Influence of water recharge on heat transfer in a semi-infinite aquifer

Pergamon Geothermics Vol. 26, No. 3, pp. 365-378, 1997

(~) 1997 CNR Published by Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0375~505/97 $17.00 + 0.00

PII: SO375--6505(96)OOO48-X

INFLUENCE OF WATER RECHARGE ON HEAT TRANSFER IN A SEMI-INFINITE AQUIFER

FRANCESCO MONGELLI and PIETRO PAGLIARULO

Department of Geology and Geophysics, University of Bari, Campus, 70125 Bari, Italy

(Received February 1996; accepted for publication July 1996)

Abstraet--A simple model is proposed for the temperature field within an unconfined semi-infinite thick aquifer, with groundwater flowing parallel to the terrestrial heat flow in the recharge zone, and perpendicular to heat flow in the other parts of the aquifer. The results enable evaluation of the extent of the influence of the recharge zone, once the thermal and hydrological parameters are known. Beyond this zone, where thermal equilibrium between water and rock is reached, water temperature reaches the constant highest value, and water movement cannot be revealed by temperature measurements. More- over, on the basis of the range of influence of the recharge zone, the regional water pore velocity can be inferred. © 1997 CNR. Published by Elsevier Science Ltd.

Key words: water recharge, heat transfer, modelling, Murge, Italy.

INTRODUCTION

It is well known that groundwater movement can influence terrestrial heat flow toward the earth's surface to a point where the geothermal gradient is zero or even inverted. Hence, a perturbation in the thermal gradient may be a powerful indicator of water flow. Apart from regions where very high heat flow can cause convective phenomena, groundwater movement, either in the aquifer or inside a borehole, can be the cause of advective perturbation to the otherwise conduction-dominated thermal regime.

The main interests in studying this problem are to evaluate the advective perturbation in order to obtain the undisturbed geothermal gradient; to find the portion of aquifer in which heat flow is not perturbed; and to forecast water temperature for low enthalpy utilisation.

Vertical infiltration in recharge or discharge areas is the simplest case of groundwater flow; a one-dimensional energy equation with a simple exponential solution efficiently describes the phenomenon (e.g. Bredehoeft and Papadopulos, 1965; Mansure and Reiter, 1979; Lachenbruch and Sass, 1977). Horizontal (or sub-horizontal) water movement through confined or unconfined aquifers requires a 2D or 3D energy equation, which is more complicated to solve. Approximations of the 2D energy equation are given by,

365

366 F. Mongelli and P. Pagliarulo

among others, Kappeimeyer (1957), Haenel and Mongelli (1988), and Maggiore and Mongelli (1991). Some authors, such as Bodvarsson (1969), Kilty and Chapman (1980), Brott etal., (1981), and Ziagos and Blackwell (1981), have investigated specific physical or hydrogeological conditions of the 2D case. In the meantime, numerical methods have been developed (e.g. Smith and Chapman, 1983). Although analytical solutions, unlike numerical models, cannot accommodate every possible situation, they are, nevertheless, powerful tools for giving an immediate physical insight of the systems and for simplifying applications. Domenico and Palciauskas (1973) give an exact solution for the temperature field in a steady state flow in a rectangular aquifer with homogeneous, isotropic thermal and hydraulic conductivity. They assume the upper boundary of the flow field to be the water table and approximate it by an analytical function. In a recent paper, Mongelli (1994) studied the heat transfer in a semi-infinite aquifer with uniform properties when the discharge zone is very far removed from the recharge zone. He assumed the temperature of the aquifer 's surface to be equal to the annual mean temperature of the surface at an almost constant level. He also imposed the same mean temperature at the vertical in-flow cross-section.

The aim of the present paper is to study the more realistic case of a temperature increase with depth in the recharge zone and its influence on the flow in the semi-infinite aquifer.

