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ЗАДАЧА ГИДРОДИНАМИЧЕСКОЙ ДИАГНОСТИКИ СКВАЖИН: РАССЛОЕННЫЙ ДВУХФАЗНЫЙ ПОТОК С ПРОМЕЖУТОЧНЫМ СЛОЕМ
Виталий Николаевич Доровский
Новосибирский технологический центр компании «Бейкер Хьюз», 630090, Россия, г. Новосибирск, ул. Академика Кутателадзе, 4а, доктор физико-математических наук, технический советник, тел. (383)332-94-43, e-mail: Vitaly.Dorovsky@BakerHughes.com
Юрий Вадимович Перепечко
Новосибирский технологический центр компании «Бейкер Хьюз», 630090, Россия, г. Новосибирск, ул. Академика Кутателадзе, 4а, кандидат физико-математических наук, научный сотрудник, тел. (383)332-94-43, e-mail: Yury.Perepechko@BakerHughes.com
Артем Викторович Манаков
Новосибирский технологический центр компании «Бейкер Хьюз», 630090, Россия, г. Новосибирск, ул. Академика Кутателадзе, 4а, кандидат технических наук, научный сотрудник, тел. (383)332-94-43, e-mail: Artem.Manakov@BakerHughes.com
Основная задача геофизических исследований, связанных с эксплуатацией нефтяных скважин, направлена на определение гидродинамических параметров пластов, в связи с чем рассматривается расслоенный двухфазный поток с промежуточным слоем в почти горизонтальных скважинах. Показано: природа промежуточного слоя обязана неустойчивости тангенциального разрыва скоростей. Представлены уравнения теории двухфазного потока, численная схема. Иллюстрируются численные решения с промежуточным слоем.
Ключевые слова: двухфазный поток, промежуточный слой, неустойчивость тангенциального разрыва.
HYDRODYNAMIC WELL TESTING PROBLEM:
STRATIFIED TWO-PHASE FLOW WITH INTERMEDIATE LAYER
Vitaly N. Dorovsky
Novosibirsk Technology Center «Baker Hughes», 630090, Russia, Novosibirsk, 4a Kutateladze St., D. Sc., Technical Advisor, tel. (383)332-94-43, e-mail: Vitaly.Dorovsky@BakerHughes.com
Yury V. Perepechko
Novosibirsk Technology Center «Baker Hughes», 630090, Russia, Novosibirsk, 4a Kutateladze St., Ph. D., Scientist, tel. (383)332-94-43, e-mail: Yury.Perepechko@BakerHughes.com
Artem V. Manakov
Novosibirsk Technology Center «Baker Hughes», 630090, Russia, Novosibirsk, 4a Kutateladze St., Ph. D., Scientist, tel. (383)332-94-43, e-mail: Artem.Manakov@BakerHughes.com
The main goal of the geophysical surveying related to oil well operation, is determination of hydrodynamic parameters of a formation. In this regard, this paper considers a stratified two-phase flow with an intermediate layer in almost horizontal wells. Our study has shown that the nature of the intermediate layer is defined by the instability of tangential velocity discontinuity. The paper presents the equations of the two-phase flow theory and the numerical scheme. It also contains the numerical solutions of the said problem.
Key words: two-phase flow, intermediate layer, tangential discontinuity instability.
Oil well operation control is one of the most important directions of geophysical surveying of wells, where analysis of hydrodynamic parameters of water-oil mixture flows remains one of the key diagnostic instruments to monitor water-oil flow in reservoirs. The main goal of such diagnostics is determination of water/oil mass ratio in a hydrodynamic flow, which requires knowing such two-phase flow parameters as viscosity and friction coefficient. The two-phase hydrodynamic theory (with no relation to particular flow structure) in combination with kinetic parameters enables one to calculate hydrodynamic velocity profiles and the phase ratios. Specialized sensors to determine the flow parameters at the selected points of a well's cross section make to possible to solve inverse hydrodynamic problems to determine the kinetic coefficients. However, one needs to be reminded that in horizontal wells, two-phase flows become stratified, and the character of this stratification is determined by borehole deviation and integral phase parts. It is necessary to understand the physical reason of the stratification, which requires developing of a specialized two-phase hydrody-namic theory based on a stationary model of a stratified flow with pronounced flow interfaces. Such interfaces will contain the discontinuity of two-phase flow tangential velocities. One should study the instability of this interface in order to determine its increment as a wave number function. As a result, the characteristic scale of the instability that develops with a minimum time can be defined. This scale determines the size of the water bubbles penetrating the upper oil phase, as well as the size of oil bubbles in the water flow. Those bubbles concentrate near the surface and stabilize the flow. Consequently, the bubble zone has a finite size and forms the intermediate layer. Numerical calculation of the obtained equations should validate the physical mechanism the theory describes. This mechanism paves the way for proper hydrody-namic control of horizontal reservoirs.
The current paper presents a two-phase theory that, with no relation to a particular phase interaction in a system, describes a stratified two-phase flow with unstable interface between the phases; calculates the instability increment; determines the characteristic scale of bubble generation in the intermediate flow; proves numerically that the equations contain solutions corresponding the stratified two-phase flow with intermediate bubble layer; paves the way for proper hydrodynamic control of horizontal reservoirs.
The conservation laws of energy, mass, momentum and two-velocity continuum entropy allow for unambiguous determination [1] of an equation describing a fluid moving with the velocity v and oil bubbles moving with the velocity u with relation to a particular phase interaction in the system:
(1)
where w = u-v, p denotes the pressure, p,, ps, , ^ - the phases' particle densi-
2 2 ties and viscosities; ""=dA+8'uk -ï8'*divu ; =8*v'+dvk -;2s"divv ; b - the interphase friction coefficient, g - the free fall acceleration.
