Download Influence of Convection on Production from Borehole Heat

PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering
Stanford University, Stanford, California, February 11-13, 2013
SGP-TR-198
INFLUENCE OF CONVECTION ON PRODUCTION FROM
BOREHOLE HEAT EXCHANGERS
Carina Bringedal, Inga Berre, Jan Nordbotten
Department of Mathematics
University of Bergen
5020 Bergen, Norway
e-mail: [email protected]
ABSTRACT
Convection cells in a porous medium form when the
medium is subjected to sufficient heating from below
(or equivalently, cooling from above) or when cooled
or heated from the side. In the context of geothermal
energy extraction from a sealed system, we are
interested in how the convection cells transport heat
when the porous medium is subjected to a cool,
sealed borehole, also known as a borehole heat
exchanger. Using pseudospectral methods together
with domain decomposition we consider two
scenarios; one system initialized with constant
temperature and one system initialized with a vertical
temperature gradient in the rock surrounding the
borehole heat exchanger. We find the convection
cells to have a positive effect on the heat extraction
for a constant initial rock temperature, but a negative
effect for some of the systems with an initial
temperature gradient in the rock: Convection gives a
negative effect when the borehole temperature is
close to the rock temperature in the borehole, but
gradually provides a positive effect when the
borehole temperature gets colder and the convection
stronger.
INTRODUCTION
Borehole heat exchangers (BHEs) are utilized for
production of shallow and medium depth geothermal
energy. The subsurface unit of a BHE consists of a
sealed vertical pipe containing a production fluid that
is circulated inside the pipe; cold fluid is pumped
downwards and gets heated by the warmer ground,
while the heated water is pumped back up giving a
net energy profit. Many buildings have shallow
boreholes producing local heating (and cooling) in
combination with heat pumps, and deep boreholes are
used for direct heating (Rybach and Hopkirk, 1995;
Kohl et al., 2002).
In the porous medium surrounding the borehole,
several thermal processes can be present. Conduction
will always transfer energy from regions with higher
temperature towards regions with lower temperature.
If the porous medium is saturated with fluid, typically
water, that is allowed to flow in the porous medium,
convection is also possible. As the density of water
depends on temperature, buoyancy forces can
generate groundwater flow without any outside force;
this is called natural convection. Natural convection
creates convection currents, which are circulation
patterns for the saturating fluid. These currents will
also transfer heat, but how and how much is
dependent on flow properties of the fluid and the
geometry of the porous medium.
In this paper we focus on how induced convection
currents affect the production from a BHE. When a
BHE is producing heat, the borehole is filled with
fluid having a temperature that is lower than the
surrounding porous medium. This temperature
difference will trigger convection currents in the
surrounding porous medium as a horizontal
temperature gradient always causes fluid motion
(Vadasz et al, 1993). It is not obvious how the
convection currents will contribute to the heat
extraction from a BHE, as they may either contribute
to the heat production by transporting more energy
towards the borehole, or give a negative effect by
transporting the heat energy away from the borehole.
To isolate the effect of the natural convection, we
assume the system to have no net groundwater flow.
A forced advective flow would also affect the heat
production, and has been investigated by e.g.
Eskilson (1987), Chiasson et al. (2000) and Diao et
al. (2004), but will not be considered here.
MODEL FORMULATION
We study an idealized geothermal system containing
a single heat producing BHE. The borehole is sealed
in the sense that there is no injection or production in
the reservoir, which is typical for shallow and
medium depth systems used mainly for heating and
cooling applications. Our borehole is of a coaxial
type where the water flows downwards in the outer
tube and upwards in the inner tube. The coaxial
borehole is used for both shallow and deep systems,
as seen in Kohl et al. (2002) and Diersch et al.