THEORY

The model we discuss is shown in Fig. 1, which represents an idealised rectangular semi-infinite aquifer whose boundaries are: z = d the water table at temperature T0 z = 0 the impermeable base at temperature T, + Gd = TI x = 0 the vertical boundary at temperature T(z), where G is the undisturbed geothermal gradient and d is depth (or aquifer thickness).

Water enters the system through the surface So and flows through the cross-section S~ into the aquifer. The flow is irrotational, at steady-state and continuity is required (i.e. incompressible water).

For simplicity we assume that the heat gained along the horizontal path in the recharge zone is very small compared to the heat gained with depth. This limits the validity of the

z,

So T = T 0

0 T! = To + e~d x

Fig. 1. Reference system of an unconfined semi-infinite aquifer; vv= vertical velocity of the groundwater: vo horizontal velocity of the groundwater: T. = temperature of the water table at d (aquifer thickness); S o = surface of the recharge area; To + Gd = temperature of the impermeable

base. Streamlines are also drawn.

Heat Transfer in a Semi-Infinite Aqui fer 367

model to the case of a recharge zone where the velocity field is dominated by the vertical component, and/or where the vertical path is far longer than the horizontal one. This means a thick aquifer with a narrow recharge area.

In the recharge zone (x < 0), where water moves vertically, beneath the layer where temperature is influenced by seasonal variations, the energy equation governing heat transfer at steady-state is

02T 0T 2 &2 = pCpVvO -~z (1)

where

T is the temperature /l is the matrix thermal conductivity p is water density Cp is water specific heat capacity at constant pressure Vv is water pore velocity, directed vertically downward 0 is porosity.

The solution to equation (1) with the boundary conditions

T (z = 0) = T1

T (z = d) = T0

is given by (Bredehoeft and Papadopulos, 1965)

T ' ( z ' ) - exp[Pev(1 - z')] - 1 exp(Pe 0 - 1 (2)

where Pev = pcpv~Od/2 is the Peclet number for flow in porous media and

T ' - T - Tc_________Zj z ' = z / d Tl - To

are dimensionless temperature and depth, respectively. In the horizontal part of the aquifer (x > 0), where the water moves horizontally with

horizontal pore velocity Vo, and the same values of O, 2 and Cp, the energy equation is

[ O 2 T 00__~ O T + = 0--; (3)

where T(x , z) is temperature and the following boundary conditions apply:

T(x = O) = T( z ) T ( z = O) = T, T ( z = d) = To (4)

By choosing the dimensionless variables

x ' = x / d z ' = z / d T ' - T - T~ T I -- r o

equation (3) becomes


O2T ' a T ' O2T ' - - - P e o + = 0 OX '2 c)X'

(5)

where

Pe , , =-pCpv°Od 2

and the boundary conditions (4) become

T ' ( x ' = O) = T'v(z') = exp[Pev(1 - z')] - 1 e x p ( P e v ) - 1

(6)

T ' ( z ' = 0) = T ' (z' = 1) = 0

A part icular solution to equat ion (5) is

U ( x ' , z ' ) = 1 - z '

Let us then consider

W ( x ' , z ' ) = T ' ( x ' , z ' ) - (1 - z ' ) = \~ A,, exp(¢~,,x') sin n x z '

I 1 - [

(7)

where

a n - - Pe,, /Pe~, + rl2~ 2

2 V 4

[1 t'1 [exp[Pe,(1 _z , ) ] _ 1 A,, = 2 W ( x ' , z ' ) sin n ~ z ' d z ' = 2

.o . I , [ e x p ( P e , , - 1)

2 j n2,rr2[exp Per - 1 ( - 1)"] +

ns-r [(n2~ 2 + Pe~,)[exp(Pe, - 1)] ( - 1 ) " - 1 1}

e x p ( P e , ) - 1

- (1 - z') 1 sin mTz ' dz '

such that

W ( x ' . O) = 0

W ( x ' , 1) = o

W ( O , z ' ) = T ' (0 , z ' ) - (1 - z ')

