Morten Jakobsen

Elastic properties of hydrate‐bearing sediments using effective medium theory

Accurate and detailed models of the seismic velocity structure of gas hydrate-bearing sediments may be determined by careful analysis of controlled source seismic data. However, interpretation of these velocities in terms of hydrate saturation of the pore space has hitherto relied on semiempirical formulas and/or simple effective medium theory. We develop a rigorous theoretical scheme to relate the seismic properties of a clay-rich hydrate-bearing sediment to its porosity, mineralogy, microstructural features and hydrate saturation. We consider separately the two possible end-members for the distribution of hydrate in the pore space: (1) hydrates are unconnected and located in the pore voids without appreciable grain contact and (2) connected hydrates are forming cement binding around the grains. The scheme is transversely isotropic, to allow for anisotropy due to alignment of clay platelets, and is based on a combination of a self-consistent approximation, a differential effective medium theory, and a method of smoothing for crystalline aggregates. We have applied the scheme to lithological and seismic velocity data from Ocean Drilling Program Site 995 on the Blake Ridge (southeastern U.S. continental margin) to make estimates of the hydrate saturation. It was found that the hydrates are probably unconnected, and their volume concentration varies between approx. 0% at 100 m below the seabed and approx. 9% at 400 m depth, just above the “bottom simulating reflector”, if the clay platelet orientation distribution resembles the function we have used.

Anisotropic approximations for mudrocks: A seismic laboratory study

An experimental technique for determining the in‐situ elastic properties of mudrocks with (horizontal) alignments in the microstructure is used to study the accuracy of a set of three nested scalar anisotropic approximations for transversely isotropic (TI) media. Each subsequent approximation adds one more velocity parameter and includes the previous as a special case. These approximations are convenient and robust because of their close relationship to standard geophysical measurements. There exists no good theory to predict the effects of an imposed stress on the elasticity of mudrocks. In this study, the tensor of elastic moduli of a single test specimen of mudrock subjected to an anisotropic stress field is determined from ultrasonic group velocity measurements involving pointlike transducers. The mechanical characterization (performed at constant pore pressure) is accompanied by detailed microscopic observations and analysis. The method was used to obtain accurate elastic constants for five well‐defi...

Unified theory of global flow and squirt flow in cracked porous media

Approximations for frequency-dependent and complex-valued effective stiffness tensors of cracked porous media (saturated with a single fluid) are developed on the basis of an inclusion-based model (the T-matrix approach to rock physics) and a unified treatment of the global-flow and squirt-flow mechanisms. Essentially, this study corrects an inconsistency or error related to fluid-mass conservation in an existing expression for the t-matrix (wave-induced deformation) of a communicating cavity, a cavity that is isolated with respect to stress propagation (through the solid matrix) but that can exchange fluid mass with other cavities because of global and/or local pressure gradients associated with passage of a long viscoelastic wave. An earlier demonstration of Gassmann consistency remains valid because the new theory of global flow and squirt flow (which also takes into account solid mechanical effects of stress interaction by us-ing products of communicating t-matrices associated with two-point correlati...

Rock physics modelling of shale diagenesis

A model for estimating the effective anisotropic properties of cemented shales is presented. The model is based on two mathematical methods for estimation of effective properties of a composite medium; a self-consistent approximation and a differential effective medium model. In combination these theories allow approximation of a shale with connected clay minerals and cement, and disconnected pores and quartz grains, which can be compared with the conditions in a real cemented shale. A strategy is also presented for estimation of stiffnesses in the transition zone from mechanical compaction to chemical compaction dominated diagenesis. Combining these theories with a shale compaction theory, enables modelling of the effective elastic stiffnesses for shales from deposition and mechanical compaction to deep burial and chemical compaction/cementing. Results from the model were compared with velocity data from three wells, showing good fit for velocity predictions, following the main velocity trends with increased temperature and depth.

