Jun 10, 1990 - convoluted, as in (7), for a finite duration pulse and may obscure problems with the test procedure. Thus it is not well suited to an analysis of ...
JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 95, NO. B6, PAGES 8697-8704, JUNE 10, 1990
Borehole Determination of Formation Thermal Conductivity Using a Thermal Pulse From Injected Fluid S. E. $ILLIMAN Department of Civil Engineering, University of Notre Dame, Notre Dame, Indiana
C. E. NEUZIL Water Resources Division, U.S. Geological Survey, Reston, Virginia
Thermal conductivity (A) of a formation can be determined in situ by pouring or injecting liquid at an arbitrary temperature into a thermally equilibrated borehole and analyzing dissipation of the resulting thermal pulse. We used this technique in a shale by pouring diesel fuel into a borehole and obtainedvaluesfor A within a few percent of the mean value obtainedfrom cores. The test showedthat the technique is relatively simple to apply and can readily be used to test multiple horizons simultaneously. A priori estimation of formation specific thermal capacity was necessary to obtain good results, apparently becausethe form of the thermal pulse was not well characterized in our test. This resulted from adding liquid to the borehole in a poorly controlled manner; injecting liquid using a tremie pipe would provide a better characterizedpulseand is probably advisablewhen implementing this technique.
INTRODUCTION
Knowledge of thermal conductivity values and the structure of the thermal conductivity field in the subsurface is required for heat flow determinations [e.g., Sass et al., 1981; Villinger and Davis, 1987], for analyzing geothermal energy storageand extraction [Murphy and Lawton, 1977], and for designof high-level radioactive waste repositories[Pollock, 1986]. The possibility of extracting information on past climates from geothermal data [Lachenbruch and Marshall, 1986] suggeststhat knowledge of thermal conductivity may be important for these endeavorsas well. Thermal conductivity A in geologic media varies by approximately an order of magnitude [Roy et al., 1981]. In commonly encountered lithologies the range is smaller, and often knowledge of the lithology or experience permits an estimate which is accurate within a factor of 2. Nonetheless, thermal conductivities in similar lithologies can vary by a factor of 4 or 5 [Roy et al., 1981], and estimates, even when soundly based, may not fulfill regulatory or engineering requirements. Thus thermal conductivity measurementscan be a practical necessity in a variety of situations. Most commonly, A is determined in a laboratory using core samples[Roy et al., 1981] or occasionallydrill cuttings [Sass et al., 1971]. However, situations may arise where samples are not available or laboratory testing is inconvenient or inadequate. Laboratory determinationsusing shale cuttings, for example, are not very accurate (J. Sass, U.S. Geological Survey, written communication, 1988). In these situations, in situ borehole measurement of A is required. Besides obviating the need for samples, in situ procedures offer the advantagesof (1) measurementunder appropriate moisture, stress,and temperatureconditions;(2) avoidance, to a large extent, of sampling-inducedalteration of the medium, particularly if the medium is unconsolidated or Copyright 1990 by the American Geophysical Union. Paper number 90JB00131. 0148-0227/90/90JB-00131 $05.00
"incompetent" [Beck et al., 1971]; and (3) measurement at a mesoscale,which may be more meaningful for many applications than the small scale of laboratory tests. The latter is true particularly if the formation is heterogeneouson a small scale. In such instances numerous laboratory tests may be required to characterize the mesoscale or large-scale conductivity, whereas relatively few in situ determinations would
suffice.
In situ measurementof thermal conductivity has generally been accomplishedwith a heater probe either dropped to penetrate soft submarine sediments [Bullard, 1954; Lister, 1979; Villinger and Davis, 1987] or, for measurements on land, lowered into a preexisting borehole [e.g., Beck et al., 1956, 1971;Beck, 1965]. These techniques have been highly developedover the years [seeBeck et al., 1971;Davis, 1988] and usually provide good results. However, they also have disadvantages. Particularly when making measurements in boreholes on land, care must be taken to minimize or to account for problems related to contact resistance [Beck et al., 1956, 1971;Davis, 1988], convection in the borehole fluid [Beck et al., 1971], nonradial heatflow [Blackwell, 1954; Christoffel and Calhaem, 1969], and time lags in the measured thermal response [Beck et al., 1956; Davis, 1988]. In addition, probes may be long, heavy, and unwieldy, and supplyingsufficientpower may be inconvenient[Becket al., 1971]. The drawbacks of using heater probes make alternative approachesfor terrestrial measurementsdesirable. Indeed, some techniqueswhich do not require heaters have been discussedand, in a few instances, applied; Beck et al. [1956] filled cement cylinders with water and analyzed the dissipationof the thermal pulse, and Murphy and Lawton [1977] describe a technique which utilizes temperature changesin water flowing along a borehole. Our purposein this paper is to describe a heaterlesstest procedurethat we have used to measureh at multiple levels in a 260-m borehole. The test, which provided promising results,useda proceduresimilar to that suggestedby Beck et al. [1956] and involves thermally perturbing the system by
8697
8698
SILLIMAN
AND NEUZIL:
BOREHOLE
DETERMINATION
addingliquid to the borehole. Besidesrequiring no heater, it is relatively simple to analyze because the mathematical model is closely approximated. The procedure has the additional advantage of permitting simultaneous testing of multiple horizons in the borehole. We performed the test in a shale for which comparable laboratory determinations of A from core sampleswere available. Comparison of the results suggests that this in situ technique can provide a useful measure of thermal conductivity. In considering our test, we will emphasize a general limitation of in situ testing: measurementof thermal conductivity of porous media is usually feasible only when relatively little heat is advected by fluid movement in the formation. In a laboratory the specimencan be confined to prevent flow. This is not possible in situ, so borehole thermal conductivity tests of this type must be restricted to environments of relatively low permeability. The rate of groundwater flow which can be tolerated during in situ testing varies somewhat with conditions. It can be calculated by requiring advective heat flow to be an arbitrarily small fraction of the conductive heat flow induced by the test. As we will show later, for the test we conducted, advective heat flow would remain acceptably small for groundwater flow rates (Darcy
OF FORMATION
Ti
I
I
THERMAL
I
I
I
'
I
I
I
I
I
I
I
I
o tl
CONDUCTIVITY
I
t2
n TIME
Fig. 1. Diagram illustrating treatment of a noninstantaneous thermal pulse. The actual temperature history is depicted with the heavy curve. The test is begun at time to, when the pulse begins. The pulse is not completed until tn, when temperature begins to
returntoequilibrium. TimeoriginforT(t')andT*(t')istakentobetn, the moment the perturbation ceases.