(2011). Other borehole structures, such as a U-type,
could be applicable for the same uses, but is not
studied here. The borehole is surrounded by a threelayered porous medium and produces heat only from
the middle layer. The middle layer is permeable and
saturated with water, while the top and bottom layer
is assumed to be heat conducting, unsaturated rock or
soil, acting as heat reservoirs for the saturated layer.
For simplicity we assume the three porous layers to
be homogeneous and isotropic.
We consider two different models: In the first model
the initial temperature variations in the surrounding
porous medium are neglected. This is the case if the
fluid inside the borehole is much colder than
temperature in the reservoir, making the initial
temperature variations in the reservoir less important.
For example, if the initial ground temperature in the
ground varies between 20C and 25C while the
production fluid has an inlet temperature of 5C or
lower, then this model is applicable. The other model
is initialized with a vertical temperature gradient and
represents a system where the initial temperature
variations are too large, compared to the borehole
temperature, to be neglected. If the initial ground
temperature varies between 45C and 65C, while the
inlet temperature in the borehole is 35C, which is
the case studied by Kohl et al. (2002), the initial
temperature variations in the porous medium must be
taken into account. For the rest of the article these
two models will be known as the constant
temperature (CT) model and the temperature gradient
(TG) model, respectively.
Our domain is shaped as an annular cylinder and is
sketched in Figure 1. The three porous layers 1, 2 and
3 have heights ,
and , respectively. We only
model the downward flow inside the BHE, which is
contained between
and
. Assuming the
casing between the inner and outer tube to be
insulated, the flow in the inner tube does not affect
the heat production. Also, the BHE is assumed to
only extract heat from layer 2.
Figure 1: Cross-section of domain.
Governing equations
To describe the fluid flow in the saturated layer, we
assume Darcy’s law,
(
)
(1)
where v is the fluid velocity, is the permeability of
the porous medium,
and
are the viscosity and
density of the groundwater, respectively,
is the
gravitational acceleration, and k is the vertical unit
vector pointing upwards. The density is given by
[
(
)],
(2)
where
at some reference temperature
,
and is the thermal expansion coefficient. Note that
the density of water would normally also be
dependent on pressure, but table 4 in Fine and
Millero (1973), reveals that for temperature and
pressure domains relevant for geothermal energy,
density variations due to temperature is the most
dominating. Even though density variations are
present, they are so small that we apply the
Boussinesq approximation, which states that density
differences in the fluid can be neglected unless they
occur together in terms multiplied with the gravity
acceleration . Hence, we can apply the mass
conservation equation for an incompressible fluid;
(3)
We further assume energy conservation for both fluid
and solid in layer 2; that is
(
)
(
)
(4)
In the above equation, subscript refers to the fluid
and
to the medium. Furthermore, ( ) is the
overall heat capacity per unit volume where is the
specific heat, and
is the overall thermal
conductivity of the fluid and solid combined. When
we use the overall heat capacity and the overall
thermal conductivity, we are using porosity-weighted
averages of the heat capacities and thermal
conductivities of the fluid and solid. Finally, is the
temperature of fluid and solid. No equations are
needed to describe the fluid flow in the lower and
upper layer; hence, only
(
)
(5)
has to be solved here. The subscript refers to the
solid. The heat conductivity of the solid is in this
equation denoted
to emphasize that it could be
different from the heat conductivity of the solid in
layer 2. The temperature inside the outer tube of the
borehole is modeled using
(
)
(
)
(6)
The fluid velocity inside the borehole is assumed to
be equal to the injection velocity. To estimate the
effect of convection on the heat production, we
calculate the heat flux into the borehole using
(7)
where is the heat flux, is an outward unit normal
for the borehole, and
is the surface element on the
borehole. The integral is to be taken over the whole
borehole facing layer 2.
