EAGE Conference on Petroleum Geostatistics

Although faults traditionally have been modelled as membrane-like surfaces, the flow pattern through a fault is affected in a volumetric region. The physical properties of the fault rock will be different from what they were prior to the faulting process. Defining specific Fault Facies only present in the close vicinity of a fault gives a possibility to model the flow through faults more detailed than by conventional modelling. The Fault Facies will depend on both the pre-faulted facies, and the strain affecting the rocks when faults are created.

A workflow has been created which demonstrates that the concept can be utilized in realistic reservoir modelling, starting from a conventional reservoir model where the fault is defined as a surface. A fault zone is defined in a small volume around the fault surface. It has a finer grid than the original model. First, all Fault Facies are modelled, followed by petrophysical modelling accounting for the fact that the greatest deformation occurs near the centre of the fault zone. The suggested concept produces direct modelling of vertical flow in the faults, making the unphysical non-neighbouring connections obsolete.

We present an overview of the practical application of the emerging geostatistical approach called multiple-point geostatistics (MPS), with an emphasis on its applications to reservoir modeling in the oil industry. MPS uses quantitative, pixel-based templates, called training images, to help us build geocellular models. MPS differs from traditional variogram-based or object-based geostatistical approaches. MPS has the virtues of being able to honor multiple types of absolute and probabilistic constraints while reproducing the features in the training image. In this paper, we highlight the significance of using training images to drive reservoir modeling, the pros and cons of pixel-based MPS vs. object-based modeling, and the utility of training image catalogs for the MPS workflow. In addition to using MPS at reservoir scales to build facies or petrophysical models, the technique can be used to reconstruct pore-scale features. We believe that within a few years MPS will be a key modeling technique, applicable to multiple scales of geological features.

Wavelet extraction is a fundamental step in linking imaged seismic data and well-logs. This step provides time-to-depth estimates that help locate seismic data more precisely in depth, and the wavelet that connects the low-resolution seismic data to the fine-scale properties typical of geocellular models. Noise estimates from well ties are the most important parameter controlling the extent to which seismic data constrain these fine-scale models.

In usual practice, many subjective judgements are made regarding issues like wavelet phase, span, rock--physics models, imaging quality etc, all of which make the results less objective than desirable. It is also widely under-appreciated that these subjective decisions have strong impacts on the most important outputs of the extraction process. They propagate far downstream into reservoir prediction or forecasting, and their influence can easily dominate development decisions.

We show that model selection choices relating to wavelet span, rock-physics models, segmentation etc, can be made more objectively using Bayesian model-selection criteria. We present some case studies showing how strongly parameter estimates and uncertainties are coupled to model choice, and thus why a more objective model-selection process is crucial to the wavelet extraction workflow.

Geostatistics has been commonly used in forward modeling and in inverse modeling to integrate seismic information in stochastic fine gride models. The quality of seismic and the downscaling of seismic attributes to the fine grid of the well measurements are still challenges to which existing geostatistical methods only give partial answers.
In this paper an iterative inversion methodology is proposed based on a direct sequential simulation and co-simulation approaches. Several images of acoustic impedances of entire field are simulated in a first step. Afterwards, co-simulations are used for the global transformation of images of acoustic impedances in an iterative process: after the convolution, local areas of best fit of the different images are selected and “merged” into a secondary image for the direct co-simulation of the next iteration. The iterative and convergent process continues until a given match with an objective function is reached. Spatial dispersion and patterns of acoustic impedances (histograms and variograms) are reproduced at the final acoustic impedance cube.

Seismic inversion is usually treated as a deterministic problem. However, since the seismic amplitude data contains noise and the frequency resolution is limited, high and low frequencies will be uncertain. For a consistent treatment of these uncertainties, a geostatistical inversion method can be used.

We have used a Bayesian linearised AVO inversion method for a turbiditic channel system reservoir containing two offset stacks. In this Bayesian approach, the earth model parameters Vp, Vs, and ρ are given by a multi-normal distribution where spatial coupling is imposed by correlation functions. A linearised relationship between the model parameters and the AVO data, allows us to obtain the posterior distribution for the earth model parameters analytically.

The posterior distribution represents a laterally consistent seismic inversion where the solution in each location depends on the solutions in all other locations. The distribution contains the best estimate of the model parameters as well as their associated uncertainties. Using kriging, full frequency information from well data are spread in a volume around the wells; and from the posterior uncertainty we generate full frequency solutions for the entire volume.

The Bayesian approach is fast and the inversion gave good match with well log data.

We present an efficient stochastic seismic inversion technique aimed at overcoming the band-limited nature of deterministic inversion methods by generating multiple realisations of elastic properties in fine scale stratigraphic grids. Our method uses a Bayesian framework and a linearised, weak contrast approximation of the Zoeppritz equation to estimate a log-Gaussian posterior distribution for P- and S-wave impedances. This distribution is constrained to reproduce the observed seismic data, within specified noise-dependent tolerance limits and to also honour conditioning well data. After elastic inversion, multiple realisations of P- wave and S-wave impedances can be used for cascaded stochastic simulation of petrophysical reservoir properties, lithology classification and uncertainty analysis. The technique has been successfully tested on different real data sets and we demonstrate results on a large model of more than 30 millions grid cells.

The Ensemble Kalman Filter (EnKF) is a statistical method to update dynamic models by sequential data assimilation. Recently EnKF has gained popularity in the reservoir simulation community as efficient history matching tool.
The validity of EnKF update equations relies on the analytical solution of linear inverse problem with Gaussian prior. This assumption is critical in dealing with reservoir facies models. Variables associated to facies are better represented by multimodal distributions than normal priors which are used in the EnKF update scheme.
In this paper we propose to model multimodal variables related to facies by Gaussian Mixture Models (GMM) and to modify EnKF for updating Gaussian Mixture (GM) distributions. First we derived the posterior distribution for a linear inverse problem assuming GM priors and the analytical solution we obtained shows that this posterior is again a GM. Using this result we then revisited the EnKF updating and we reformulated the update equations when the priors is assumed to be a GMM.
We show two simple examples that give evidence of a good flexibility of GMM in managing multimodal distributions even though some computational issues linked to large scale applications are worth of a deeper investigation.

An alternative method for modelling facies within a reservoir by using a level set formulation is proposed. A simple example of a sedimentary channel with two injectors and two producers has been modelled. It is shown how a combination of the level set method with a Gaussian random field results in a versatile and useful tool for the modelling and history matching of facies within a reservoir.

Well test data and traditional log measurements give information about the effective permeability in the reservoir on very different scales.
The variable representing the well test permeability can be regarded as a spatial weighted average of the ordinary permeability variable. The kriging procedure thus involves covariances that are averaged over the well test volume. This gives an inverse block kriging problem.
Solving the expressions for the averaged covariances by straightforward computations on large grids in 3D, involves computations of big triple sums. Transforming the spatial averaging to the Fourier domain, and using the convolution theorem, means that the heavy summation is replaced by simple cell by cell multiplications. This combined with the Fast Fourier Transform algorithm gives a highly efficient method for solving the inverse block kriging problem.