whereTf is the temperature and Kthe thermaldiffusivityof
flux rates)of 10-9 to 10-8 m s-• or less.This condition the formation. (See the notation list at the end of the paper wouldgenerally bemetwherehydraulic conductivity is 10-8 for explanationsof symbols.) The temperature changein the
borehole resulting from heat flow between the borehole fluid and the formation is described by
m s-• or smaller. TEST METHODOLOGY
AND THEORETICAL
BASIS
rwCOT/Ot= 2A OTf/Or
In situ determination of thermal conductivity A using a singleborehole requires the addition or removal of heat from the borehole and observation of the ensuing heat flow between the borehole and the rock as manifested by the subsequentchangesin borehole temperature. The technique consideredhere adds or removes heat by adding liquid of an arbitrary temperature to the borehole, displacingthe fluid (either water or air) already present. The general test methodology may be outlined as follows. Prior to the test, temperature sensorsare suspendedin the hole at horizons of interest, and the borehole fluid, sensors, and surrounding rock are permitted to reach approximate thermal equilibrium. The test is initiated by introducing liquid of an arbitrary temperature into the borehole. The liquid may be poured from the surfaceif the borehole is dry or pumped in using a small tremie pipe. Once a sufficiently large thermal perturbation is indicated by all sensors, the flow of liquid into the borehole is stopped, and the return to thermal equilibrium in the borehole is monitored while mixing the liquid over a short vertical distance. The mixing prevents formation of horizontal thermal gradients in the borehole
where T is the borehole fluid temperature and c its specific thermal capacity. The boundary conditions which apply in this case are
Tf(rw,t) = T(t)
t> 0
(3a)
Tf(•, t)= Ti Tf(r, O)= Ti
(3b) r > rw
(3c)
T(0) = To
(3d)
where TO is the borehole temperature at its maximum excursion from equilibrium. Equation (3a) implies perfect thermal connection between the formation and the liquid. Equations (3b) and (3c) require a radially extensive formation in comparison with the penetration of the perturbation. Condition (3d) implies that the borehole fluid temperature is
perturbedinstantaneouslyby T0. The solutionof (1) at r = rw provides the borehole temperature. Constrainedby (2) and (3), it can be expressed as
r(t) -
fluid.
Continuous monitoring of the thermistors in the borehole can be readily accomplished with currently available data loggers to provide an essentially continuousrecord of temperature showing the initial thermal equilibrium, the perturbation and maximum temperature excursion, and reequilibration (see Figure 1). These data permit computationof the near-field
(2)
= T*(t)=f(a,
13)
(4)
where Ti and To are the initial borehole temperatureand temperature at the maximum excursion, respectively; and T(t) is the subsequent, time-dependent borehole temperature. In this case
value of A.
