A constitutive equation and generalized Gassmann modulus for multimineral porous media

Geophysics ◽  
2005 ◽  
Vol 70 (2) ◽  
pp. N17-N26 ◽  
Author(s):  
José M. Carcione ◽  
Hans B. Helle ◽  
Juan E. Santos ◽  
Claudia L. Ravazzoli
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Strain Energy ◽  
Saturated Porous Medium ◽  
Low Frequency ◽  
P Wave ◽  
Stress Strain Relation ◽  
P Wave Velocity ◽  
Two Phases ◽  
The Time Domain

We derive the time-domain stress-strain relation for a porous medium composed of n − 1 solid frames and a saturating fluid. The relation holds for nonuniform porosity and can be used for numerical simulation of wave propagation. The strain-energy density can be expressed in such a way that the two phases (solid and fluid) can be mathematically equivalent. From this simplified expression of strain energy, we analogize two-, three-, and n-phase porous media and obtain the corresponding coefficients (stiffnesses). Moreover, we obtain an approximation for the generalized Gassmann modulus. The Gassmann modulus is the bulk modulus of a saturated porous medium whose matrix (frame) is homogeneous. That is, the medium consists of two homogeneous constituents: a mineral composing the frame and a fluid. Gassmann's modulus is obtained at the low-frequency limit of Biot's theory of poroelasticity. Here, we assume that all constituents move in phase, a condition similar to the dynamic compatibility condition used by Biot, by which the P-wave velocity is equal to Gassmann's velocity at all frequencies. Our results are compared with those of the Berryman-Milton (BM) model, which provides an exact generalization of Gassmann's modulus to the three-phase case. The model is then compared to the wet-rock moduli obtained by static finite-element simulations on digitized images of microstructure and is used to fit experimental data for shaly sandstones. Finally, an example of a multimineral rock (n > 3) saturated with different fluids is given.

Iterative P-wave velocity inversion using Markov chain Monte Carlo method

2020 ◽  
Author(s):  
Hyunggu Jun ◽  
Hyeong-Tae Jou ◽  
Han-Joon Kim ◽  
Sang Hoon Lee
Keyword(s):  
Wave Velocity ◽  
Seismic Data ◽  
Velocity Structure ◽  
Low Frequency ◽  
P Wave ◽  
Velocity Analysis ◽  
Seismic Section ◽  
Traveltime Tomography ◽  
P Wave Velocity ◽  
Frequency Components

<p>Imaging the subsurface structure through seismic data needs various information and one of the most important information is the subsurface P-wave velocity. The P-wave velocity structure mainly influences on the location of the reflectors during the subsurface imaging, thus many algorithms has been developed to invert the accurate P-wave velocity such as conventional velocity analysis, traveltime tomography, migration velocity analysis (MVA) and full waveform inversion (FWI). Among those methods, conventional velocity analysis and MVA can be widely applied to the seismic data but generate the velocity with low resolution. On the other hands, the traveltime tomography and FWI can invert relatively accurate velocity structure, but they essentially need long offset seismic data containing sufficiently low frequency components. Recently, the stochastic method such as Markov chain Monte Carlo (McMC) inversion was applied to invert the accurate P-wave velocity with the seismic data without long offset or low frequency components. This method uses global optimization instead of local optimization and poststack seismic data instead of prestack seismic data. Therefore, it can avoid the problem of the local minima and limitation of the offset. However, the accuracy of the poststack seismic section directly affects the McMC inversion result. In this study, we tried to overcome the dependency of the McMC inversion on the poststack seismic section and iterative workflow was applied to the McMC inversion to invert the accurate P-wave velocity from the simple background velocity and inaccurate poststack seismic section. The numerical test showed that the suggested method could successfully invert the subsurface P-wave velocity.</p>

Stability of Bioconvection in Suspensions of Gyrotactic Microorganisms in Porous Media

