Overview of Wellbore Fluid Flow and Heat Transfer Modelling with Applications in the Oil Industry
Figure 1: Discretised Wellbore System and Formation, and Formation Boundary Conditions with Geometric Spacing in Radial Direction for the Formation Part
r z T=Tsurface In 1981, 1982 and 1985, Farouq Ali,23 Fontanilla and Aziz24 and Yao,25
respectively, proposed two simultaneous ordinary differential equations for estimating the steam pressure and quality and solved these equations using the fourth-order Runge–Kutta method. The major differences between these models lie mainly in the type of correlations used to describe the multiphase flow inside the wellbore and the techniques to evaluate the formation temperature.
In 1989, Sharma et al.26 with a downhole electrical heater and in 1990 Wu and Pruess27 Qloss
modified Ramey’s model for production wells suggested
presented a simplified two-phase method for hand calculations using field data. An analytical model to determine the flowing fluid temperature inside the well was developed by Hasan and Kabir29
an analytical solution for wellbore heat transmission in a layered formation with different thermal properties without Ramey’s assumptions. A year later, Sagar (1991)28
in T=Treservoir
The variety of applications can be summarised by viewing the wellbore simulator as a tool to study diverse scenarios for wellbore completion and cement placement job designs, circulation of mud and cuttings, wellbore equipment, flow-assurance issues, surface pipelines and facilities design, tubular, etc. during a field design, development, production and optimisation.
History of Wellbore Flow Modelling As early as 1937 Schlumberger and Doll10
and Bird,12 pointed out the benefit of
fluid temperature measurement in the wellbore. In 1953 and 1954, Nowak11
presented a flowing temperature gradient chart in gas lift operation. However, the first mathematical model to estimate fluid temperature as a function of production time and well depth was proposed by Ramey (April 1962)16 1962)17
temperature logs to obtain information about the direction and amount of the fluid that is flowing in a wellbore. Later, procedures for predicting wellbore fluid temperature in flowing gas wells (Lesem et al., 1957)13 water injection wells (Moss and White, 1957),14 Kirkpatrick (1959)15
for hot water injection and mud circulation, respectively.
Since then, Ramey’s work has been referred to by many subsequent studies modelling wellbore heat loss and pressure drop. In fact, many researchers just tried to add more terms to Ramey’s formulations and relax some of Ramey’s assumptions (steady-state flow of incompressible single phase, fixed fluid and formation properties, no frictional loss and kinetic energy effect, etc.16
) to model more complex wellbores.
improved Ramey’s analytical model by considering a depth-dependent overall heat transfer coefficient and phase- and temperature-dependent fluid properties. A year later, Holst and Flock (1966)19
In 1965, Satter18 respectively, proposed using an interpretation of nd were proposed.
1994. These authors started with a steady-state energy balance equation and combined it with the definition of fluid enthalpy in terms of heat capacity and the Joule–Thompson coefficient. Using some simplifications, they then converted the original partial differential equation to an ordinary differential equation and solved it with appropriate boundary conditions. Several recent publications by these same authors involving further modifications of the original model appeared in 2002, 2005 and 2007.8,30,31
In 2004, Hagoort32
developed a comprehensive numerical non-isothermal multiphase wellbore model. After their initial attempts to solve the fully coupled conservation equations, they decoupled the wellbore energy balance equation from the mass balance equation in most of their investigations. They reported that the decoupling can be justified when the change in density of each phase with respect to temperature is much less than that with respect to pressure. Additionally, they found that this decoupling approach can decrease the computational time of simulations without violating stability.
Ramey’s model in order to find applicable scenarios for this model. Many researchers, including Hagoort, found that Ramey’s model works for late time (typically more than a week) temperature estimation but can cause serious errors for early time temperature distribution. In 2008, Livescu et al.9
and Edwardson et al. (November
Common Formulations for the Wellbore Flow Simulators Almost all wellbore flow models have three conservation equations in common; however, some terms in these equations may be slightly or even sometimes completely different from each other. These three equations are mass, momentum and energy balance equations.2,8,9,23,30,31,33–36
∂z ∂∂)+mws (ρgvsg+ρL
144 dp dz
Qloss 3600 Ati
added the friction loss and kinetic energy effects to Ramey’s and Satter’s models. In a subsequent study, Willhite (1967)20
vsL
=+ dz +
ρmgcos(θ) gc
= ++ρpαp Ep 2gc
∂ ∂t
Σ p proposed
a method for the estimation of an overall heat transfer coefficient that has since widely been used. Pacheco and Farouq Ali (1972)21 Herrera et al. (1978)22
and proposed wellbore models for simulation of steam injection process and validated their models with field data. 62
⎧ ⎩
⎧ ⎩
hp
11 2
v2 p Jc
1 + 2 2gc
v2 p Jc
⎫ ⎭
⎫ ⎭
fmv2 2dti
mρm gc
ρmvm gc
∆z p
∂(mE)cv ∂t
‘ +Σ ρmvsp gc +mwa
= (ρm ∂t
) (1) dvm +
ρm gc
- Σρpvsp
∂ ∂z
p
gcos(θ) Jc
+ mH (3)
In the above mass balance equation, it is assumed that there is only zone component and two phases inside the wellbore. However, if there is more that one component inside the wellbore, one mass balance
EXPLORATION & PRODUCTION – VOLUME 9 ISSUE 1
∂vm ∂t
(2)
conducted a comprehensive study on
Wellbore Tubing
Cementing Formation
Insulation Annulus Casing
No heat flux boundary
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