The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

History matching is a type of inverse problem in which observed reservoir behavior is used to estimate reservoir model variables that caused the behavior. Obtaining even a single history-matched reservoir model requires a substantial amount of effort, but the past decade has seen remarkable progress in the ability to generate reservoir simulation models that match large amounts of production data. Progress can be partially attributed to an increase in computational power, but the widespread adoption of geostatistics and Monte Carlo methods has also contributed indirectly. In this review paper, we will summarize key developments in history matching and then review many of the accomplishments of the past decade, including developments in reparameterization of the model variables, methods for computation of the sensitivity coefficients, and methods for quantifying uncertainty. An attempt has been made to compare representative procedures and to identify possible limitations of each.

Recent progress on reservoir history matching: a review

One major cause of elastic wave attenuation in heterogeneous porous media is wave-induced flow of the pore fluid between heterogeneities of various scales. It is believed that for frequencies below 1 kHz, the most important cause is the wave-induced flow between mesoscopic inhomogeneities, which are large compared with the typical individual pore size but small compared to the wavelength. Various laboratory experiments in some natural porous materials provide evidence for the presence of centimeter-scale mesoscopic heterogeneities. Laboratory and field measurements of seismic attenuation in fluid-saturated rocks provide indications of the role of the wave-induced flow. Signatures of wave-induced flow include the frequency and saturation dependence of P-wave attenuation and its associated velocity dispersion, frequency-dependent shear-wave splitting, and attenuation anisotropy. During the last four decades, numerous models for attenuation and velocity dispersion from wave-induced flow have been developed with varying degrees of rigor and complexity. These models can be categorized roughly into three groups according to their underlying theoretical framework. The first group of models is based on Biot’s theory of poroelasticity. The second group is based on elastodynamic theory where local fluid flow is incorporated through an additional hydrodynamic equation. Another group of models is derived using the theory of viscoelasticity. Though all models predict attenuation and velocity dispersion typical for a relaxation process, there exist differences that can be related to the type of disorder periodic, random, space dimension and to the way the local flow is incorporated. The differences manifest themselves in different asymptotic scaling laws for attenuation and in different expressions for characteristic frequencies. In recent years, some theoretical models of wave-induced fluid flow have been validated numerically, using finite-difference, finite-element, and reflectivity algorithms applied to Biot’s equations of poroelasticity. Application of theoretical models to real seismic data requires further studies using broadband laboratory and field measurements of attenuation and dispersion for different rocks as well as development of more robust methods for estimating dissipation attributes from field data.

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Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks — A review

Gas geochemistry and seismotectonics: a review

Heterogeneity can be dealt with by defining homogeneous equivalent properties, known as averaging, or by trying to describe the spatial variability of the rock properties from geologic observations and local measurements. The techniques available for these descriptions are mostly continuous Geostatistical models, or discontinuous facies models such as the Boolean, Indicator or Gaussian-Threshold models and the Markov chain model. These facies models are better suited to treating issues of rock strata connectivity, e.g. buried high permeability channels or low permeability barriers, which greatly affect flow and, above all, transport in aquifers. Genetic models provide new ways to incorporate more geology into the facies description, an approach that has been well developed in the oil industry, but not enough in hydrogeology. The conclusion is that future work should be focused on improving the facies models, comparing them, and designing new in situ testing procedures (including geophysics) that would help identify the facies geometry and properties. A world-wide catalog of aquifer facies geometry and properties, which could combine site genesis and description with methods used to assess the system, would be of great value for practical applications.

/pdf/dealing-with-spatial-heterogeneity-4d67dw7xwy.pdf

Dealing with spatial heterogeneity

A fast Fourier transform (FFT) moving average (FFT-MA) method for generating Gaussian stochastic processes is derived. Using discrete Fourier transforms makes the calculations easy and fast so that large random fields can be produced. On the other hand, the basic moving average frame allows us to uncouple the random numbers from the structural parameters (mean, variance, correlation length, ... ), but also to draw the randomness components in spatial domain. Such features impart great flexibility to the FFT-MA generator. For instance, changing only the random numbers gives distinct realizations all having the same covariance function. Similarly, several realizations can be built from the same random number set, but from different structural parameters. Integrating the FFT-MA generator into an optimization procedure provides a tool theoretically capable to determine the random numbers identifying the Gaussian field as well as the structural parameters from dynamic data. Moreover, all or only some of the random numbers can be perturbed so that realizations produced using the FFT-MA generator can be locally updated through an optimization process.

