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2.3 Wave Spreading From the Vibrating Surface of the Reservoir Matrix Into the Saturated Medium
ОглавлениеCurrently, 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, V0 is 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:
where v is kinematic viscosity factor. Equation (2.22) is similar to the diffusion and heat-conductivity equations. Boundary conditions of the problem are
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]:
where
(2.25)
Stokes solved this problem for a case of h → ∞:
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
Here, the value may be called the “vibration penetration depth”.
At viscosity factor v = (1.007–1.519)·10−2 cm2/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 2.3 includes 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 sh2c + ch2c = 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 mth harmonic, 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).
If the fracture surface is making random periodic vibrations in two mutually perpendicular directions in two mutually perpendicular directions lying on its surface, the liquid velocity in each of these directions are determined independent from one another. But, the aforementioned qualitative patterns remain valid. In particular, if periods of mutually perpendicular vibrations coincide, then the liquid particles’ trajectories at a sufficient distance from the fracture surface approach ellipses, and the size of these ellipses is exponentially declining. In a case when periods are substantially different, the liquid motion at distancing from the fracture surface approaches rectilinear harmonic vibrations coinciding in the direction and frequency with longer period vibrations. In a manner of speaking, the liquid layer has properties of a high frequency filter.
The quoted results are comparable with the solution of a problem of fading the agitations caused by a spherical fractured surface in the process of its harmonic vibrations along some direction in an incompressible liquid.
For instance, we will review for a case of Reynolds’ small numbers a spherical surface within which 95% of the total energy loss in a viscous incompressible liquid is dispersed (Figure 2.4a). Assume that R0 is radius of vibration penetration and r0 is radius of the spherical surface. Then, the value δ = R0 − r0 may be considered the appropriate depth of vibrations’ penetration into the liquid. Then, correlation δ = R0 − r0 may be treated as the appropriate depth of vibrations’ penetration into the liquid. For the correlation δ/r0 vs. the parameter we obtain a solid line diagram in Figure 2.4b. At that, is the same value with the dimension of length as in Equation (2.27). The same diagram within the considered limits of β variation may be approximated as follows:
(dashed line in Figure 2.3). Upon reducing by r0, this equality is exactly equal to (2.27).
Thus, there is practically total coincidence of the results for a case of the flat and spherical fracture surfaces’ vibrations. The study result is also close if the agitation area from the spherical surface is defined not by the energy dissipation, but by leveling of the pressure field or by fading of the velocity field as it has been done for a case of a flat fracture. The agitation zone in the case of a spherical surface’s incremental advance in a viscous liquid under the problem’s conditions is by an order of magnitude greater in size than in the case of its vibration. This conclusion is in the qualitative agreement with correlations of Equations (2.27) and (2.28), which indicate an expansion of the agitation zone with the decline in the vibration frequency.
In conclusion, one may note that under the assumption of incompressibility of liquids, a case of lateral fracture surface vibrations is trivial as the liquid vibrates in this direction together with the fracture surface. If compressibility is present, then the process is described by a wave equation which includes an addend accounting for the dissipation of the vibration energy.
Figure 2.4 Vibration in a liquid volume at progressive vibrations of the fracture’s spherical surface.
Figure 2.5 Schematic advance of a mixture of a wetting and nonwetting liquids in a unit pore volume in a field of elastic vibrations. Ar and Aa are the relative and absolute displacement amplitudes of the nonwetting liquid; εr and εa are the relative and absolutes dislocations.
In the case of pore volume filled up with a wetting liquid with floating droplets of nonwetting liquid, the vibration penetration depth may be estimated treating the mixture equivalent to some liquid with viscosity greater than that of the wetting liquid with droplets of a nonwetting liquid. In order to estimate the absolute and relative amplitudes of displacements, force, and energy expense needed for providing vibrations in a reservoir, we will review a unit pore volume restricted by the rock matrix. Let us assume that a selected volume is filled up with a wetting liquid with floating droplets of a non-moisturizing liquid. To this volume are applied harmonic vibrations with a frequency ω with the maximum displacement amplitude A (Figure 2.5). Walls of the pore volume and the wetting liquid are vibrating in phase, coherently with the displacement amplitude:
where t is the vibration time.
