Hans B. Helle

White's model for wave propagation in partially saturated rocks: Comparison with poroelastic numerical experiments

We use a poroelastic modeling algorithm to compute numerical experiments of wave propagation in White’s partial saturation model. The results are then compared to the theoretical predictions. The model consists of a homogeneous sandstone saturated with brine and spherical gas pockets. White’s theory predicts a relaxation mechanism, due to pressure equilibration, causing attenuation and velocity dispersion of the wavefield. We vary gas saturation either by increasing the radius of the gas pocket or by increasing the density of gas bubbles. Despite that the modeling is two dimensional and interaction between the gas pockets is neglected in White’s model, the numerical results show the trends predicted by the theory. In particular, we observe a similar increase in velocity at high frequencies (and low permeabilities). Furthermore, the behavior of the attenuation peaks versus water saturation and frequency is similar to that of White’s model. The modeling results show more dissipation and higher velocities than White’s model due to multiple scattering and local fluid-flow effects. The conversion of fast P-wave energy into dissipating slow waves at the patches is the main mechanism of attenuation. Differential motion between the rock skeleton and the fluids is highly enhanced by the presence of fluid/fluid interfaces and pressure gradients generated through them.

Determination of facies from well logs using modular neural networks

Zonation of well logs and the correlation of zones between wells are primary tasks in sub-surface geological and engineering analysis. We propose in this paper an artificial neural network (ANN) approach for objective clustering and identification of facies from well logs. The method relies upon combining back-propagation neural networks in ensembles and modular systems, where the multi-class classification problem of facies identification has effectively been reduced to a number of two-class problems. Based on the neural network responses using synthetic logs from a realistic model, we optimized the architecture and training procedure of the component networks in the modular system, where the building blocks are simple three-layer back-propagation ANNs. Ensembles of ANNs are trained on disjoint sets of patterns using soft overtraining to ensure diversity and generalization. Recurrent ANNs are shown to enhance the facies continuity by effectively removing ambiguous or spurious classifications. The performance of the technique was demonstrated using synthetic data and it was then used to detect four different facies within the Ness Formation in the North Sea. An average hit rate of above 90% in wells not used for training the network is slightly to significantly better than the performance published for similar classification experiments.

Effects of attenuation and anisotropy on reflection amplitude versus offset

To investigate the effects that attenuation and anisotropy have on reflection coefficients, we consider a homogeneous and viscoelastic wave incident on an interface between two transversely isotropic and lossy media with the symmetry axis perpendicular to the interface. Analysis of PP and PS reflection coefficients shows that anisotropy should be taken into account in amplitude variation with offset (AVO) studies involving shales. Different anisotropic characteristics may reverse the reflection trend and substantially influence the position of the critical angle versus offset. The analysis of a shale-chalk interface indicates that when the critical distance is close to the near offsets, the AVO response is substantially affected by the presence of dissipation. In a second example, we compute reflection coefficients and synthetic seismograms for a limestone/black shale interface with different rheological properties of the underlying shale. This case shows reversal of the reflection trend with increasing offset and compensation between the anisotropic and anelastic effects.

The physics and simulation of wave propagation at the ocean bottom

We investigate some aspects of the physics of wave propagation at the ocean bottom (ranging from soft sediments to crustal rocks). Most of the phenomena are associated to the presence of attenuation. The analysis requires the use of an anelastic stress‐strain relation and a highly accurate modeling algorithm. Special attention is given to modeling the boundary conditions at the ocean‐bottom interface and the related physical phenomena. For this purpose, we further develop and test the pseudospectral modeling algorithm for wave propagation at fluid‐anelastic solid interfaces. The method is based on a domain‐decomposition technique (one grid for the fluid part and another grid for the solid part) and the Fourier and Chebyshev differential operators. We consider the reflection, transmission, and propagation of seismic waves at the ocean bottom, modeled as a plane boundary separating an acoustic medium (ocean) and a viscoelastic solid (sediment). The main physical phenomena associated with this interface are ...

Pore pressure estimation in reservoir rocks from seismic reflection data

A method is used to obtain pore pressure in shaly sandstones based upon an acoustic model for seismic velocity versus clay content and effective pressure. Calibration of the model requires log data—porosity, clay content, and sonic velocities—to obtain the dry-rock moduli and the effective stress coefficients as a function of depth and pore pressure. The seismic P-wave velocity, derived from reflection tomography, is fitted to the theoretical velocity by using pore pressure as the fitting parameter. This approach, based on a rock-physics model, is an improvement over existing pore-pressure prediction methods, which mainly rely on empirical relations between velocity and pressure. The method is applied to the Tune field in the Viking Graben sedimentary basin of the North Sea. We have obtained a high-resolution velocity map that reveals the sensitivity to pore pressure and fluid saturation in the Tarbert reservoir. The velocity map of the Tarbert reservoir and the inverted pressure distribution agree with the structural features of the Tarbert Formation and its known pressure compartments.

