Читать книгу Acoustic and Vibrational Enhanced Oil Recovery - George V. Chilingar - Страница 28

Vibration excitation in oil reservoirs is best studied in cases of power load application directly to the land surface. It is implemented for a directional vibroseismic action on shallow oil accumulations and for a solution of seismic exploration tasks. Of greatest interest are issues of coordinating the sources with the medium from a point of view of the best transformation of the source power in the power of irradiated waves. Reviewed have been the models of spherical sources and sources distributed over the surface of elastic half-space [25]. A solution of the problem of creating efficient method of forming highly directional radiation suggests that the use of acoustic antenna theory constructed for ideal uniform acoustic medium in a case of stratified media results in significant errors. Taking into account vertical nonuniformity of the medium plays an important role for obtaining acceptable practical results. Distribution studies of energy supply arriving in multilayer elastic half-space from a harmonic surface power loading established that, in the multilayer half-space, may exist the frequency range wherein emerge reflected waves with the opposite direction of phase and group velocities.

An analysis of energy flow lines, which may be strongly involuted up to generating the areas where energy is circulating on closed trajectories, indicated the following. In the vicinity of resonance frequencies, the power flow of these internal circular flows may substantially exceed the flow coming from the source. This indicates a possibility of a vibration source energy accumulation with an unlimited amplitude increase in certain areas of semi-infinite laminated space. In these studies, the actual rocks have been modeled by an elastic medium. The solutions of dynamic problems for elastoporous media are known where, applying the Frenkel-Bio equations was conducted an analysis of the wave field excitation by a load varying in time according to the harmonic law. At that, the excited field energy distribution by the wave type has been estimated.

Similar tasks related to the half-space force excitement had a practical value for selecting efficient vibroseismic action regimes on shallow oil reservoirs using powerful surficial vibration sources developing force up to 10⁶H.

The energy transfer processes from well to a reservoir are associated with substantial difficulties. These difficulties are associated with well geometry and paucity of a well as an irradiator compared with the excited wave length. If a source is creating vibration in the well fluid, then the field generated in the surrounding reservoir is equivalent to the field of an infinitely long cylinder on the surface of which one of possible is excited vibration modes. In this case, a solution of the wave equation is

where and are Hankel functions of the first and second kind, m = k_yR_c ; R_c is the well radius; k_r, k_y, and k_z are wave numbers.

The first term in this equation corresponds to a wave converging to the cylinder axis, and the second term corresponds to the wave expanding from the cylinder. For the expanding running waves, the k_r number must be a real value, i.e., the condition must be observed. If k_z ˃ k_r, then k_r is a purely imaginary value, and in such a case, the field sharply exponentially declining with distancing from the cylinder surface. No irradiation occurs in such a case. The length of a flexural wave in the axial direction turns out smaller than a sound wave in the encircling medium regardless of another wave number k_y = m/R_e, which corresponds to the circular system of nodal lines. If we introduce the angle θ = arcsin (ξ/k), k = ωR_e/c as the incidence angle or reflection angle of cylindrical wave, normal or modulated by front, then at incidence angles smaller than θ₁ = arcsin(c/ c_p), the energy enters the reservoir, and at the angle larger than θ₁, it returns in the well for forming head-waves in the liquid. Within the angle range between a critical value θ₁ to the second critical value θ₂ = arcsin(c/c_s), the energy flow into the medium is caused only by irradiation of the shear waves. At high frequencies, such that we have cos θ ≈ 1, and the waves irradiated by the cylinder spread perpendicular to its surface. With the k decline, the θ angle is increasing, and at k = k_z, the wave is spreading only tangentially to the cylinder’s surface. As soon as the axial flexural wave length becomes shorter than the sound wave length in the enclosing medium, appropriately k becomes smaller than k_z, and the cylinder-well is not radiating in the enclosing medium at all. Irradiation in a case is studied in great detail, the results are practically applied in geophysical acoustic studies in wells and at high-frequency thermoacoustic action on the reservoir’s bottomhole zone.

Energy interrelations at wave irradiation from wells by an axisymmetric source have been studied in [10]. Connections have been identified between field energy parameters and well’s acoustic impedance. Also studied has been the nature of frequency and amplitude distribution on the irradiation surface onto the energy transmission into the massif. At the fixed spatial frequency ξ of a source u₀ harmonic amplitude offset distribution, equations have been derived for per-unit impedance of the well-loading massif. Radial resonance frequencies of a ring liquid layer between the emitter body and the casing and antiresonance frequencies also have been determined. The energy release into massif at the resonance frequency turned out to be most efficient.

