(2.20)
Each of these displacements may be directed by integrodifferential equation of Equation (2.11) type but with different multiplier before the terms including time derivatives due to the differences in the high frequency sound velocity limit for these two wave types. The integral terms creep cores also differ (displacement creep function and triaxial creep function). However, their time correlation is identical if identical processes lead to local relaxation. For this reason, the methods designed for scalar equations may be applied with every scalar component of every polarization displacement vector simply by scaling the derived solutions.
2.2 Effect on the Wave Spread in the Oil Accumulations by the Geologic-Geophysical Conditions
As an indicator of efficiency at a certain frequency may serve encompassment radius within which are maintained certain interrelations between the threshold values of vibration parameters—vibratory displacements ξ and vibratory accelerations
where ρC is the wave resistance or acoustic impedance of the medium.
Numerical modeling was conducted by Sherifulling et al. [26] in order to determine the vibrations’ space-energy distribution. This enabled the computation of the energy picture of the wave distribution field for the assigned vibration frequency and of the vibratory accelerations and offsets field in the reservoir accounting for petro-physical properties of the top and base of the reservoir. Modeling was conducted using a method of the wave spreading statistical testing in the plane dissecting the reservoir with the plane-parallel boundaries [20]. The source of harmonic waves was distributed along the circumference of a well with the radius Rc with the center in the origin. The top and base of a reservoir having thickness H was positioned parallel to the X axis. At calculating by the statistical testing method, the spreading of the low-frequency harmonic waves is modeled using acoustic “quanta” flying out from the source in a random direction. Acoustic energy is assigned to each “quantum”, the energy equal to surficial density of a cylindric pulsator with the pressure amplitude P0:
This quantum “is traveling” in the reservoir along a ray trajectory defined by the indicatrix of reflection from the top and base. The quantum’s energy is continuously declining to the local patterns of absorption in the reservoir rocks, wave divergence and angular parameters of reflection coefficients on the porous boundaries. When the quantum is flowing out of the studied portion of the reservoir, a transition is implemented to playing out the subsequent history of the acoustic quantum.
For the evaluation of acoustic properties of the saturated porous medium and description of two-dimensional processes of wave transfiguration on the boundaries, one can use the approach proposed by Sharifullin based on Bio’s linear theory. The offsets of solid and liquid phases (appropriately, u and V) may be determined from Bio’s equations:
Here, λ1, λ2, Q, and R are Bio’s elastic moduli for the medium; b = m2μ/k is Bio’s resistance coefficient; μ is the fluid’s dynamic viscosity; k the permeability of porous medium; m is porosity; ρ11, ρ12, and ρ22 are Bio’s density parameters expressed by rock matrix density ρ1 and fluid density ρ2 as follows:
where ρ12 is the so-called “reduced density”.
The Bio’s moduli may be expressed directly by the measured parameters: rock matrix triaxial compression modulus K, shear modulus G for a “dry” porous medium, and compressibility factors β1 and β2.
On assuming that in plane OXZ on the separation boundary of two saturated porous media (the boundary coinciding with the OX axis), first or second kind compressional wave or shear wave incidence angle to OZ axis is α. The OZ axis is directed into the second medium. In the general case, six types of waves are generated on the boundary, three are reflected and three are refracted.
This approach was implemented for computing phase velocities, free space attenuation factors, angular wave transformation factors, and energy reflection parameters at reservoir’s boundaries. These parameters are needed for a “raffle” of the acoustic “quanta” trajectories.
The same way was modeled a great number of the acoustic quanta trajectories flowing from the source in all possible directions. A field intensity was computed in 2.5 to 5 thousand of the reservoir points using the energy flux vector:
where A is amplitude of wave potential in the medium and Cm is the phase velocity of m-wave.
The exit angle of each “quantum” trajectory ray of 0° to 180° was “raffled” using pseudorandom number sensor for the numbers uniformly distributed in the zero to one interval. The field near every of 5,000 observation points was averaged on the surface of a cell with the side H/50, where H is the reservoir thickness. Thus, phase displacement at the quanta superposition in a given point was taken into account by that separate cells have been provided for accumulating two orthogonal components of the wave vector at every point of the reservoir. Modeling was being performed in the mode of continuous information accumulation with releasing results over a certain number of histories— “acoustic quanta exits” from the source. The result at each release was normalized per a number of hits in every reservoir cell. For calculating the hit numbers, a special two-dimensional massif was included in the memory.
Computations have been conducted of the spatial-energy distribution of the well vibration source field in the oil-saturated productive reservoirs restricted by a low permeability water-saturated top and base. At constructing of the reservoir and enclosing rocks’ design model have been used real field-geologic property data of the reservoirs, oils, reservoir water and enclosing rocks. The reservoir thickness varied in the range of 5 to 30 m. As source parameters have been taken average parameters for each type of boundary of porosity, permeability, solid phase and saturating fluid density, rock triaxial compression, and shear moduli. The top and base rocks have been assumed practically impermeable and saturated with reservoir water having density of 1.188·103 kg/m3 and viscosity of 0.05 cP or gas with density of 70 kg/m3 and viscosity of 0.015 cP. In Figure 2.1, relative fields of vibration intensity for various frequencies in an oil-saturated reservoir, 20-m thick, are shown. The obtained spatial distributions of vibrational energy flow density from the vibration source in the reservoir are wave pictures with interference maxima and minima illustrating a substantial influence by the top and base depending on frequency.