The solution to equat ion (5) with the boundary condit ions (6) is

T' = 1 - z ' + ~ [ A,, exp(a, ,x ' ) sin n ~ z '

n I

After substituting, the solution to equat ion (3) with boundary conditions (4) can be written a s

- G-d- - 1 - + , A, ,exp(a , ,x ' ) sin n ~ z ' (8)

1

• ~-

0.7~

0.6

,,~ 0.5

0.4

0.3

0.2

0.1

o~ 50

Heat Transfer in a Semi-Infinite Aquifer

T* = 0.1

. / f 0.2

/ z

/ J 0.3

0.6 /

i ~ 0 . 8

0.9

100 150 200

369

½Pe o : 2 5 0 ; Pe v : 2 5 x/d

[ b 0.9 ~ T* = 0.1 I

0.8 L ~ / 0.2

0.7 ~ ~ 03

0.6[ ~ J 0 4

o.4 05 • ~ 0.6

0.3 ~ J 0.7

0.8 0 . 2 ~ J ~ ~

0.1 ~- 0 9

O0 20 40 60 8'0 x/d

100 120 140 160

½Pe o = 250; Pe v = 2.5

Fig. 2. (a) Dimensionless isotherms in an unconfined aquifer for ½ Peo =250 and Per = 25; (b) Per = 2.5. Pe o = Peclet number for horizontal flow of water; Pev = Peclet number for vertical flow of water

in the recharge area; z/d, x/d = adimensional distance (see text); see text for T*.

If isotherms of the dimensionless temperature T* are plotted on a graph showing the dimensionless distances x ' and z ' , the solution has only two parameters which can vary, the two Peclet numbers.

To underline the influence of the recharge zone on the thermal state of the aquifer it is convenient to calculate isotherms only as functions of Pev, taking therefore Peo to be constant. For example, Fig. 2a and b show the calculated isotherms for Peo/2 = 250 and Per = 2.5, 25. Note that the influence of Per on the shape of isotherms is dominant: for low values of Pev the isotherms start at different points along the z-axis and are only slightly


curved, whereas for values of Pe~ equal to 25 they start at the origin and markedly curve. Thereafter , the isotherms tend to run parallel to the earth's surface. Results obtained for Per -> 25 are consistent with those obtained by assuming temperature on the vertical boundary to be T0 = const. (Mongelli, 1994).

PERTURBED AND UNDISTURBED GEOTHERMAL GRADIENT

In order to understand what can be expected or found by measuring temperature in boreholes, we have studied some aspects of equation (8). By taking x = x ' d = f ld in equation (8), we obtain

T(fld, z) = To + G d 1 - -~ + A,, exp(c~fl) sin nJr~ (9) t t = l

which represents the geotherms at a distance f ld from the origin. For example, Fig. 3a and b show the results for T0 = 0°C, G = 10 K/km, d = 1000 m,

Peo/2 = 25 and Pe~ = 2.5, 25; note that the geotherms are curved close to the origin and tend to become linear as the distance from the origin increases. Moreover, they are more and more disturbed as Pe~ increases. In all cases at distances corresponding to fl > 100, the geothermal field is almost undisturbed.

This aspect can be evaluated better by studying the temperature gradient at the surface. From equation (8), the surface gradient is

() / z t OT = G - 1 + ( - l ) " n ~ A , , e x p a,, (10) Z z = d I t = l

Figure 4 shows the results for d = 1000 m, G = 10 K/km, (Peo/2 = 250) and Per = l, 5, 25. It should be noted that gradients are very different close to the origin (x ~ 0), depending on the values of Pev. In particular, for Per >~ 25, the gradient at the origin is zero, as when temperature on the vertical boundary is T0 (Mongelli, 1994). Surface gradients increase with the distance from the origin; the higher the Pev values, the greater the increase, so that at great distances their difference is negligible.