Fluid saturation from well logs using committee neural networks

Neural computing has made a major step forward by the introduction of multi-net systems in practical applications. In this study we developed and tested a modular artificial neural network system for predicting underground fluids, water, oil and gas, and their partial saturations, directly from well logs, without explicit knowledge of the fluid and rock properties as required by conventional methods. Based on laboratory data on relative permeability for alternative fluid systems –oil–water or gas–oil – respectively, relative permeability logs may also be provided for input to reservoir simulation while drilling. Simple three-layer back-propagation artificial neural networks (ANN) constitute the building blocks of a modular system, where the input logs are sonic, density, neutron porosity and resistivity. By numerical experiments using synthetic logs we have determined the optimal architecture of the ANN. We find that the overtraining strategy is a suitable technique for bias reduction and an unconstrained optimal linear combination is the best method of combining outputs in the committee neural net. The accuracy of the net is restricted only by accuracy of data. Comparison between ANN predictions of fluid saturation with those of conventional petrophysical analysis, in wells unknown to the network, indicates a standard error of less than 0.03.

Edge and Tip Diffractions: Theory and Applications in Seismic Prospecting

In Edge and Tip Diffractions: Theory and Applications in Seismic Prospecting (SEG Geophysical Monograph Series No. 14), the theoretical framework of the edge and tip wave theory of diffractions has been elaborated from fundamental wave mechanics. Seismic diffractions are inevitable parts of the recorded wavefield scattered from complex structural settings and thus carry back to the surface information that can be exploited to enhance the resolution of details in the underground. The edge and tip wave theory of diffractions provides a physically sound and mathematically consistent method of computing diffraction phenomena in realistic geologic models. In this book, theoretical derivations are followed by their numerical implementation and application to real exploration problems. The book was written initially as lecture notes for an internal course in diffraction modeling at Norsk Hydro Research Center, Bergen, Norway, and later was used for a graduate course at Novosibirsk State University in Russia. The material is drawn from several previous publications and from unpublished technical reports. Edge and Tip Diffractions will be of interest to geoscientists, engineers, and students at graduate and Ph.D. levels.

3D diffraction modeling of singly scattered acoustic wavefields based on the combination of surface integral propagators and transmission operators

We present an improved method for modeling 3D acoustic wavefields scattered at smooth curved interfaces. The approach is based on a high-frequency approximation of surface integral propagators and a correct description of their boundary values in terms of transmission operators. The main improvement is a uniform local approximation of these operators in the form of effective reflection and transmission coefficients. We show that the effective coefficients represent a generalization of the plane-wave coefficients widely used in conventional seismic modeling, even for the case of curved reflectors, nonplanar wavefronts, and finite frequencies. The proposed method is capable of producing complex wave phenomenas, such as caustics, edge diffractions, and head waves. Seismograms modeled for even simple models reveal significant errors implicit in the plane-wave approximation. Comparison of modeling based on effective coefficients with the analytic solution reveals errors less than 4% in peak amplitude at seismic frequencies.

Wave simulation in partially frozen porous media with fractal freezing conditions

A recent article [J. M. Carcione and G. Seriani, J. Comput. Phys. 170, 676 (2001)] proposes a modeling algorithm for wave simulation in a three-phase porous medium composed of sand grains, ice, and water. The differential equations hold for uniform water (ice) content. Here, we obtain the variable-porosity differential equations by using the analogy with the two-phase case and the complementary energy theorem. The displacements of the rock and ice frames and the variation of fluid content are the generalized coordinates, and the stress components and fluid pressure are the generalized forces. We simulate wave propagation in a frozen porous medium with fractal variations of porosity and, therefore, realistic freezing conditions.

Theory of borehole stability when drilling through salt formations

We present a mathematical analysis of borehole stability when drilling through rock salt. First, we consider an elastic transversely isotropic medium and find the optimal mud weight as a function of the vertical overburden and horizontal tectonic stresses. Then, the Zener and Maxwell mechanical models are used to model the effects of transient and steady-state creep flow, respectively, in isotropic media. Under certain conditions such as the absence of dilatational anelasticity, the Burger model can be used to describe the steady-state flow, including transient creep effects. The type of creep is regulated by critical octahedral-stress values that depend on temperature and pressure. A typical drilling results in conditions of plane strain, whose solution is given by Kirsch’s equations. In this case, the borehole is subject to minimum and maximum horizontal stresses, which differ from the vertical stress. The analysis provides expressions for the shape of the borehole-cross section, the boreholewall closure time, and the optimal mud weight to avoid wall collapse or expansion. It is shown that an anisotropic state of tectonic stress may require mud pressures exceeding the overburden stress and that the calculation should consider the joint optimization of the shape and area of the borehole cross section.