Based on a solution of the dispersion equation for Rayleigh waves in the reservoir, a case has been reviewed when the irradiation was totally disappeared. The well acoustic impedance in this case becomes equal to zero and the well becomes absolutely “soft”. The total energy flow through the well’s walls at that becomes equal to zero, although in a near-surface layer quite significant densities are created of the kinetic and potential energy. At θ ˃ θ_R, the acoustic energy may enter the reservoir from the well only in a case of normal fading of waves in the liquid. This moment corresponds with a case of an infinitely long cylinder surface-wave mode of which is excited by a low-frequency action within the well Therefore, at the wave radiation from a well in the reservoir, there are critical frequencies, below which the irradiation in the conventional sense is absent.

Wave excitement form well considering reservoir elastoporosity was studied by Rakhmanov et al. (1985) [19]. The boundary conditions at the surface of a liquid-filled cylindrical well were assigned in the form of a tension in the solid phase and pressure in the liquid generated by moving at a constant speed load charge as follows:

Restricted solutions of the Bio equations satisfying the irradiation conditions allows one to derive equations for calculating offsets and tensions in the solid phase as well as the pressure in a fluid-porous medium.

White [23] reviewed the absorption mechanism of normal waver spreading in the well fluid caused by fluid pulsation flow through the permeable walls of an uncased well. His results were appropriate for low frequency, when the tube-wave length is large compared with the well diameter. If to evaluate the contribution of such fading by a normal wave in its radiation at subcritical frequencies, then with the frequency decline, approximately from 100 Hz and to very low frequencies, the fraction of vibration energy irradiated into the massif monotonously grows to some value determined by the media permeability. Regardless of permeability, at frequency above 100 Hz, this phenomenon make practically no contribution to the irradiation of vibration energy. These results apply to uncased wells and did not take into account the mechanism of wave energy reradiation through a well fluid.

In order to increase the efficiency of energy introduction into the reservoir at low frequency, it is necessary to find a possibility to increase substantially the irradiation from a well into the reservoir by changing loading parameters and optimally utilize action of resonance properties of the well and reservoir systems. The existence of resonance regimes of vibration present in a well, which are associated with parameters of the enclosing medium and emerging in a pre-critical frequency area, is experimentally confirmed. For instance, if to run in a fluid-filled well a receiver of sound vibrations and to measure the energy spectrum of the noise, then in a fluid-saturated reservoir the resonance frequency may be identified.

The resonance excitation in a well may be achieved at the regime of high-frequency radial resonance of the fluid layer as well as a lengthwise resonance of the fluid column at low frequency. The radial resonance is determined by the well radius and emerges at high frequency on the order of dozen kilohertz and higher. Thus, coordination of the operation regime of the vibration source and the reservoir achieved, so that practically the entire power of the vibration source is transferred in the reservoir. By varying the frequency and load distribution on the surface of the vibration source, it is possible to control field energy structure in the reservoir. However, practical utilization of the radial resonance is substantially complicated for the following reasons. The frequency even of the first radial resonances at the existing well radiuses are too high both for favorable manifestation of elastic vibration action mechanisms on the reservoir and for the frequency coordination with the reservoir excitation resonance regimes. Besides, high-frequency elastic waves experience a strong fading in the reservoir. As the motions of porous medium at such resonances are radial, the generation of high-pressure amplitudes in wells is restricted by the acceptable radial displacement of the casing and cement column.

At lengthwise resonance, the vibration frequency of the fluid column in a well is defined mostly by the distance between reflecting surfaces in the well. As the lower reflecting surface is usually taken the dib hole and as the upper, the fluid and gas contact next to wellhead. At long distances between these boundaries, the vibration resonance frequency may reach the value on the order of 1 Hz and lower. Nevertheless, efficient utilization of lengthwise resonances at low frequencies is associated with certain difficulties. For creating the resonance lengthwise vibration of the entire fluid column in a well, it is necessary to know the fluid level in the well and the phase velocity of a tube wave propagation in it. It is practically impossible to support a constant fluid level in well in most technological operations. On taking the reflection properties from the lower boundary (the dib hole), they are clearly low because the wave resistance in the cement and in the surrounding reservoir are only slightly different from the wave resistance of a fluid. That is a reason why, due to a significant length of the fluid column, plenty of energy is expended on wave fading in the fluid itself and uselessly irradiated into the nonproductive zones above the reservoir.

That is why various submersible resonance devices are used, in which a set frequency vibration source is rigidly tied with acoustic reflecting filters [17]. These devices include a generator and a resonator enclosed in the body. At that, the fluid column filling the cavity of a resonator chamber forms a vibration contour in which a standing wave is generated. The resonance excitation of vibrations is achieved if the reflector length L and generator operating frequency ν_o are tied as follows:

where c is the wave propagation phase velocity.