2002 ◽  
Author(s):  
A. V. Kuznetsov ◽  
A. A. Avramenko
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Saturated Porous Medium ◽  
Critical Value ◽  
Numerical Results ◽  
Velocity Fluctuations ◽  
Gyrotactic Microorganisms ◽  
Upward Direction ◽  
Upward Velocity ◽  
Fluid Saturated Porous Medium

In this paper, a model of bioconvection in a suspension of gyrotactic motile microorganisms in a fluid saturated porous medium is suggested. The microorganisms considered in this paper are heavier than water and gyrotactic behavior results in their swimming towards the regions of most rapid downflow. Because of that, the regions of downflow become denser than the regions of upflow. Buoyancy increases the upward velocity in the regions of upflow and downward velocity in the regions of downflow, thus enhancing the velocity fluctuations. The experiments performed by Kessler (1986) and the numerical results of Kuznetsov and Jiang (2001) indicate that if the permeability of porous medium is sufficiently small it will prevent the development of convection instability. However, for practical purposes, in order to maximize the flux of the cells in the upward direction it is desirable to have the permeability of the porous medium as high as possible. The aim of this paper is to investigate the value of critical permeability. If permeability is smaller than this critical value bioconvection does not occur and microorganisms simply swim in the upward direction.

scholarly journals Transient Fluid Motions in Saturated Porous Media

1957 ◽  
Vol 10 (1) ◽  
pp. 43 ◽  
Author(s):  
JR Philip
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Heat Conduction ◽  
Exact Solutions ◽  
Kinematic Viscosity ◽  
Saturated Porous Medium ◽  
Steady Motion ◽  
Potential Gradient ◽  
Saturated Porous Media ◽  
Approximate Analysis

The transition from rest to steady motion on the sudden application of a potential gradient to the fluid contained in a saturated porous medium is investigated. An approximate analysis gives the result that the time of the effective establishment of the steady motion is proportional to the permeability and inversely proportional to the kinematic viscosity. Two exact solutions (one of them new) for simple cases suggest that the approximate analysis is remarkably accurate. An analogy between this problem and one in heat conduction makes the relevant results in that field immediately applicable here.

scholarly journals Experimental Study on Thermal Damage and Energy Evolution of Sandstone after High Temperature Treatment

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Rong-rong Zhang ◽  
Lai-wang Jing ◽  
Qin-yong Ma
Keyword(s):  
Strain Energy ◽  
Thermal Damage ◽  
Peak Stress ◽  
Temperature Treatment ◽  
P Wave ◽  
High Temperature Treatment ◽  
Energy Evolution ◽  
P Wave Velocity ◽  
Total Input ◽  
Evolution Characteristics

Thermal damage and energy evolution characteristics in process of impact failure of sandstone after high temperature treatment were studied by split Hopkinson pressure bar (SHPB) system. The ultrasonic P-wave velocity, density, porosity, peak stress, ET/E0, thermal damage, fracture, and energy evolution characteristics of sandstone with temperature during the experimental process were explored. Results show that, with the increase of temperature, the ultrasonic P-wave velocity and density decrease, while the porosity increases. It is found that the peak stress and ET/E0 decrease with the increase of temperature, and the decreasing trend is fitted with the simple cubic equation. Above 600°C, dynamic peak stress and ET/E0 decrease rapidly. The thermal damage of rock increases with the increase of temperature, which is in accordance with the logistic curve model. The thresholds of damage strain energy release rate are 200°C and 800°C in this research. Its total input strain energy decreases with the increase of processing temperature and decreases sharply when the temperature is over 600°C. The variation of total input strain energy has small change at the range from 400°C to 600°C.

