Upscaling in orthorhombic media: Behavior of elastic parameters in heterogeneous fractured earth

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. C113-C126 ◽  
Author(s):  
Yuriy Ivanov ◽  
Alexey Stovas
Keyword(s):  
Seismic Waves ◽  
Homogeneous Medium ◽  
Transversely Isotropic ◽  
Processing Parameters ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Weak Anisotropy ◽  
Fluid Saturation ◽  
Fractured Medium ◽  
Anisotropic Layers

A stack of horizontal homogeneous elastic arbitrary anisotropic layers in welded contact in the long-wavelength limit is equivalent to an elastic anisotropic homogeneous medium. Such a medium is characterized by an effective average description adhering to previously derived closed-form formalism. We have used this formalism to study three different inhomogeneous orthorhombic (ORT) models that could represent real geologic scenarios. We have determined that a stack of thin orthorhombic layers with arbitrary azimuths of vertical symmetry planes can be approximated by an effective orthorhombic medium. The most suitable approach for this is to minimize the misfit between the effective anisotropic medium, monoclinic in that case, and the desirable orthorhombic medium. The second model is an interbedding of VTI (transversely isotropic with a vertical symmetry axis) layers with the same layers containing vertical fractures (shales are intrinsically anisotropic and often fractured). We have derived a weak-anisotropy approximation for important P-wave processing parameters as a function of the relative amount of the fractured lithology. To accurately characterize fractures, inversion for the fracture parameters should use a priori information on the relative amount of a fractured medium. However, we have determined that the cracks’ fluid saturation can be estimated without prior knowledge of the relative amount of the fractured layer. We have used field well-log data to demonstrate how fractures can be included in the interval of interest during upscaling. Finally, the third model that we have considered is a useful representation of tilted orthorhombic medium in the case of two-way propagation of seismic waves through it. We have derived a weak anisotropy approximation for traveltime parameters of the reflected P-wave that propagates through a stack of thin beds of tilted orthorhombic symmetry. The tilt of symmetry planes in an orthorhombic medium significantly affects the kinematics of the reflected P-wave and should be properly accounted for to avoid mispositioning of geologic structures in seismic imaging.

scholarly journals Pore-fluid effects on seismic waves in vertically fractured earth with orthotropic symmetry

Geophysics ◽  
2010 ◽  
Vol 75 (6) ◽  
pp. T185-T200 ◽  
Author(s):  
James G. Berryman
Keyword(s):  
Seismic Waves ◽  
Vertical Plane ◽  
Transversely Isotropic ◽  
Pore Fluid ◽  
System Response ◽  
Orthorhombic Symmetry ◽  
Detailed Model ◽  
Vertical Fracture ◽  
Fractured Medium ◽  
Vertical Fractures

For elastically noninteracting vertical-fracture sets at arbitrary orientation angles to each other, a detailed model is presented in which the resulting anisotropic fractured medium generally has orthorhombic symmetry overall. Some of the analysis methods and ideas of Schoenberg are emphasized, together with their connections to other similarly motivated and conceptually related methods by Sayers and Kachanov, among others. Examples show how parallel vertical-fracture sets having HTI (horizontal transversely isotropic) symmetry transform into orthotropic fractured media if some subsets of the vertical fractures are misaligned with the others, and then the fractured system can have VTI (vertical transversely isotropic) symmetry if all of the fractures are aligned randomly or half parallel and half perpendicular to a given vertical plane. An orthotropic example having vertical fractures in an otherwise VTI earth system (studied previously by Schoenberg and Helbig) is compared with the other examples treated and it is finally shown how fluids in the fractures affect the orthotropic poroelastic system response to seismic waves. The key result is that fracture-influence parameters are multiplied by a factor of [Formula: see text], where [Formula: see text] is Skempton’s second coefficient for poroelastic media. Skempton’s [Formula: see text] coefficient is itself a measurable characteristic of fluid-saturated porous rocks, depending on porosity, solid moduli, and the pore-fluid bulk modulus. For heterogeneous porous media, connections between the present work and earlier related results of Brown and Korringa are also established.

