George V. Chilingar - Acoustic and Vibrational Enhanced Oil Recovery

Figure 2.2 Distribution of relative vibration intensity averaged over the reservoir thickness vs. the distance to the vibration source R and frequency: 1, 12 Hz; 2, 50 Hz; 3, 120 Hz; 4, 1,000 Hz; and 5, 10,000 Hz.

Phenomena similar to the described ones are also observed at different reservoir thickness values. For example, in a 5-m-thick reservoir a “strong” normal vibration mode near the frequency of 480 Hz is present. The derived vibration intensity fields for assigned frequencies enable computation of the fields of vibrational displacements and vibrational accelerations.

Currently, the problem of wave propagation from the vibrating surface of the reservoir matrix into the various media has not yet been satisfactorily solved.

The motion of a viscous incompressible liquid filling half-space over the flat surface performing extension vibrations has been the earliest considered by Stokes.

A corresponding solution of the linear problem is easily generalized for a case of periodic vibrations [11].

We will review first a case of a horizontal fracture in the reservoir when it is performing straight-linear incremental harmonic vibrations in the same plane:

(2.21)

where η is the vibration velocity of the horizontal surface, V 0is its amplitude, and ω is the vibration frequency.

For the velocity of uncompressible liquid over a vibrating surface V = V ( x, t ) from the Navier-Stokes equation:

(2.22)

where v is kinematic viscosity factor. Equation (2.22)is similar to the diffusion and heat-conductivity equations. Boundary conditions of the problem are

(2.23)

where h is the facture width.

The first equality is the condition of a liquid sticking to the surface. The stationary solution of Equation (2.22)satisfactory to the conditions Equation (2.23)is [2, 3]:

(2.24)

where

(2.25)

Stokes solved this problem for a case of h → ∞:

(2.26)

It follows from Equation (2.26)that the facture’s surface involves the liquid in a vibratory motion with the same frequency ω and with the amplitude rapidly (exponentially) declining with distancing from the surface. The thickness δ of the liquid’s layer involved in vibration by the fracture surface due to liquid’s viscosity may be described by a distance at which the velocity amplitude is equal to 5% of its value on the fracture’s surface. This s a value determined, according to Equation (2.26), as

(2.27)

Here, the value may be called the “vibration penetration depth”.

At viscosity factor v = (1.007–1.519)·10 −2cm 2/s (which corresponds to water temperature change from 20° to 50°) and the vibrations frequency 2.5 to 5 Hz, the penetration depth is about 1 mm, which, is quite commensurate with the fracture width.

Figure 2.3includes distribution diagrams of nondimensional vibratory velocity amplitudes of liquid vs. the width drawn corresponding with Equations (2.24)and (2.26). At the fracture width h ˂ δ , the velocity distribution with fracture height may significantly differ from the free surface ( h → ∞ ). If, however, h ˃ δ (i.e., βh ˃ 3), then this difference is small. In such a case, Equations (2.24)and (2.26)provide close results. For this reason, liquid velocity in a δ -wide fracture may be determined with an accuracy sufficient for practical calculations from a simple Stokes formula Equation (2.26)considering the liquid outside of this layer immobile. If, however, h ˂ δ, (βh ˂ 0.5), the calculations should be conducted using a more complex formula Equation (2.24). Liquid layers for which practically move together with the fracture surface (as a solid body).

Figure 2.3 Vibration velocity distribution in a fracture (solid curves) and over a free surface (dashed line).

In a finite thickness layer above the harmonically vibrating fracture surface, for which βh ≥ c ≈ 3.69 (where c is the root of equation sh 2 c + ch 2 c = 400), and the penetration depth determined as previously is somewhat greater than the value δ = 3/ β , but does not exceed the value c / β which it assumes at βh = c . In layers where βh = c, the 20-fold decline in the velocity amplitude is not reached.

Due to linearity of the problem, the above results are easily generalized for cases of rectilinear periodic arbitrary vibrations and periodic vibrations in two mutually perpendicular directions on the fracture surface.

At periodic rectilinear motion of the fracture surface (represented by a sum of harmonics with the frequency kω , where k 1, 2, …), the penetration depth for some m thharmonic, according to Equation (2.27), will be times lower than for the first one. For this reason, at sufficient distance from the fracture surface, liquid vibration is determined by the first harmonic; in other words, it approaches harmonic character regardless of the pattern of periodic vibrations (of course, on condition that h ≥ δ and the harmonic amplitudes do not grow with the increase of their numerical order).