Thermal losses from fluids upwelling in a conduit, and the effect on surface heat flow

ELSEVIER

Tectonophysics 257 ( 1996) 39-53

Thermal losses from fluids upwelling in a conduit, a on surface heat flow C.R.B. Lister

’

Received 2 May 1994; accepted 25 January 1995

Abstract The therml disturbmx to fluid flow in a fracture conduit of limited extent can he madelled well by conduction from a cylindrical pipe. At u depth large compnrcd to the largest dimension of flow, as seen in cross-section, the hc the surroundings becomes numcricnlly about equal to the thermal conductivity of the rock. There are no fre the flow approaches an isothermal surface ut 90°, there is a somewhat greater loss near the surface. This is disturb the strong and nearly linear correlation bctwecn tcmpcraturc and tlaw rate expected for one-way up othermi\l gradient, in the rungc of tlow mtcs bctwcen 0. I and I(3 kg/s. The distuncc over which flows npp npcn~turc rise (or Ml, il‘ downwclling) is i\pplUXi~lIiWly NSS flow X specific h&It =+ thermal ~~~n~‘uctivity~The w rise is thitt lcnprh X the gc&crM rwrhunce to the surlhsc hart flow around II upwclling varies inve c)y with radius out 1 about 20eg of the depth und than drops 01’1’ more s~cply, The purnmctcr is tcmpcruturc rise sonductivity + 2 s changes the heat 190~ by more hn IO% out tn u r:rdius of’ I km, if the source vertical gradient mcusured in i\ borehole is disturbed substantii\Hy to u depth equal to th borehole. The sr)lution could be applied to diffusion of a chemicul species if that process could be tnuted lin~~tr~~~

fforts have been made to measure heat flow in tectsnie zones, since the thermal state, if‘ fix the history and mechanics sf the motion. Tectsnism is usually cxxmected to therefore there is likely to be a component of fluid transport that is upward et‘fus~~ in addition to the iTlowsdriven by rainfall on elevated terrain and ltny convective G below. If the flows were unitbrm and diffuse over la e areas, the problem could k~ one-dimensional differential equntion. However, the fie information is that mineral sp discrete places, even though kese 119ay be located on fault zones. Thus the two~dirn~~~iQ~~

’Deceased. 0040-1951/96/$lS.O0 0 1996 Elsevier ScienceB.V. All rights reserved SSDI 0040- I95 I (95)OO I 19-o

is unlikely to apply in the near-surface re 8 cyiindricni conduit.

ion,

and

it

may be better to consider the other end-member:

The abstraction of the pro em to be solved in this pri r is best described by a diagram _-_ Fig. 1. The similarity to an oil well tappin an oil sand is obvious, and ,idiszar ( 1958) onuiysed that specific problem. He studied the tx1_..Amt heat-flo oiutions for im infinite cyii icai hole in Jaeger ( 1942) and Jaeger and Ciarkt c 1942) and decided thrrt he could approximute the heat loss as a fixed thcrtM resistance. However, at the long conduction loss could npproach steady-state flow to the surface. This , which is limited to piu+ticuiilrparmeter sets and Sorsy ( runsicnt and numerical steady-state rcsuits, but u ~~~919r(~xi~~~~~te m~iytic solution for the steady-state i%42),This COUI~ then he ~xt~i~d~~l to estilni\tc the ~~i~t~6rb~~~~~~ 01‘wfirce hut fiiw duo tar Iha( ~~~~~~~~ rruclhcip IOw&riW whcthcr such distortions of surfwe hcut t1ow CrN&lbe detected,

C. R.B. Lister / Tuctmophysics

257 f I9961 39-53

41

f;ig. 2. (A) Equivalence of tlow through a IateraHy limited crack, or collection ot’cracks, to Ilow through a cylindrical conduit, for the purpose of cidculuting conduction losses. An increase in the local resistance can be modelled by reducing ;he cqkv;tlenc radius (1. (

he

cylindrical geometry for the solutions of Corsli\w and Jaeger ( 1947): ,q increases upward from 0 to 1. while s represents u small increment of final path to the surfxe,

(I and b iwe the inside and outside radii of il cylindrical region of conductingsolid.