The transversely isotropic poroelastic wave equation including the Biot and the squirt mechanisms: Theory and application

Geophysics ◽  
1997 ◽  
Vol 62 (1) ◽  
pp. 309-318 ◽  
Author(s):  
Jorge O. Parra
Keyword(s):  
Wave Equation ◽  
Fluid Flow ◽  
Analytical Solution ◽  
Seismic Waves ◽  
Transversely Isotropic ◽  
P Wave ◽  
Fluid Properties ◽  
Permeability Anisotropy ◽  
Attenuation And Dispersion ◽  
Maximum Permeability

The transversely isotropic poroelastic wave equation can be formulated to include the Biot and the squirt‐flow mechanisms to yield a new analytical solution in terms of the elements of the squirt‐flow tensor. The new model gives estimates of the vertical and the horizontal permeabilities, as well as other measurable rock and fluid properties. In particular, the model estimates phase velocity and attenuation of waves traveling at different angles of incidence with respect to the principal axis of anisotropy. The attenuation and dispersion of the fast quasi P‐wave and the quasi SV‐wave are related to the vertical and the horizontal permeabilities. Modeling suggests that the attenuation of both the quasi P‐wave and quasi SV‐wave depend on the direction of permeability. For frequencies from 500 to 4500 Hz, the quasi P‐wave attenuation will be of maximum permeability. To test the theory, interwell seismic waveforms, well logs, and hydraulic conductivity measurements (recorded in the fluvial Gypsy sandstone reservoir, Oklahoma) provide the material and fluid property parameters. For example, the analysis of petrophysical data suggests that the vertical permeability (1 md) is affected by the presence of mudstone and siltstone bodies, which are barriers to vertical fluid movement, and the horizontal permeability (1640 md) is controlled by cross‐bedded and planar‐laminated sandstones. The theoretical dispersion curves based on measurable rock and fluid properties, and the phase velocity curve obtained from seismic signatures, give the ingredients to evaluate the model. Theoretical predictions show the influence of the permeability anisotropy on the dispersion of seismic waves. These dispersion values derived from interwell seismic signatures are consistent with the theoretical model and with the direction of propagation of the seismic waves that travel parallel to the maximum permeability. This analysis with the new analytical solution is the first step toward a quantitative evaluation of the preferential directions of fluid flow in reservoir formation containing hydrocarbons. The results of the present work may lead to the development of algorithms to extract the permeability anisotropy from attenuation and dispersion data (derived from sonic logs and crosswell seismics) to map the fluid flow distribution in a reservoir.

Reflection moveout approximations for P-waves in a moderately anisotropic homogeneous tilted transverse isotropy layer

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. C175-C185 ◽  
Author(s):  
Ivan Pšenčík ◽  
Véronique Farra
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
3D Space ◽  
Axis Of Symmetry ◽  
Relative Errors ◽  
Symmetry Tests ◽  
Ti Media

We have developed approximate nonhyperbolic P-wave moveout formulas applicable to weakly or moderately anisotropic media of arbitrary anisotropy symmetry and orientation. Instead of the commonly used Taylor expansion of the square of the reflection traveltime in terms of the square of the offset, we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. No acoustic approximation is used. We specify the formulas designed for anisotropy of arbitrary symmetry for the transversely isotropic (TI) media with the axis of symmetry oriented arbitrarily in the 3D space. Resulting formulas depend on three P-wave WA parameters specifying the TI symmetry and two angles specifying the orientation of the axis of symmetry. Tests of the accuracy of the more accurate of the approximate formulas indicate that maximum relative errors do not exceed 0.3% or 2.5% for weak or moderate P-wave anisotropy, respectively.

scholarly journals Seismic AVOA Inversion for Weak Anisotropy Parameters and Fracture Density in a Monoclinic Medium

2020 ◽  
Vol 10 (15) ◽  
pp. 5136 ◽  
Author(s):  
Zijian Ge ◽  
Shulin Pan ◽  
Jingye Li
Keyword(s):  
Fractured Rock ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
P Wave ◽  
Inversion Method ◽  
Fracture Density ◽  
Weak Anisotropy ◽  
Amplitude Versus Offset ◽  
Porosity And Permeability ◽  
Transverse Isotropic

In shale gas development, fracture density is an important lithologic parameter to properly characterize reservoir reconstruction, establish a fracturing scheme, and calculate porosity and permeability. The traditional methods usually assume that the fracture reservoir is one set of aligned vertical fractures, embedded in an isotropic background, and estimate some alternative parameters associated with fracture density. Thus, the low accuracy caused by this simplified model, and the intrinsic errors caused by the indirect substitution, affect the estimation of fracture density. In this paper, the fractured rock of monoclinic symmetry assumes two non-orthogonal vertical fracture sets, embedded in a transversely isotropic background. Firstly, assuming that the fracture radius, width, and orientation are known, a new form of P-wave reflection coefficient, in terms of weak anisotropy (WA) parameters and fracture density, was obtained by substituting the stiffness coefficients of vertical transverse isotropic (VTI) background, normal, and tangential fracture compliances. Then, a linear amplitude versus offset and azimuth (AVOA) inversion method, of WA parameters and fracture density, was constructed by using Bayesian theory. Tests on synthetic data showed that WA parameters, and fracture density, are stably estimated in the case of seismic data containing a moderate noise, which can provide a reliable tool in fracture prediction.