In such a test, where radial thermal gradients far exceed those in the vertical, the conduction-dominated heat flow in the rock bordering the borehole possessesapproximate radial symmetry and can be described by
t([(O2Tf/Or 2)+ (1/r)(OTf/Or)] = OTf/Ot
(1)
.g(., a) 4a f0o• exp (-21•u2/a)
f(a,,8)=•-5
du
(5)
where
g(/t,O•)----[/tJo(//)- o•Jl(/t)]2 q-[/tYo(/t)- o•Yl(/t)]2 (6)
SILLIMAN
AND NEUZIL:
BOREHOLE DETERMINATION
and J0(u), Jl(U), Y0(u), and Yi(u) are zero-order and firstorder Bessel functions of the first and second kind. Apparently first obtained by Jaeger in 1940 [Jaeger, 1956], (5) was presentedby Bullard [1954], Jaeger [1956], and Carslaw and Jaeger [1959, p. 342], among others. Generally, (5) must be evaluated numerically; tabulations have been presented by Bullard [1954], Jaeger [1956], Cooper et al. [1967], Papadopulos et al. [1973], Bredehoeft and Papadopulos [1980], and Davis [1988], among others. The dimensionless quantity c• is twice the ratio of the
OF FORMATION
THERMAL
CONDUCTIVITY
8699
theactualT*(• iscomputed usingtherecorded valuesof To, Ti, andT(t). Havingestimated co (thusspecifying a), A is obtainedby comparing the theoreticalresponseT*(t) for the
observed perturbation, adjusting A to obtain the best match. The matching procedure is iterative and may be formalized and automated or, as in our case, accomplishedmanually by graphical comparison. An alternative procedure, often used to analyze in situ heater tests, does not explicitly consider TOand Ti. Knowledge of the heater output is used to compute A from the late specific thermal capacity of the rock, co, to that of the time temperature response to the heater pulse. When a • 2 boreholefluid, c, definedby c• = 2co/c. The dimensionless and dimensionlesstime 13/a> 5 the relation between T and quantity/3 dependsdirectly on thermal conductivity A and is f(a, 13) is approximately linear and depends only on the definedby /3 = At/Crw 2. Because the temperature behavior amount of heat added to the borehole and A [Lister, 1979; dependson both co and A, it is possible, in principle, to Villinger and Davis, 1987; Davis, 1988]. This approach obtain both quantities through an analysis of a test. Howpresents problems of nonuniqueness when the solution is ever, for reasons we discuss below, it is advantageous to convoluted, as in (7), for a finite duration pulse and may obscureproblems with the test procedure. Thus it is not well estimateco a priori. Equation (3d) applies when the perturbation from equilib- suited to an analysis of tests when the perturbation is rium is instantaneous. The time required to apply the ther- significantly noninstantaneous. Additional impediments to mal pulse in our test ranged from 3 min to nearly 13 min, the use of the linear approximation are related to the nature which is probably representative for this type of procedure. of the procedure under consideration. For example, in the This violates the prescribed instantaneity of the pulse suffi- test we conductedand discussbelow, a did not approximate ciently that the response is not well described by (5). This 2, and, because of the relatively large borehole used, large nonideal aspect of the procedure may contribute to error in dimensionlesstimes were not attained; 13/aremained less the determination of A, but the analysis can be readily than 1 at the end of the 4-hour test. Discussing the linear modified to account for it. Because (1) is linear, a noninstan- approximation technique, Davis [1988, p. 241] states that taneous displacement from equilibrium may be approxi- reliable values cannot be obtained under these conditions. mated by several small instantaneoustemperature changes Boreholes available for testing, particularly those of any offset in time as shown in Figure 1. The actual temperature depth, are apt to be sufficientlylarge that attaining 13/a> 5 change during reequilibration can then be approximated by would require inconveniently long test periods ranging from hours to 1 day or more. Even if the test were extended summingf(c•,/3) for each componentperturbation, that is sufficiently, the heat flow occurring at such large dimensionless times is relatively small and sensitive to heat advection by any water movement in the borehole and between the
T*(h =T(h Ti_It= • •To ATk To --Ti - Tiflc(a' /3) i>0 (7)
borehole
wherefk(a,/3) is the functionf(a,/3) initiated at time tk with
instantaneous temperature changeATe.For T(• and T*(• thetimeoriginis takento be tn, andt is defined by i = t tn (see Figure 1). This formulation is applicableto perturbations applied at arbitrary and varying rates. The approximation provided by (7) can be made arbitrarily accurate by refining the discretization of the pulse. However, our experience in using (7) suggeststhat discretization beyond approximately 5 time increments makes little difference in the result. We have used (7) to analyze the data presented in this
and the formation.
Because
of this and the accurate
values of To and Ti available from the temperature record, we analyzed our data using (7) as outlined above. APPLICATION
AND RESULTS
Using the procedure outlined above, we conducted a test in a borehole
in the Pierre
Shale
in central
South
Dakota.
The Pierre Shale in this region is a massive, highly montmorillonitic clay stone of approximately 30% porosity. It is approximately 300 m thick at the drill site and has a surficial paper. weathered zone of some 15 m. The shale's hydraulic and An important advantage of the technique we tested is the mechanicalproperties and pore fluid behavior at the drill site close correspondencebetween the test and the relatively have been investigated in studies by Bredehoeft et al. [1983], simple mathematical description embodied by (7). The well- Neuzil and Pollock [1983], Neuzil et al. [1984], and Nichols stirred borehole liquid is in contact with both the thermistors et al. [1986]. These studies have shown that the unweathered and the borehole face; time lag in the thermistor response shale is saturated and, at the scale of a borehole, of quite low and contact resistance should be negligible. Assuming that permeability. At such a subregional scale the hydraulic
the borehole temperature history between to and tn is accurately known, (7) has the additional advantage of correctly accounting for the heat added to or removed from the borehole during the perturbation. In proceduresusing heaters, the heat input is known from the heater characteristics. In the present procedure, which uses no heater, the heat input must be computed, as in (7), from the boreholethermal history and the borehole fluid specific thermal capacity c. Using the thermal history of the perturbationand response as depicted in Figure 1, the data can be analyzed as follows:
conductivity of theshaleis no higherthan10-l• m s-• and is probablycloserto 10-13 m s-1 [Bredehoeft et al., 1983; C. E. Neuzil, manuscript in preparation, 1990]. A 13.4-cm-diameter, 253-m borehole was drilled using air rotary equipment and completed in 2 days. Steel surface casing was eraplaced in the upper 18 m and grouted to seal out water inflow from the weathered zone. Although the shale is saturated, this borehole, like others we have completed, remained dry. This is partly attributable to the low permeability and subhydrostaticpore fluid pressurespresent
8700
SILLIMAN
AND NEUZIL:
BOREHOLE DETERMINATION
OF FORMATION
in the shale [Neuzil and Pollock, 1983] but probably also reflects near-field fluid pressurereduction causedby inelastic dilatation
'•' ['•
of the shale.