A solution for the general temperuture is uvtiili\ble for the solid 01 boundary I’= n is held at .f’(t ) while all other surfaces tire held at zero letting h tend to infinity and obtaining the radial gri\dient of ternpcraturc ut I* to:

where y( I) is the heat loss per unit length of pipe at height t, and the K’s are Bessel functions of the fou~h kind. The series is not absolutely convergent unless f’(:) goes smoothly to zero at ;: = 0, 1 and the _ reason for this is clear. If j’( z) +-0 tit z = 0,1, then points at r = CIon the top and bottom boundaries have two prescribed temperatures: j’(z) and 0. The summed heat flow throu these points is infinite. t it is useful to see wh be handled by changing the boundary conditions, but for the mo from this relatively simple formula by considering only the terms in small H, which eonv These terms contain the physically reasonable heat loss front ihe conduit at depth.

When the arguments of the Bessel functions are either very small, or very large, there are simple asymptotic formulae for them (Boas, 1966). It is presumed that in the real world I s a, so that for small n: and !$,(~)=ln(&)

(5)

and the result for the heat loss can be rearranged to look like: ‘1

(6) The value of this can be seen if j’( z) is replaced by its Fourier transform:

whereupon y becomes:

dd = 2nk

sin

Apart fromthe 1 arithm factor, this is very sitnilar to the early terms in expansion (71, and we have already $&?en thnt the vu1 N. So for smooth tcmperrrture cs little with rrr~umcnt when I

functions we hr\vc valiclsrtcd that:

y eondwtive me, rend not the yuusi-iarsubtive one t it is ilt 2330 t~~~~~~~~t~l~~ is not il problem bWJ\J!X a ny other change.) Perhaps the easiest wiry to illustrate lsss

4 from

t = 0 to z = I:

the ef‘kst

of this is to put

C.R.B. Lister J Tectonophysics 257 (I 996) 39-53

43

and, taking the logarithm as the value for the first term:

16Zk Q=

/ ?TIn -

( na )

? c

1

=a (2m + 1)? I”

z

16Ek

d

2nlk

(12) **n(G)

7-z

In(&)

where the sum of the (infinite) series, rr2/8 = 1.23, shows the dominance of the first term, and validates the removal of (2~ + 1) from the logarithm. Finally, this result can be converted to the one for the insulated base by dividing the region 0 < z < I into two symmetrical halves. This is done by setting I = 21’and halving the

total heat loss from the conduit:

. (13) showing that the effect of the insulated base increases the logarithm by about IO%, and decreases the heat loss by the same, modest, factor. The above demonstrates that the general heat loss from a conduit at right angles to a cold surface is, to

geophysical accuracy:

proportional simply to the temperature difference distance. If the spring has mass flow rate III and crmal gradient is h, and the flow is in int scale), then the t~~llpcratur~ in the rising

between the conduit surface and rock horizontally at a specific heat c, assumed constant (Le., no boilin thermal contact with the walls (a sutisfics the dit’fercntial equation

and this has the solution:

The temperature of the sprin settles toward ht above ambient after rising a distance 1, from the SOUKX area (Fig. 3). Using the general parameters k = 2.5 W/m”C, c = 4180 J ’ 1 = &wJo fn 00 ~~bv a h one can calculate representa geechemical thermometric temperature at the source), and b = 0.035°C/ /S, c1 s 0.4 XII,and ii 1 resulrs for a small spring, CII= r one, III= 0.76 kg/s, Q = 4 m. Theye values cover the range of raw flow fo s near Scuol, Switzerla that unfortunately cannot be discussed filcher in this paper (but see Wexteen et al,, I%#), and the CI’Sare pure guesses based on the sort displayed by uncurbed seeps there. The results are II,= I93 and 1300 m, bL = 4.8 and 46°C. a stro mperature such as reported by Bodvarsson ( 1950) but not shown by the between flow rate ing the guessed parameuzr u over wide limits has relatively little effect, provided the mentioned above, u criterion is adhered to. The solution presented so far has infinite heat loss near the surface, and this is a re iQ” where the dcptb below surface (s in Fig. 2B) s not necessarily large compared to Q. Clearly, the near-sur e problem be shirked, and the infinity must be removed by more realistic boundary conditions. These are available. and