From a mathematical point of view, one can infer that the surface and the undisturbed gradients never coincide. Consequently, it may be important to calculate the distance X from the origin at which the ratio between surface and undisturbed gradient is 0.95 (5% is a reasonable accuracy for gradient measurements).

From equation (10) we obtain

r = (~zT)z_d = G { - I + ~ ( - 1 ) n n ~ A , , e x p ( a n X)} (11)

n - - I

Figure 5a and b show the variation of X / d versus d, assuming r = 0.95, Pev -= 2.5, 25 and different values of Peo/d.

Figure 5a and b are essentially identical, indicating that the penetration distance X / d is mainly a function of Peo, not Per. In fact, the ratio of the surface gradient to the undisturbed gradient, equation (11), will be essentially equal to one when X / d is large enough such that all of the exponential terms are zero. Since the magnitudes of the terms

Heat Transfer in a Semi-Infinite Aquifer 371

~ 0.5 N

0

0

x=5 Od

t r i _ ~ _ t I i i i ~

1 2 3 4 5 6 7 8 9 I0

T ( K )

Per=2.5 ½Peo=250~ G=10K/km

1 t ' b ' ' - ' ' ' ' ' '

=

i 2 3 4 5 6 7 8 9 i0

T ( K )

Pev=25 ½Peo=250; G=10K/km

Fig. 3. Geotherms in an unconfined aquifer at different distances x from the origin (the recharge zone) with aquifer thickness of 1000 m (d). (a) Pe v = 2.5; (b) Pe v = 25.


l 0

4

Pev= 1 "" - J " - / " " / / / *

/ / / , / /

/ / " /// / /

Pe v : 5 / / / / : /

// /

// /' / /

/ / 'Pe v />25 / / / / //

/ / // / / /

0 ~0 I(X) 150 200 250

x(hn)

i

3('13

½ P e o = 2 5 0 G = 1 0 K / k i n d = 1 k m

Fig. 4. Surface temperature gradient for different values of Pc numbers and aquifer thickness of 1 km. Pe,. = Peclet number for vertical flow of water in the recharge area. See text for X/d: for other

symbols see Fig. 2.

a,, increase with n, we need only consider the term n = 1. An exponential term becomes negligible if its argument is < - 4 [i.e. e x p ( - 4 ) = 0.02], so we need

X al-~= -4

o r

X [Peo' - " ,2 - - = - - 4 ~v Pe ,, + ~2]

d (~2)

where Pe~, = ~ Peo. N o w assume that Pe~, is large, say Pe~, >> :r 2. In this case we can approximate equation

(12) by

[PeL - PeL x/ i + ~2/Pe/, l X = - 4

o r

Pe~ , - PeL(I + 2Pe'~,,/] d - 4

hence


104

10:*

102 x

101

8

- 0 . 2 5 c ~ / / - - J , /apeo ~j__.._~

._~j.._1 - ..... ~..00"25 L~- .......... ~

102 1 (P I (P

d(m)

104

10 ~

Pev = 2 .5

102 X

10'

J ~. (3.25 6 J J t / ~ j

/ /

f -

ltf 10 2 10 ~

d(m)

Pev = 25

104

Fig. 5. The distance from the origin (recharge zone) at which the ratio between observed and undisturbed temperature gradient is 0.95 for Pev = 2.5 (a) and Per = 25 (b).

and

X _ 8Pe~) _ 4Peo d $T 2 7l .2

(13)

On the other hand, if Pe" is small, we can approximate equation (12) by

X - z r - ~ = - 4

o r

X 4

d (14)


EXAMPLE OF APPLICATION

The Murge plateau (Apulia, southern Italy) is part of the Apulian foreland, extending NW-SE between the Adriatic Sea and the Apennine foredeep (Fig. 6). The Murge relief is made of a thick sequence of carbonate rocks consisting of limestone and dolomite, referred to the Cretaceous age (Ricchetti et al., 1988). The main structural features are represented by a wide anticlinorium, trending NW-SE, affected by several faults trending in the same direction that are the result of extensional tectonics. Topography is characterised by moderate elevations gently sloping to the sea, reaching 400-500 m a.s.l, in the internal part of the tableland.