had been attained
between
the borehole
• 180 rn
22
2O
uJ
and the
surrounding rock. The test was begun when a thermal perturbation was produced by pouring diesel fuel into the open borehole from the top. Filling progressedas the diesel fuel free-fell
CONDUCTIVITY
24
Within 24 hours of completion, four thermistors on a 200-m cable were suspendedin the hole at 180 m, 150 m, 120 m, and 90 m depth. In addition to the thermistors, plastic mixing bladeswere affixed to the cable at 2-m intervals over the length of the cable. After 18 hours in the borehole the thermistors indicated that air temperatures were nearly stabilized (showing changes of less than 0.01øC in the preceding hour) and were also quite close to those at comparable depths in a nearby thermally equilibrated borehole. We interpreted this as indicating that approximatethermal equilibrium
THERMAL
"" 20
or ran down the borehole wall. Diesel fuel was
used becauseit is nonwetting and preserved the borehole for future use. Diesel fuel has the additional advantageof having a smaller specific thermal capacity than water, which reduced the time necessary for completing the test. During filling and for the duration of the test (approximately 4 hours) the thermistor cable was cyclically raised and lowered 2 m approximately every 15 s, causingthe mixing bladesto keep the borehole fluid locally well mixed. This prevented formation of radial thermal gradients within the borehole, a requirement implicit in (3), and kept the thermistors in thermal equilibrium with the fluid. The mixing length was maintained at approximately 2 m to prevent vertical mixing over a longer distance. The temperature record at each of the four thermistors is plotted in Figure 2. The direction of the temperature perturbation (positive or negative) was determined by the ambient formation temperature. The temperature of the diesel fuel as it entered the borehole varied between approximately 19.5øC and 22.0øC. This was lower than ambient formation temperatures at 180 m and 150 m and higher than those at 120 m and
18,
,
,
9
I
,
11
I
13
15
TIME, IN HOURS
Fig. 2. Temperaturehistory at four test levels in the boreholein the Pierre Shale. Plots from the shallow and deep test horizons are
separatedfor clarity. The initial temperature(Ti) at each level is indicated by arrows. To (see also Figure 1) is the maximum departurefrom equilibrium and occurs at the minimum value in the upper two curves and at the maximum in the lower two curves. The
timeoriginfor T*(t')is takenat tn (i.e., •' = t - tn).Theapparent temperature drop shortly after 1000 CST was caused by changing data loggersand was removed from the data before analysis. The time when mixing in the borehole ceased, at 1315 CST, is clearly visible
as inflections
in the curves.
borehole. The temperatureof the dieselfuel was closeto the ambient 150-m temperature, and the large temperature de90 m. crease measured during filling may have been caused by The maximum departuresfrom equilibrium in Figure 1 can evaporative cooling of the tip by the pooled fuel. The large be seen to be displacedin time, reflectingthe time required temperature excursion caused by the evaporative cooling to fill the borehole. Filling beganat 0914:30 Central Standard was reducedrelatively abruptly when the thermistorbecame Time (CST) and was completed by approximately 0935 CST. submergedand began showingthe actual temperature of the The thermistors indicated small perturbations almost imme- dieselfuel. Becausethe form of the temperatureperturbationat diately, with the initiation of the main pulse (to in Figure 1) 150m was poorly defined,thesedata were not analyzed. within 1.5 min, except at 120 m where nearly 5 min passed Data from the remaining three thermistors were analyzed before it appeared. The duration of the pulsesindicated by to estimate,• at 180m, 120 m, and 90 m depth. We estimated the thermistors(correspondingto tn - to in Figure 1) were, the value of a on the basis of the circumstancesand design from deepestto shallowest, 3.7, 2.6, 12.2, and 12.7 min. The of our test. Measurementsreported by Nichols et al. [1986], irregularity in these times and in to presumablyreflectsthe obtainedfrom dessicatedshale cores, indicate Csis 2.30 J local conditions at each level, such as the proximity of the cm-3 øC-]. Forthesaturated 30%porositywhichprevailsin thermistor to the wall and vagaries in the route followed by the shale [Nichols et al., 1986] its bulk specific thermal the incoming diesel fuel. This suggeststhat because of the capacityco can be computedusing poorly controlled manner of filling the borehole, the characco- (1 - n)Cs+ ncf (8) terizationof the thermalpulsebetweento and tn providedby the thermistors
in this test was not as accurate as one would
Assuming thatcf = 4.20Jcm-3øC-2forwater, thevalue
wish.
of cofrom(8) is 2.86J cm-3 øC-•. The specific thermal
At 150 m the temperature minimum was anomalouslylow, and the return to thermal equilibrium was anomalously steplike. We attribute this response to the fact that the housingfor this thermistor was inadvertently oriented facing up, causing diesel fuel to pool around the thermistor tip
refiner,was 1.75J cm-3 øC-•. The corresponding a hasa valueof 3.27.Usingvaluesof To, Ti, and T(• fromthe temperature record,the observed T*(• wascomputed and compared withthe theoretical T*(• for the samevaluesof
before
To, Ti, and a by plotting on semilogarithmicgraph paper.
the diesel fuel
itself
had risen to that level
in the
capacity of the diesel fuel, based on information from the
SILLIMAN
AND NEUZIL'
BOREHOLE DETERMINATION
For each set of data, A was adjusted to obtain a reasonable match. Although a formal fitting procedure could have been used, we do not believe it would have materially improved the results; our matches were obtained subjectively. The degree of fit is illustrated by Figure 3, which shows the computed response compared with the data from the 90-m thermistor.
