Displacement of Oil from Porous Bed by the Oscillating Flow of Polymer Solution
Volodymyr G. Pogrebnyak^{1}, Iryna V. Perkun^{1}, Andriy V. Pogrebnyak^{2}
^{1}Department of Environmental Engineering, National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine
^{2}Faculty of Equipment and Technical Service, Kharkiv State University of Food Technology, Kharkiv, Ukraine
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Volodymyr G. Pogrebnyak, Iryna V. Perkun, Andriy V. Pogrebnyak. Displacement of Oil from Porous Bed by the Oscillating Flow of Polymer Solution. American Journal of Science, Engineering and Technology.Vol. 1, No. 2, 2016, pp. 53-57. doi: 10.11648/j.ajset.20160102.16
Received: November 1, 2016; Accepted: December 9, 2016; Published: January 12, 2017
Abstract: Aqueous solutions of polymer have been studied under the conditions of oscillating hydrodynamic field with stretching. The investigations have confirmed the opinions as to nonlinearity oscillating flow of polymer solutions according to the mechanism the basis for which is provided by the concepts of a strong strain action of the hydrodynamic field on coiled macromolecules. The effects causing the highly dissipative regime in oscillating flow of polymer solutions through porous media play an important role in optimization of the process of oil displacement of a porous bed using polymer solutions.
Keywords: Oscillating Flow, Solution of Polymer, Velocity Gradient, Oil Displacement, Porous Bed, Viscosity, Relaxation Time
1. Introduction
Experiments were conducted with the aim of elucidating the nature of the mechanism of an "anomalously" high degree displacement of oil from porous bed using the polymer solution. The most contradictory opinions as to the nonlinearity of filtration flow of polymer solutions are being stated; most researchers believe that the nonlinearity is caused by the adsorption of macromolecules onto the porous-medium walls [1-3]. There is an opinion that the deviations from Darcy’s law are due to the elastic-strain effects manifesting themselves in the case of flow of solutions of flexible-chain polymers in a porous medium [4-6]. Below, we will be dealing with experiments on checking the mechanism of an "anomalously" high degree displacement of oil from porous bed using the polymer solutions; the basis for this mechanism is provided by the concepts of a strong strain action of the hydrodynamic field on coiled macromolecules in filtration flow [7].
The mechanism of deviations from Darcy’s law lies in the self-oscillating regime of reversible processes of unrolling of macromolecules under the action of the longitudinal velocity gradients quasiregularly arising in the porous medium and in the influence of unrolling macromolecules at both the molecular and supermolecular levels on the structure of the filtration flow. The oscillation of the strains of macromolecules and dynamic supermolecular formations arising under the action of stretching flows results in the higher-than-average energy dissipation filtration and the flow nonlinearity [5]. The latter ensures the equalization of the front of advance and filling of inhomogeneous laminar beds with polymer fluid and hence a wider coverage of the bed and hence a high degree displacement of oil.
From the above mechanism of nonlinearity of the filtration of polymer solutions and from the theory of elastic-strain effects in the case of flow with stretching of polymer solutions [4-6], it follows that the higher degree displacement of oil from porous bed must be attained in oscillating injection of a polymer solution into it. To substantiate this prediction we studied the aqueous solutions of polyethylene oxide (PEO) under the conditions of oscillating hydrodynamic field with stretching.
2. Materials and Methods
In realizing the model regime of injection of a solution polymer into a porous medium (bed), the solution under study was forced through a flow chamber (module) under the action of a constant pressure difference (see Figure 1). This is an unusual kind of viscosimeter in which the hydroresistors are short capillaries.
In the stationary regime, flow through the hydroresistors was carried out by creation of a pressure difference at the viscosimeter inlet and outlet, whereas in the oscillating regime, it was carried out by superposition of a constant pressure difference and the pulsating pressure arising in the vessel between the capillaries due to the vibrations of the membrane of an electromagnetic vibrator. The amplitude of membrane vibrations was monitored by a special device. It was constant at different frequencies in each run of experiments and was selected such that the rate of flow (caused by the membrane vibrations) of the polymer solution through the capillaries was lower than the flow rate duet the constant pressure difference. The oscillating component of the flow velocity in the capillaries that was caused by the pressure pulsations in the inter capillary volume was smaller than the flow velocity in the capillary due to the constant pressure difference in the viscosimeter. The flow-rate velocity of the polymer solution through the capillaries was higher than the critical value [7]. The diameter of the capillaries was 0.2·10^{−3} m.
