scholarly journals Horizontally isotropic double porosity convection

2019 ◽  
Vol 475 (2221) ◽  
pp. 20180672 ◽  
Author(s):  
B. Straughan
Keyword(s):  
Porous Medium ◽  
Nonlinear Stability ◽  
Vertical Component ◽  
Vertical Direction ◽  
Transversely Isotropic ◽  
Double Porosity ◽  
Solid Skeleton ◽  
The Difference

We analyse instability and nonlinear stability in a layer of saturated double porosity medium. In a double porosity or bidisperse porous medium, there are normal pores which give rise to a macroporosity. But, there are also cracks or fissures in the solid skeleton and these give arise to another porosity known as micro porosity. In this paper, the macropermeability is horizontally isotropic, in the sense that the vertical component of permeability is different to the horizontal one which is the same in all horizontal directions. Thus, the permeability is transversely isotropic with the isotropy axis in the vertical direction of gravity. We also allow the micro permeability to be horizontally isotropic, but the permeability ratios of vertical to horizontal are different in the macro- and micro-phases. The effect of the difference of ratios is examined in detail.

scholarly journals Horizontally isotropic bidispersive thermal convection

2018 ◽  
Vol 474 (2213) ◽  
pp. 20180018 ◽  
Author(s):  
B. Straughan
Keyword(s):  
Thermal Conductivity ◽  
Nonlinear Stability ◽  
Thermal Convection ◽  
Vertical Direction ◽  
Interaction Coefficient ◽  
Linear Instability ◽  
Solid Skeleton ◽  
Stationary Convection ◽  
Symmetric Tensors ◽  
Stability Threshold

A bidispersive porous material is one which has usual pores but additionally contains a system of micro pores due to cracks or fissures in the solid skeleton. We present general equations for thermal convection in a bidispersive porous medium when the permeabilities, interaction coefficient and thermal conductivity are anisotropic but symmetric tensors. In this case, we show exchange of stabilities holds and fluid movement will commence via stationary convection, and additionally we show the global nonlinear stability threshold is the same as the linear instability one. Attention is then focused on the case where the interaction coefficient and thermal conductivity are isotropic, and the permeability is isotropic in the horizontal directions, although the permeability in the vertical direction is different. The nonlinear stability threshold is calculated in this case and numerical results are presented and discussed in detail.

scholarly journals Anisotropic bidispersive convection

2019 ◽  
Vol 475 (2227) ◽  
pp. 20190206 ◽  
Author(s):  
B. Straughan
Keyword(s):  
Porous Medium ◽  
Stability Analysis ◽  
Rayleigh Number ◽  
Nonlinear Stability ◽  
Thermal Convection ◽  
Linear Instability ◽  
Solid Skeleton ◽  
Nonlinear Stability Analysis ◽  
Cell Structures ◽  
Linear Instability Analysis

This paper investigates thermal convection in an anisotropic bidisperse porous medium. A bidisperse porous medium is one which possesses the usual pores, but in addition, there are cracks or fissures in the solid skeleton and these give rise to a second porosity known as micro porosity. The novelty of this paper is that the macro permeability and the micro permeability are each diagonal tensors but the three components in the vertical and in the horizontal directions may be distinct in both the macro and micro phases. Thus, there are six independent permeability coefficients. A linear instability analysis is presented and a fully nonlinear stability analysis is inferred. Several Rayleigh number and wavenumber calculations are presented and it is found that novel cell structures are predicted which are not present in the single porosity case.

scholarly journals Nonlinear stability of two-layer shallow water flows with a free surface

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190594
Author(s):  
Francisco de Melo Viríssimo ◽  
Paul A. Milewski
Keyword(s):  
Free Surface ◽  
Initial Data ◽  
Nonlinear Stability ◽  
Mathematical Proof ◽  
Passive Layer ◽  
Physical Parameters ◽  
Immiscible Fluid ◽  
Dimensional System ◽  
The Cauchy Problem ◽  
The Difference

The problem of two layers of immiscible fluid, bordered above by an unbounded layer of passive fluid and below by a flat bed, is formulated and discussed. The resulting equations are given by a first-order, four-dimensional system of PDEs of mixed-type. The relevant physical parameters in the problem are presented and used to write the equations in a non-dimensional form. The conservation laws for the problem, which are known to be only six, are explicitly written and discussed in both non-Boussinesq and Boussinesq cases. Both dynamics and nonlinear stability of the Cauchy problem are discussed, with focus on the case where the upper unbounded passive layer has zero density, also called the free surface case. We prove that the stability of a solution depends only on two ‘baroclinic’ parameters (the shear and the difference of layer thickness, the former being the most important one) and give a precise criterion for the system to be well-posed. It is also numerically shown that the system is nonlinearly unstable, as hyperbolic initial data evolves into the elliptic region before the formation of shocks. We also discuss the use of simple waves as a tool to bound solutions and preventing a hyperbolic initial data to become elliptic and use this idea to give a mathematical proof for the nonlinear instability.

