Research
Fluids Dinamics
Temperature profiles due to continuous hot water injection into
homogeneous fluid-saturated porous media through a line source
A. Medina1
F. J. Higuera2
M. Pliego3
G. Gómez4
1ESIME Azcapotzalco, Instituto Politécnico
Nacional, Av. de las Granjas 682, Col. Sta. Catarina, Azcapotzalco 02250 CDMX,
México. ETSIAE, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3,
28040 Madrid, Spain. e-mail: amedinao@ipn.mx
2ETSIAE, Universidad Politécnica de Madrid,
Plaza Cardenal Cisneros 3, 28040 Madrid, Spain.
3DCB, Instituto Tecnológico de Querétaro, Av.
Tecnológico s/n esq. M. Escobedo, Col. Centro, 76000, Querétaro, Qro.,
Mexico.
4ESIME Zacatenco, Instituto Politécnico
Nacional, Av. Miguel Othon de Mendizabal SN, La Escalera, CDMX,
México.
Abstract
We report a theoretical study aimed at determining the temperature distribution
of a homogeneous, fluid-saturated porous medium initially at a low temperature,
into which a constant flow rate of hot water is injected. The size of the heated
region is assumed to be large compared to the radius of the injection pipe,
which is idealized as a line source of mass and heat. The temperature
distribution is found to be self-similar and to depend on a single dimensionless
parameter which is a Peclet number.
Keywords: Heat flow in porous media; flows through porous media; analytical and numerical techniques of heat transfer
PACS 44.30.+v; 47.55.Mh; 44.05.+e
1.Introduction
In this paper, we theoretically determine the temperature distribution of a
homogeneous fluid-saturated porous medium due to the injection of hot water through
a perforated pipe embedded in the medium. Physically, this is a way of heating the
porous matrix and the fluids therein. The problem is ubiquitous in technological
areas like heavy and extra-heavy oil recovery 1-6 and for the production and storage of energy in geothermal
systems 7-12, among others.
The injection of hot water into a porous medium does imply the simultaneous inflow of
mass and thermal energy (enthalpy), both by conduction and advection. In the
theoretical treatments, researchers commonly have treated the injection problem as
one where the temperature of the injected fluid and the initial temperature of the
fluid-saturated porous medium are known, while the spatial and temporal profiles of
the temperature of the medium around the injection pipe are computed 1-11.
In the current work, we assume that the hot water has penetrated the porous medium a
distance large compared to the radius of the injection pipe, which is then idealized
as a line source of mass and heat 13. However, we neglect the effects of gravity, which would
come into play at later times. This leaves aside buoyancy-driven flows in miscible
and immiscible fluids 14. Such
cases correspond, for instance, to the early stages of the injection of hot water
into heavy and extra-heavy oil reservoirs (immiscible fluids) and to the injection
of hot or cold water into geothermal aquifers (miscible fluids). In these cases, the
densities of the fluids involved have similar values, and thermal convection can be
assumed to play a secondary role. We also neglect finger-like instabilities.
The basis for the study of problems of hot fluid injection is the energy conservation
equation for fluid-saturated porous media. For transient problems, this is a partial
differential equation for the temperature as a function of time and the spatial
coordinates, which involves advection and conduction heat transfer and initial and
boundary conditions that depend on the specific problem considered.
Hot water flooding is a thermal method used in petroleum engineering. It is useful to
enhance oil recovery because the high viscosity of the in situ oil
drastically decreases when its temperature is increased 1-6. In the case of geothermal systems, cold or hot water
can be injected depending on the purpose. Cold water is injected to extract heat
from the hot rock by having the water heated before the waterfront reaches the
production wells 7-11. Hot water is pumped into shallow
permeable layers of rock for seasonal heat storage into reservoirs 12.
In the models, commonly, rock and fluid properties such as the specific heat,
density, and thermal conductivity, are considered to be constant in the reservoir.
This assumption is valid when the change of temperature in the porous media is
small.