Quartic moveout coefficient: 3D description and application to tilted TI media

Geophysics ◽  
2003 ◽  
Vol 68 (5) ◽  
pp. 1600-1610 ◽  
Author(s):  
Andres Pech ◽  
Ilya Tsvankin ◽  
Vladimir Grechka
Keyword(s):  
Heterogeneous Media ◽  
Symmetry Axis ◽  
High Sensitivity ◽  
Transversely Isotropic ◽  
Seismic Inversion ◽  
P Wave ◽  
Individual Values ◽  
Weak Anisotropy ◽  
Essential Information ◽  
Ti Media

Nonhyperbolic (long‐spread) moveout provides essential information for a number of seismic inversion/processing applications, particularly for parameter estimation in anisotropic media. Here, we present an analytic expression for the quartic moveout coefficient A4 that controls the magnitude of nonhyperbolic moveout of pure (nonconverted) modes. Our result takes into account reflection‐point dispersal on irregular interfaces and is valid for arbitrarily anisotropic, heterogeneous media. All quantities needed to compute A4 can be evaluated during the tracing of the zero‐offset ray, so long‐spread moveout can be modeled without time‐consuming multioffset, multiazimuth ray tracing. The general equation for the quartic coefficient is then used to study azimuthally varying nonhyperbolic moveout of P‐waves in a dipping transversely isotropic (TI) layer with an arbitrary tilt ν of the symmetry axis. Assuming that the symmetry axis is confined to the dip plane, we employed the weak‐anisotropy approximation to analyze the dependence of A4 on the anisotropic parameters. The linearized expression for A4 is proportional to the anellipticity coefficient η ≈ ε − δ and does not depend on the individual values of the Thomsen parameters. Typically, the magnitude of nonhyperbolic moveout in tilted TI media above a dipping reflector is highest near the reflector strike, whereas deviations from hyperbolic moveout on the dip line are substantial only for mild dips. The azimuthal variation of the quartic coefficient is governed by the tilt ν and reflector dip φ and has a much more complicated character than the NMO–velocity ellipse. For example, if the symmetry axis is vertical (VTI media, ν = 0) and the dip φ < 30°, A4 goes to zero on two lines with different azimuths where it changes sign. If the symmetry axis is orthogonal to the reflector (this model is typical for thrust‐and‐fold belts), the strike‐line quartic coefficient is defined by the well‐known expression for a horizontal VTI layer (i.e., it is independent of dip), while the dip‐line A4 is proportional to cos4 φ and rapidly decreases with dip. The high sensitivity of the quartic moveout coefficient to the parameter η and the tilt of the symmetry axis can be exploited in the inversion of wide‐azimuth, long‐spread P‐wave data for the parameters of TI media.

scholarly journals Analysis of fluid substitution in a porous and fractured medium

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA157-WA166 ◽  
Author(s):  
Samik Sil ◽  
Mrinal K. Sen ◽  
Boris Gurevich
Keyword(s):  
Water Saturation ◽  
Transversely Isotropic ◽  
P Wave ◽  
Reflection Coefficients ◽  
Fluid Substitution ◽  
Fracture Density ◽  
Maximum Increase ◽  
Amplitude Variation With Offset ◽  
Fractured Medium ◽  
Medium Change