Non-Darcy Mixed Convection With Thermal Dispersion in a Saturated Porous Medium

2009 ◽  
Vol 132 (1) ◽  
Author(s):  
V. V. Sobha ◽  
R. Y. Vasudeva ◽  
K. Ramakrishna ◽  
K. Hema Latha
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Porous Media ◽  
Nusselt Number ◽  
Mixed Convection ◽  
Vertical Plate ◽  
Saturated Porous Medium ◽  
Thermal Dispersion ◽  
Convection In Porous Media ◽  
Infinite Vertical Plate

Thermal dispersion due to local flows is significant in heat transfer with forced convection in porous media. The effects of parametrized melting (M), thermal dispersion (D), inertia (F), and mixed convection (Ra/Pe) on the velocity distribution, temperature, and Nusselt number on non-Darcy, mixed convective heat transfer from an infinite vertical plate embedded in a saturated porous medium are examined. It is observed that the Nusselt number decreases with increase in melting parameter and increases with increase in thermal dispersion.

Modeling of magmatic channel formation in thin lithosphere areas

2020 ◽  
Author(s):  
Yuri Perepechko ◽  
Konstantin Sorokin ◽  
Georgiy Vasilyev
Keyword(s):  
Mathematical Model ◽  
Porous Medium ◽  
Porous Media ◽  
Saturated Porous Medium ◽  
High Stress ◽  
Marked Change ◽  
Porous Matrix ◽  
Small Scale ◽  
Magma Intrusion ◽  
Magma Flow

<p>The aim of the research is to construct a mathematical model of the formation of a fracture system in magma intrusion in the permeable zones of the lithosphere and on this basis to study the formation of magmatic channels in the lithosphere and crust. The lithosphere substrate is modeled by a saturated porous medium in which the processes of small-scale destruction in the mantle magma intrusion lead to the formation of faults and, consequently, to a magmatic channel. Destruction and occurrence of micro-fracture fields can be associated with both magma flow and external seismic effect leading to the rock breaking. The process of small-scale destruction is described within the framework of the dynamics of the elastoplastic fracture-porous medium and causes variations in the rheological properties of the lithosphere substrate. A feature of this process is the destruction substrate in the compression zone represented by a narrow area with a sharply changing concentration of micro-fractures. The micro-fracture accumulation provides the conversion of the broken area into a macro-fissure. The elastoplastic porous matrix in the destruction zone contains both broken and intact substrate, the relative content of which is determined by relaxation of deformations, the speed of which depends on stress and yield stress point according to the power law. The obtained mathematical model provides investigation of currents in fractured-porous media and their effect on the small-scale destruction. Based on the TVD-Runge Kutta method numerical simulation of the compressible fluid infiltration into the fracture-porous permeable channel has shown that stresses in the compression domain can reach stress limits of breaking and result in fracture formation. Change in relaxation time does not result in a marked change in stress fields. The concentration of maximum stresses is observed in the channel center leading to an increase in its fracture porosity. The computational results show the appearance of high stress values in the compression domain in the process of a liquid phase injection, for instance, magma, into a low-permeable fracture-porous layer. The introduction of the destruction criterion will help to associate the occurrence of such regions to the local breaking of the porous matrix. Thus, the proposed micro-fracture generation mechanism can be used to describe the formation of fracture or channels in micro-fracture porous media. Work is done on state assignment of IGM SB RAS with partial support from the Russian Foundation for Basic Research, grants No. 16-29-15131, 19-05-00788.</p>

Forced Convection With Laminar Pulsating Counterflow in a Saturated Porous Channel

2009 ◽  
Vol 131 (10) ◽  
Author(s):  
D. A. Nield ◽  
A. V. Kuznetsov
Keyword(s):  
Porous Medium ◽  
Nusselt Number ◽  
Forced Convection ◽  
Saturated Porous Medium ◽  
Applied Pressure ◽  
Low Frequency ◽  
Constant Heat Flux ◽  
Perturbation Approach ◽  
Dimensionless Frequency ◽  
Flux Boundary Conditions