can not

44

C.K.B. Lister / T~?~to,rt~ph!‘sics257 f 1996) 39-53

ise that 11surface must rise in temperature in order to shed heat into air or water. For small surface losses iire lineilrly proportional to temperature rise whether the mechanism is convection tern 01”rrrahtion, Mid ili% h%St ~~~~~S~~~~tit!4 D t~~~~i~~i~l ta?ncc(W/m2”C) divided by the condu Then the boundary condition bccames

the boundaries at 5 =

CR&

Lister / Tectonophysics 257 (1996) 39-53

45

a result that is still exact, and still not intuitively usable. A small simplification can be made by using the (good) approximation that h! 3 1:

but now further analytic progress can be made only by obtaining crude sums for groups of terms. This has been done by multiplying the number of significant terms by the average term. There are three groups: the early terms, the terms where sin’( $a,, S) is near 1, and the terms where sin( cy,,s) is near 1. After the first maximum, the sine terms oscillate rapidly enough for those parts of the series to contribute negligibly to the sum. The procedure is crude enough to leave some uncertainty in the numerical scale of the final formula, so this has been “recalibrated” by letting s become large and comparing the result to Eq. (13):

Q(s) = 15ak

2,628s ‘23928 23843 2.2267 I ,9602 I .7919 PD I a6757 I .S%43 I .4290 I ,2974 I,2280 1.1559 I.1 I87 I .0803 I SKM I ,048’~ I .0409 I .0308

C.R.B. Lister/ Tectonopi~ysics 257 ( 1996139-53

45

The recalibration is also in agreement with the intuitive result 2mzkhs when s is very small ( +C a), and flow through the rock comer is limited only by h. The centre term only adds significantly to the result when hs is small, very close to the outle;, where the theoretical geometry and the assumption of intimate contact between spring water and wall both break down in the real world. For all practical purposes it can be left out (though it was needed for the two numerical calibrations to agree simultaneously). The result (2 1) is likely to be more useful to most readers when they are shown how well behaved the ratio K,/K, is. Hence it is given in Table 1, together with asymptotic formulae for both large argument and small argument. There is only a small region, between arguments 0.18 and 0.5, where one or the other of the formulae does not fall within 10% of the real function. For springs that show a significant temperature rise, Its is large enough to be removed by assuming it to be large: h is at least 5 m-l (still air) and the effective value for s (see below) at least 100 m. This converts Eq. (21) into: KI( +J

KI( Y)

K,,( ;;)

+ K[,( 7)

U’s = l’

(22)

I z (s/nn) ln(s/na), to within 1% (Table l), and the he ratio K,(~TYI/s)/K,,(T~/.s) K’s with argument ntr/&s can be replaced by their small-argument asymptotes with an even greater accuracy.

This criterion is not as strict as was required t’or q. ( 14) to be seen physically to apply, because now the solution explicitly deals with the proximity of the surface. Hence:

re the only remin s und (I is within the 1 ms, and the second term within the kcts is a correction 4) can now be compared q. (14): s replaces 1, and the slightly smiler numericul factor, 5.0 instead of 6.3, is cmpensuted by the loss factor of two inside the urithm. A question thut arises immediately is whether the heat losses in the on governed by the valu f II are nitkant for even CLsmulI sprin For cx1~mple,one cstn compute the loss per “C temperature rise t?orcl = 0.3 m, s = 3 m, k = 2.5 W/m”C kindfor k = 5 in - ’ (air) or 1000 in - ’ (water): thut the vnluc of Ir is not important (the only purpose of the third signil’icmt f spri the tm’mgdj lass rate is 8. /u+T ov~*rthe lust 3 ni, cotnp;lr We IIIIWexpect fi to vrrry IV3~ the vuk chosen for s, and the inevitable assunrp the dther hand, we know fkm Eq. (161 that the spring t~~~~p~~~t~~~ settles to 1 constant y ) in 11distuurcet = rm*,k/ (c being the specific heat of the fluid), The solved fur the varying q as the surfxe is approached, but, in view of the other approximations required, a

