J. Fluid Mech. (2012), vol. 707, pp. 467481. c Cambridge University Press 2012 467doi:10.1017/jfm.2012.291
On the periodic injection of fluid into, and itsextraction from, a porous medium for seasonal
heat storage
Peter Dudfield and Andrew W. Woods
BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK
(Received 28 March 2012; revised 18 May 2012; accepted 10 June 2012;first published online 26 July 2012)
We examine the oscillatory motion of fluid which spreads under gravity alonga horizontal impermeable boundary through a porous medium controlled by theperiodic injection and extraction of fluid from a horizontal well. Over the first fewcycles the volume of injected fluid exceeds that which is extracted owing to thegravitational spreading of the current. However, after many cycles, these volumesconverge and the flow develops into two regions. Near the source there is a zone0 < x . xC = 2.4 (Q 2S/)1/3 in which the depth of the fluid varies periodicallywith each cycle, where Q is the fluid injection rate, is the injection or extractiontime, S is the speed of the buoyancy-driven flow and is the porosity. The currentattains its maximum depth, 1.8 (Q2/2S)1/3 at the source, where the minimum depthequals zero. At long times, the current depth at x = xC is approximately constant,1.15 (Q2/2S)1/3, and beyond this point, the current spreads horizontally, driven byan effective flux Ql 0.54Q (t/)1/2, so that the length of the current increases asxnose 1.73 (Q 2S/)1/3 (t/)1/2. We confirm these predictions with new experimentsusing a Hele-Shaw cell. We also model the evolution of the thermal front whichdevelops if the injected fluid is hotter than the formation temperature. We findconditions under which all the extracted fluid is hot but owing to the mismatchbetween the volume of injected and extracted fluids, not all the injected thermal energyis recovered, and the surrounding rock heats up.
Key words: convection in porous media, gravity currents, porous media
1. IntroductionPower stations, some industrial plant and large buildings with significant cooling
load, including hospitals and data centres, generate large amounts of low-gradewaste thermal energy. Given the seasonal fluctuations in the demand for heatingand initiatives for energy efficiency, there is interest in the potential for inter-seasonal storage and recovery of this thermal energy in shallow permeable layersof rock (Mackay 2008). The thermal energy may be transferred by the injection of hotwater into the system in the summer, and the extraction of this water in the winter asillustrated in figure 1(a, b).
Email addresses for correspondence: [email protected], [email protected]
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468 P. Dudfield and A. W. Woods
Summer Winter
Power station
Power station HousesHouses
h
k y
x
Hot fluid
Cold fluid
(a)
(c)
(b)
FIGURE 1. (Colour online) Schematic illustration of pumping (a) in (summer) and (b) out(winter) of an aquifer. (c) Schematic illustration of fluid in aquifer, showing xnose(t) andxTnose(t). The height of the fluid and thermal fronts are shown as h and hT respectively.
The effectiveness of such systems depends on the detailed pattern of flow andthermal energy transport through the permeable rock. The flow pattern depends onthe balance of pressure to buoyancy forces, whether the shallow permeable rock isoriginally saturated or unsaturated, and on any heterogeneities in the system whichcan disperse the injected fluid. In addition, owing to the thermal inertia of a porousmedium, the thermal front associated with the hot injectate typically travels moreslowly than the fluid (Phillips 1991) and this can lead to complex effects in anoscillatory flow.
In this contribution, in order to establish some of the principles, we focus on anidealized problem in which we model oscillatory flow in an originally unsaturated,homogeneous, horizontal shallow aquifer of large lateral extent. We examine theflow arising from injection and extraction at a line source associated with either ahorizontal well or a planar fracture around a vertical well. We assume buoyancyforces are significant in controlling migration of the fluid and that initially the effectsof transverse thermal diffusion are not dominant in controlling the migration of thethermal energy. We also assume that the temperature of the fluid does not affect thephysical properties of the fluid.