The calcareous-dolomitic formations of the Murge form a huge aquifer, characterised by secondary permeability due to the presence of fissures, solution channels and other voids of karst origin; water circulation takes place in several layers, interbedded with impervious strata that are less affected by fissures and karst processes.

The piezometric head varies from zero (sea or base level) to 200 m corresponding to the major topographic elevations. General hydraulic conductivity is low-to-moderate, due to the presence of less fissured and karstified rock bodies.

The various aquifer levels are in any case interconnected and form, on a regional scale, a single groundwater system (Grassi et al., 1977; Maggiore, 1991), with an average porosity 0 = 5 x 10 -2. The base of the groundwater system consists of an impervious dolomitic layer at a depth of 1000-1500 m; the total thickness of the aquifer system is about 1000 m (Quarto and Schiavone, 1994).

The groundwater system is fed by a considerable recharge of rainwater. The average

1 0 1100 km

Manfre~nm I ~ " " r, ADRIATIC SEA

, ' / , / / /

a c . . . . . s d

Fig. 6. Geological sketch of part of southern Italy including the example area. (a) Mesozoic carbonate formations of the Apulian foreland; (b) Plio-Quaternary sequences of the Apenninic

foredeep; (c) Apenninic thrust sheets; and (d) location of the area.


1•m 7 km

5

Fig. 7. Block diagram (not to scale) showing main physiographical and hydrogeological features of the Murge karst aquifer in the considered area.Wl-W6: water wells.

precipitation is about 650-700 mm per year. The main recharge area is located in the internal part of the plateau, at more than 300 m elevation; toward the coast, infiltration of rainwater is less important and a less developed system of streams is present to discharge the surface runoff (Fig. 7).

On the Adriatic side of the tableland, groundwater flow is directed towards the northeast, in a direction approximately perpendicular to the coastline (Fig. 8).

Depending on the spatial distribution of fracturing and karst development, flow velocities, both horizontal and vertical, are spatially variable. Using tracing techniques, local values of horizontal velocity ranging between 2.7 x 10 -5 and 6.9 x 10 -5 m/s have been noted in the area. Vertical flow velocity is in the range 1.2 x 10 -3 to 3.7 x 10 -3 m/s (Grassi et al., 1977).

Accurate temperature measurements were performed in wells located at different distances from the recharge area (Pagliarulo, 1994). Temperature gradients were very low, varying between 0.35 and 9.1 °C/km, showing a regular increase in the flow direction. Assuming p = 1000 kg/m 3, cp = 4.186 x 103 J/(kg.K), 0 = 5 x 10 -2, d = 1000 m, and ~. = 2.3 W/(m.K), the mean value of Pev is of the order of 105. This value is so high (>>25) that the heat transfer regime is consistent with that obtained by assuming temperature on the vertical boundary to be To = const. On the other hand, the undisturbed gradient in the area varies between 15 and 35°C/kin.

A fit of observed gradients with different sets of curves (Fig. 9), obtained by fixing Peo and varying G, gives G ~ 23 K/km and Peo = 200, which corresponds to v = 2 x 10 "6 m/s.

The resulting heat flow Q = 2G for this area is 53 mW/m 2, which is consistent with a Mesozoic stable area.


A D R I A T I C S E A ~

JS- Y '

• W5 .~o~'~ ~.. Palo ;~'Colle

.Wf

o @ 1 0 k i n

Fig. 8. Dis t r ibut ion of the piezometr ic head in the Murge area. Con tours represen t iso ) lezometr ic lines in metres above sea level (af ter G r a s s i e t al., 1977); arrows indicate main direct ions of

g roundwate r flow. W I - W 6 : water wells with t empera tu re data (from Pagliarulo, 1994).