The simplest case is with nonwetting liquid droplets being balls of equal diameter d = 2r suspended in the wetting liquid and the volume concentration C within the study volume of no greater than 5%. It such a case, the distances between the droplets are greater than 2 or 3 their diameters and their mutual influence may be disregarded. On assuming further that the Reynolds’ acoustic numbers are much smaller than a unit:
here, ν is the kinematic viscosity of the wetting liquid.
A solution for the nonwetting liquid displacement amplitude relative to the surrounding wetting liquid is [6]:
and the absolute amplitude of the displacement is
The force F needed for the occurrence of harmonic vibrations of such unit volume is:
where
ρ = ρs − ρ′ is the effective density of liquids’ mixture at vibration;
ρs = ρe(1 − C) + Cρn = ρc[1 + C(Δ − 1)] is the static mixture density;
is the effective vibration damping factor;
Δ = ρn/ρc is the density ratio of the nonwetting to the wetting liquids.
Due to a greater mobility of the nonwetting liquid relative to wetting one, the effective density of liquid mixture at vibration lower than the density in a dormant state by a positive value ρ′. Power that needs to be expended for the vibration support of a unit pore volume is N = KA2/2. For the most practically important cases . The equations for the displacement amplitudes are:
These equations show that non-wetting liquid’s relative displacement maximum amplitudes Ar are greater, the greater the difference between its density and the density of wetting phase, the non-wetting liquid’s vibration is leading in phase the vibration of the wetting liquid. At absolute motion, of the non-wetting liquid, the absolute amplitudes of displacement exceed those of wetting one.
As the equivalent viscosity at certain conditions may exceed by one order of magnitude (and even greater) the wetting liquid’s viscosity, the vibration penetration depth for a liquids’ mixture may be significantly greater. Various equations have been proposed for calculating effective dynamic viscosity factor. For example, at 30% content of nonwetting phase, the effective dynamic viscosity factor is almost three times greater than the similar viscosity factor for a wetting liquid. Appropriately, the mixture’s kinematic viscosity is 1.4 times greater than for the wetting phase. Thus, the penetration depth is 1.4 times greater than in wetting phase.
Some qualitative patterns (regularities) in vibration’s penetration into a unconsolidated reservoir for the simplest case of circular vibrations follow from the solution of motions by a material particle touching a flat surface. If the surface vibration acceleration
where ω is the frequency, A is the radius of circular vibration trajectory, f is the slipping friction factor, and g is the gravity acceleration, then the particle is moving together with the vibrating surface.
At Aω2 ˃ fg, the particle acceleration W, velocity U, and trajectory radius r at steady absolute motion are defined by the following equations:
(2.29)
i.e., the trajectory velocity and radius rapidly decline with growing frequency.
At sufficiently large surface vibration frequency, the particle remains practically immobile in space. This pattern is manifested to some extent also at vibratory motions of unconsolidated reservoir layers. There is some range of the surface vibration accelerations, within which vibration penetrates the reservoir in a certain most efficiently.
A basic assumption is explaining the motion of one relative to another of the elementary layers composing the reservoir is that the dry friction factor f between the layers is considered a function of the normal overburden pressure, i.e., the weight of the overlying layer. This is supported by a series of direct and indirect experiments by Yamshchikov [27]. Designating the overlying layer weight per unit area as G, the relationship between the dry friction force F (“resistance force to a relative layers’ displacement”) and G and f values is as follows:
(2.30)
where f(G), according to our assumption, is some increasing G function. Then, at a growth of vibrating surface vibration acceleration Aω2, the motion looks as follows.