Using the simplification that the flow is parallel to the boundary of the aquifer,(Bear 1988), we develop a theoretical model to describe the migration of the fluid andthermal energy during both the injection and extraction phases, and explore how the
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Periodic injection of fluid into, and its extraction from, a porous medium 469
system evolves over many such cycles. We test the model with some new laboratoryexperiments. In 5 we discuss the simplifications of the model.
2. Fluid frontWe consider the periodic injection of fluid of density and viscosity into, and
its extraction from, an unsaturated aquifer, with period 2 and of volume flow rate perunit length 2Q(t), as shown in figure 1(c). The aquifer has porosity and permeabilityk, and has an impermeable bottom layer. We focus on flows with a small aspect ratio(depth/length) 1, in which case the flow is approximately one-dimensional andparallel to the boundary and, in the absence of capillary effects (see 5), the pressureis assumed to be hydrostatic.
2.1. Governing equations of the fluid frontThe current occupies the region 0 6 y 6 h(x, t) and xnose(t) 6 x 6 xnose(t) whereh(xnose(t), t)= 0 and by symmetry we only consider the region x > 0. We assume theaquifer is of sufficient vertical extent, H, so that h(x, t) < H for all time.
Using the assumption that the flow has a small aspect ratio, the pressure isapproximately hydrostatic and combining this with Darcys law (Barenblatt, Entov& Ryzhik 1990) gives
u= k
p
x=kg
h
x. (2.1)
Local conservation of fluid then implies
h
t= S
x
(hh
x
)(2.2)
where S = kg/ is a scale for the buoyancy-driven flow speed. The boundarycondition at x= 0 has the form
Shhx= Q(t) if h> 0, where Q(t)=
{Q if 2n < t < (2n+ 1)Q if (2n+ 1) < t < (2n+ 2),
(2.3a)h
t= 0 if h= 0 and (2n+ 1) < t < (2n+ 2) (2.3b)
where n is an integer. This condition expresses the difference between injection, at aconstant volume flow rate per unit length Q, and the extraction phase, in which fluid iswithdrawn at the same rate Q until the current depth at x= 0 falls to zero, after whichthe extraction flux decreases and is determined as part of the solution. Also h(x, t)= 0when t = 0, as no fluid has been injected into the system.
It is convenient to introduce scalings for the flow in order to work withdimensionless variables. The natural time scale in this problem is , driving a volumeinjected per unit length over one phase of Q . If the fluid has depth h and lateralextent L, then these are controlled by the gravity-driven flow speed Sh/L leading tothe scalings
Q LhL
Sh
L
L (Q 2S/)1/3
h(
Q2
2S
)1/3.
(2.4)
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470 P. Dudfield and A. W. Woods
In order that the current is long and thin as required by our assumption of parallelflow, (2.4) requires that
h
L=(
Q
S2
)1/3 1. (2.5)
Using (2.4), we re-cast the system of equations in the dimensionless form
h=(
Q2
2S
)1/3F(, t) where = x
(Q 2S/)1/3, t = t
(2.6)
and define the dimensionless leading edge of the current to have position
nose(t)= xnose(t)
(Q 2S/)1/3where F(nose, t)= 0. (2.7)
We also define the dimensionless volume of the current, V (t), as
V (t)= V(t)Q=
xnose0
h dx
Q= nose
0F d. (2.8)
Equation (2.2) then becomes
F
t=
(FF
). (2.9)
We seek solutions of (2.9) for which F = 0 at t = 0 with the boundary conditionsFF
=1 if 2n< t < 2n+ 1
FF
= 1 if 2n+ 1< t < 2n+ 2 and F > 0
F
t= 0 if 2n+ 1< t < 2n+ 2 and F = 0
at = 0 (2.10a)
and F 0 as . (2.10b)We solve this problem numerically using a finite-difference method (Ames 1965),
which was tested for numerical convergence and accuracy. We reduced the grid sizefor t and until the results were independent of grid size. We then tested for accuracyby comparing the numerical prediction of the current profile during the first injectionphase, Fn, with the similarity solution found by Huppert & Woods (1995), Fs, andc
