The surface conditions for viscous displacement in a homogeneous porous medium

1979 ◽  
Vol 57 (10) ◽  
pp. 1738-1745 ◽  
Author(s):  
T. J. T. Spanos
Keyword(s):  
Boundary Layer ◽  
Porous Medium ◽  
Continuum Theory ◽  
Immiscible Displacement ◽  
Surface Conditions ◽  
Fluid Displacement ◽  
Two Fluids ◽  
Mathematical Abstraction ◽  
Homogeneous Porous Medium ◽  
Macroscopic Shape

The viscous surface conditions between two fluids are considered for fluid displacement in a homogeneous porous medium. The term viscous surface is defined here as a mathematical abstraction used to approximate the macroscopic shape of the boundary layer between two fixed saturations of displacing fluid by continuum theory. In the limit as the saturations approach each other, one then obtains a convergence of the viscous surface to a constant saturation contour. This yields a mathematical description of immiscible displacement in porous media which contains a theory of viscous fingering.

The equations of immiscible viscous displacement in a porous medium

1981 ◽  
Vol 59 (1) ◽  
pp. 45-56 ◽  
Author(s):  
T. J. T. Spanos
Keyword(s):  
Porous Medium ◽  
Statistical Theory ◽  
Boundary Conditions ◽  
Initial Conditions ◽  
Continuum Theory ◽  
Immiscible Displacement ◽  
Macroscopic Scale ◽  
Inertial Terms ◽  
Inhomogeneous Porous Medium ◽  
Relative Motions

A statistical theory for the construction of the equations of viscous displacement in a porous medium is considered. This yields a continuum theory for immiscible displacement which can be applied to either a homogeneous or inhomogeneous porous medium. The relative motions of the fluid are considered in terms of the motion of surfaces of constant saturation which are smoothed surfaces at the macroscopic scale considered. The boundary conditions and initial conditions at the injection boundary are considered as well as the boundary conditions and breakthrough conditions at the recovery boundary and the side boundary conditions. The inertial terms are included in the equations and shown to be of importance in describing these initial conditions and the breakthrough conditions.

A general stability analysis for immiscible viscous displacement in a porous medium

1981 ◽  
Vol 59 (5) ◽  
pp. 678-687 ◽  
Author(s):  
T. J. T. Spanos
Keyword(s):  
Porous Medium ◽  
Stability Analysis ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Immiscible Displacement ◽  
Stability Criteria ◽  
Macroscopic Scale ◽  
Two Fluids ◽  
General Stability ◽  
Inhomogeneous Porous Medium

A perturbation of an immiscible displacement process causes relative motion of the two fluids involved. At the macroscopic scale such relative motions are considered to propagate throughout the porous medium in the form of fluid waves. A description of these waves is given on surfaces of constant saturation in a similar fashion to the description of a surface wave propagating on the interface between two fluids. In the porous medium, however, the wave propagation is not restricted to the surface of constant saturation and as a result one obtains a wave equation that is both dissipative and diffusive.A stability analysis is also considered for the immiscible displacement process. Here, a characteristic time for instability to occur can be calculated when the inertial terms are included in the equations of motion. Also a generalization of the wave equations and stability criteria are considered for an inhomogeneous porous medium.

Modelling Of Transport And Capture Of Particles In A Porous Medium

2019 ◽  
pp. 56-60
Keyword(s):  
Porous Medium ◽  
Asymptotic Solution ◽  
Retention Mechanism ◽  
Inverse Filtering ◽  
Underground Structures ◽  
Loose Soil ◽  
Monodisperse Suspension ◽  
Homogeneous Porous Medium ◽  
Model Equations ◽  
Pass Through

The study of the transport and capture of particles moving in a fluid flow in a porous medium is an important problem of underground hydromechanics, which occurs when strengthening loose soil and creating watertight partitions for building tunnels and underground structures. A one-dimensional mathematical model of long-term deep filtration of a monodisperse suspension in a homogeneous porous medium with a dimensional particle retention mechanism is considered. It is assumed that the particles freely pass through large pores and get stuck at the inlet of small pores whose diameter is smaller than the particle size. The model takes into account the change in the permeability of the porous medium and the permissible flow through the pores with increasing concentration of retained particles. A new spatial variable obtained by a special coordinate transformation in model equations is small at any time at each point of the porous medium. A global asymptotic solution of the model equations is constructed by the method of series expansion in a small parameter. The asymptotics found is everywhere close to a numerical solution. Global asymptotic solution can be used to solve the inverse filtering problem and when planning laboratory experiments.

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