Nondimensional equations and the Rayleigh
number
To nondimensionalize the equations, we use the
following coordinate transform:
(8)
(
)
( ) is the thermal diffusivity, and
Here,
( ) ( ) is the ratio of the volumetric heat
capacities of medium and fluid. The two
temperatures
and
are reference temperatures
and should represent a typical temperature difference
in the system. In the CT model,
and will be the
initial temperatures in the porous medium and in the
borehole, respectively. In the TG model,
and
are the initial temperatures at the bottom and top of
the saturated layer. The superscript denotes that the
variable has no dimension. Substituting the above
nondimensional variables into the governing
equations, introduce the dimensionless Rayleigh
number
(
)
(9)
where
is the kinematic viscosity of the
saturating fluid. The Rayleigh number works as a
measure of the strength of the convection; a Rayleigh
number of zero corresponds to the saturated layer
being impermeable and provides no convection. A
Rayleigh number larger than zero would always
provide convection due to the horizontal temperature
gradient (Vadasz et al., 1993). If there is no
horizontal temperature gradient, convection currents
can only develop if the Rayleigh number is larger
than some critical Rayleigh number, see (Bringedal et
al., 2011) for a further discussion.
Substituting the dimensionless variables into our
model equations yields a new system of equations.
Darcy’s law (1) is transformed into
(10)
the mass conservation equation (3) becomes
(11)
the energy conservation equation for layer 2,
equation (4), becomes
(12)
while for the lower and upper layer, equation (5), is
(13)
The energy conservation equation for the borehole
(6) is now
(14)
Initial and boundary conditions
To solve the equations, initial and boundary
equations are required for temperature, and also
boundary equations for velocity. The CT model is
initialized with the constant temperature
in
the porous domain and with
in the borehole.
Hence,
is also the injection temperature in the
borehole at all times. All outer boundaries around the
porous medium is held perfectly heat conducting at
, while the top of the borehole is kept at
and otherwise insulated. In the connection
between borehole and porous medium, and also in the
connection between the layers, continuity in
temperature and heat flux is required. The TG model
is in the porous medium initialized with a linear
temperature distribution corresponding to
at
the bottom of layer 2 and
at the top of layer 2.
The borehole is initialized with an injection
temperature
. The outer boundaries
belonging to the porous domain and the top boundary
of the borehole are all kept perfectly heat conducting
corresponding to their initial condition. The left hand
side and bottom of the borehole is kept insulated, and
we require continuity in temperature and in heat flux
at the internal boundaries as in the CT model. For
both CT and TG model, we require boundaries of
layer 2 in the porous domain to satisfy a no-slip
condition for the velocity. In the borehole, the
velocity is given by
( ) and satisfies no-slip
condition at the vertical boundaries.
NUMERICAL SOLUTION APPROACH
An unsteady 3D-solver that approximates the
solution of the nonlinear equations (10)-(14) has been
written using pseudospectral methods in space and
MATLAB’s built-in package ODE15s in time.
Pseudospectral methods are higher-order numerical
methods known for their good convergence
properties. We have chosen this method to obtain
high resolution of the temperature distribution near
and inside the borehole, which is essential for
investigating small changes in the heat flux into the
borehole. In earlier work we have successfully
applied these methods to investigate onset and
stability of convection currents (Bringedal et al.,
2011). Here we give a short review of the
pseudospectral methods and refer to Boyd (2001) for
a more thorough introduction.
Pseudospectral methods is a collocation methods that
search for the function values in the collocation
points, and is in that sense related to finite difference
methods. However, pseudospectral methods obtain
very high accuracy by placing the collocation points
in a specific manner, and the optimal choice of grid
points depends on the geometry of the domain. As we
use cylindrical coordinates, we apply the Chebyshev
nodes in the radial and vertical direction and the
Fourier nodes in the azimuthal direction. These
choices will provide the fastest possible convergence,
as explained by Boyd (2001). In the resulting matrix
equation when discretizing with pseudospectral
methods, each line represents an equation for the
function value in one of the collocation points.
Applying boundary equations is then only a matter of
localizing collocation points at the boundaries and
insert a discrete version of the boundary condition.