The hydrodynamic efficiency of the polymer solutions under study was evaluated by the value of the efficient viscosity η_{ef} which was determined by comparison of the flow-rate velocities of the solution under study and calibration viscous Newtonian fluids at the same pressure difference. The entire unit was placed in a thermostat, where the temperature of the solutions under study was monitored by an electronic heat controller.
Figure 3 shows the experimental cell represents a layer of quartz compacted in a hermetically sealed space is filled with oil between two glass plates having inlet and outlet pipes. The inlet pipe was connected either directly to a measuring vessel, into which the aqueous polymer solution had been poured in advance, or via a module with a vibrator. The measuring vessel is equipped with a system for recording the flow-rate of the solution injected into the model of natural bed. The solution is injected due to the excess air pressure produced in the measuring vessel; the pressure is monitored by a manometer. The structure of the experimental setup enables us to visualize the process of advance of the front of the polymer solution in the porous bed and to qualitatively evaluate the efficiency of different regimes of oil displacement from porous bed by the flow patterns.
PEO having the viscosity-average molecular weight of =4.5⋅10^{6} was used as a polymeric additive in water and characteristic of a viscosity 1,72 m^{3}/kg. The aqueous solutions of polymer had concentration C_{PEO}=0.001÷0.050%. The viscosity of the oil ranged from 10^{-2} to 5⋅10^{-2} Pa⋅sec.
3. Results and Their Discussion
The results of the experiment are presented in Figure 2a, where η_{ef} is the effective viscosity of the solution under the conditions of stationary out flow and ∆η_{ef} is the addition resulting under the oscillating action on the mainstream flow. The curves correspond to different velocities of mainstream flow through the capillaries. It is seen that the critical frequency of action ω_{cr′} on the stationary flow exists; beginning with this frequency, flow becomes more dissipative.
The quantity ω_{cr′} grows with average flow-rate velocity, whereas ∆η_{ef} decreases. With lower concentrations of PEO in the solution and the same flow-rate velocity, the value of the critical frequency is higher.
When polymer systems (PEO solutions in our case) are strained, changes in the strains ε and the stresses σ occur in different phases [8, 9]. The angle δ′ characterizing this difference is a composite function of the frequency of actions ω′. Strain in this case is carried out under the action of the stress following the law
(1)
where – stationary tension, – time.
Taking into account that and , and excluding the time from these equalities we obtain the equation
(2)
Expression (2) represents the equation of an ellipse whose area is equal to the work A_{0} done in a cycle of harmonic vibrations and irreversibly lost (dissipating) under strain. Then the dissipation function W is computed as the product of A_{0} and the number of cycles per unit time:
(3)
From relation (3) it is seen that, under the same conditions of strain, the dissipative function only depends on the polymer characteristics expressed in the value of δ′.
Changing the stresses by the harmonic law, we may track a change in the strain rate. The latter is related to the strain ε as follows:
(4)
The value of the ratio is a complex dynamic viscosity η^{∗}. This quantity may be represented in the form of the real and imaginary components η^{∗}=η′−iη′′, where η′=(σ_{0}⁄ε_{0}ω′)sinδ′ and η′′=(σ_{0}⁄ε_{0}ω′)cosδ′. Introducing η′ into relation (3) and performing transformations, we obtain
(5)
Dynamic viscosity manifests itself in the system under strain, and the increment of the effective viscosity ∆η_{ef} under the conditions of superposition of harmonic actions on mainstream flow is attributable to the appearance of η′. Since the amplitude value of the strain ε_{0} remains constant in the experiment, we may use scaling and rewrite (5) in the form
(6)
For the experimental curves (Figure 2a), we have obtained dissipative-function curves (Figure 2b) using expression (6). An analysis of these curves shows that, as the average velocity of flow of the aqueous PEO solution through the capillaries increases, i.e., as the fraction of macromolecules (because of the presence of the molecular-weight distribution) that have experienced unrolling [6, 7] and are unrolled to a higher degree, which causes the induction of rigidity in the macromolecules, increases, the dissipation of pulsating-flow energy decreases. The threshold value of the frequency, beginning from which the flow becomes distinctly dissipative, grows for less "flexible" and stretched molecules. As the induced rigidity of macromolecules increases, the dissipative curve acquires a more pronounced extremum character.
From the data obtained it follows that oscillating filtration flow of polymer solutions must ensure a higher degree of filling and coverage of a porous bed and thus a higher degree displacement of oil from a natural bed.