The
values
of A we obtained
are 1.34 W m
OF FORMATION
THERMAL
1.0•--•.,,••.•
CONDUCTIVITY
;, I
0.8 -
8701
I
' I
'
'\••X•
-
-1
øC-• at 180m, 1.41W m-t øC-• at 120m, and 1.34W m-•
0.6-
\'t
-
øC-• at 90 m. DISCUSSION
OF RESULTS
We were fortunate that laboratory measurements of A from a core of Pierre Shale from this site were reported by $ass and Galanis [1983]. The 1.5-m-long core from approximately 50 m depth yielded six values for horizontal A
0.4-
than 10% in the stratum we tested. Therefore
we believe that
comparison of our in situ values from various depths with the mean laboratory value is meaningful. The computed values of A fall within a few percent of the mean laboratory value and are bracketed by the extreme laboratory values. This very close agreement is probably fortuitous; fitting the computed responsesto the data involves an estimated error of plus or minus 10-15%. This error estimate may be judged by examining Figure 4, which displayscurvesfor A values 15% higherand lower than the fit 1.0
0.8
0.4
•x
10o
I
I
I
10•
102
103
, I 104
-
• 105
TIME (t), IN SECONDS
Fi•. 4, •atch between observed (circles) and computed (cu•cs) temperature responsein borehole for • = 3.27:90 m depth. The curve on the left representsthe convolutcd solution for • = 1.54
W m-] øC-]' the curveon the d•ht represents the convolutcd
solution for• = 1.14W m-• øC-•;circles represent obsc•cd temperatures. These values of • are 15% hi•hcr•d lower than the subjectively Qttcd best value of 1.34 W m-' øC . in Figure 3. Except for the upper third of the observed response, these curves bracket the data. It is instructive to consider the discrepancy in the observed and theoretical behavior at early times. Examination of Figures 3 and 4 shows that the theoretical response underestimates the observed T* during the first several minutes. This was true at all three analyzed depths. The early responseis sensitiveto the circumstancesat the initiation of the test, specificallythe duration of the pulse (tn
_
The discrepancy is not due to the noninstantaneousquality of the pulse, a fact clearly illustrated in Figure 3. This figure shows both instantaneouspulse solution results computed with (4) and noninstantaneouspulse solution results computed with (7) for the 90-m level. Two instantaneouspulse approximations of the test are shown; curve A is the response to an instantaneous pulse which duplicates the
observedtemperatureexcursion(To - Ti), and curve B is the responseto an instantaneouspulse which duplicates the heat input computed by applying (7). The noninstantaneous pulse analysis provides a much closer fit to the data but still overestimates the early response. The fact that the early behavior
0.0
_
- to in Figure 1) and the form of the pulsebetweento and tn.
0.6÷
0.2
0.0
-
•
0.2
ranging from1.25to 1.49W m-• øC-• witha meanof 1.38W m-• øC-•. Despitethisrangeof valuesona scaleof several centimeters, geothermal profiles measured in this borehole (S. E. Silliman, unpublished data, 1987), and a nearby borehole [Sass and Galanis, 1983] indicate homogeneous thermal properties on a scale of meters, with A varying less
•••
_
is not well described
even when the noninstanta-
neous nature of the pulse is accounted for suggeststhat the 101 102 103 104 l0 s 100 poorly controlled filling process and the resulting poor TIME (t), IN SECONDS definition of the pulse between to and tn are responsible. Fig. 3. Match between observed and computed temperature Becauseof dieselfuel streamingdown the borehole walls the -1 response in borehole for a = 3.27:90 m depth, A = 1.34 W m temperature perturbation at the borehole face could have øC-1. Circlesrepresent observed temperatures, andthesolidcurve been more abrupt and sustained than indicated by a therrepresentsthe analytical solutionconvolutedusing(7) to accountfor mistor suspended away from the borehole wall. The rethe time (870 s) between to and tn at this level. Dashed curve A represents the response to an instantaneouspulse at t n which sponsefollowingtime tn would then be slowerthan expected duplicatesthe observedtemperatureexcursion(To • Ti) for the from the thermistor response. same a and A values. Dashed curve B representsthe responseto an Another potential source of the error lies in the necessity instantaneouspulse at the midpoint between to and tn which duplicatesthe test heat input (obtainedby summingall ATk) for the of keepingthe boreholeliquid well stirred. It is implicit in the same a and A values. technique that no radial thermal gradients exist within the
8702
SILLIMAN
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BOREHOLE DETERMINATION
OF FORMATION
THERMAL
CONDUCTIVITY
whether co can generally be estimated independentlywith sufficient accuracy to obtain good results with the test. The least certain componentsof c0 in saturatedformations are specificthermalcapacityof the solid, cs, and porosityn. Accordingto Roy et al. [1981], cs for mostgeologicmaterials
1.0
0.8
lies within 20% of 2.3 J cm-3 øC-1, and the extensive 0.6
0.4
0.2
O.qo 0 '
101
102
103
104
105
TIME (t), IN SECONDS
Fig. 5. Matches between observed (circles) and computed (curves) temperature responsein borehole at 120 m with two values of a: the solid curve uses the a priori estimate a = 3.27 and yields A = 1.41 W m -1 øC-l' the dashed curve uses a = 0.327 which
provides a subjectively goodfit andyieldsA = 3.14W m-1 øC-1.