The FFT Moving Average (FFT-MA) Generator: An Efficient Numerical Method for Generating and Conditioning Gaussian Simulations

Although P- and S-wave dispersion is known to be important in porous/cracked rocks, theoretical predictions of such dispersions have never been given. The authors report such calculations and show that the predicted dispersions are high in the case of low aspect ratio cracks ({le}10{sup {minus}3}) or high crack density ({ge}10{sup {minus}1}). Their calculations are derived from first-principle computations of the high- and low-frequency elastic moduli of a rock permeated by an isotropic distribution of pores or cracks, dry or saturated, with idealized geometry (spheres or ellipsoids). Henyey and Pomphrey developed a differential self-consistent model that is shown to be a good approximation. This model is used here, but as it considers cracks with zero thickness, it can not account for fluid content effects. To remove this difficulty, one combines the differential self-consistent approach with a purely elastic calculation of moduli in two cases: that of spherical pores and that of oblate spheroidal cracks with a nonzero volume. This leads to what the authors call the extended differential, self-consistent model (EM). When combining these EM results with the Gassmann equation, it is possible to derive and compare the theoretical predictions for high- and low-frequency effective moduli in the case of amore » saturated rock. Since most laboratory data are ultrasonic measurements and in situ data are obtained at much lower frequencies, this comparison is useful for interpreting seismic data in terms of rock and fluid properties. The predicted dispersions are high, in agreement with previous experimental results. A second comparison is made with the semi-empirical model of Marion and Nur, which considers the effects of a mixed porosity (round pores and cracks together).« less

High- and low-frequency elastic moduli for a saturated porous/cracked rock-Differential self-consistent and poroelastic theories

An effective medium approach (differential self-consistent scheme) has been used to calculate elastic wave velocities in rocks containing spherical pores and ellipsoidal cracks with two fluid phases. The results are applied to the case of imbibition/drainage processes performed in the laboratory (at high frequencies). For that purpose, two scales of heterogeneities are accounted for. The first one corresponds to the size of pores or cracks. It is the scale of a dispersion mechanism termed “local flow” because it depends on local variations in pore/crack compressibility. The second scale, much larger, is controlled by the size of the heterogeneities of the fluid phase distribution. These heterogeneities are assimilated to pockets whose saturation is different from that of the surrounding medium. The pocket scale involves a second dispersion mechanism, often called “global flow” or “pocket flow,” which depends on fluid pressure equilibration among the regions which have distinct saturation levels. Our calculations include both mechanisms, each having its own characteristic scale. Imbibition is described through simple relations involving two parameters. The same is true for drainage. The predicted behavior for P wave velocities against saturation appears to be in good agreement with experimental data reported in the literature, for both drainage and imbibition.

Elastic wave velocities in partially saturated rocks: Saturation hysteresis

A method of predicting petrophysical characteristics of an underground reservoir by constructing a geologic model consistent with seismic measurements is disclosed which permits optimization of oil reservoir development schemes. A geologic model, from which seismic data are simulated in depth, is constructed in depth. The geologic model is made consistent with seismic measurements acquired in time by minimization of an objective function by comparing the seismic measurements with seismic data simulated from the geologic model and converted to time. During minimization, the interval velocities used for conversion are updated by comparing, within the objective function, an observed thickness in time ΔT obs m,n between two markers with a thickness in time ΔT sim m,n simulated from the seismic data, and by modifying simulation parameters such as the error e sim m,n on the thickness of the two markers estimated in depth.

Method for updating a geologic model by seismic data

https://pastel.archives-ouvertes.fr/pastel-00581082/document

Mickaele Le Ravalec

Papers

The FFT Moving Average (FFT-MA) Generator: An Efficient Numerical Method for Generating and Conditioning Gaussian Simulations

High- and low-frequency elastic moduli for a saturated porous/cracked rock-Differential self-consistent and poroelastic theories

Elastic wave velocities in partially saturated rocks: Saturation hysteresis

Method for updating a geologic model by seismic data

History matching of petroleum reservoir models by the Ensemble Kalman Filter and parameterization methods