Modeling reservoir temperature transients,
and matching to permanent downhole gauge (PDG) data for reservoir parameter
estimation
Obinna Duru
1. Background and
Motivation
Over the last decade, permanent download gauges (PDGs) have
been used to provide a continuous source of downhole data in the form of
pressure, temperature and sometimes flow rate. The tools provide access to data
acquired continuously over a large period of time and containing reservoir
information at a much larger radius of investigation than conventional wireline testing.
The behavior of pressure transients in reservoir and
wellbore flow has been studied extensively, and applied in conventional well
test analysis for reservoir description, parameter estimation for
characterization and evaluation of well performance. In recent times, with
convolution and deconvolution, filtering and tuning
the data, conventional pressure transient analytical methods have also been
applied to pressure data from permanent downhole gauges.
However, in the development of pressure transient
analytical methods, it has been assumed that the temperature distributions in
the reservoir and production zone are isothermal. The temperature changes
associated with fluid flow had been considered to be relatively small and hence
negligible for any consideration in the analysis of flow behavior of most fluids.
An analysis of temperature measurements, at a finer scale
using continuous data from PDGs, has shown that the temperature of the fluids
responds to changes in flow conditions in the reservoir and production zone.
Generally, the flow is not isothermal when the scale of observation and
resolution of the temperature data is reduced. This research is motivated by the
possibility of identifying the underlying physical phenomena responsible for
this temperature transient behavior and application to reservoir
characterization and evaluation of well performance.
2. Literature
Review
Several authors have studied the thermodynamics of flow
through porous media, especially in the context of heat convection and
conduction. One of the earliest works in this regard was by Ramey (1962). He
developed a model for the prediction of wellbore fluid temperature as a
function of depth for injection wells. He expanded this to give the rate of
heat loss from the well to the formation, assuming steady state in the wellbore
and unsteady radial conduction in heat transfer to the earth. Horne and Shinohara (1979) presented
single-phase heat transmission equations for both production and injection
geothermal well systems by modifying Ramey’s model. Shiu
and Beggs(1980)
presented another modification of Ramey’s model to predict the wellbore
temperature profile for a producing well, where the temperature of fluid
entering the wellbore from the reservoir is known. These wellbore models
considered heat transfer as strictly convection and conduction phenomena with
no internal generation in the reservoir.
Izgec et al. (2006) presented a model that applied to coupled wellbore and reservoir systems. They provided a
transient wellbore temperature simulator coupled with a variable-earth-temperature
scheme for predicting wellbore temperature profiles in flowing and shut-in
wells. Again, their study looked at the
mechanism of heat transfer without consideration for possible heat sources and
sinks such as internal generation.
Sagar, Doty and Schmidt (1991), developed a steady-state
two-phase model for the wellbore temperature distribution accounting for
Joule-Thomson effects due to heating/cooling caused by pressure changes within
the fluid during flow. They considered Joule-Thomson effects as a possible
source/sink of heat during fluid flow and applied the model to estimate heat
losses in gas flow.
Valiullin et al. (2004) showed that indeed, adiabatic and
Joule-Thomson effects as well as effects due to heat of phase transition (gas
liberation from oil) are present during fluid flow in a hydrocarbon saturated
porous medium. They designed experiments
to estimate the thermodynamic coefficients, namely the Joule-Thomson
coefficient and adiabatic coefficients.
Ramazanov and Parshin (2006) went on to develop
an analytical model that described the formation temperature distribution in a
reservoir, while accounting for phase transitions. They solved a steady-state convective
thermal flow model with constant flow rate and extended it to cases with phase
changes.
Ramazanov and Nagimov (2007) presented a
simple analytical model to estimate the temperature distribution in a saturated
porous formation with variable bottomhole pressure. Their investigation showed that for a
single-phase fluid in a homogenous reservoir, temperature-pressure effects such
as Joule-Thomson can cause the temperature in the reservoir to change
significantly when reservoir pressure is changing in time. They again solved the convective thermal
transport model with variable pressure but constant flow rate.
An attempt to solve the full energy balance equation for
the temperature distribution in a reservoir was made by Dawkrajai
(2006) and Yoshioka (2007). Both presented equations for reservoir and wellbore
heat flow and developed prediction models for the temperature and pressure. In
an inversion step, they showed a means for the identification of water and gas
entry into a well. Both approaches made considerations for Joule-Thomson and
frictional heating effects but assumed a constant flow rate, and steady-state
conditions in arriving at the solution to their models.