At Aω2=(Aω2)1 ˂ gfl, where fl is the value of friction factor at the upper boundary of the layer, all layer’s particles move together with the vibrating surface as one solid body. After the acceleration exceeds, the Aω2 value, relative sliding begins of the upper layer’s part on the lower one. It continues moving together with the vibrating surface. At further increase of the acceleration, the relative motion expands over the underlying layers and at
reaches the lower boundary of the reservoir (subscript m belongs with values related to this boundary). At Aω2> (Aω2)2, the entire reservoir is sliding relative the vibrating surface (sliding variously at different levels). The particles’ absolute velocity at this layer is decreasing from the lower boundary to the upper one. Velocity moduli at the points with certain G values are found as
If the particles’ material is uniform in the reservoir, then the value u,
(where x is absolute coordinate, Gm is the total weight of the larger per vibrating surface unit area, and h is the reservoir thickness), is a relative coordinate counted measured vertically down from the reservoir upper boundary. If the correlation of friction factor f with G is linear, then Equation (2.31) becomes
(2.32)
where ξ = fm − f0/f0>0 is the relative difference of friction factors at the lower and upper reservoir boundaries. At these boundaries
(2.33)
The former equation is equivalent to Equation for a particle velocity if the friction factor f is replaced with its f0 value for the upper boundary of the reservoir. The other patterns of vibration’s penetration for the subject model of an unconsolidated reservoir are also similar to those established for the particle. Maximum vibration acceleration of the reservoir particles
is reached at its lower boundary at the same value of the vibration acceleration of the surface Aω2= (Aω2)2. At the further increase of Aω2 increase, the reservoir particles accelerations remain unchanged, whereas the velocity and the trajectory radius at a constant vibration amplitude decline with the increase in frequency. Thus, with increasing surface acceleration vibrations Aω2˂ (Aω2)1 = gf0, the vibrations completely penetrate the entire reservoir thickness, and at Aω2˃ (Aω2)1, they do only partially. The “penetrating acceleration” with its increasing value Aω2 stabilize at the gf0 level for the upper reservoir’s boundary, and “the velocity and amplitude penetration” declines with increasing frequency, in proportion to ω and ω2.
The maximum amplitude of relative migrations close to the amplitude of absolute surface vibrations occurs at a sufficiently large value of acceleration Aω2, when particles of an unconsolidated reservoir remain practically immobile in space.
A case of rectilinear extensional vibration of the surface displays the same patterns of vibration penetration as the considered case of circular vibrations. Solution of the problem here is more complex than in the case of circular vibration, especially for the second model, because the particle motion occurs with two long or instantaneous stops in each vibration period [5].
A different situation occurs with the vibration penetration process in individual reservoir layers in a case of transverse surface vibration of one of the layers. Experimental data indicate exponential nature of particles’ vibration amplitude decline with certain distance from the vibrating surface of one of the layers. Let us denote v a pulsating component of the reservoir particle velocity and introduce a function
where R is the velocity recovery factor at a strike, and V0 is some characteristic velocity value. This function will satisfy equation of heat conductivity type equation, i.e., will play the role of quasi-temperature [9]. Figure 2.6 shows diagrams of the function Ux/U0 at various values of the recovery factor R. Fading of the velocity of reservoir particles has exponential nature, which agrees with the above mentioned experimental results.
Figure 2.6 Velocity distribution in unconsolidated reservoir at transverse vibrations of its confining surface.
Certain specificity is typical of the vibration penetration dynamics in the cement grouts. The cement grouts are a multi-component medium composed of large and small filling aggregate, cohesive material, and water especially selected by the composition. In the process of mixture preparation, air unavoidably is getting into it. The amount of air in the mixture is relatively small (and is changing in the preparation process). Physicomechanical properties of the mixture and of the solidified cement substantially depend on its contents. As result, there is a rather complex rheological property of the cement mixture. Most important among them are resistance of a dry friction type and also the viscous friction and elasticity (the elasticity is mostly defined by the presence of air component). The vibration causes transformation of the mixture rheological parameters and properties, in particular, its pseudo-fluidization, which provides for the efficient compaction necessary for obtaining a cement with high strength parameters. For this reason, the cement mixture is modeled in theoretical studies as a continuous medium with elastic, plastic, and viscous properties.