Figure 3 shows the patterns of oil displacement from of a model porous bed with a laminar in homogeneity using polymer solution.
In the first version (see Figure 3a), displacement of oil of the porous pace using polymer solutions was carried out in the stationary regime, whereas in the second version (see Figure 3c), it was carried out in the pulsating regime. The structure of the experimental setup enables us to visualize the process of advance of the front of the polymer solution in the porous bed and to qualitatively evaluate the efficiency of different regimes of oil displacement from porous bed by the flow patterns.
Calculation of the injection regimes is based on the use of the Debora (De) number characterizing the beginning of the development of a higher-than-average hydrodynamic resistance: De_{cr}=θ_{c}., where θ_{c}, relaxation time of the solution; , strain rate.
The relaxation time (θ_{c}) may be determined experimentally or from the known molecular characteristics of the polymer [10, 11]:
, k < 1; , k> 1, .
where θ_{0} - relaxation time of weakly strained coiled macromolecules, k - criterion of the degree of concentration of the solution, [η]_{0} - characteristic viscosity, C, concentration of the polymer in the solution.
The longitudinal velocity gradient realized in the pore cross sections in flowing of the polymer solution defies an accurate calculation. However, we may evaluate it as a first approximation, solving the problem on flow of a Newtonian fluid in a convergent channel [5] as an element of a pore channel: , where V_{f} - filtration rate, K - coefficient of porosity of the medium, d_{p} - average pore size. For the ranges of PEO molecular weight and concentrations of practical importance, the Debora number may be assumed to be equal to unity for k < 1 and to 2.5 for k > 1 in accordance with experimental data.
Passing from the model to an actual filtration flow and on the basis of the results obtained, we may state that oscillating filtration flow of polymer solutions ensures a higher degree of coverage of the bed and thus a higher degree displacement of oil from a natural bed (Figure 3c) than stationary flow (Figure 3a) and stationary regime of oil displacement at supercritical velocities (Figure 3b).
4. Conclusions
1. The experiments conducted have confirmed the correctness of the prediction of the theory of elastic-strain effects in tensile flow of the higher degree of oil displacement from of a porous bed in the oscillating regime of injection of a solution polymer into a porous medium (bed) and hence the mechanism of deviations from Darcy’s law the basis for which is provided by the concepts of a strong strain action of the hydrodynamic field on coiled macromolecules.
2. The effects causing the highly dissipative regime in oscillating flow of polymer solutions through porous media play an important role in optimization of the process of oil displacement from porous (especially with a laminar in homogeneity) bed using polymer solutions. The injection of a solution polymer into a porous medium (bed) must occur in the oscillating regime, when elastic strains appear, and the frequency of pulsating action on the filtration flow must correspond to the maximum of the dissipative function W(ω)≈.
5. Notation
A_{0}, strain work done in a cycle of vibrations, J; C, concentration of the polymer in the solution, %, kg/m^{3}; De, Debora number; De_{cr}, critical Debora number; d_{p}, average pore size, m; d_{p1} and d_{p2}, average pore size in the first and second porous media, m; i, imaginary unit; k, criterion of the degree of concentration of the solution; K, coefficient of porosity of the medium; M, molecular weight of the polymer, a.m.u; t, time, sec; , average velocity of flow through a capillary, m/sec; V_{f}, filtration rate, m/sec; W, dissipative function, rad^{2}⋅Pa/sec; δ′, phase difference between stress and strain fluctuations, rad; ∆η_{ef}, increment of the effective viscosity that arises in oscillating flow, Pa⋅sec; ε, relative strain in stationary flow; ε_{0}, amplitude value of the relative strain in no stationary flow; , strain rate, sec^{−1}; η_{ef}, effective viscosity, Pa⋅sec; η^{∗}, complex dynamic viscosity, Pa⋅sec; η′, dynamic viscosity, Pa⋅sec; η′′, imaginary part of the complex dynamic viscosity, Pa⋅sec; [η]_{0}, characteristic viscosity, m^{3}/kg; θ_{c}, relaxation time of the solution, sec; θ_{0}, relaxation time of weakly strained coiled macromolecules, sec; σ, stress in stationary flow, N/m^{2}; σ_{0}, amplitude value of the stress in oscillating flow, N/m^{2}; ω′, frequency of action on the flow, rad/sec; ω_{cr}′, critical frequency of action on the flow, rad/sec. Subscripts: c, concentration; cr, critical value; 0, amplitude value; p, porous medium; p_{1} and p_{2}, first and second porous media; f, filtration; ef, efficient.
References