Showing papers in "Geophysics in 2004"

Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies

Abstract: Prestack migration and/or inversion may be implemented in either the time or the frequency domain. In the frequency domain, it is possible to discretize the frequencies with a much larger sampling interval than that dictated by the sampling theorem and still obtain an imaging result that does not suffer from aliasing (wrap around) in the depth domain. The selection of input frequencies can be reduced when a range of offsets is available; this creates a redundancy of information in the wavenumber coverage of the target. In order to optimize the use of this information, we define a new discretization strategy that depends on the maximum effective offset present in the surface seismic survey: the larger the range of offsets, the fewer frequencies are required. The strategy, exact in a homogeneous 1D earth, selects frequencies by making use of the well-known effect of image stretch in normal-moveout (NMO) correction and in migration (usually considered detrimental for the imaging). The strategy is also useful in more general earth models: we apply it to the 2D Marmousi model and recover a continuous range of wavenumbers using only three input frequencies. The Marmousi inversion result accurately predicts all other data frequencies, demonstrating the redundancy of the data.

New insights into magnetic derivatives for structural mapping

Abstract: Magnetic and gravity derivatives can be likened to seismic attributes in that they can help define/estimate the physical properties of the source structure causing the anomaly. This contribution looks at the tilt derivative, first reported in 1994 and more recently used to derive the local wavenumber (1997). We will show that the combination of the tilt derivative and its total horizontal derivative are highly suitable for mapping shallow basement structure and mineral exploration targets and that they have distinct advantages over many conventional derivatives. We provide the simple theory behind the derivatives, use a range of simple 2D models to illustrate their response, and apply them to mapping a mineral target in Namibia.

Layered and laterally constrained 2D inversion of resistivity data

Abstract: In a sedimentary environment, quasi-layered models often can represent the actual geology more accurately than smooth minimum-structure models. We present a 2D inversion scheme with lateral constraints and sharp boundaries (LCI) for continuous resistivity data. All data and models are inverted as one system, producing layered solutions with laterally smooth transitions. The models are regularized through lateral constraints that tie interface depths or thicknesses and resistivities of adjacent layers. A priori information, used to resolve ambiguities and to add, for example, geological information, can be added at any point of the profile and migrates through the lateral constraints to parameters at adjacent sites. Similarly, information from areas with well-resolved parameters migrates through the constraints to help resolve areas with poorly constrained parameters. The estimated model is complemented by a full sensitivity analysis of the model parameters supporting quantitative evaluation of the inversion result. A simple synthetic model proves the need for a quasilayered, 2D inversion when compared with a traditional 2D minimum-structure inversion. A 2D minimum-structure inversion produces models with spatially smooth resistivity transitions, making identification of layer boundaries difficult. A continuous vertical electrical sounding field example from Sweden with a depression in the depth to bedrock supports the conclusions drawn from the synthetic example. A till layer on top of the bedrock, hidden in the traditional inversion result, is identified using the 2D LCI scheme. Furthermore, the depth to the bedrock surface is easily identified for most of the profile with the 2D LCI model, which is not the case with the model from the traditional minimumstructure inversion.

Minimum weighted norm interpolation of seismic records

Abstract: In seismic data processing, we often need to interpolate and extrapolate data at missing spatial locations. The reconstruction problem can be posed as an inverse problem where, from inadequate and incomplete data, we attempt to reconstruct the seismic wavefield at locations where measurements were not acquired.We propose a wavefield reconstruction scheme for spatially band‐limited signals. The method entails solving an inverse problem where a wavenumber‐domain regularization term is included. The regularization term constrains the solution to be spatially band‐limited and imposes a prior spectral shape. The numerical algorithm is quite efficient since the method of conjugate gradients in conjunction with fast matrix–vector multiplications, implemented via the fast Fourier transform (FFT), is adopted. The algorithm can be used to perform multidimensional reconstruction in any spatial domain.