C.R.B. Lisler/ Tectonophysics 257 (1996) 39-53

47

reasonable approximate result is to select s = L and then estimate spring emanation temperature as bL (b being the geothermal gradient). For a small spring, a = 0.4 m, s = 100 m, rn = 0.083 kg/s, 6 = 0.035”C/m: Q(100)=312ak=312W/“C;

L= 110m; 6L=4OC

which compares to the 68°C calculated before. Similarly, for the large spring where m = 0.76 kg/s: Q( 1000) = 309 ak = 309OW/“C; L = 103Om; bL = 36°C

(26) a =

4 m, s = loo0 m, (27)

This is less than the 46°C calculated before, but the reduction ratio is smaller than for the small spring, enough to negate the effect of the larger a and make the temperature appear approximately proportional to flow rate over this range. In effect, near-surface losses sharpen the transition from the near-ambient temperature of small springs to the near-source temperature of large springs. For the special simplification of holding a comtant, the near-surface equilibration length L can be plotted against flow rate, showing the deviation from what bvould, without the near-surface losses, be constant thermal loss (Fig. 4).

3. Transient effects Since it takes many millennia for conductive equilibrium to become established for a deep-source spri (I > 1 km), the obvious question arises whether transient cooling could not be much greater for a relativ young channel. The time range of most interest is the one short enough to allow the use of the solutions for the infinite conducting solid bounded internally by a cylinder of radius a. When heat has not diffused t’tlrfrom the channel, the presence of the additional boundaries in Fig. 2B has little effect, and the enormous col~plex~ty of the full solution can be avoided. For the most realistic case of a constant, non-zero temperature established at ives the s~~l~~ti~~)~~ t = 0 on the surface r = a, the rest of the solid initial temperature zero, Jite 8kt Ez

l(O.1 J)

where y is Euler’s constant 0.57722, Immediately, one can do a “reasonability” test by finding the time at which the asymptotic solutists b~c~~~~~~

so the transient heat leakage in the infinite solid decays to the steady value with a cold upper boundary whamsth reaches about half the depth of burial. This is reasonable, and sets an upper limit on the

diffusion length 6;

L=

m

) and the first term ablt: frm I when the first term alone only does so at T> 100,

C. R. B. Listw / Tectonophysics 257 f 19%) 39-53

first

large

term only

’approx

1

l/a

100

..-..a..-..w

..mm”W,,

.

upproxirnation itself breuks down for ream e wlues of l/a Appropri values of 100 and loo0 ureincludedin Fi to show how the ~~~r~~i steady-state one, Only at very short times, T nt heat one, and it takes 11T of arbswt0.1 to increase it by an order of ma nitude. The unit ;ind for CJ = 4 111and K = 10-’ n+/s it is about I/ es in the path of spring waters. Most observed springs have historica iues of T > 100 are appropriate, und the transient heat losses ure no except t’ora very deeply buried new channel. 4. Preliminary

of

conelushs

The conclusion of this relatively careful analysis of the amductive temperature I

by circulation from a considerable temperatures re The value of 1: Sor sprin medium-size sp very close to equilibrium, even if of recent o other hand, springs of greater than 10 kg/s up whether these are surface aquifers or buried dewatering zo changes in conduit size or path cannot change this result, nor c;an transient cooling f’sr existence times of

years or more. Intermediate-sized springs between 0, I and 10 kg/s should show a correlation between exit temperature atid flow rate, the relation being somewhat steeper than linear because of the component of near-surface cooling that is greater than the almost-constant loss at depth. In this range springs are very susceptible to additional quasi-advective cooling if they traverse an aquifer with independent transverse flow. Another source of additional cooling is flow from the elevated end of a conduit (such as one sealed by chemical barrier deposition) laterally downwards towards a deep river valley. This may well apply to the Swiss springs mentioned above as they show some mixing with local surface waters, in spite of having higher densities than fresh water (based on the chemistry in Wexteen et al., 1988). On the other hand, the logarithmic temperature drop that develops in cylindrical geometry (2). ensures that interaction between neighbouring springs is confined eneral influence on the geothermal gradient.