tabulations of rock thermal properties presented by these workers make available values of cs for many specific lithologies. Thus in most instances the uncertainty in cs should be relatively small. The greatest uncertainty will generally lie in estimating n, the formation porosity. Using (8), the effects of errors in porosity can be examined. In our case, rather large errors (estimated porosity values of 0.10 and 0.50 compared with the actual 0.30) yield a values of 2.84 and 3.70, respectively. The correspondingerrors in computed A are approximately 10%. Thus more realistic error ranges in porosity estimates (e.g., 10%) can be expected to result in errors of a few percent in the computed A. This test procedure, like others for use in vertically oriented boreholes, provides a measure of horizontal A. Vertical A may differ, particularly in bedded formations. However, Sass et al. [1971] state that when present, anisotropy is often only 10% or 20%. Thus borehole tests would be helpful even if vertical A is the quantity of interest. In the case of the Pierre Shale the laboratory tests indicated an
anisotropy of 16%with a meanverticalA of 1.20W m-• borehole. The strongest tendency to form such gradients øC-• [Sassand Galanis,1983]. occurs at the beginning of the test when heat flow between IN SITU THERMAL TESTING AND GROUNDWATER FLOW the borehole and the formation is greatest. Inadequate mixing could cause a delay in the early response and may Except for water flow in the borehole itself, discussedby contribute to the early time discrepancieswe observed. Beck et al. [1956, 1971] and Murphy and Lawton [1977], the The suspicionthat the early responsecould not be readily role of heat advection by groundwater has received relaanalyzedled us to subjectivelyweightthe later responsemore tively little attention in the context of in situ thermal testing. heavily during our analysis. Our approach in this regard is Fluid flux within the formation and between the borehole similar to that of Bullard [1954] and experimentalistsconduct- and the formation during testing is potentially disruptive. ing analogoushydraulic tests [Trimmer et al., 1980]. The fact Significant components of both horizontal and vertical that accurate values of A were obtained despite these proce- pore fluid flow exist in many settings. In the present context dural difficultiessuggeststhat the techniqueis fairly robust. the horizontal flow component is most significant because Although the procedural shortcomingsappear to have had borehole fluid in formations with ambient horizontal flow is little effect on the computed A, they appear to have contrib- steadily exchanged. Depending on the rate of exchange, uted to the necessity of obtaining an independent estimate of dissipationof a thermal pulse can be accelerated,indicating c0. In each of our tests, different matchescan be obtainedby a thermal conductivity which is too large. permitting a to assume values different than the estimate of Our test provided an opportunity to compute the rate of 3.27. This is demonstrated in Figure 5, which compares ambient horizontal groundwater flow which could be tolercomputed curves for a = 0.327 and a = 3.27 with the data for ated without degradingthe response. By referring to Figure 120 m. The value of 0.327 for a, which is 10 times smaller 2, it can be seenthat the smallest average rate of temperature than our estimate of the actual value, clearly represents change, dT/dt, is near the end of the test and is approxiunrealistic specific thermal capacity values for the system mately0.04øC h-• . The corresponding fluxof heatbetween componentsbut provides a responsewhich fits the data quite the borehole fluid and the formation can be computed using well. The values of A computed from the fitted solutionsin
Figure5 are 1.41W m-• øC-• whena = 3.27and3.14W m-• øC-• whena = 0.327,thelattervaluebeingapparently erroneous. This clearly points up the relative insensitivity of the test to storage properties, a characteristic recognized from earlier thermal tests [Beck et al., 1956, 1971; Lister, 1979] and well known from hydraulic tests [Cooper et al., 1967]. It also shows that in the absence of an independent estimate of a the test can provide an erroneous result. The relatively extensive suite of data available for the shaleat this site allowed us to computec0 with a highdegree of confidence. In most situations, of course, such sitespecific data are not available. It is useful to consider
F = «rwc(dT/dt)
(9)
whereF is the heat flux per area of the boreholeface. For the
testweconducted, rw = 6.7cmandc = 1.75J cm-3 øC-1. WithdT/dt= 0.04øCh-• , (9) gives0.65W m-2 for F. In contrast, the advective heat flux per area of the borehole face is given by
F a = qc(Ar)
(10)
where q is the horizontal groundwater flux and T is the temperature difference between the borehole fluid and the formation fluid. Equation (10) ignores groundwaterflow per-
SILLIMAN
AND NEUZIL:
BOREHOLE DETERMINATION
OF FORMATION THERMAL CONDUCTIVITY
turbationsnear a borehole [e.g., Freeze and Cherry, 1979, p. 430] but is order-of-magnitudecorrect. Making the conservative assumptionsfor our site of a unit hydraulicgradientand a
Liquid pumped in
Tremie pipe •....,,
hydraulic conductivity of 10-• m s-• q is 10-11m s-• Using AT= 4øCandc -- 4.2Jcm-3 øC-•,representing water,then(5) indicates thatFa is 1.7x 10-4 W m-2. ,
8703
ß
Comparison of F a and F leads to the conclusion that
Thermistors
advective heat flow was, at most, a small fraction of 1% of the total heat flow during the test. By arbitrarily specifying
that an advective componentof up to 1% of total heat flow
Mixing blades
(Fa = 6.5 x 10-3 W m-2) is tolerable, thecorresponding q is 3.8 x 10-lø m s-1. Becausehydraulicgradientsare usually smaller than 1 and often less than 0.1, this suggests
thata hydraulicconductivity of 10-9 to 10-8 m s-1 could have been tolerated, with the possibilityof somedegradation of the late portions of the response. ASSESSMENT AND IMPLEMENTATION OF THE TECHNIQUE
Our results suggest that the technique we utilized is capable of providing reasonably accurate (error less than 10-15%) in situ determinations of horizontal A. In our view the principal advantagesof the technique are (1) the absence of a heater and (2) the easewith which multiple horizons can be tested simultaneously.Certain limitationsof the technique are apparent as well. It cannot be applied in boreholeswith axial flows and generally shouldonly be used in relatively low permeability formations. However, the latter restriction should be consideredwhen usingother techniquesas well.