Angle‐domain common‐image gathers for migration velocity analysis by wavefield‐continuation imaging

Abstract: We analyze the kinematic properties of offset‐domain common image gathers (CIGs) and angle‐domain CIGs (ADCIGs) computed by wavefield‐continuation migration. Our results are valid regardless of whether the CIGs were obtained by using the correct migration velocity. They thus can be used as a theoretical basis for developing migration velocity analysis (MVA) methods that exploit the velocity information contained in ADCIGs.We demonstrate that in an ADCIG cube, the image point lies on the normal to the apparent reflector dip that passes through the point where the source ray intersects the receiver ray. The image‐point position on the normal depends on the velocity error; when the velocity is correct, the image point coincides with the point where the source ray intersects the receiver ray. Starting from this geometric result, we derive an analytical expression for the expected movements of the image points in ADCIGs as functions of the traveltime perturbation caused by velocity errors. By applying this ana...

Experimental design: Electrical resistivity data sets that provide optimum subsurface information

Abstract: Although multielectrode electrical‐resistivity systems have been commercially available for more than a decade, resistivity imaging of the subsurface continues to be based on data sets recorded using one or more of the standard electrode arrays (e.g., the Wenner or conventional dipole‐dipole array). To exploit better the full capabilities of multielectrode acquisition systems, we have developed an experimental design procedure to identify suites of electrode configurations that provide subsurface information according to predefined optimization criteria. The experimental design algorithm includes a goodness function that ranks the sensitivity of every possible electrode configuration to changes in the subsurface parameters. To examine the potential and limitations of the new algorithm, comprehensive data sets that included data from all standard and nonstandard electrode configurations were (a) generated for a complex 2D resistivity model and (b) recorded across a well‐studied test site in Switzerland. Im...

Diffraction imaging by focusing‐defocusing: An outlook on seismic superresolution

Abstract: Diffractions always need more advertising. It is true that conventional seismic processing and migration are usually successful in using specular reflections to estimate subsurface velocities and reconstruct the geometry and strength of continuous and pronounced reflectors. However, correct identification of geological discontinuities, such as faults, pinch-outs, and small-size scattering objects, is one of the main objectives of seismic interpretation. The seismic response from these structural elements is encoded in diffractions, and diffractions are essentially lost during the conventional processing/migration sequence. Hence, we advocate a diffraction-based, data-oriented approach to enhance image resolution—as opposed to the traditional image-oriented techniques, which operate on the image after processing and migration. Even more: it can be shown that, at least in principle, processing of diffractions can lead to superresolution and the recovery of details smaller than the seismic wavelength. The so-called reflection stack is capable of effectively separating diffracted and reflected energy on a prestack shot gather by focusing the reflection to a point while the diffraction remains unfocused over a large area. Muting the reflection focus and defocusing the residual wavefield result in a shot gather that contains mostly diffractions. Diffraction imaging applies the classical (isotropic) diffraction stack to these diffraction shot gathers. This focusingmuting-defocusing approach can successfully image faults, small-size scattering objects, and diffracting edges. It can be implemented both in model-independent and modeldependent contexts. The resulting diffraction images can greatly assist the interpreter when used as a standard supplement to full-wave images.

Finite‐difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid

Abstract: We describe the application of the rotated staggered-grid (RSG) finite-difference technique to the wave equations for anisotropic and viscoelastic media. The RSG uses rotated finite-difference operators, leading to a distribution of modeling parameters in an elementary cell where all components of one physical property are located only at one single position. This can be advantageous for modeling wave propagation in anisotropic media or complex media, including high-contrast discontinuities, because no averaging of elastic moduli is needed. The RSG can be applied both to displacement-stress and to velocity-stress finite-difference (FD) schemes, whereby the latter are commonly used to model viscoelastic wave propagation. With a von Neumann-style anlysis, we estimate the dispersion error of the RSG scheme in general anisotropic media. In three different simulation examples, all based on previously published problems, we demonstrate the application and the accuracy of the proposed numerical approach.