/

G ._ '35 ~/'Iy'~__ -- lJ

20 l 25 - ~ J

20 _ 15[

[ ab __--

10~ , ~ ~ J

5 10 15 20 25 30 35 40 45 50

km

Pe o = 2 0 0 Pe v >> 25

Fig. 9. Curve fitting of observed gradients. Pc,, = 20(I, Per >> 25. Pe o = Peclet n u m b e r for hor izontal flow of water; Pe , = Peclet n u m b e r for vertical flow.


CONCLUSIONS

It is well known that the thermal influence of groundwater flow is greatest close to the recharge zone. In this paper we propose a method to evaluate the extent of the zone of influence and its magnitude for a given set of proper thermal and hydrological parameters, for a thick aquifer with a narrow recharge area.

We used mean values for our parameters and selected Peclet numbers throughout our calculations to gain a physical insight of the system behaviour.

Pe > 25 values produce situations almost equal to the case T(z) = To on the vertical axis, i.e. equal to external temperature (Mongelli, 1994). This is the "worst" possible situation since water needs to follow the longest horizontal path before the conductive regime is reached.

At long distances from recharge, and all other conditions being equal, different Per values no longer produce significantly different conditions. At such large distances, when isotherms become almost horizontal, and the superficial gradient almost equal to the undisturbed gradient, the horizontal water movement no longer influences the aquifer's thermal state: this means that at long distances water flow is no longer revealed by measuring temperatures. Moreover, the model can indicate the depth and distance from the recharge area where water can be found at higher temperature and can be exploited for low enthalpy utilisation.

Lastly, if the distance at which the measured gradient is almost equal to the undisturbed gradient is known, the regional value of water pore velocity can be inferred.

The results of this work are still valid when we deal with a discharge zone that mirrors the recharge area.

Acknowledgements--The authors are very grateful to S. Bachu and R. W. Zimmermann for their useful comments and suggestions.

REFERENCES

Bodvarsson, G. (1969) On the temperature of water flowing through fractures. Journal of Geophysical Research 74, 1987-1992.

Bredehoeft, J. D. and Papadopulos, I. P. (1965) Rate of vertical groundwater movement estimated from Earth's thermal profile. Water Resources Research 1(2), 325-328.

Brott, C. A., Blackwell, D. D. and Ziagos, I. P. (1981) Thermal and tectonic implications of heat flow in Eastern Snake River Plain. Journal of Geophysical Research 86, 11,709- 11,734.

Domenico, P. A. and Palciauskas, V. V. (1973) Theoretical analysis of forced convective heat transfer in regional groundwater flow. Geological Society of America Bulletin 84, 3803-3814.

Grassi, D., Tadolini, T., Tazioli, G. S. and Tulipano, L. (1977) Ricerche sull'anisotropia dei caratteri idrogeologici delle rocce carbonatiche mesozoiche della Murgia nord- occidentale. Geologia Applicata e ldrogeologia XII(I), 187-213.

Haenel, R. and Mongelli, F. (1988) Thermal exploration methods. In Handbook of Terrestrial Heat Flow Density Determination, ed. R. Haenel, L. Rybach and L. Stegena, pp. 353-390. Kluwer, Utrecht.

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Kilty, K. and Chapman, D. S. (1980) Convective heat transfer in selected geologic situations. Groundwater 18(4), 386-394.

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Maggiore, M. and Mongelli, F. (1991) Hydrogeothermal model of groundwater supply to San Nazario Spring (Gargano, Southern Italy). Proceedings International Conference on Environmental Changes in Karst Areas, IGU-UIS.

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Quarto, R. and Schiavone, D. (1994) Hydrogeological implications of the resistivity distribution inferred from electrical prospecting data from the Apulian carbonate platform. Journal of Hydrology 154, 219-244.

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Influence of water recharge on heat transfer in a semi-infinite aquifer

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