ct on surfaceheat The solution for the geometry of Fig. 213from C&law and Jaeger (1947, p. 191). with h allowed to tend to infinity, can be differentiated in the vertical direction:

(32)

Hc)w this series is summed de

nds on the value of I’. It is useful to recapitulute the approxim; tions for K,,

C.R.B. Lister/ Tectonophysics 257 (19%) 39-43

Sl

subject to recalibration. According to Eq. (36), each ring 27rr dr puts out flux 2k dr, but each segment of conduit d z has loss only k d z, from Eq. (14) if ln(2l/na) = 2~. From the general way the flux lines go (Fig. l), each segment of conduit should feod a similar ring (d 1 = dp), so the recalibrated result is: k for r < I (in practice < 0.11) 4(r) -2.,,,., (37) where f( 5 ) = 1 has now been replaced by fl[ ) = AT, the temperature rise of the spring above ambient.

---L--B

(b E 20 K/km)

--w--l,

For values of r Larger than 0.1 I, the series must be summed term by term. When 7~r/l > 1, the large argument approximation can be used for the numerator Bessel fun&on, while the denominator remains in the logarithmic range for the significant terms: (2~ + 1)7rr

(

J&J

I

-

i

(21M+ l)nct

41

(

I

Z-

1 6

(

(2m

exp - -

+

1)nr

1

1

(38)

1

where the value l/6 garithm implies I/cr = 1270. This value increases for increasing 1~ but is Bearing in mind that the value of a is uncertain and could easily vary with compensated by the 1 c represented accurately enough by a declining geometric series based on the depth, later terms can incrementing exponential. In the transition region 0.1 I < r < 0.31 actual values of K,, need to be used for up to four terms ancl the remainder estimated from the geometric series. Representative results have been plotted for a spring with a 10°C temperature rise in Fig. 6. The choice of I = 10 km might seem large, but this solution is symmetrical and applies to an isothermal boundary at depth (flowing aquifer). For the more likely case of a stagnant aquifer, or insulated boundary, the effective source depth is only 5 km. Heat flow near the spring is perturbed substantially, the ambient flux of 60 mW/m2 for this increased by IIIOTCthan 1 % out to nearly 1 km (825 m). The overall flux picture provides another The heat advected throu h the surface by the spring waters themselves is nearly 15 kW; the extra the 1 km radius circle of significant disturbance is 30 kW, and the total amount of heat heat conducted advected away nant aquifer at 5 km depth is 146 kW.

in the nuturcrlworld, and a region where one is known may ot’tenhave many more. The problem is complicated by the ftlct that most springs of deep origin do not appear at the surf&e, but are dischar ed into an aquifer. If the waters contain subs ial reduced iron, they are capable of forming impermeable b icrs by precipitation, This can inqpedesprea diffusion into the aquifer and may allow an identifiable spr to appear where the aquifer crcii q out, Many others may not be so identifiab and yet their transport of heat from depth is still presen%.If Letitf’lowis measured by borehole logging, the diont below the last significant aquifer is used, and this is increased by any cryptic springs that may be present. Detection of the contribution of springs or seeps to the heat f’lowis possible in three ways, If there are known contilining reduced iron, there is a high probability of active metamorphism below the ternlined by nugrhy boreholes varies substantially, and there are no other plausible ex t’or this, such as ~.onesof rock which ~~~w~~~h~s conductivity, then the influence of springs can be inferred. Finally, C)IW should suspect fluid upwelli if the heat flow at successive depth intervals in the borehole es, cspeciulty if it dccreasus with d Both for the case of a borehole ~~~~lle~i~~g a conduit at SOIIX distance und fhe ctlse of myriad tiny seeps ~oneadime~~sion~~l porous flow), the apparent vertieal flux will decrease with depth, but, in the case of the nearby conduit, to much greater depths than for simple porous flow. A natural conduit IWY follow u tortuous path, especially if the direction of the rock fabric ch;lnges with depth-so the solutions presented here are only a starting point, demonstrating the magnitud:: of the effects.