Gassmann's equation and fluid‐saturation effects on seismic velocities

Abstract: Gassmann’s (1951) equations commonly are used to predict velocity changes resulting from different porefluid saturations. However, the input parameters are often crudely estimated, and the resulting estimates of fluid effects can be unrealistic. In rocks, parameters such as porosity, density, and velocity are not independent, and values must be kept consistent and constrained. Otherwise, estimating fluid substitution can result in substantial errors. We recast the Gassmann’s relations in terms of a porosity-dependent normalized modulus Kn and the fluid sensitivity in terms of a simplified gain function G. General Voigt-Reuss bounds and critical porosity limits constrain the equations and provide upper and lower bounds of the fluid-saturation effect on bulk modulus. The “D” functions are simplified modulus-porosity relations that are based on empirical porosity-velocity trends. These functions are applicable to fluid-substitution calculations and add important constraints on the results. More importantly, the simplified Gassmann’s relations provide better physical insight into the significance of each parameter. The estimated moduli remain physical, the calculations are more stable, and the results are more realistic.

Expanded uncertainty quantification in inverse problems: Hierarchical Bayes and empirical Bayes

Abstract: A common way to account for uncertainty in inverse problems is to apply Bayes' rule and obtain a posterior distribution of the quantities of interest given a set of measurements. A conventional Bayesian treatment, however, requires assuming specific values for parameters of the prior distribution and of the distribution of the measurement errors (e.g., the standard deviation of the errors). In practice, these parameters are often poorly known a priori, and choosing a particular value is often problematic. Moreover, the posterior uncertainty is computed assuming that these parameters are fixed; if they are not well known a priori, the posterior uncertainties have dubious value.This paper describes extensions to the conventional Bayesian treatment that assign uncertainty to the parameters defining the prior distribution and the distribution of the measurement errors. These extensions are known in the statistical literature as “empirical Bayes” and “hierarchical Bayes.” We demonstrate the practical applicati...

Seismic low-frequency effects in monitoring fluid-saturated reservoirs

Abstract: There is a complex relationship between seismic attributes, including the frequency dependence of reflections and fluid saturation in a reservoir. Observations in both laboratory and field data indicate that reflections from a fluid-saturated layer have an increased amplitude and delayed traveltime at low frequencies, when compared with reflections from a gas-saturated layer. Comparison of laboratory-modeling results with a diffusive-viscous-theory model show that low (<5) values of the quality factor Q can explain the observations of frequency dependence. At the field scale, conventional processing of time-lapse VSP data found minimal changes in seismic response of a gas-storage reservoir when the reservoir fluid changed from gas to water. Lowfrequency analysis found significant seismic-reflectionattribute variation in the range of 15‐50 Hz. The field observations agree with effects seen in laboratory data and predicted by the diffusive-viscous theory. One explanation is that very low values of Q are the result of internal diffusive losses caused by fluid flow. This explanation needs further theoretical investigation. The frequencydependent amplitude and phase-reflection properties presented in this paper can be used for detecting and monitoring fluid-saturated layers.