An analytical solution is obtained for forced convection in a parallel-plate channel occupied by a layered saturated porous medium with counterflow produced by pulsating pressure gradients. The case of asymmetrical constant heat-flux boundary conditions is considered, and the Brinkman model is employed for the porous medium. A perturbation approach is used to obtain analytical expressions for the velocity, temperature distribution, and transient Nusselt number for convection produced by an applied pressure gradient that fluctuates with small amplitude harmonically in time about a nonzero mean. It is shown that the fluctuating part of the Nusselt number alters in magnitude and phase as the dimensionless frequency increases. The magnitude increases from zero, goes through a peak, and then decreases to zero. The height of the peak depends on the values of various parameters. The phase (relative to that of the steady component) decreases as the frequency increases. The phase angle at very low frequency can be π/2 or −π/2 depending on the degree of asymmetry of the heating and the values of other parameters.

Natural Convection Experiments in a Stratified Liquid-Saturated Porous Medium

1986 ◽  
Vol 108 (3) ◽  
pp. 660-666 ◽  
Author(s):  
D. C. Reda
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Porous Media ◽  
Natural Convection ◽  
Rayleigh Number ◽  
Saturated Porous Medium ◽  
High Permeability ◽  
Transfer Rates ◽  
Variable Porosity ◽  
Radial Extent

Natural convection heat transfer from a constant-flux cylinder, immersed vertically through a stratified (two-layer) liquid-saturated porous medium, was investigated experimentally. Measured radial temperature profiles and heat transfer rates agreed well with numerical predictions based on the work of Hickox and Gartling. The 1:6 permeability-ratio interface existing between the two layers was found to effectively trap buoyancy-driven fluid motion within the high-permeability region, beneath the interface. Within this high-permeability region, Nusselt number versus Rayleigh number data were found to correlate with previously measured results, obtained for the same basic geometry, but with a fully permeable upper-surface hydrodynamic boundary condition. In both cases, the vertical and radial extent of the region under study were large compared to the radius of the heat source. Combined results indicate that, for a given Rayleigh number in the Darcy-flow regime, heat transfer rates from cylinders immersed vertically in uniform liquid-saturated porous media of large vertical and radial extent potentially approach limiting values. Variable-porosity effects which occur in unconsolidated porous media adjacent to solid boundaries were investigated numerically for cases where the particle-to-heater diameter ratio was small (≈ 10−2). Results showed variable-porosity effects to have a negligible influence on the thermal field adjacent to such boundaries under conditions of Darcy flow.

scholarly journals Oscillatory motion of an electrically conducting viscoelastic fluid over a stretching sheet in a saturated porous medium with suction/blowing

2006 ◽  
Vol 2006 ◽  
pp. 1-14 ◽  
Author(s):  
K. Rajagopal ◽  
P. H. Veena ◽  
V. K. Pravin
Keyword(s):  
Magnetic Field ◽  
Porous Medium ◽  
Porous Media ◽  
Skin Friction ◽  
Viscoelastic Fluid ◽  
Stretching Sheet ◽  
Saturated Porous Medium ◽  
Oscillatory Motion ◽  
Power Series Method ◽  
Viscoelastic Parameter

The effect of an oscillatory motion of a viscoelastic fluid over an infinite stretching sheet through porous media in the presence of magnetic field with applied suction has been studied. The surface absorbs the fluid in a porous medium in the presence of magnetic field and the velocity oscillates depending on the stretching rate(b). Analytical expressions for the velocity and the coefficient of skin friction have been studied, first by the perturbation method and then by power series method. The effect of viscoelastic parameterk1, porous parameterk2, magnetic parameter Mn, and the vertical distancexin the presence of suction/blowing on the velocity and the flow characteristics are discussed. The velocity of the viscoelastic fluid is found to decrease in the presence of magnetic field and porous media, as compared to the study of viscous fluid. It is also found that the effect of unsteadiness in the wall velocity and skin friction are found to be appreciable in the presence of suction/blowing parameter.

Sign in / Sign up

Export Citation Format

Share Document