Fluid flow over and through a regular bundle of rigid fibres

2016 ◽  
Vol 792 ◽  
pp. 5-35 ◽  
Author(s):  
Giuseppe A. Zampogna ◽  
Alessandro Bottaro
Keyword(s):  
Porous Medium ◽  
Fluid Flow ◽  
Flow Field ◽  
Stokes Equations ◽  
Transversely Isotropic ◽  
Navier Stokes ◽  
Leading Order ◽  
Macroscopic Scale ◽  
Navier Stokes Equations ◽  
Homogenized Model

The interaction between a fluid flow and a transversely isotropic porous medium is described. A homogenized model is used to treat the flow field in the porous region, and different interface conditions, needed to match solutions at the boundary between the pure fluid and the porous regions, are evaluated. Two problems in different flow regimes (laminar and turbulent) are considered to validate the system, which includes inertia in the leading-order equations for the permeability tensor through a Oseen approximation. The components of the permeability, which characterize microscopically the porous medium and determine the flow field at the macroscopic scale, are reasonably well estimated by the theory, both in the laminar and the turbulent case. This is demonstrated by comparing the model’s results to both experimental measurements and direct numerical simulations of the Navier–Stokes equations which resolve the flow also through the pores of the medium.

Normal moveout from dipping reflectors in anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 268-284 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Weak Anisotropy ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
The Difference

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

scholarly journals DOUBLE POROSITY MODELS FOR LIQUID FILTRATION IN INCOMPRESSIBLE POROELASTIC MEDIA

2010 ◽  
Vol 20 (04) ◽  
pp. 635-659 ◽  
Author(s):  
ANVARBEK MEIRMANOV
Keyword(s):  
Pore Space ◽  
Double Porosity ◽  
Homogenization Theory ◽  
Darcy Law ◽  
Solid Skeleton ◽  
Fractured Reservoir ◽  
Liquid Filtration ◽  
Liquid Domain ◽  
Poroelastic Media ◽  
The Common

Double porosity models for the liquid filtration in a naturally fractured reservoir is derived from the homogenization theory. The governing equations on the microscopic level consist of the stationary Stokes system for an incompressible viscous fluid, occupying a crack-pore space (liquid domain), and stationary Lame equations for an incompressible elastic solid skeleton, coupled with the corresponding boundary conditions on the common boundary "solid skeleton-liquid domain". We assume that the liquid domain is a union of two independent systems of cracks (fissures) and pores, and that the dimensionless size δ of pores depends on the dimensionless size ε of cracks: δ = εr with r > 1. The rigorous justification is fulfilled for homogenization procedure as the dimensionless size of the cracks tends to zero, while the solid body is geometrically periodic. As the result we derive the well-known Biot–Terzaghi system of liquid filtration in poroelastic media, which consists of the usual Darcy law for the liquid in cracks coupled with anisotropic Lame's equation for the common displacements in the solid skeleton and in the liquid in pores and a continuity equation for the velocity of a mixture. The proofs are based on the method of reiterated homogenization, suggested by Allaire and Briane. As a consequence of the main result we derive the double porosity model for the filtration of the incompressible liquid in an absolutely rigid body.

3D PETROPHYSICS FOR HAWE: CASE STUDIES

2021 ◽  
Author(s):  
Alexander Kolomytsev ◽  
◽  
Yulia Pronyaeva Pronyaeva ◽  
Keyword(s):  
Case Studies ◽  
Seismic Data ◽  
Reservoir Characterization ◽  
Vertical Direction ◽  
Geological Structure ◽  
Current Layer ◽  
Radial Model ◽  
The Difference ◽  
Geological Concept ◽  
Property Changes

Most conventional log interpretation technics use the radial model, which was developed for vertical wells and work well in them. But applying this model to horizontal wells can result in false conclusions. The reasons for this are property changes in vertical direction and different depth of investigation (DOI) of logging tools. DOI area probably can include a response from different layers with different properties. All of this complicates petrophysical modeling. The 3D approach for high angle well evaluation (HAWE) is forward modeling in 3D. For this modeling, it is necessary to identify the geological concept near the horizontal well section using multiscale data. The accuracy of modeling depends on the details of the accepted geological model based on the data of borehole images, logs, geosteering inversion, and seismic data. 3D modeling can be applied to improve the accuracy of reservoir characterization, well placement, and completion. The radial model is often useless for HAWE because LWD tools have different DOI and the invasion zone was not formed. But the difference between volumetric and azimuthal measurements is important for comprehensive interpretation because various formations have different properties in vertical directions. Resistivity tools have the biggest DOI. It is important to understand and be able to determine the reason for changes in log response: a change in the properties of the current layer or approaching the layers with other properties. For this, it is necessary to know the distance to the boundaries of formations with various properties and, therefore, to understand the geological structure of the discovered deposits, and such information on the scale of well logs can be obtained either by modeling or by using extra deep resistivity inversion (mapping). The largest amount of multidisciplinary information is needed for modeling purposes - from images and logs to mapping and seismic data. Case studies include successful examples from Western Siberia clastic formations. In frame of the cases, different tasks have been solved: developed geological concept, updated petrophysical properties for STOIIP and completion, and provided solutions during geosteering. Multiscale modeling, which includes seismic, geosteering mapping data, LWD, and imagers, has been used for all cases.

Sign in / Sign up

Export Citation Format

Share Document