Crosshole IP imaging for engineering and environmental applications

Abstract: Induced polarization (IP) imaging is a promising tool in engineering and environmental studies. Application of this technique for near-surface investigations has previously been limited by incomplete understanding of the physicochemical controls on the IP response, together with a lack of appropriate methods for data inversion. As laboratory studies have shown, description of IP in terms of complex electrical conductivity enables access to various structural characteristics pertinent to practical issues such as subsurface lithology definition, hydraulic permeability estimation, or hydrocarbon contaminant mapping. In particular, analysis in terms of real and imaginary conductivity components offers improved lithological characterization, since surface polarization effects are separated from electrolytic and surface conduction effects. An Occam-type IP inversion algorithm based on complex algebra is described which accounts for these advances in IP interpretation by directly solving for complex conductivity. Results from crosshole applications at two case study sites demonstrate the suitability of the IP imaging approach for subsurface characterization. In the first case study, the imaging results correlate with the observed complex sequence of Quaternary sediments at a waste disposal site. Characterization of the polarizability of these sediments offers significant value in lithological differentiation. In the second case study, the results of IP imaging at a hydrocarbon-contaminated site illustrate the potential of the method in environmental studies. The hydrocarbon location is clearly evident from the IP image, and a markedly different response is observed at an uncontaminated region of the site. By adopting empirical structural‐electrical relationships, images of textural and hydraulic properties are estimated as a step toward improved quantitative characterization. The success of the method for these contrasting applications supports further investigation into understanding the physical and chemical processes that control observed IP.

Three-dimensional regularized focusing inversion of gravity gradient tensor component data

Abstract: We develop a new method for interpretation of tensor gravity field component data, based on regularized focusing inversion. The focusing inversion makes its possible to reconstruct a sharper image of the geological target than conventional maximum smoothness inversion. This new technique can be efficiently applied for the interpretation of gravity gradiometer data, which are sensitive to local density anomalies. The numerical modeling and inversion results show that the resolution of the gravity method can be improved significantly if we use tensor gravity data for interpretation. We also apply our method for inversion of the gradient gravity data collected by BHP Billiton over the Cannington Ag-Pb-Zn orebody in Queensland, Australia. The comparison with the drilling results demonstrates a remarkable correlation between the density anomaly reconstructed by the gravity gradient data and the true structure of the orebody. This result indicates that the emerging new geophysical technology of the airborne gravity gradient observations can improve significantly the practical effectiveness of the gravity method in mineral exploration.

Inversion of 3D electromagnetic data in frequency and time domain using an inexact all-at-once approach

Abstract: We present a general formulation for inverting frequencyor time-domain electromagnetic data using an all-at-once approach. In this methodology, the forward modeling equations are incorporated as constraints and, thus, we need to solve a constrained optimization problem where the parameters are the electromagnetic fields, the conductivity model, and a set of Lagrange multipliers. This leads to a much larger problem than the traditional unconstrained formulation where only the conductivities are sought. Nevertheless, experience shows that the constrained problem can be solved faster than the unconstrained one. The primary reasons are that the forward problem does not have to be solved exactly until the very end of the optimization process, and that permitting the fields to be away from their constrained values in the initial stages introduces flexibility so that a stationary point of the objective function is found more quickly. In this paper, we outline the all-atonce approach and apply it to electromagnetic problems in both frequency and time domains. This is facilitated by a unified representation for forward modeling for these two types of data. The optimization problem is solved by finding a stationary point of the Lagrangian. Numerically, this leads to a nonlinear system that is solved iteratively using a Gauss-Newton strategy. At each iteration, a large, indefinite matrix is inverted, and we discuss how this can be accomplished. As a test, we invert frequency-domain synthetic data from a grounded electrode system that emulates a field CSAMT survey. For the time domain, we invert borehole data obtained from a current loop on the surface.

Nonlinear rock physics model for estimation of 3D subsurface stress in anisotropic formations: Theory and laboratory verification

