G. G. Tsypkin

Instabilities of uniform filtration flows with phase transition

New mechanisms of instability are described for vertical flows with phase transition through horizontally extended two-dimensional regions of a porous medium. A plane surface of phase transition becomes unstable at an infinitely large wavenumber and at zero wavenumber. In the latter case, the unstable flow undergoes reversible subcritical bifurcations leading to the development of secondary flows (which may not be horizontally uniform). The evolution of subcritical modes near the instability threshold is governed by the Kolmogorov-Petrovskii-Piskunov equation. Two examples of flow through a porous medium are considered. One is the unstable flow across a water-bearing layer above a layer that carries a vapor-air mixture under isothermal conditions in the presence of capillary forces at the phase transition interface. The other is the vertical flow with phase transition in a high-temperature geothermal reservoir consisting of two high-permeability regions separated by a low-permeability stratum.

Mathematical model for dissociation of gas hydrates coexisting with gas in strata

1. Fields of natural gas hydrates discovered in various world regions have turned out to be so significant that they are considered as potential sources of natural gas. Presently, the development of technologies for gas extraction from natural deposits of gas hydrates has become a principal problem [1]. In [2, 3], simple models were proposed for hydrate formation and decomposition in natural conditions, which allow for basic governing physical mechanisms. Further investigations demonstrated that the consideration of new physical processes leads to principle changes in the mathematical structure of the solution to the problem of gashydrate decomposition in natural reservoirs [4–6].

A Phenomenological Model of the Increase in Solute Concentration in Ground Water Due to Evaporation

We present a new phenomenological model of the evaporation of ground water containing a polluting material in the dissolved form. Only the one-dimensional case is treated. It is assumed that there exists a sharp evaporation front separating between the dry and the water-saturated soil. The water-saturated soil is assumed to occupy a semi-infinite domain x > X(t), where x is a vertical coordinate directed downward and X(t) is a position of the evaporation front. The mathematical description is based on four linear diffusion equations coupled through four boundary conditions, one of which is nonlinear, on the free moving evaporation front. We use a similarity solution of the governing equations and analyze it qualitatively showing that the solute concentration increases in the upward vertical direction and reaches its maximum on the evaporation front. The dependence of the solute concentration at the evaporation front and of the velocity of the front on the initial solute concentration and the temperature of the ground surface are computed. It is shown that for not high values of the initial solute concentration that are below the concentration value cd at which a deposition of the pollutant sets in, the solute concentration on the evaporation front can reach values that are above the deposition value cd. These results point to a possible mechanism of pollutant deposition in ground water caused by the evaporation.

Stability of the evaporation and condensation surfaces in a porous medium

The stability of a phase transition interface which separates the soil regions saturated with water and humid air, respectively, is investigated. The humid air region contacting with the atmosphere is assumed to be located above the water-saturated region. Water flows through the porous medium in the lower region, while diffuse vapor transfer is implemented in the upper region. Two cases corresponding to water evaporation and vapor condensation are considered. In the first case water flows out from the porous aquifer, evaporates, and comes out into the atmosphere. In the second case, during condensation, the atmospheric moisture saturates soil. The problem is solved in the steady-state case. The investigation of linear stability carried out by means of the normal mode method shows that the evaporation surface can be unstable in both nonwettable and wettable soils in the presence of the capillary pressure gradient. Depending on the parameters, the condensation surface can be unstable also in the neutral medium.

A mathematical model of carbon dioxide flooding with hydrate formation

The injection of carbon dioxide into a reservoir that contains methane and water in a free state is investigated. A mathematical model of this process is proposed that suggests the formation of the CO2 hydrate on the surface of the phase transition separating regions of methane and carbon dioxide. The conditions on the interface are derived, and an asymptotic solution of the problem is found. Critical diagrams are obtained that define parameter ranges in which there is full or partial transition of gaseous carbon dioxide to a hydrate state.

Regimes of haline convection during the evaporation of groundwater containing a dissolved admixture

The problem of formation of salt concentration profile in high-permeability soil duringwater evaporation and solution upflow is considered. The numerical experiments performed showed that the salt concentration profile may be either stable or unstable. As instability develops, there arises natural haline convection whose different regimes are described and analyzed. If the evaporation intensity is moderate, in soil the curvilinear upward or circulatory flow that fills the entire layer is established. The intense evaporation leads to the formation of a small-scale structure of salt “fingers”. Boundaries between regimes are determined.

Rayleigh-Taylor instability of an interface in a nonwettable porous medium

The diffusion of vapor through the roof of an underground structure located beneath an aquifer is considered. In the process of evaporation, an interface between the upper water-saturated layer and the lower layer containing an air-vapor mixture is formed. A mathematical model of the evaporation process is proposed and a solution of the steady-state problem is found. It is shown that in the presence of capillary forces in the case of a nonwettable medium the solution is not unique. Using the normal mode method, it is shown that Rayleigh-Taylor instability of the interface can develop in the nonwettable porous medium. It is found that there are two scenarios of loss of stability corresponding to the occurrence of the most unstable wavenumber at zero and at infinity, respectively. It is shown that for zero wavenumber the stability limit is reached at the same time as the solution of the steady-state problem disappears.

Stability of a water-vapor phase transition interface in geothermal systems

The problem of normal stability of a phase transition interface for vertical flows in a geothermal reservoir with the water layer lying above the vapor layer is solved. The problem is formulated with account for convective energy transfer, which ensures applicability for arbitrary permeability values. Typical examples of variation of the phase transition interface stability parameters in response to changes in permeability are considered for various geothermal reservoirs.

Evolution of a Condensation Surface in a Porous Medium near the Instability Threshold

We consider the dynamics of a narrow band of weakly unstable and weakly nonlinear perturbations of a plane phase transition surface separating regions of soil saturated with water and with humid air; during transition to instability, the existing stable position of the phase transition surface is assumed to be sufficiently close to another phase transition surface that arises as a result of a turning point bifurcation. We show that such perturbations are described by a Kolmogorov–Petrovskii–Piskunov type equation.

Filtration-Flow Fragmentation in Medium with Capillary-Pressure Gradient

The deformation of a water-saturated region during filtration in a porous medium with variable capillary pressure is considered. The numerical calculations show that the capillary-pressure gradient directed along the liquid-gas interface leads to deformation of the region, which is eventually divided into two disconnected regions. It is assumed that the process under consideration describes one of the possible mechanisms of fragmentation of filtration flows.