Using of similar devices with fixed parameters in practice is not sufficiently efficient because the elastic wave phase velocity in a well fluid depends on the reservoir elastic properties and immersion depth and besides may change from one well to the next [4]. If the wave running in a productive interval of a well with perforation holes in the casing, then its phase velocity also depends on the reservoir porosity, permeability, compressibility, and well fluid viscosity [23]. All this results in a substantial instability of the existing devices resonance regime because these devices have been designed according to simplified rules without considering the aforementioned factors. Besides, these devices cannot be too long, and this in restricting the frequency regime of vibration excitation. As a result, resonance lengthwise vibrations of the fluid within the device are created, and at irradiation into the well, the vibration energy is passed along the column of the well fluid beyond the productive interval limits and is irradiated into overlying and underlying unproductive rocks. A possibility of achieving a well fluid effective lengthwise resonance was studied using “sliding” reflecting filters installed within the productive interval.

For instance, a well’s resonance excitation regime may be achieved using hollow downhole reflector-filters filled with gas and fit on the production tubing. Such design does not require a hermetic contact with casing walls. Before running into a well, positions of the reflecting filters along the well length and relative the vibration generator may be changed. Distances between the reflectors are selected so that most energy from the generator was concentrated in the productive interval. The field excited in the enclosing rocks is close in its character to a finite length pulsating cylinder.

At the operation of the downhole vibration source within the limits of productive interval, a vibrating energy may be additionally irradiated which creates the emergence of pulsating fluid flows in the well’s perforation holes. As the perforation channels are narrow, the fluid motion velocity in then is high compared with the fluid flows within the well proper. Within some volume small compared with the wave length is alternatively created the excess or shortage of a given medium’s matter; its surface is permeable for a given medium and is a sovereign type radiator. That is why every perforation channel of radius r_k and length l_k at low frequency may be considered a point monopole exciting a spherically symmetric wave in the enclosing rocks. This wave is mostly defined not by geometric size of the perforation channel but by the size of generator’s created outflowing fluid flow.

The vibration energy irradiated by an individual perforation channel, in considering of its geometrical parameters, may be presented as follows:

(2.34)

where V is the velocity amplitude of fluid motion in a perforation channel, is the channel’s cross-sectioned area, and ρ is the fluid density.

The irradiation efficiency of each individual perforation channel is low mostly due to a low resistance to the irradiation at low frequency. However, if a system is available of closely spaced monopole irradiators and a condition is satisfied where L is the distance between irradiators and λ is the wave length in the medium, then these irradiators interact between themselves. At that, every individual irradiator is working in the pressure field of all other irradiators, which is equivalent to increased resistance to irradiation. As a result, active power increases and passive power, decreases.

At synphase operation of a group of similar irradiators, the total irradiation power is equal to the power of every one of them multiplied by squared number of the irradiators [16]. If for a perforation interval H_p th condition is satisfied, then accrual of the estimated irradiation power in the well should be expected to be as follows:

(2.35)

here, n is the perforation density (number of holes per unit length of the perforation interval).

For studying additional energy of individual channel, it is necessary to provide a sufficiently efficient transition mechanism of pulsating pressure in wells into vibration velocity fluid motion in channels. This process is accompanied by a high extent of tube wave fading in the well fluid caused by a low absorption on the casing walls and in the fluid as well as by the elevated acoustic irradiation of the perforation channels. Most of the casing wave energy is absorbed through acoustic irradiation in channels in the reservoir perforation interval.

A similar mechanism is operating in a well with a fluid and perforation channels as an oscillator with lumped parameters. A low basic frequency of vibration in such as oscillator is reached either by the participation in vibration of two media with drastically different properties (for instance, gas bubbles, and liquid) or by the realization of mechanism present in a Helmholtz resonator (where at a small actual mass of the vibrating medium it is possible to create a large amount of effective mass).

When a vibration source is operating in a well, fluid flows are generated in perforation holes. Due to a narrow channels’ diameter, the motion flow in them is high compared with the fluid’s velocity in the well. The kinetic energy

(2.36)

is concentrated in perforation channels despite that the actual fluid mass in the well is much greater than the fluid mass in the channels. Whereas the elastic energy is concentrated within the well. Therefore, as the potential and kinetic energy are localized in different media (channel medium and in the well medium), the well may be considered an analog of Helmholtz resonator. On coordinating the generator operating frequency with eigen frequency of such resonator effective transmission of the generator energy in to kinetic energy of a fluid in the perforation channels occurs. The perforation channels represent monopole sources having purely active resistance at resonance frequency. The resonance vibration frequency of such an oscillator is [8]:

(2.37)

where S_k and l_k are, respectively, the area and length of a channel, respectively; ρ and β are the density and compressibility of a vibrating fluid, respectively; and Ω is the volume of fluid amount possessing potential elastic energy.