Abstract: We develop a rock physics model based on nonlinear elasticity that describes the dependence of the effective stiffness tensor as a function of a 3D stress field in intrinsically anisotropic formations. This model predicts the seismic velocity of both P- and S-waves in any direction for an arbitrary 3D stress state. Therefore, the model overcomes the limitations of existing empirical velocity-stress models that link P-wave velocity in isotropic rocks to uniaxial or hydrostatic stress. To validate this model, we analyze ultrasonic velocity measurements on stressed anisotropic samples of shale and sandstone. With only three nonlinear constants, we are able to predict the stress dependence of all five elastic medium parameters comprising the transversely isotropic stiffness tensor. We also show that the horizontal stress affects vertical S-wave velocity with the same order of magnitude as vertical stress does. We develop a weakanisotropy approximation that directly links commonly measured anisotropic Thomsen parameters to the principal stresses. Each Thomsen parameter is simply a sum of corresponding background intrinsic anisotropy and stressinduced contribution. The stress-induced part is controlled by the difference between horizontal and vertical stresses and coefficients depending on nonlinear constants. Thus, isotropic rock stays isotropic under varying but hydrostatic load, whereas transversely isotropic rock retains the same values of dimensionless Thomsen parameters. Only unequal horizontal and vertical stresses alter anisotropy. Since Thomsen parameters conveniently describe seismic signatures, such as normal-moveout velocities and amplitudevariation-with-offset gradients, this approximation is suitable for designing new methods for the estimation of 3D subsurface stress from multicomponent seismic data.

Amplitude and kinematic corrections of migrated images for nonunitary imaging operators

Abstract: Obtaining migrated images with meaningful amplitudes is a challenging problem when the migration operator is not unitary. One possible solution to this problem is iterative inversion. However, inversion is an expensive process that can be rather difficult to apply, especially with 3D data. In this paper, I propose estimating migrated images similar to the least-squares inverse images by approximating the inverse Hessian, thus avoiding the need for iterative inversion. The inverse Hessian is approximated with a bank of nonstationary matching filters. These filters are not exact impulse responses and are limited in their ability to mimic the full effects of least-squares inversion. Tests on two data sets show that this filtering approach gives results similar to iterative least-squares inversion at a lower cost. This technique is flexible enough to be applied to images migrated from zero-offset or angle-domain common-image-point gathers.

The Fresnel volume and transmitted waves

Abstract: In seismic imaging experiments, it is common to use a geometric ray theory that is an asymptotic solution of the wave equation in the high-frequency limit. Consequently, it is assumed that waves propagate along infinitely narrow lines through space, called rays, that join the source and receiver. In reality, recorded waves have a finite-frequency content. The band limitation of waves implies that the propagation of waves is extended to a finite volume of space around the geometrical ray path. This volume is called the Fresnel volume. In this tutorial, we introduce the physics of the Fresnel volume and we present a solution of the wave equation that accounts for the band limitation of waves. The finite-frequency wave theory specifies sensitivity kernels that linearly relate the traveltime and amplitude of band-limited transmitted and reflected waves to slowness variations in the earth. The Fresnel zone and the finite-frequency sensitivity kernels are closely connected through the concept of constructive interference of waves. The finite-frequency wave theory leads to the counterintuitive result that a pointlike velocity perturbation placed on the geometric ray in three dimensions does not cause a perturbation of the phase of the wavefield. Also, it turns out that Fermat’s theorem in the context of geometric ray theory is a special case of the finite-frequency wave theory in the limit of infinite frequency. Last, we address the misconception that the width of the Fresnel volume limits the resolution in imaging experiments.

Detection and estimation of gas hydrates using rock physics and seismic inversion: Examples from the northern deepwater Gulf of Mexico

Abstract: Natural gas hydrates, composed primarily of water and methane, are solid, crystalline, ice-like substances found in permafrost areas and deepwater basins around the world. As the search for oil and gas extends into ever-deeper waters, particularly within the northern Gulf of Mexico, gas hydrates are becoming more of a focus in terms of both safety and as a potential energy resource.