C. R. B. Lister / I’ectonophysics 257 f 1996) 39-53

53

7. Conclusions Given that the flow of a spring at depth can be modelled usefully by flow in a cylindrical conduit, &e analytic mathematics offers some strong constraints on the thermal history of the waters. The thermal gain or loss from a conduit at depth, in flux per degree temperature rise per unit length, is numerically about equal to the rock thermal conductivity in the same units. The transition between a negligible temperature rise and the ability to preserve the temperature of the source area occurs between the flow rates of 0.1 and 10 kg/s for rock of conductivity 3 W/mYZ. Disturbance to the surface heat flow around a one-way upwelling follows the out to a radius r of 0.2 of the source depth. This assumes that the source is a largely relation AT 9k/2nr stagnant aquifer that approximates a no perturbation-flux boundary. At greater radii, the surface heat flux disturbance falls off more steeply than l/r. Perhaps more importantly, the factor cos(rznz/l) in IEq. (32) shows that the vertical gradient in a measurement borehole parallel to a natural flowing conduit can be affected to a substantial fraction of source depth 1. The major disturbance extends to a depth roughly equal to the separation between conduit and borehole, as can be seen from the flow lines in Fig. 1. Large one-way flows acquire a large temperature rise relative to their surroundings, so that significant heat flow perturbations could extend to 1 km or more beneath the last significant aquifer encountered by a drill hole. The solutions can also be applied to the diffusion of chemical species if such diffusion along boundaries or through rock porosity can be modelled as a linear process. The critical flow rates that allo spring to acquire the local rock signature are then reduced by the proportion that chemical diffusivity X labil element concentration falls below the thermal conductivity. Discrete conduit flows that approach equilib~um with the rock by diffusion rather than direct chemical dissolution are likely to be small indeed.

Acknowledgements

eferences Boas, M.L., 1966. MathematicalMethods in the Physical Sciences.Wiley, New York, NY, 5’43 pp. Bodvursson, G., 1950. Geophysicalmethodsin prospectingfor hot wtiter in Icelund.Timarit V~rkfri~~din~~f~li~~sIsl., 35: 49-59, Boldiszur, T., 19513.The distribution of temperaturein flowing wells. Am. J. Sci.. 225: 427-435. Carsluw, H.S. and Jaeger,J.C., 1947. Conductionof Heat in Solids. oxford, 1st ed., 386 pp. Jueger,J.C., 1942. Heat flow in the region boundedinternully by u circular cylinder. Proc. R, Sot, Edinku

Jaeger,J.C. irnd Clarke, M., t942. A short tublr: of jr(e-‘“’

,/[Jlf(tr)+

Y,f(rr))Hdrc/ II), Proc. H. Sot. Edt

h, A., 61: ~~9~~~~~~

Lowell, IU.. 1975. Circulation in fri’actures,hot spr&s und convectivehe;tt trunsport on mid-oce;mridge crests. Geophys=J. II. Altron, Sot., 40: 3s I -x?s. Sorey, M.L., 1975. Numerzculmodelling of liquid g,Uhcrmul systems. Ph.D. Thesis, Univ. Calif.. Berkeley, CA. Truesdell, A.M., Nuthenson, M. imd Rye, I&Q., 1977. The effects of subsurfucc boiling and dilution OEI the i~olopic c(~rn~~l~~(~n of Yellowstone thermul waters, J. Geophys.Kes., 82: 3694-3704. Wextecn, P., Juffd, F,C. und Muzor, E., 1988. Geochemistryof cold C02-rich springs of the Scuol-Ruq~ rc~i~~l~~ ~WW ~n~~~~lifl~~ ?&v&t* Alps. J. Hydrol,, 104: 77-92.