A steady flow in a plane channel of incompressible fluid is determined by a system of equations for the zones divided by a tangential-velocity discontinuity surface:
i i \ du ps Pib (u - v ) + = VP, dy P
ppb (
u - v
\ d 2v p, dy P
(3)
where v = [v (y) ,0,0], u = \_u (y) ,0,0], Vp = {dp/dx ,0,0) = const. Here, the boundary
conditions are absence of normal and tangential velocities at the external boundaries of the domain, while at the interface the condition of equal velocities for carrying and
disperse phases in neighboring zones du/ dy + ^ dv/ dy = 0 is fulfilled. The stationary
solution of the system (3) v 0, u0 is studied for instability relative to the disturbances
x, t) of the surface separating the flows. The equations of velocity disturbances
are obtained by linearization of the system of equations (1), (2):
(1)
evolution v(1) , u
v = v0 + v(1), u = u
u
(1)
v 0 » v
(1)
(1)
0 , u , , 0 ^ T , u0 >> u(1). The equations defining the velocity disturbances and the interface surfaces are linear ones. The solution is structured as:
ç « exp
—kaIm(Q(ka)) t I-exp i — kaRe(Q(ka))t + kx
a J VK a ))
(4)
Here, a denotes the size of the layer, v* - the characteristic velocity of the forced medium flow, k - the wave vector of the perturbation wave, ka Im(Q(ka)) - the
instability increment (inverse time of instability development), t - the time. Figure 1 presents the calculated instability increment as a function of the wave vector of a running perturbation wave.
350
300
250
200
150
100
Fig. 1. Calculated instability increment as a function of the wave vector of a running perturbation wave
The performed study has demonstrated.
1. The instability increment takes values above zero. Infinitely small perturbations applied at the initial moment of time to the interface between the two-phase media increase infinitely with time.
2. The instability increment takes the maximal value at k = k*. A perturbation with the characteristic wavelength = 2n/k* has the minimum development time U =(a/v* )(ka ImQ(ka)).
3. The latter means that the interface between the media sliding past each other while instability is developing, produce bubbles of water in oil as a carrier component and bubbles of oil in water as a carrier component (the characteristic size of particles d is determined by the surface tension at the interface within the water-oil system
and is calculated using the formula d = n/k*).
4. The further scenario of instability development is as follows: instability is determined by the discontinuity in tangential velocity. When there is no discontinuity, there is nothing to cause instability. The bubble generation leads to an increase of the drag coefficient and, as a result, to stabilization of instability; the bubble generation stops when their concentration reaches a certain level (this situation is determined by the size of the transitional layer between the two interphase flows.
The computational algorithm for numerical analysis of the two-velocity continual hydrodynamic system is based on the method of controlled volume [3, 4]. The scheme used is entirely implicit with respect to time. When approximating convective sum-mands to compute the flows via the faces of control volumes, the Hybrid Linear/Parabolic Approximation (HLPA) scheme of the second order [6] is realized, which satisfies the quantitative accuracy requirement and the convection boundedness criterion (CBC). When approximating diffusive summands, the central difference scheme is used. To compute the pressure field coordinated with the flow field, a scheme similar to the Inter-Phase Slip Algorithm (IPSA) [2] is realized. When approximating the summands determining the forces between interacting phases, an entirely implicit scheme is used. To solve numerically the systems of linear algebraic equations for discrete counterparts of the difference equations and pressure-correction equation, the alternating direction method and the PARDISO solver (from the Intel MKL [5]) are used.
Figure 2 illustrates the result of numerical computation for the complete system of two-phase 2D hydrodynamic equations [6] representing distribution of phase parts in the upper, lower and intermediate mixed two-phase layers, where the upper layer is pure oil, and the lower one is pure water. The horizontal axis is the longitudinal coordinate of the 'plane' pipe with 5-degree deviation relative to the horizon. Figure 3 demonstrates the state of the system at the initial time moment: from the left boundary is inflow of stratified fluid with the both phases traveling with equal velocity. In the course of time the mixed layer has been evolving: the layer thickness reduces abruptly, the phase velocities change, and the oil phase shrinks significantly. The system produces the stratified flow with characteristic size of phase zones and of intermediate layer.
x 10
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Fig. 2. Stratified two-phase flow: state at time t=50 s
x 10
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Fig. 3. Stratified two-phase flow: initial state
Thus, the presented equations demonstrate their fundamental property: they describe (at small deviation angles) a stratified flow with intermediate two-phase layer. The equations of the theory do no assume there is any particular kind of interaction in the system, they rely exclusively upon the conservation laws, where flows are determined only by the first law of thermodynamics [3]. This makes it possible for one to utilize a strictly formalized approach to the geophysical methods of hydrodynamic monitoring.
REFERENCES
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2. Darwish M., Moukalled F. A unified formulation of the segregated class of algorithms for multifluid flow at all speeds // Numerical Heat Transfer Part B. - 2001. - Vol. 40. - P. 99-137.
3. Dorovsky V., Perepechko Yu., Sorokin K. Two-velocity flow containing surfactant // Journal of Engineering Thermophysics. - 2017. - Vol. 26. - P. 371-379.
4. Patankar S. Numerical Heat Transfer and Fluid Flow. - CRC Press, 1980. - 197 p.
5. Schenk O., Gartner K. Solving unsymmetric sparse systems of linear equations with PARDISO // Journal of Future Generation Computer Systems. - 2004. - Vol. 20 (3). - P. 475-487.
6. Zho J.A. Low-diffusive and oscillation free convection scheme // Communications in applied numerical methods. - 1991. - Vol. 7. - P. 225-232.
© B. H. ffopoecKUU, №. B. nepeneHKO, A. B. ManaKoe, 2017
