Models of filtration flow

Complex and irregular character of porous space structure does not allow studying flow of fluids by direct solution of the equations of viscous liquid motion for each porous channel or for each fracture. However, it is known, that with the increase in number of separate microscopic movements that constitute macroscopic filtration movement, we can discover statistical dependencies that describe the whole fluid movement, though these dependencies do not hold for one or several porous channels.

So, in hydrodynamics, the movement of separate molecules is not studied, we consider average thermodynamic parameters of a liquid as a continuous medium.

In mechanics of continuous medium the flow of liquids and gases is described by three conservation laws: 1) conservation of mass, 2) conservation of kinetic momentum or of impulse, 3) conservation of energy. When we study filtration flow in underground hydromechanics we use only first two laws. The change of temperature of a fluid is neglected due to small velocities of flow and significant heat exchange with a skeleton of rocks. Owing to considerable surface of contact with rock the temperature of fluid is not changed. Thus, the flow process is supposed isothermal. In some cases when detailed studying near bottom-hole zones, (e.g. thermal methods of an intensification of extraction of fluids) temperature variation is also taken into account.

The presence of the periods of change of flow parameters in time (e.g. wells’ start-up and wells’ shut-down, operation when stimulating inflow) is typical for the processes that occur during oil-and-gas formations development. Such processes are called non-steady state (non-stationary). The models which describe processes with parameters that do not change on time, are called stationary (steady-state).

Modeling filtration flow in relation to spatial change of parameters can be carried out in one-dimensional statement i.e. when parameters are the function of only one variable (flow along a straight line or a curve); in two-dimensional statement (flat flow), and in three-dimensional statement (flow in space).

Fluids vary on a degree of compressibility. So natural gas considerably changes its volume with pressure change, while water and oil are practically incompressible in a wide range of pressure variation (approximately up to 20 MPa). At high pressures the fluids have elastic properties. In connection with the specified factors there are individual models of compressible, incompressible and elastic medium.

In the area of contact of fluids when displacing one fluid by another or when escaping of one fluid from another, each microscopic volume contains two or more fluids, occupying separate precisely distinct volumes (bubbles of gas in a liquid, drops or films in gas) and interacting at the boundary surfaces. Such systems are called heterogeneous or multiphase systems in contrast to multicomponent mixes (natural gas, oil) in which interaction occurs at a molecular level and it is impossible to indicate a separating boundary. In hydrodynamics such multicomponent mixes are called single-phase or homogeneous.

During movement, fluids undergo various deformations (compression, torsion, a stretching, etc.) caused by variation of loading (friction of the neighbor volumes, external forces). The value of this loading per unit-area is called a stress. The relation between deformation or speed of change of deformation and a pressure is a rheological relationship or rheological law. Most frequently with reference to liquids, Newton’s law is applied to describe influence of tangent component of pressure on shift deformation. Rather frequently the movement of fluids does not obey the given law. For example, when rock oil starts moving, certain non-zero pressure must be applied to break off colloid structures formed by rock water. Such medium is called non-Newtonian, and appropriate model is called a model of nonNewtonian flow.

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