Multiple reflection and transmission phases in complex layered media using a multistage fast marching method

Abstract: Traditional grid-based eikonal schemes for computing traveltimes are usually confined to obtaining first arrivals only. However, later arrivals can be numerous and of greater amplitude, making them a potentially valuable resource for practical applications such as seismic imaging. The aim of this paper is to introduce a grid-based method for tracking multivalued wavefronts composed of any number of reflection and refraction branches in layered media. A finite-difference eikonal solver known as the fast marching method (FMM) is used to propagate wavefronts from one interface to the next. By treating each layer that the wavefront enters as a separate computational domain, one obtains a refracted branch by reinitializing FMM in the adjacent layer and a reflected branch by reinitializing FMM in the incident layer. To improve accuracy, a local grid refinement scheme is used in the vicinity of the source where wavefront curvature is high. Several examples are presented which demonstrate the viability of the new method in highly complex layered media. Even in the presence of velocity variations as large as 8:1 and interfaces of high curvature, wavefronts composed of many reflection and transmission events are tracked rapidly and accurately. This is because the scheme retains the two desirable properties of a single-stage FMM: computational speed and stability. Local grid refinement about the source also can increase accuracy by an order of magnitude with little increase in computational cost.

A parallel finite-difference approach for 3D transient electromagnetic modeling with galvanic sources

Abstract: A parallel finite-difference algorithm for the solution of diffusive, three-dimensional (3D) transient electromagnetic field simulations is presented. The purpose of the scheme is the simulation of both electric fields and the time derivative of magnetic fields generated by galvanic sources (grounded wires) over arbitrarily complicated distributions of conductivity and magnetic permeability. Using a staggered grid and a modified DuFort-Frankel method, the scheme steps Maxwell's equations in time. Electric field initialization is done by a conjugate-gradient solution of a 3D Poisson problem, as is common in 3D resistivity modeling. Instead of calculating the initial magnetic field directly, its time derivative and curl are employed in order to advance the electric field in time. A divergence-free condition is enforced for both the magnetic-field time derivative and the total conduction-current density, providing accurate results at late times. In order to simulate large realistic earth models, the algorithm has been designed to run on parallel computer platforms. The upward continuation boundary condition for a stable solution in the infinitely resistive air layer involves a two-dimensional parallel fast Fourier transform. Example simulations are compared with analytical, integral-equation and spectral Lanczos decomposition solutions and demonstrate the accuracy of the scheme.

Stochastic reservoir characterization using prestack seismic data

Abstract: Reservoir characterization must be based on information from various sources. Well observations, seismic reflection times, and seismic amplitude versus offset (AVO) attributes are integrated in this study to predict the distribution of the reservoir variables, i.e., facies and fluid filling. The prediction problem is cast in a Bayesian setting. The a priori model includes spatial coupling through Markov random field assumptions and intervariable dependencies through nonlinear relations based on rock physics theory, including Gassmann's relation. The likelihood model relating observations to reservoir variables (including lithology facies and pore fluids) is based on approximations to Zoeppritz equations. The model assumptions are summarized in a Bayesian network illustrating the dependencies between the reservoir variables. The posterior model for the reservoir variables conditioned on the available observations is defined by the a priori and likelihood models. This posterior model is not analytically tra...

Shear waves in acoustic anisotropic media

Abstract: Acoustic transversely isotropic (TI) media are defined by artificially setting the shear‐wave velocity in the direction of symmetry axis, VS0, to zero. Contrary to conventional wisdom that equating VS0 = 0 eliminates shear waves, we demonstrate their presence and examine their properties. Specifically, we show that SV‐waves generally have finite nonzero phase and group velocities in acoustic TI media. In fact, these waves have been observed in full waveform modeling, but apparently they were not understood and labeled as numerical artifacts.Acoustic TI media are characterized by extreme, in some sense infinite strength of anisotropy. It makes the following unusual wave phenomena possible: (1) there are propagation directions, where the SV‐ray is orthogonal to the corresponding wavefront normal, (2) the SV‐wave whose ray propagates along the symmetry axis is polarized parallel to the P‐wave propagating in the same direction, (3) P‐wave singularities, that is, directions where P‐ and SV‐wave phase velocitie...