The objective of this study is to understand the heat transfer mechanisms in
homogeneous porous media through the theoretical modeling of heat transfer coupled
with fluid flow. The paper is organized as follows. The physical ingredients of the
problem are reviewed in Sec. 2. The mathematical problem is formulated and solved in
Sec. 3, showing that the temperature distribution is self-similar and depends on a
single dimensionless parameter, a Peclet number that measures the ratio of advection
to conduction heat transfer. The features of the temperature profile for slow,
medium, and fast injection are discussed. Finally, Sec. 4 summarizes the main
conclusions of the work.
2. Physical model
The temperature field around the injection pipe will be analyzed to arrive at
conclusions of practical interest.
The hot water to be injected flows through a pipe immersed in the porous medium and
leaves the pipe through an array of uniformly distributed orifices in the pipe wall.
Owing to the low viscosity of the water, the pressure drop in the pipe can be
neglected so that the injection pressure is uniform along the pipe and the flow
ensuing in the porous medium can be taken to be two-dimensional in planes
perpendicular to the pipe, and purely radial at distances from the pipe large
compared to its radius.
The porous medium is homogeneous, with permeability K and porosity
φ, and is initially saturated with a quiescent liquid of density nearly equal to the
density p of the water and temperature T∞ smaller than the temperature of the hot water in the pipe,
Tw. For example, this liquid could be a heavy or extra-heavy oil or
fresh/saline water, all of which have densities very similar to the density of the
hot water. The liquid is assumed to be immiscible with the injected water at the
time scale of the process. We also assume that the fluid and solid phases of the
porous medium have the same local temperature (local thermal equilibrium. The
pressure in the pipe is such that a constant flow rate of water, q
per unit length of the pipe, is injected.
The injected water cools down while moving radially out by transferring heat to the
solid matrix and surrounding liquid. The density change of the water during this
process is small so that the approximation of constant p is
applicable.
3.Analytical model
Figure 1 is a sketch of the problem with the
idealizations described in the previous paragraphs. With these idealizations, the
radial filtration velocity in the porous medium is v=q/(2πr), where r is the distance to the injection pipe, idealized as a line
source at r = 0 (from the continuity equation r-1∂(rv)/∂r=0). The radius of the cylindrical region occupied by the injected water at
a time t after the onset of injection is
rf(t)=qtπ,
(1)
from the condition drf/dt=v(r=rf) with rf(0)=0.
The temperature of the porous medium obeys the energy equation 10,15,16
(ρc)m∂T∂t+(ρc)fq2πr∂T∂r=1r∂∂rrkm∂T∂r,
(2)
where (ρc)f is the volumetric heat capacity of the liquid and the properties of the
solid matrix, and the liquid are integrated into the average volumetric heat
capacity and thermal conductivity
(ρc)m=(1-φ)(ρc)s+φ(ρc)f and km=(1-φ)ks+φkf.
(3)
These magnitudes will, in general, be discontinuous at r=rf. In what follows, to simplify the analysis, the values of the volumetric
heat capacity and thermal conductivity are taken to be the same for both liquids.
Removing this assumption does not affect the self-similar character of the solution
discussed in the remainder of the paper.
Defining
q˜=ρcfρcmq
(4)
and writing (2) in conservative form, we have
∂T∂t+1r∂∂rq˜2πT-αmr∂T∂r=0,
(5)
where αm=km/(ρc)m is the thermal diffusivity of the rock/liquid system.
The quantity (ρc)f/(ρc)f v would be the velocity of the thermal front in the absence of heat
conduction 17. Since typically
(ρc)s>(ρc)f, this front would lag behind the liquid front r = r
f
17,18.
Equation (5) must be solved with the initial and boundary conditions
T=T∞ at t=0q˜2π(T-T∞)-αmr∂T∂r=Φ2π(ρc)m for r→0T=T∞ for r→∞
(6)
where Φ is the heat injected per unit time per unit length of the source, which
is taken to be constant (see comments following Eq. (9) later in this section) and
appears as the sum of the advection and conduction fluxes in the medium 19.
The solution of (5) and (6) is self-similar, of the form
T-T∞=Φ2πkmθ(η) with η=rαmt.