To improve quantitative interpretation of seismic data, we analyze the effect of fluid substitution in a porous and fractured medium on elastic properties and reflection coefficients. This analysis uses closed-form expressions suitable for fluid substitution in transversely isotropic media with a horizontal symmetry axis (HTI). For the HTI medium, the effect of changing porosity and water saturation on (1) P-wave moduli, (2) horizontal and vertical velocities, (3) anisotropic parameters, and (4) reflection coefficients are examined. The effects of fracture density on these four parameters are also studied. For the model used in this study, a 35% increase in porosity lowers the value of P-wave moduli by maximum of 45%. Consistent with the reduction in P-wave moduli, P-wave velocities also decrease by maximum of 17% with a similar increment in porosity. The reduction is always larger for the horizontal P-wave modulus than for the vertical one and is nearly independent of fracture density. The magnitude of the anisotropic parameters of the fractured medium also changes with increased porosity depending on the changes in the value of P-wave moduli. The reflection coefficients at an interface of the fractured medium with an isotropic medium change in accordance with the above observations and lead to an increase in anisotropic amplitude variation with offset (AVO) gradient with porosity. Additionally, we observe a maximum increase in P-wave modulus and velocity by 30% and 8%, respectively, with a 100% increase in water saturation. Water saturation also changes the anisotropic parameters and reflection coefficients. Increase in water saturation considerably increases the magnitude of the anisotropic AVO gradient irrespective of fracture density. From this study, we conclude that porosity and water saturation have a significant impact on the four studied parameters and the impacts are seismically detectable.

Analytic study of the geometrical spreading of P-waves in a layered transversely isotropic medium with a vertical symmetry axis

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1305-1315 ◽  
Author(s):  
Hongbo Zhou ◽  
George A. McMechan
Keyword(s):  
Isotropic Medium ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Analytical Formula ◽  
Transversely Isotropic ◽  
P Wave ◽  
Geometrical Spreading ◽  
Transversely Isotropic Medium ◽  
Weak Anisotropy ◽  
Special Cases

An analytical formula for geometrical spreading is derived for a horizontally layered transversely isotropic medium with a vertical symmetry axis (VTI). With this expression, geometrical spreading can be determined using only the anisotropy parameters in the first layer, the traveltime derivatives, and the source‐receiver offset. Explicit, numerically feasible expressions for geometrical spreading are obtained for special cases of transverse isotropy (weak anisotropy and elliptic anisotropy). Geometrical spreading can be calculated for transversly isotropic (TI) media by using picked traveltimes of primary nonhyperbolic P-wave reflections without having to know the actual parameters in the deeper subsurface; no ray tracing is needed. Synthetic examples verify the algorithm and show that it is numerically feasible for calculation of geometrical spreading. For media with a few (4–5) layers, relative errors in the computed geometrical spreading remain less than 0.5% for offset/depth ratios less than 1.0. Errors that change with offset are attributed to inaccuracy in the expression used for nonhyberbolic moveout. Geometrical spreading is most sensitive to errors in NMO velocity, followed by errors in zero‐offset reflection time, followed by errors in anisotropy of the surface layer. New relations between group and phase velocities and between group and phase angles are shown in appendices.

Quartic moveout coefficient for a dipping azimuthally anisotropic layer

Geophysics ◽  
2004 ◽  
Vol 69 (3) ◽  
pp. 699-707 ◽  
Author(s):  
Andrés Pech ◽  
Ilya Tsvankin
Keyword(s):  
Reservoir Characterization ◽  
Fractured Reservoirs ◽  
Symmetry Plane ◽  
Exact Expression ◽  
Naturally Fractured Reservoirs ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Fracture Density ◽  
Weak Anisotropy ◽  
Velocity Anisotropy

Interpretation and inversion of azimuthally varying nonhyperbolic reflection moveout requires accounting for both velocity anisotropy and subsurface structure. Here, our previously derived exact expression for the quartic moveout coefficient A4 is applied to P‐wave reflections from a dipping interface overlaid by a medium of orthorhombic symmetry. The weak‐anisotropy approximaton for the coefficient A4 in a homogeneous orthorhombic layer is controlled by the anellipticity parameters η(1), η(2), and η(3), which are responsible for time processing of P‐wave data. If the dip plane of the reflector coincides with the vertical symmetry plane [x1, x3], A4 on the dip line is proportional to the in‐plane anellipticity parameter η(2) and always changes sign for a dip of 30○. The quartic coefficient on the strike line is a function of all three η–parameters, but for mild dips it is mostly governed by η(1)—the parameter defined in the incidence plane [x2, x3]. Whereas the magnitude of the dip line A4 typically becomes small for dips exceeding 45○, the nonhyperbolic moveout on the strike line may remain significant even for subvertical reflectors. The character of the azimuthal variation of A4 depends on reflector dip and is quite sensitive to the signs and relative magnitudes of η(1), η(2), and η(3). The analytic results and numerical modeling show that the azimuthal pattern of the quartic coefficient can contain multiple lobes, with one or two azimuths of vanishing A4 between the dip and strike directions. The strong influence of the anellipticity parameters on the azimuthally varying coefficient A4 suggests that nonhyperbolic moveout recorded in wide‐azimuth surveys can help to constrain the anisotropic velocity field. Since for typical orthorhombic models that describe naturally fractured reservoirs the parameters η(1,2,3) are closely related to the fracture density and infill, the results of azimuthal nonhyperbolic moveout analysis can also be used in reservoir characterization.