Source-independent full waveform inversion of seismic data

Abstract: A set of seismic trace data is collected in an input data set that is first Fourier transformed in its entirety into the frequency domain. A normalized wavefield is obtained for each trace of the input data set in the frequency domain. Normalization is done with respect to the frequency response of a reference trace selected from the set of seismic trace data. The normalized wavefield is source independent, complex, and dimensionless. The normalized wavefield is shown to be uniquely defined as the normalized impulse response, provided that a certain condition is met for the source. This property allows construction of the inversion algorithm disclosed herein, without any source or source coupling information. The algorithm minimizes the error between data normalized wavefield and the model normalized wavefield. The methodology is applicable to any 3-D seismic problem, and damping may be easily included in the process.

A comparison between one-way and two-way wave-equation migration

Abstract: Results for wave-equation migration in the frequency domain using the constant-density acoustic two-way wave equation have been compared to images obtained by its one-way approximation. The two-way approach produces more accurate reflector amplitudes and provides superior imaging of steep flanks. However, migration with the two-way wave equation is sensitive to diving waves, leading to low-frequency artifacts in the images. These can be removed by surgical muting of the input data or iterative migration or high-pass spatial filtering. The last is the most effective. Iterative migration based on a least-squares approximation of the seismic data can improve the amplitudes and resolution of the imaged reflectors. Two approaches are considered, one based on the linearized constantdensity acoustic wave equation and one on the full acoustic wave equation with variable density. The first converges quickly. However, with our choice of migration weights and high-pass spatial filtering for the linearized case, a real-data migration result shows little improvement after the first iteration. The second, nonlinear iterative migration method is considerably more difficult to apply. A real-data example shows only marginal improvement over the linearized case. In two dimensions, the computational cost of the twoway approach has the same order of magnitude as that for the one-way method. With our implementation, the two-way method requires about twice the computer time needed for one-way wave-equation migration.

Quantitative estimation of compaction and velocity changes using 4D impedance and traveltime changes

Abstract: In some hydrocarbon reservoirs, severe compaction of the reservoir rocks is observed. This compaction is caused by production, and it is often associated with changes in the overburden. Time-lapse (or 4D) seismic data are used to monitor this compaction process. Since the compaction causes changes in both layer thickness and seismic velocities, it is crucial to distinguish between the two effects. Two new seismic methods for monitoring compacting reservoirs are introduced, one based on measured seismic prestack traveltime changes, and the other based on poststack traveltime and amplitude changes. In contrast to earlier methods, these methods do not require additional empirical relationships, such as, for instance, a velocity-porosity relationship. The uncertainties in estimates for compaction and velocity change are expressed in terms of errors in the traveltime and amplitude measurements. These errors are directly related to the quality and repeatability of time-lapse seismic data. For a reservoir at 3000-m depth wit h9mo fcompaction, and assuming a 4D timeshift error of 0.5 ms at near offset and 2 ms at far offset, we find relative uncertainty in the compaction estimate of approximately 50‐60% using traveltime information only.

Induced polarization measurements on unsaturated, unconsolidated sands

Abstract: Induced polarization (IP) measurements were obtained on unsaturated, unconsolidated sediments during (1) evaporative drying and (2) pressure drainage followed by subsequent imbibition (water reentry). Porous ceramic discs were used with existing laboratory IP instrumentation to permit accurate IP measurements on unsaturated samples. Polarization magnitude during evaporative drying approximates a power law dependence on saturation. Saturation exponents for the polarization term were consistently less than Archie conduction exponents, although no clear relationship between the exponents was observed. The polarization measured over a pressure drainage and imbibition cycle exhibits a complex (yet similar between tested samples) saturation dependence, being a function of saturation range and saturation history. Polarization is observed to increase with saturation over certain saturation intervals, yet decrease with saturation over others. High polarization observed during sample imbibition is consistent with a...