Certain aspectsof the resultsraise questionswhich cannot be answered definitively without further testing. The inability to match observed early time responseappears to have resulted from a deficiency in procedure, namely, filling the borehole with liquid in a poorly controlled manner. This apparently little affected the determined A but probably is the primary reason independent estimation of a was important in our analysis. A more accurately characterized pulse would permit more accurate tests and perhaps eliminate the need for a prior estimate of a. For this reason a variation of the procedure we followed may prove desirable for implementing this technique. We suggesta modifiedtest procedureas outlined
below.
Figure 6 depicts a borehole ready for testing. The hole may be dry or filled with water. In addition to the sensorsa tremie pipe is suspendedin the hole. The thermal massof the pipe, if significant, can be accountedfor by incorporatingit in the heat capacity of the borehole fluid. To conduct the test, water is pumped down the tremie. The pumped water displaceswater or air already present from the bottom up, producing a regular and easily characterized thermal pulse. For example, in a borehole already filled with water, addition of cool water through the tremie will cool the lower sensors,while warm water will be displaced upward, warming the higher sensors. The test can be repeated, if necessary, to provide usuable temperature pulses at all the hori-
Fig. 6. Configuration for proposed test procedure. The assembly must be capable of being raised and lowered during the test to mix the borehole liquid. The tremie pipe shouldbe small to minimize the thermal mass of the liquid inside.
radial advective heat transport if the flow is occurring over the entire section. Measuring the geothermal profile in the boreholefluid prior to testingshouldindicatewhether a preexisting axial flow is presentwhich couldjeopardize the test. Our tests were conducted
borehole.
How-
a metal casing.Steel is approximately 102 times more conductive than most formations; no significant thermal gradients would be generated across the casing wall. The casing thermal mass would be aggregatedwith that of the borehole fluid, and the borehole radius would be effectively equal to the outer radius of the casing. In practice, however, testing through casing may present problems. Casings are often in poor contact with the rock, and grout is sometimes injected to fill annular voids. Poor thermal contact between the formation and the casing, and the presenceof grout, may provide misleading results. Beck et al. [1956] discussed some of these problems and suggestedthat they may decrease accuracy or make some resultsunusable.Murphy and Lawton [ 1977]useda borehole with grouted casing and concluded that the grout had little effect
on the results
because
it and the formation
had a
similar A. They also suggestedthat the thermal perturbation from the test penetrated deeply enoughthat the properties of the grout did not affect the test. In any event, tests in cased borehole should be viewed circumspectly. CONCLUDING
zons of interest.
It may be advisable to monitor the liquid level in the well during the test, particularly if significant amounts of liquid are added to initiate the test. A declining liquid level signals loss to the formation as a result of increased hydraulic head in the borehole. If sufficientlyrapid, it may cause errors by (1) axial flow and heat advection along the borehole, particularly if the flow is to one or a few discrete fractures, and (2)
in an uncased
ever, many boreholes are cased over the intervals of interest. Casing provides the desirable assurance of no fluid exchange between the borehole and the formation, and in principle, in situ testswould be unaffectedby the presenceof
Few
borehole
determinations
REMARKS
of formation
thermal
con-
ductivity have been attempted with heaterless techniques. Results of the test described in this paper indicate that fluid injection into a borehole can be a viable alternative to heater testing. The relative convenience of the injection procedure and its ability to test multiple horizons simultaneously may make it an attractive option in some applications.
8704
SILLIMAN
AND NEUZIL:
BOREHOLE DETERMINATION
Our results suggest that the injection procedure is not sensitive to nonideal conditions.
Inconsistencies
between the
observedand the theoreticalthermal responsesat an early time in our test may have resulted from a procedural deficiency, namely, addingthe liquid to the boreholein a poorly controlled manner. Resolution
of these inconsistencies
and refinement
of
the procedurewill require further testing. NOTATION
C borehole fluidspecific thermalcapacity(M T -2 L -1 øC-•). cb bulkformation specific thermalcapacity (M T -2 L -1 oc-•). cs formation solidsspecific thermalcapacity (M T -2 L-1 oc-1).
cœformation fluidspecific thermal capacity (M T-2 L -1 oc-1).
n porosity (dimensionless).
q groundwater specific discharge (L T-l). r
rw t to tn t F
radial distance from borehole center (L).
well or borehole radius (L). time (T). time of initiation of pulse (T). time of completionof pulse (T). t - tn, time since completion of pulse (T). total heat flux, per area of borehole face, between
borehole andformation(M/T3). F a advective heat flux, per area of boreholeface,
betweenboreholeandformation(M/T3). T
temperature of borehole fluid after completion of pulse (øC).
Tf temperature in theformation(øC). Ti
initial or equilibrium temperature of borehole fluid (oc). To temperature of fluid in borehole at maximum displacementfrom thermal equilibrium (øC). T* = (T - Ti)/ (To - Ti) (dimensionless). a - 2c•/c (dimensionless).