Interpretation of synthetic common‐depth‐point gathers for a single anisotropic layer

Geophysics ◽  
1982 ◽  
Vol 47 (3) ◽  
pp. 323-335 ◽  
Author(s):  
Stuart Crampin ◽  
Barbara J. Radovich
Keyword(s):  
Angular Velocity ◽  
Elastic Constants ◽  
Arrival Time ◽  
Transversely Isotropic ◽  
Orthorhombic Symmetry ◽  
Isotropic Layer ◽  
Anisotropic Layer ◽  
Wide Range ◽  
Velocity Variations ◽  
Anisotropic Layers

Analysis of synthetic traveltime gathers shows that anisotropy may have a large enough effect on P, SH, and SV propagation to alter significantly the interpretation of the subsurface below the anisotropic layers. Consequently, if anisotropy exists below a seismic line, it is important to estimate the anisotropic parameters correctly. We discuss the effects of anisotropy on seismic waves and present a method for estimating the elastic constants of a transversely isotropic layer from P and SH arrival‐time gathers. The technique may be extended to more general anisotropic symmetries by analyzing gathers from several azimuths. To illustrate the possible effect of anisotropy on exploration surveys, P, SH, and SV velocity variations are calculated for several types of anisotropic sedimentary fabrics. Alignments due to bedding, shale lithology, and dry parallel cracks may have similar velocity variations. Fabrics with other configurations of cracks may still possess overall transversely isotropic symmetry, but they have a wide range of angular velocity variations with different polarities and periodicities. Synthetic gather curves are generated for a range of models with an anisotropic layer over an isotropic substrate. They show departures from hyperbolas, and erroneous depth determinations, that depend upon the elastic constants of the anisotropic layer. The elastic constants of the anisotropic layers are estimated from the synthetic gather curves by means of approximate equations for the angular velocity variations, which are linear in the elastic constants. Formulas are developed which relate tangents to the gather curves directly in terms of the elastic constants. These are tested for single‐layer transversely isotropic models and allow the five elastic constants to be estimated by drawing three tangents to P and SH synthetic arrival‐time gathers in [Formula: see text] space. Comparisons of estimated with original elastic constants are good for a number of different types of transversely isotropic fabrics. Gathers are also calculated at two azimuths in an anisotropic layer with orthorhombic symmetry and are analyzed with some success.

Processing‐induced anisotropy

Geophysics ◽  
2002 ◽  
Vol 67 (6) ◽  
pp. 1920-1928 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Seismic Data ◽  
Symmetry Axis ◽  
Homogeneous Medium ◽  
Transversely Isotropic ◽  
P Wave ◽  
Well Logs ◽  
Velocity Analysis ◽  
Induced Anisotropy ◽  
Vertical Heterogeneity ◽  
Vti Media

Processing of seismic data is often performed under the assumption that the velocity distribution in the subsurface can be approximated by a macromodel composed of isotropic homogeneous layers or blocks. Despite being physically unrealistic, such models are believed to be sufficient for describing the kinematics of reflection arrivals. In this paper, we examine the distortions in normal‐moveout (NMO) velocities caused by the intralayer vertical heterogeneity unaccounted for in velocity analysis. To match P‐wave moveout measurements from a horizontal or a dipping reflector overlaid by a vertically heterogeneous isotropic medium, the effective homogeneous overburden has to be anisotropic. This apparent anisotropy is caused not only by velocity monotonically increasing with depth, but also by random velocity variations similar to those routinely observed in well logs. Assuming that the effective homogeneous medium is transversely isotropic with a vertical symmetry axis (VTI), we express the VTI parameters through the actual depth‐dependent isotropic velocity function. If the reflector is horizontal, combining the NMO and vertical velocities always results in nonnegative values of Thomsen's coefficient δ. For a dipping reflector, the inversion of the P‐wave NMO ellipse yields a nonnegative Alkhalifah‐Tsvankin coefficient η that increases with dip. The values of η obtained by two other methods (2‐D dip‐moveout inversion and nonhyperbolic moveout analysis) are also nonnegative but generally differ from that needed to fit the NMO ellipse. For truly anisotropic (VTI) media, the influence of vertical heterogeneity above the reflector can lead to a bias toward positive δ and η estimates in velocity analysis.

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