As we have internal boundaries both between the
layers in the porous medium and between the
borehole and the porous medium, we apply domain
decomposition to ensure that we have collocation
nodes also at the boundaries. We divide in four
subdomains equal to the four rectangles sketched in
Figure 1; one for each layer and one for the borehole.
The decomposing also enables us to use a finer grid
in any of the layers or in the borehole if necessary.
The subdomains are stitched together using the
continuity requirements described within initial and
boundary conditions.
RESULTS AND DISCUSSION
The governing equations (10)-(14) are solved by
timestepping the energy equations (12)-(14) and
updating the velocity field in the porous medium
using Darcy’s law (10) and the mass conservation
equation (11) in each time step. The production
period for a BHE depends on geological conditions
and usage, but is expected to be less than 50 years for
deeper boreholes, while shallow boreholes used for
direct heating are normally closed or applied for
cooling purposes after each winter season. In
dimensionless time, both these time spans correspond
to
. All following results are given at
.
We need to apply values for the parameters listed in
equations (10)-(14). For , ,
and we apply
values realistic for groundwater and ground, while
for the Rayleigh number we use a range of
representative values. In the papers by Diersch et al.,
(2011), Lazzari et al. (2010) and Kohl et al. (2002)
values for injection velocity
for both shallow and
deep systems can be found. Translated to
dimensionless injection temperature, this corresponds
to
. The same three papers provide
values for the borehole diameter, which in
dimensionless variables results in
. The values of
, and
are chosen to be
so large such that the outer boundaries will not have a
significant effect on the results in the relevant time
period. For the TG model, a range of values for the
injection temperature
is needed. For a complete
list of values of parameters, see Table 1.
Table 1: Values or ranges of values used for parameters in
simulations.
Parameter
(
)
(only for TG model)
Value/Range
(
)
The CT model
In the CT model the convection currents, when
present, would always distribute such that hot
groundwater is transported towards the upper half of
the borehole, then down along the borehole while
heat diffuses into the borehole and cools the
groundwater, and then transported away from the
borehole at the bottom of layer 2. See Figure 2 for
temperature distribution near the borehole for a
convective and a non-convective case.
Figure 3 the obtained heat fluxes
Rayleigh numbers is showed.
for various
Figure 3: Nondimensional heat fluxes into the borehole for
various Rayleigh numbers.
From Figure 3 we see that the heat flux increases
with increasing Rayleigh number, but the difference
is not large. We have also performed simulations
where we have tried other values for
,
, ,
and
corresponding to other scenarios, but the
results in heat flux where qualitatively the same; the
heat fluxes still increase with increasing Rayleigh
number.
The TG model
The convection currents distribute in the same
manner in the TG model; groundwater flow towards
the borehole in the upper half of layer 2 and away
from the inner cylinder in the lower half. See Figure
4 for temperature distribution near the borehole for a
convective and non-convective case.
Figure 2: Temperature distribution for Ra = 0 (top) and Ra
= 100 (bottom). The colored lines are isotherms
where red is colder and yellow warmer. The
arrows indicate groundwater velocity.
The simulations shown in Figure 2 were made using
,
,
,
and
(
)
,
corresponding to a shallow borehole. From Figure 2
it is clear that convection currents will provide
slightly larger heat production in the upper half of
layer 2 as warm fluid is transported towards the
borehole here. In the lower half of layer 2 the heat
flux becomes slightly smaller, but altogether the
convection currents gives larger heat production. In
Figure 5: Nondimensional heat fluxes into the borehole for
various Rayleigh numbers when
.
