Summary
An improved nonisothermal mathematical model considering the nonequilibrium
effects involved during rapid multiphase flow in wells is presented. The extent
of the nonequilibrium effect on deviation from the equilibrium-model
predictions is delineated at various flow rates, fluid saturations, and
temperatures. Applications for two-phase (oil/gas) and three-phase
(oil/water/gas) systems are presented. The model presented here can be coupled
with reservoir simulation for accurate representation of the well-fluid
hydraulics under non-equilibrium- and nonisothermal-flow conditions.
Introduction
In general, reservoir fluids consist of three phases, referred to as the
gas, oil, and water phases. Each of these phases has different physical
behaviors, and each phase interacts differently with the others as the
reservoir fluid travels along the production pipe or a pipeline from the inlet
to the outlet. At atmospheric or standard conditions, the pipe fluid stream can
be separated physically into three parts, known as the pseudocomponents. These
are the gas, oil, and water pseudo-components, and they are different
substances from the gas, oil, and water phases. The pseudocomponents have been
widely studied so that the physical properties of the phases can be estimated
from the pseudocomponent’s properties.
Multiphase flow in pipes is a phenomenon yet to be modeled accurately. When
of two or more phases flow in a mixture in a closed environment they flow at
different volumetric flow rates. The densities of the phases differ greatly in
oil and gas wells; thus, the gas phases and liquid phases flow at different
velocities in a pipe with constant cross-sectional area. This causes a slippage
of the gas phases past the liquid phases.
Under no-slip conditions, the volume fraction for each phase is the
volumetric flow ratio (Sg , So , and
Sw ). This is the ratio of the volumetric flow rate of that
phase to the volumetric flow rate of the mixture. Because the phases flow at
different velocities, these two quantities are not equal. It has been observed
that for wells, the volumetric fraction of the liquid phases
(HL ) is larger than its volumetric flow rate
(SL ). This characteristic is called liquid holdup.
The density of the mixture at any point can be calculated by taking the
weighted average of each phase density with the volumetric fraction of each
phase as the weighting factor. It is the main property that predicts the
pressure drop in a well. Therefore, the estimation of the liquid holdup is of
paramount importance to the pressure drop prediction. However, for all models
proposed so far the accuracy of predicting liquid holdup is not acceptable.
Several procedures and techniques have been developed to describe the liquid
holdup in multiphase flow. In describing this flow, some dimensionless
parameters have been defined to assert its nature. Multiphase flow has been was
observed experimentally to occur in various modes, such as bubble flow, plug
flow, slug flow, froth flow, and mist flow, depending on the way the liquid and
gas phases distribute in a representative elemental volume.
Apparently, each kind of flow has a characteristic physical behavior. These
behaviors may indicate that the laws governing the liquid holdup are different
for each kind of flow. This assumption has been accepted widely by the most
recent and accurate techniques available today. For instance, such analysis for
predicting the liquid holdup was carried out by Ros (1961).
Each kind of flow has a different model to predict liquid holdup in every
proposed technique. This makes the liquid-holdup prediction become
discontinuous when the flow changes from one kind to another. There is no
smooth adjustment for transition between different kinds of flow. Therefore,
the liquid-holdup prediction is extremely sensitive to the prediction of the
changing point.
The assumption that each kind of flow is governed by different laws might
not be an accurate representation of the slippage phenomenon. Assuming a
general law for all kinds of flows would yield a continuous curve for the
liquid-holdup prediction. Then, the change from one kind of flow to another
would be smooth and continuous, and liquid holdup would be insensitive to the
changing point. The current study makes the assumption that the governing
equations for the liquid holdup are the same regardless of the kind of flow
occurring inside the well.
The mass fraction of the liquid phases and the gas phase for a mixture can
be determined by empirical correlations that have been accepted widely for gas
and oil wells. Of course, these correlations predict the mass fraction of the
phases once the mixture has reached an equilibrium state. Because of the liquid
holdup, the mass fraction of the phases in a fluid in motion differs from the
equilibrium state. This suggests that the liquid holdup is characteristic for
flow under a nonequilibrium condition.
As a multiphase-fluid system, the reservoir fluid flow can be described by
means of the fundamental equations governing the flow of fluids in conduits.
These equations represent the mass, momentum, and energy conservation laws for
the multiphase-fluid system. In addition to these previous laws of
conservation, the mass conservation for the gas phase is considered to find a
relationship between the dryness at the equilibrium state and at the
nonequilibrium state.
A pressure drop occurs along the well during the motion of the multiphase
fluid from the bottom of the hole to the surface. The gas-phase generation
occurs when the pressure of the system falls below the bubblepoint pressure of
the liquid phases (oil and water). At this condition, a mass transfer takes
place between the gas phase and the liquid phases (oil and water) across the
interface between the gas phase and the liquid phases. Generally, this
interface mass-transfer is assumed to happen instantaneously. However, in
reality, the mass transfer process requires a finite period of time to be
completed. This phenomenon is referred to as relaxation. The multiphase-fluid
system is said to be at an equilibrium state when the gas mass transfer across
the gas/liquid interface has been completed; otherwise, the multi-phase-fluid
system is said to be at a nonequilibrium state or undergoing a flashing
phenomenon.
The interface mass transfer is bidirectional (i.e., reversible, involving
the separation and dissolution of the gas phase). In this study, a cumulative
separation is assumed to describe the bidirectional gas mass transfer. In
addition to consideration of the flashing phenomenon, the energy losses by heat
exchange with the surroundings and the temperature change by the
multiphase-fluid expansion are also taken into account in the formulation of
the wellbore hydraulics. The pressure drop, caused by friction between the
flowing fluid and pipewall, causes the multiphase-fluid to expand. This
expansion spreads the internal energy into a greater volume thus causing a
temperature drop. This process is known as the Joule-Thompson effect, and it
can be modeled by the fundamentals of thermodynamics. The heat exchange can be
estimated by knowing the temperature of the surroundings. Usually, it is
assumed that the temperature of the surroundings or external temperature,
varies with a constant slope, referred to as the thermal gradient. For wells,
this gradient is called the geothermal gradient. The rate of the heat exchange
is set mainly by the completion technique for wells and by the coating
technique for pipelines.
Several studies have been carried out to simulate and predict the flow
conditions in oil and gas wells. Ayala and Adewumi (2003) successfully
simulated the fluid transmission in pipelines with liquid holdup.
Cazaraez-Candia and Vásquez-Cruz (2005) simulated the flow in vertical wells
under the transient-flow regime. However, the heat loss to the ambient medium
was ignored in both studies. Hagoort (2005) considered the heat losses in
vertical wells for a single-phase fluid (gas). Yoshioka et al. (2005) described
multiphase flow in horizontal wells with heat losses. However, these studies
ignored the effect of gas flashing from the liquid phases to the gas phase in
the mass-transfer model and considered constant-inclination wells. Civan (2006)
included the effect of gas flashing and demonstrated its significance.
The primary objective of this paper is the consideration of the thermal
effects and nonequilibrium gas separation from the liquid phases in the
description of the hydraulics for wells. The model presented in this paper is
an improvement over the model presented by Civan (2006). The flow of the
multiphase fluid is described with the differential versions of the previously
mentioned conservation laws. A numerical solution for the set of differential
equations is obtained for the steady-state case. The procedure for solving the
system of equations for any particular set of data is explained. A number of
scenarios are presented to demonstrate the importance of a new relationship
provided in the new modeling of the liquid holdup.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
19 October 2006
- Meeting paper published:
24 September 2006
- Revised manuscript received:
5 July 2007
- Manuscript approved:
30 August 2007
- Version of record:
20 May 2008
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