3. Problem
Description
Many attempts at developing interpretation method for
temperature profiles in wellbore-reservoir systems have remained largely
qualitative. Most of the analyses have concentrated on wellbore thermal
exchanges due to conduction and convection, assuming that the produced fluid
enters the wellbore at the geothermal temperature (Maubeuge,
1994). Others have attempted the study of thermometric fields in reservoirs and
porous systems, but have constrained the analyses to convective effects only in
steady-state formulations. A few have considered the effects of heating or
cooling of the produced fluid before it enters the wellbore due to factors like
the Joule-Thomson effect, adiabatic expansion and viscous dissipation.
This work aims to solve the problem of modeling the
reservoir thermometric effects, as convective, conductive and transient
phenomena. The contributions of compressibility and viscous dissipation are
also included.
4. Progress
to Date
Maximal correlation between temperature and
flow rate
In
order to proceed with the study of the physics behind the observed temperature
response to changes in flow rate and pressure, a nonlinear, nonparametric regression
analysis tool namely the Alternating Conditional Expectation algorithm (ACE [Breiman and Friedman, 1985]) was used to estimate the
transformations that may lead to the maximal multiple correlation between the
response variable (temperature in this case) and the set of predictor variables
(pressure, rate and time). These
transformations are useful in establishing the existence of a functional
relationship between the response and predictor variables.
Using
a field data set obtained from PDGs, the ACE method was applied to the
pressure, rate and temperature data to establish the existence or otherwise of
a correlation and functional form for their relationship. Figures 1, 2, and 3,
show the rate, pressure and temperature data, and Figure 4 shows the plot of
the regression on the optima transformations.
Figure 1: Rate data Figure 2: Pressure data
Figure 2: Temperature data Figure 4: ACE regression
The
optimal transformation functions showed a correlation coefficient of 0.99. The
nature of the forms of the optimal transformations also showed that a
functional relationship form may exist between the temperature, and rate and
pressure and this functional form can be extracted from any representative data
set.
Integrated
modeling of coupled reservoir and wellbore thermal system
PDGs
are usually located some hundreds of feet above the perforation/production zone
of a reservoir. The PDG placement constraint is one that is imposed by the
design of the completions, and the optimal location for pressure and
temperature data management would be a position as close enough to the perforation
as possible, to give data that are comparable with their sandface
values.
The
approach adopted in the formulation of the physical model describing the
behavior of the temperature distribution involved a coupling of a wellbore
thermal model to a reservoir thermal transient model.
Wellbore
temperature transient model
The
wellbore model used here followed the dynamic model developed by Izgec, et al. (2006). The solution to the model was
modified to account for heat transfer at shut-in times when flow rate is zero
and heat transfer in the wellbore is only by conduction into the formation.
Reservoir
temperature transient model (reservoir thermometry)
Theory:
The
thermodynamics of fluid flow in porous media has been studied over the
years. Bejan
(2004) presented a comprehensive thermodynamic approach to obtaining a
representative model. He allowed considerations for spatial and temporal
reservoir temperature distribution in the physical modeling step.
In
a flowing well, the pressure and flow rate measurements by PDGs are not
constant. For gauges placed close enough to the sandface
area, these changes reflect the dynamics of the flow in the reservoir. These flow dynamics cause a temperature field
to evolve in the reservoir, driven by thermodynamic effects such as the
Joule-Thomson heating (or cooling), adiabatic expansion, and heat of phase
transitions. Other effects namely the viscous dissipation, equal to the
mechanical power needed to extrude the viscous fluid through the pore, as well
as frictional heating between the fluid and rock matrix during the fluid flow
are also factors that contribute to the
evolution of a nonuniform temperature field.
Joule-Thomson effect and viscous
dissipation: The
Joule-Thomson effect is the change in the temperature of a fluid due to
expansion or compression of the fluid in a flow process involving no heat
transfer or work (constant enthalpy).
This effect is usually due to a combination of the effects of
compressibility and viscous dissipation. The Joule-Thomson effect due to the
expansion of oil in a reservoir or wellbore results in the heating of the fluid
because of the value of the Joule-Thomson coefficient of oil – it is negative
for oil and positive and more prominent in gas.
Combined
with other factors, on expansion of the fluid and subsequently flow of liquid
oil and/or water out of the reservoir, the wellbore and near wellbore areas in
the reservoir become heated above the normal static reservoir temperature. By
convection, diffusion and further generation of heat due to these effects, the nonuniform temperature spreads into the reservoir,
increasing the temperature deeper into the reservoir. Conversely, during
no-flow conditions (shut-ins), the regions already heated lose heat to the
surrounding formation through diffusion and result in a temperature decline at
a rate determined by the thermal diffusivity of the medium.