From Figure 5 we see that the heat flux decreases
significantly with increasing Rayleigh number. We
have also performed simulations where we have tried
other values for
,
,
,
and
corresponding to other scenarios, but the results in
heat flux where qualitatively the same; the heat
fluxes still decrease with increasing Rayleigh
number. However, lowering
revealed a
significant effect. With a lower value of
, the
initial temperature variations in the ground water is
less important. Convection will still transport the
colder upper lying fluid towards the borehole, but
even the coldest groundwater will be significantly
warmer than the borehole. Especially for larger
Rayleigh numbers, the dominating effect when using
a low
is that the convection currents provide
more heat transport and give a larger heat flux into
the borehole. Heat fluxes for various Rayleigh
numbers and
is given in Figure 6.
Figure 4: Temperature distribution for Ra = 0 (top) and Ra
= 100 (bottom).
The simulations shown in Figure 4 were made using
the same parameters as for Figure 2, and with
. This value for
corresponds to the
injection temperature being very close to the coldest
ground temperature from where heat is extracted and
is an extreme case, but is not uncommon for longterm heat extraction. From Figure 4 we observe that
the convection currents now give a negative effect;
even though there is more heat transport in the
system, it is the colder, upper lying fluid that is
transported towards the borehole. In Figure 5 the
obtained heat fluxes
for various Rayleigh numbers
are showed.
Figure 6: Nondimensional heat fluxes into the borehole for
various Rayleigh numbers when
(white),
(light grey),
(dark grey) and
(black).
For borehole injection temperatures lower than 0, we
see two trends in Figure 6: First of all we get in
general larger heat fluxes for all Rayleigh numbers;
this is due to there being a larger temperature
difference between borehole and ground present.
However, if we compare heat fluxes for the same
values of
we see that for larger Rayleigh
numbers, the convection can give a positive effect on
the heat production. For gradually colder
, this
occurs for even smaller Rayleigh numbers. This is as
expected since the TG model with a very low value
of
should act as the CT model; when
becomes significantly lower than 0 the initial
temperature variations in the ground becomes less
important and we could in stead use the CT model to
describe the system. Simulations with
between
( ) and ( ) gave results showing the same trend
as observed for the CT model.
CONCLUSIONS
Using pseudospectral discretization together with
domain decomposition, we have made a solver for
modeling the heat transfer processes in a layered
porous medium when a sealed borehole is extracting
heat. Our high-order numerical simulations show
how the heat transfer into a borehole heat exchanger
is affected by the presence of natural convection
currents. For a system using a constant temperature
distribution as initial condition in the porous medium,
convection currents will give a small positive effect
to the heat extraction as the currents retrieve some
extra heat towards the borehole. Stronger convection
will always result in a larger heat flux into the
borehole. For a system having a vertical temperature
gradient as initial condition in the porous medium,
the effect of convection depends on the injection
temperature in the borehole. When the injection
temperature is close to the coldest temperature in the
saturated porous medium, the convection will give a
negative effect on the heat production as the cold
groundwater is transported towards the borehole
giving a very small heat flux. For stronger
convection, the heat flux becomes even smaller. For
colder injection temperatures in the borehole, the
situation gradually changes as the initial temperature
variations in the subsurface become less significant
and the system acts more like the model initialized
with a constant temperature in the porous medium.
For borehole heat exchangers our results affect the
choice of injection temperature in the borehole.
Using an injection temperature close to the
groundwater temperature is typical for long-term
BHEs as this results in the ground not cooling down
so quickly, but we see here that this results in less
heat production than expected if the ground is
permeable and saturated with water so that
convection currents can evolve. If using an injection
temperature much lower than the ground temperature,
such that
in the model initialized with a
temperature gradient in the porous medium, or such
that the porous medium can be considered having a
constant temperature before production from the
BHE starts, the convection currents will give a
positive effect on the heat extraction as more heat is
transported towards the borehole.
If the borehole is used for cooling purposes, such that
the borehole temperature is warmer than the ground,
the results are qualitatively the same; convection will
provide a negative effect if the injection temperature
is close to the initial ground temperature, and a
positive effect is the injection temperature is
significantly higher than the ground temperature.
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