One can now estimate the amount of fluid volume possessing a store of the potential elastic energy.

For this case, The total area of channel openings in the well length H_p is equal to Using Equation (2.37), the resonance frequency for a well with perforation channels is equal to

(2.38)

At the resonance frequency, irradiated on average the reservoir power is

(2.39)

and this is the reason why the effective absorption of the generator energy which is expended for irradiation through the perforation channels is determined by the density of perforation holes. As Equation (2.38) indicates, this value also defines the generator’s resonance frequency. Considering Equations (2.36) and (2.39), we will evaluate fading factor of the casing wave (caused by the acoustic irradiation through perforation channels) as

(2.40)

Switching to the casing wave fading factor over the well length H_p, one obtains

where δ(E) = E_k/E₀ is a relative loss of the generator energy to the irradiation over the well length; here, E₀ is kinetic energy on the generator wall. Taking Equations (2.40) and (2.38) into account one obtains an equation connecting n, H_p, and δ(E) values:

(2.41)

Of special interest is the experimental verification of these equations through modeling field conditions. The experimental testing unit was a horizontal model of the reservoir with a well and perforation channels.

The reservoir model had a cylindrical core holder filled up with a cemented porous medium. The core holder, through a steel disk with the perforation hole, was joining a special stand for testing hydraulic vibrators. The space between the steel disk and the porous medium was filled up with a cement stone imitating the well cement shell. The aforementioned special stand imitated a well with hydraulic generator and was represented by a segment of casing with flanges. It was connected with a high-pressure and high performance water-saturated reservoir. Water run-off through the hydraulic generator, a static pressure of the stand and the model was controlled by valves and the pressure was shown on a manometer. The hydraulic generator operating parameters have been determined on a gauge; elastic vibration parameters within the porous medium at the model input and at the model output have been observed using gauges and probes. The gauges have been connected with a normalizer and then the signals have been supplied to the remembering oscillograph and spectrum analyzer. The porous medium has been placed in a core holder that is 0.31-m long and 0.14 m in diameter. The porous medium has been a mixture of quartz sand and has epoxy resin. The perforation channel was 0.003 m in diameter and 0.01-m long. Between the stand and the model has been established a pivot crane intended for switching the model off the stand and measuring background noise and if necessary, for controlling the pressure fluctuations level.

The experimental methodology consisted of measuring fluctuation of vibration parameters at the input, output and within the model at a stable operation of the hydraulic generator. The measurement results have enabled evaluation of the vibration weakening amplitude at different points of the model and at different frequencies. Correlation diagrams of relative pressure fluctuation levels (registered at various points in the model) vs. frequency of the hydraulic generator were obtained. The experimental results have indicated that at transmission of vibrations into the reservoir their weakening occurred increasing with the frequency growth. At the same time, behavior of the curves has been notably different from the anticipated linear correlation of vibrations’ weakening with frequency observed for a saturated porous medium under regular conditions at low frequencies. This unexpected result has been recorded in the maximum deviation area. The recorded pressure fluctuation level in a perforation channel was lower than the relative pressure pulsation level at distant points of the porous model.

The obtained result confirmed the theoretical analysis. The behavior of experimental diagrams may be explained as follows. Within the studied frequency range defined by the parameters of the used experimental equipment the conditions have been determined with optimum conversion of the generator pressure fluctuations into the kinetic energy of liquid within the perforation channel. An elevated efficiency has been observed of the perforation channel monopole irradiation due to an increase of the fluid volumetric flow rate. A relative level of pressure fluctuations in the reservoir has turned out higher than expected; pressure fluctuations in the perforation channel have converted in fluid rate fluctuations. At that, the relative level of pressure fluctuations in the channel has been declining.

In conclusion, an estimate of vibrations’ resonance frequency according to experimental conditions. Parameters used in Equation (2.37) are as follows:

 – Cross-section area of a perforation channel, Sk = 7.065 × 10−6 m2;

 – The perforation channel length, lk = 0.01 m;

 – Fluid density, ρ = 1,000 kg/m3;

 – Fluid compressibility β = 0.455 × 10−9 Pa−1.

Substituting these data, one obtains ω = 134 Hz. Comparison of this frequency with the one measured in the experiments are in good agreement. Results obtained indicate a real possibility of increasing efficiency by introducing vibratory energy from a well into the reservoir. The reason is that the real field values of parameters τ_k and l_k in the well with perforation density of 10 to 30 holes/meter result in the excitation of resonance frequencies within a 50- to 300-Hz range.