Effects of fluids and dual-pore systems on pressure-dependent velocities and attenuations in carbonates

TheeffectsoffluidsubstitutiononP-andS-wavevelocitiesin carbonates of complex texture are still not understood fully. The often-used Gassmann equation gives ambiguous results when compared with ultrasonic velocity data. We present theoretical modelingofvelocityandattenuationmeasurementsobtainedata frequency of 750 kHz for six carbonate samples composed of calcite and saturated with air, brine, and kerosene.Although porosities 2%‐14% and permeabilities 0‐74 mD are relatively low, velocity variations are large. Differences between the highest and lowest P- and S-wave velocities are about 18% and 27% for brine-saturated samples at 60 and 10 MPa effective pressure, respectively. S-wave velocities are measured for two orthogonal polarizations;forfourofsixsamples,anisotropyisrevealed.The Gassmann model underpredicts fluid-substitution effects by 2% for three samples and by as much as 5% for the rest of the six samples. Moreover, when dried, they also show decreasing attenuationwithincreasingconfiningpressure.Tomodelthisbehavior, we examine a pore model made of two pore systems: one constitutesthemainanddrainableporosity,andtheotherismade ofundrainedcracklikeporesthatcanbeassociatedwithgrain-tograincontacts.Inaddition,thesedriedrocksamplesaremodeled to contain a fluid-filled-pore system of grain-to-grain contacts, potentiallycausinglocalfluidflowandattenuation.Forthetheoretical model, we use an inclusion model based on the T-matrix approach,whichalsoconsiderseffectsofporetextureandgeometry, and porefluid, global- and local-fluidflow. By using a dualpore system, we establish a realistic physical model consistently describingthemeasureddata.

A 3D computational study of effective medium methods applied to fractured media

This work evaluates and improves upon existing effective medium methods for permeability upscaling in fractured media. Specifically, we are concerned with the asymmetric self-consistent, symmetric self-consistent, and differential methods. In effective medium theory, inhomogeneity is modeled as ellipsoidal inclusions embedded in the rock matrix. Fractured media correspond to the limiting case of flat ellipsoids, for which we derive a novel set of simplified formulas. The new formulas have improved numerical stability properties, and require a smaller number of input parameters. To assess their accuracy, we compare the analytical permeability predictions with three-dimensional finite-element simulations. We also compare the results with a semi-analytical method based on percolation theory and curve-fitting, which represents an alternative upscaling approach. A large number of cases is considered, with varying fracture aperture, density, matrix/fracture permeability contrast, orientation, shape, and number of fracture sets. The differential method is seen to be the best choice for sealed fractures and thin open fractures. For high-permeable, connected fractures, the semi-analytical method provides the best fit to the numerical data, whereas the differential method breaks down. The two self-consistent methods can be used for both unconnected and connected fractures, although the asymmetric method is somewhat unreliable for sealed fractures. For open fractures, the symmetric method is generally the more accurate for moderate fracture densities, but only the asymmetric method is seen to have correct asymptotic behavior. The asymmetric method is also surprisingly accurate at predicting percolation thresholds.

Full waveform inversion in the frequency domain using direct iterative T-matrix methods

We present two direct iterative solutions to the nonlinear seismic waveform inversion problem that are based on volume integral equation methods for seismic forward modelling in the acoustic approximation. The solutions are presented in the frequency domain, where accurate inversion results can often be obtained using a relatively low number of frequency components. Our inverse scattering approach effectively replaces an ill-posed nonlinear inverse problem with a series of linear ill-posed inverse problems, for which there already exist efficient (regularized) solution methods. Both these solutions update the wavefield within the scattering domain after each iteration. The main difference is that the background medium Green functions are kept fixed in the first solution, but updated after each iteration in the second solution. This means that our solutions are very similar to the Born iterative (BI) and the distorted Born iterative (DBI) methods that are commonly used in acoustic and electromagnetic inverse scattering. However, we have eliminated the need to perform a full forward simulation (or to invert a huge matrix) at each iteration via the use of an iterative T-matrix method for fixed background media for the BI method and a variational T-matrix method for dynamic background media for the DBI method. The T-matrix (variation) is linearly related with the seismic wavefield data (residuals), but related with the unknown scattering potential model parameter (updates) in a non-linear manner, which is independent of the source-receiver configuration. This mathematical structure, which allows one to peel off the effects of the source-receiver configuration, is very attractive when dealing with multiple (simultaneous) sources, and is also compatible with the (future) use of renormalization methods for dealing with local minima problems. To illustrate the performance and potential of the two direct iterative methods for FWI, we performed a series of numerical experiments on synthetic seismic waveform data associated with a simple 2D model and the more complicated Marmousi model. The results of these numerical experiments suggest that the use of a fixed (e.g. smooth and ray-tracing friendly) background medium may be adequate for some applications with moderately large velocity contrasts, but the solution based on a dynamic (non-smooth and constantly updated) background medium will normally provide superiour inversion results; also in the case of low signal-to-noise ratios.