(7)
Carrying this to (5) and (6), we find
θ''+ 1-Peη+η2θ'=0 with Pe θ-ηθ'=1 for η→0 and θ=0 for η→∞,
(8)
where Pe=q˜/(2παm) and primes denote differentiation with respect to η.
The solution of (8) is
θ=1Pe 1-12Pe-1ΓPe2∫0ηηPe-1e-14η2 dη,
(9)
which is plotted in Fig. 2 for three values of
the Peclet number Pe. As can be seen, θ→1/Pe for η→0 in all the cases. The conduction heat flux tends to zero when r→0, and the injected heat Φ can be directly related to the temperature Tw of the water in
the injection pipe: Φ=(ρc)fq(Tw-T∞). Notice that this relation and the pressure variations that Darcy’s law
associates to the radial filtration velocity are time-independent. This justifies
the assumption of constant values of the injected flow rate and Φ in realistic cases.
The figure shows three very different behaviors for the quantity Pe θ by changing the Peclet number, by an order of magnitude each time: the
blue dotted curve is for Pe=0.1, which means that heat diffusion dominates over the advection of the
flow, the red dashed curve for Pe = 1 shows the case where advection and diffusion
have the same intensity and the green-continuous curve corresponds to Pe = 10, which
means that advection dominates strongly on conduction.
The importance of the dimensionless plot in Fig.
2 is that it shows different ways of cooling down the injected hot fluid
in the porous medium, depending primarily on the value of the Peclet number. For
instance, a low value of Pe (which can be given as Pe=ρcfq/2πkm) means that injection of the hot liquid is slow and/or that its
volumetric heat capacity is small, and/or conversely, the average thermal
conductivity of the water-saturated porous medium is high.
In the context of oil recovery, sand (unconsolidated) 1
2 and sandstone (consolidated)
4 reservoirs, which store large
reserves of heavy and extra-heavy oil, have been exploited by using hot water
injection. In these cases, the Peclet number can be determined using
order-of-magnitude estimates as follows: the average thermal conductivity k
m
is, in both cases, in the order of 1 W/m K in a wide range of temperatures
20,21, meaning that the heat transfer
from the water to the porous matrix is very efficient. The volumetric heat capacity
of water (ρc)f is in the order of 1 MJ/(m3K)22). Consequently, the ratio (ρc)f/km is of order 106s/m2, and it is found that for
water flooding into this type of reservoir, the best form of change Pe is by
changing the injection rate. As an example, if a process requires Pe∼10-1 then q∼10-7m2/s, and so forth and so on for the other Peclet numbers.
Geothermal aquifers have similar values of the ratio (ρc)f/km
23 and, therefore, similar
conclusions as aforementioned for Pe and q are reached.
4.Conclusions
An analysis has been carried out of the heating of an unbounded porous medium by
injection of hot water through a perforated pipe. After a short initial period that
is not analyzed, the size of the heated region becomes large compared to the radius
of the injection pipe, which can be idealized as a line source of mass and heat. On
the other hand, the temperature distribution is axisymmetric before buoyancy comes
into play. When both conditions are satisfied, the solution is self-similar and
depends only on a Peclet number. This solution has been computed, and its dependence
on the Peclet number has been discussed. In conditions typical of hot water
injection into the sand and sandstone heavy oil reservoirs, the ratio of volumetric
heat capacity of water to the average thermal conductivity of the medium is very
large, and the Peclet number is essentially a measure of the flow rate of water
injected, which determines the way the thermal energy behaves during the
injection.
Acknowledgments
A.M. acknowledges support from IPN, from Fondo de Hidrocarburos-CONACYT, and from
Universidad Politécnica de Madrid through the project “Fundamental models for the
thermal methods of steam injection in EOR”. He also appreciates the invaluable
support of Prof. A. Liñan, who guided him in part of this work. Finally, the authors
are grateful to an anonymous reviewer whose comments on an earlier version of the
manuscript helped us to clarify it and to improve the presentation of the solution
in Eq. (9).
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