The model
Performing
an energy balance through a control volume element, as well as establishing the
attending mass conservation and flow equations, the following
advection-diffusion forms of a thermal model for porous media can be derived (Bejan, 2004).
Single-phase formulation
The
thermal model in a one-dimensional radial symmetric coordinate system with a
single well takes the form:
(1.1)
On
rearrangement and assuming negligible gravity effects, this becomes
(1.2)
The
mass balance equation yields:
(1.3)
The Darcy flow
equation also yields:
(1.4)
Where:
.
Equations 1.1,
1.2 and 1.3 form the governing equations for one-dimensional thermal transport
in a homogenous porous medium. The assumptions made in deriving the equations
were:
·
The medium is
homogenous, such that the solid and fluid permeating the pores are evenly
distributed throughout the porous medium.
·
The medium is
isotropic such that permeability, k
and thermal conductivity
do not depend on the direction of the experiment.
·
At any point in
the porous medium, the solid matrix is in thermal equilibrium with the fluid in
the pores.
·
Darcy law
applies.
Two-phase formulation (oil
and water)
The assumptions for
the two-phase formulation are similar to the assumptions made in the single
phase case, with the addition of negligible capillary effects.
The
thermal model in one-dimensional radial coordinate system for the two-phase
system becomes:
(1.5)
On
rearrangement and with negligible gravity and capillary effects,
, this becomes
(1.6)
The
mass balance equation yields:
( 1.7)
The Darcy flow
equation also yields:
(1.8)
Where:
Equations 1.2
and 1.6 are advection-diffusion equations that are difficult to solve
analytically. Numerical methods also have issues with stability due in part to
the nature of the model – a combination of hyperbolic convective transport and
parabolic diffusion transport models.
In order to see
the behavior of the model with changing model parameters, a semi-analytical
solution was constructed using the method of Operator Splitting and Time Stepping (OSTS), developed for the
solution of contaminant transport in ground water hydrology.
Operator
splitting and Adaptive Time Stepping (OSTS) approach
Kacur (2002) and Remesikova (2003)
described methods of solving the convection-diffusion problem using the
operator splitting approach. The operator splitting method breaks the model into
two different parts, the transport and diffusion parts. Then, at each time
step, the nonlinear transport part and the nonlinear diffusion part are solved
separately. Holden et al. (2000) showed the theoretical basis for this
technique.
Kacur (2002) showed a precise way of solving the following
type of problems:
(1.9)
With boundary
and initial conditions:
(1.10)
(1.11)
First, the
transport part is solved, which presents a hyperbolic problem of the form:
(1.12)
With boundary
condition of the form in Equation (1.10,) and initial condition of the form:
(1.13)
Then the solution
obtained is denoted as 
. Thereafter, the diffusion part is solved, which is of the
form:
(1.14)
With the same
boundary condition and but initial condition given by the solution
of the convective part -
(1.15)
Finally the
solution is put as:
(1.16)
which is the solution of Equation (1.14). This is continued
until the solution at the last time step is obtained, which becomes the final
solution of the model.
Solution
of the thermal advection-diffusion model by Operator Splitting
and Adaptive Time Stepping (OSATS)
The OSATS
approach was used to solve the thermal model Equations 1.2 and 1.6. The methodology adopted was:
·
Decouple the model
into two parts: the convection transport part and the diffusion part.
·
At each time
step, first solve the hyperbolic convection transport part, accounting for
variable flow rate, as well as heat generation due to viscous dissipation,
frictional and Joule-Thomson effects.
·
Then solve
the diffusion part at the same time step, adaptively modifying the time step to
ensure stability if solution is numerical.
·
Continue until the
last time step.
Solution
of the Single-Phase Formulation by OSATS
Assumptions:
·
Constant fluid
Joule-Thomson and adiabatic expansion coefficient, thermal conductivity
·
Constant fluid
viscosity and formation porosity
·
Negligible
gravity effects
A. Solution
of the transport problem:
The convection
equation with its initial condition becomes:
(1.17)
(1.18)
Using the method
of characteristics, the solution yields:
(1.19)
Such that
for constant rate,
(1.20)
Another form of
Equation 1.20 starts by solving the pressure equation, and using the Darcy equation
to replace velocity in Equation 1.19. If the total compressibility of the
porous medium is considered negligible, then the pressure equation reduces to:
(1.21)
With boundary
conditions:
(1.22)
Where 
This
yields:
(1.23)
And 