T-matrix approach to seismic forward modelling in the acoustic approximation

Forward seismic modelling in the acoustic approximation, for variable velocity but constant density, is dealt with. The wave equation and the boundary conditions are represented by a volume integral equation of the Lippmann-Schwinger (LS) or Fredholm type. A T-matrix (or transition operator) approach from quantum mechanical potential scattering theory is used to derive a family of linear and nonlinear approximations (cluster expansions), as well as an exact numerical solution of the LS equation. For models of 4D anomalies involving small or moderate contrasts, the Born approximation gives identical numerical results as the first-order t-matrix approximation, but the predictions of an exact T-matrix solution can be quite different (depending on spatial extention of the perturbations). For models of fluid-saturated cavities involving large or huge contrasts, the first-order t-matrix approximation is much more accurate than the Born approximation, although it does not lead to significantly more time-consuming computations. If the spatial extention of the perturbations is not too large, it is practical to use the exact T-matrix solution which allows for arbitrary contrasts and includes all the effects of multiple scattering.

Seismic Properties of Shales During Compaction

Shales, mudstones and their unconsolidated equivalents, constitute the vast majority of all sediments on Earth. The composition of such rocks may be roughly characterized (ignoring variation in clay mineralogy) by only two parameters: the water content (porosity) and the fraction of silt relative to total solid fraction (hereafter called the solid fraction of silt). The water content will decrease during compaction of the rock while the solid fraction of silt is unchanged. For shales and mudrocks the silt grains are mainly unconnected and the load-bearing component is the clay. Furthermore, it is reasonable to assume that the porosity of the uncompacted rock will decrease with increasing solid fraction of silt. The clay consists of small flakes (platelets) which after deposition have a random orientation, but which during compaction will become gradually more horizontally aligned. The degree of alignment will be reduced when silt is present since the platelets will drape around the much larger and rounder silt grains. Any alignment will result in an elastically anisotropic rock. For shales and mudstones without high carbonate content the compaction in the upper 2 km of the sediment column is mainly a mechanical process and it is then possible to calculate the orientation distribution function of the platelets as a function of the compaction (or porosity) of the rock. Given the composition (porosity and silt fraction) and the orientation distribution of the platelets one can use a variety of existing methods to model the elastic properties of the rock. All seismically observable properties of a shale (density, vertical Pand S-wave velocities, and the three anisotropy parameters of a transversally isotropic medium) are thus given by only two parameters, the porosity and the silt fraction. This reduction in the degrees of freedom would be very appealing for seismic inversion where the problem is seriously underdetermined.

Anisotropic effective conductivity in fractured rocks by explicit effective medium methods

In this work, we assess the use of explicit methods for estimating the effective conductivity of anisotropic fractured media. Explicit methods are faster and simpler to use than implicit methods but may have a more limited range of validity. Five explicit methods are considered: the Maxwell approximation, the T-matrix method, the symmetric and asymmetric weakly self-consistent methods, and the weakly differential method, where the two latter methods are novelly constructed in this paper. For each method, we develop simplified expressions applicable to flat spheroidal “penny-shaped” inclusions. The simplified expressions are accurate to the first order in the ratio of fracture thickness to fracture diameter. Our analysis shows that the conductivity predictions of the methods fall within known upper and lower bounds, except for the T-matrix method at high fracture densities and the symmetric weakly self-consistent method when applied to very thin fractures. Comparisons with numerical results show that all the methods give reliable estimates for small fracture densities. For high fracture densities, the weakly differential method is the most accurate if the fracture geometry is non-percolating or the fracture/matrix conductivity contrast is small. For percolating conductive fracture networks, we have developed a scaling relation that can be applied to the weakly self-consistent methods to give conductivity estimates that are close to the results from numerical simulations.