in the deep intrusive basement of this geothermal field, we derived the expected amplitude versus angle (AVA) responses of the vapour reservoirs found in some intensely, but very localized, fractured volumes within the massive rocks. The information we have available limit us to build 1D elastic and isotropic models only and thus anisotropy effects related to the presence of fractures cannot be properly modelled. We analysed the velocities and the density logs pertaining to three wells which reached five deep fractured zones in the basement. The AVA response of the fractured intervals is modelled downscaling the log data to seismic scale and comparing the analytical AVA response (computed with the Aki and Richards approximation) and the AVA extracted from a synthetic common mid point (calculated making use of a reflectivity algorithm). The results show that the amplitude of the reflections from the fractured level is characterized by negative values at vertical incidence and by decreasing absolute amplitudes with the increase of the source to receiver offset. This contrasts with many observations from hydrocarbon exploration in clastic reservoirs where gas-sand reflections often exhibit negative amplitudes at short offsets but increasing absolute amplitudes for increasing source to receiver offsets. Thereby, some common AVA attributes considered in silicoclastic lithologies would lead to erroneous fracture localization. For this reason we propose a modified AVA indicator which may highlight fracture locations in this peculiar rock type

Elastic pre-stack seismic inversion through Discrete Cosine Transform reparameterization and Convolutional Neural Networks

We develop a pre-stack inversion algorithm that combines a Discrete Cosine Transform (DCT) reparameterization of data and model spaces with a Convolutional Neural Network (CNN). The CNN is trained to predict the mapping between the DCT-transformed seismic data and the DCT-transformed 2-D elastic model. A convolutional forward modeling based on the full Zoeppritz equations constitutes the link between the elastic properties and the seismic data. The direct sequential co-simulation algorithm with joint probability distribution is used to generate the training and validation datasets under the assumption of a stationary non-parametric prior and a Gaussian variogram model for the elastic properties. The DCT is an orthogonal transformation that is here used as an additional feature extraction technique that reduces the number of unknown parameters in the inversion and the dimensionality of the input and output of the network. The DCT reparameterization also acts as a regularization operator in the model space and allows for the preservation of the lateral and vertical continuity of the elastic properties in the recovered solution. We also implement a Monte Carlo simulation strategy that propagates onto the estimated elastic model the uncertainties related to both noise contamination and network approximation. We focus on synthetic inversions on a realistic subsurface model that mimics a real gas-saturated reservoir hosted in a turbiditic sequence. We compare the outcomes of the implemented algorithm with those provided by a popular linear inversion approach and we also assess the robustness of the CNN inversion to errors in the estimated source wavelet and to erroneous assumptions about the noise statistic. Our tests confirm the applicability of the proposed approach, opening the possibility to estimate the subsurface elastic parameters and the associated uncertainties in near real-time while satisfactorily preserving the assumed spatial variability and the statistical properties of the elastic parameters

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