Velocity model estimation with data-derived wavefront attributes

Abstract: Kinematic information for constructing velocity models can be extracted in a robust way from seismic prestack data with the common‐reflection‐surface (CRS) stack. This data‐driven process results, in addition to a simulated zero‐offset section, in a number of wavefront attributes—wavefront curvatures and normal ray emergence angles—associated with each simulated zero‐offset sample. A tomographic inversion method is presented that uses this kinematic information to determine smooth, laterally heterogeneous, isotropic subsurface velocity models for depth imaging. The input for the inversion consists of wavefront attributes picked at a number of locations in the simulated zero‐offset section. The smooth velocity model is described by B‐splines. An optimum model is found iteratively by minimizing the misfit between the picked data and the corresponding modeled values. The required forward‐modeled quantities are obtained during each iteration by dynamic ray tracing along normal rays pertaining to the input dat...

Incorporating geologic information into reflection tomography

Abstract: In areas of complex geology, prestack depth migration is often necessary if we are to produce an accurate image of the subsurface. Prestack depth migration requires an accurate interval velocity model. With few exceptions, the subsurface velocities are not known beforehand and should be estimated. When the velocity structure is complex, with significant lateral variations, reflection-tomography methods are often an effective tool for improving the velocity estimate. Unfortunately, reflection tomography often converges slowly, to a model that is geologically unreasonable, or it does not converge at all. The large null space of reflection-tomography problems often forces us to add a sparse parameterization of the model and/or regularization criteria to the estimation. Standard tomography schemes tend to create isotropic features in velocity models that are inconsistent with geology. These isotropic features result, in large part, from using symmetric regularization operators or from choosing a poor model parameterization. If we replace the symmetric operators with nonstationary operators that tend to spread information along structural dips, the tomography will produce velocity models that are geologically more reasonable. In addition, by forming the operators in helical 1D space and performing polynomial division, we apply the inverse of these space-varying anisotropic operators. The inverse operators can be used as a preconditioner to a standard tomography problem, thereby significantly improving the speed of convergence compared with the typical regularized inversion problem. Results from 2D synthetic and 2D field data are shown. In each case, the velocity obtained improves the focusing of the migrated image.

A classification for the pressure-sensitivity properties of a sandstone rock frame

Abstract: A three‐parameter formula is proposed to describe the sensitivity of the bulk modulus (κ) or shear rigidity (μ) for a sandstone rock frame under applied isotropic loading. The theoretical basis for this relation is the use of excess compliance as a pseudo function to describe all internal weaknesses in the rock, regardless of their origin. The law is fitted to 179 sets of laboratory measurements on unsaturated reservoir core and outcrop sandstones that have low to moderate porosity and a range of clay fractions and cementation. A stress index, defined to quantitatively rank the reservoir sands, indicates that laboratory‐derived pressure sensitivity is highest (10% per MPa maximum) for the clean, moderate‐porosity Paleocene sands from the Forties Formation and West of Shetlands, whereas the lower‐porosity, cemented, and more‐consolidated Permian sands rank as least pressure sensitive (1% per MPa maximum). Higher‐porosity sands are stress sensitive because of the overall rock‐frame compressibility. All of t...

Two‐ and three‐dimensional electrical resistivity imaging at a heterogeneous remediation site

Abstract: Geometrically complex heterogeneities at a decommissioned sour gas plant could not be adequately characterized with drilling and 2D electrical resistivity surveys alone. In addition, 2D electrical resistivity imaging profiles produced misleading images as a result of out‐of‐plane resistivity anomalies and violation of the 2D assumption. Accurate amplitude and positioning of electrical conductivity anomalies associated with the subsurface geochemical distribution were required to effectively analyze remediation alternatives. Forward and inverse modeling and field examples demonstrated that 3D resistivity images were needed to properly reconstruct the amplitude and geometry of the complex resistivity anomalies. Problematic 3D artifacts in 2D images led to poor inversion fits and spurious conductivity values in the images at depths close to the horizontal offset of the off‐line anomaly. Three‐dimensional surveys were conducted with orthogonal sets of Wenner and dipole–dipole 2D resistivity survey lines. The ...