• = Xt/cr•(dimensionless). K= a/Cb,thermaldiffusivity (L T-2). • formation thermalconductivity (L M T -3 øC-l). Acknowledgments. We wish to thank Steve Ingebritsen, John Sass, A. E. Beck, and Earl Davis for their constructive reviews.
OF FORMATION
THERMAL
CONDUCTIVITY
Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, 510 pp., Oxford at the Clarendon Press, London, 1959. Christoffel, D. A., and I. M. Calhaem, A geothermalheat flow probe for in situ measurement of both temperature gradient and thermal conductivity, J. Phys. E, Ser. 2, 2,457-465, 1969. Cooper, H. H., Jr., J. D. Bredehoeft, and I. S. Papadopulos, Response of a finite-diameter well to an instantaneouscharge of water, Water Resour. Res., 3(1), 263-269, 1967. Davis, E. E., Oceanic heat flow density, in Handbook of Terrestrial Heat-Flow Density Determination, edited by R. Haenel, L. Ryback, and L. Stegena, pp. 223-260, D. Reidel, Hingham, Mass., 1988.
Freeze, R. A., and J. A. Cherry, Groundwater, 604 pp., PrenticeHall, Englewood Cliffs, N.J., 1979. Jaeger, J. C., Conduction of heat in an infinite region bounded internally by a circular cylinder of a perfect conductor, Aust. J. Phys., 9(2), 167-179, 1956. Lachenbruch, A. H., and B. V. Marshall, Changing climate: Geothermal evidence from permafrost in the Alaskan Arctic, Science, 234, 689-696, 1986. Lister, C. R. B., The pulse-probemethod of conductivity measurement, Geophys. J. R. Astron. Soc., 57(2), 451-461, 1979. Murphy, H. D., and R. G. Lawton, Downhole measurementsof thermal conductivity in geothermal reservoirs, J. Pressure Vessel Technol. Trans. ASME, 99, 607-611, 1977. Neuzil, C. E., and D. W. Pollock, Erosional unloading and fluid pressuresin hydraulically "tight" rocks, J. Geol., 9•(2), 179-193, 1983.
Neuzil, C. E., J. D. Bredehoeft, and R. G. Wolff, Leakage and fracture permeability in the Cretaceous Shales confining the Dakota aquifer in South Dakota, in Proceedings of the First C. V. Theis Conference on Geohydrology, edited by D. G. Jorgenson and D. G. Signor, pp. 113-120, National Water Well Association, Worthington, Ohio, 1984. Nichols, T. C., Jr., D. S. Collins, and R. R. Davidson, In situ and laboratory tests of the Pierre Shale near Hayes, South Dakota--A characterizationof engineeringbehavior, Can. Geotech. J., 23(2), 181-194, 1986. Papadopulos,S. S., J. D. Bredehoeft, and H. H. Cooper, Jr., On the analysisof "slug test" data, Water Resour. Res., 9(4), 1087-1089, 1973.
Pollock, D. W., Simulation of fluid flow and energy transport processesassociatedwith high-level radioactive waste disposalin unsaturated alluvium, Water Resour. Res., 22(5), 765-775, 1986. Roy, R. F., A. E. Beck, and Y. S. Touloukian, Thermophysical properties of rocks, in Physical Properties of Rocks and Minerals, vol. II-2, edited by Y. S. Touloukian and C. Y. Ho, pp. 409-502, McGraw-Hill, New York, 1981. Sass, J. H. and S. P. Galanis, Jr., Temperatures, thermal conductivity and heat flow from a well in Pierre Shale near Hayes, South Dakota, U.S. Geol. Survey Open-File Report. 83-25, 10 p., 1983. Sass, J. H., A. H. Lachenbruch, and R. J. Munroe, Thermal conductivity of rocks from measurements on fragments and its application to heat-flow determinations, J. Geophys. Res., 76, 3391-3401, 1971.
REFERENCES
Beck, A. E., Techniques of measuring heat flow on land, in Terrestrial Heat Flow, Geophys. Monogr. Ser., vol. 8, edited by W. H. K. Lee, pp. 24-51, AGU, Washington, D.C., 1965. Beck, A. E., J. C. Jaeger, and G. Newstead, The measurementof the thermal conductivities of rocks by observationsin boreholes, Aust. J. Phys., 9(2), 286-296, 1956. Beck, A. E., F. M. Anglin, and J. H. Sass, Analysis of heat flow data--In situ thermal conductivity measurements, Can. J. Earth Sci., 8(1), 1-19, 1971. Blackwell, J. H., A transient-flow method for determination of thermal constants of insulating materials in bulk, I, Theory, J. Appl. Phys., 25(2), 137-144, 1954. Bredehoeft, J. D., and S. S. Papadopulos,A methodfor determining the hydraulic properties of tight formations, Water Resour. Res., 16(1), 233-238, 1980. Bredehoeft, J. D., C. E. Neuzil, and P. C. D. Milly, Regional flow in the Dakota Aquifer: A study of the role of confining layers, U.S. Geol. Surv. Water Supply Pap., 2237, 45 pp., 1983.
Bullard, E., The flow of heat throughthe floor of the Atlantic Ocean, Proc. R. Soc. London, Ser. A, 222(1150), 408-429, 1954.
Sass, J. H., D. D. Blackwell, D. S. Chapman, J. K. Costain, E. R. Decker, L. A. Lawver, and C. A. Swanberg, Heat flow from the crust of United States, in Physical Properties of Rocks and Minerals, Vol. II-2, edited by Y. S. Touloukian and C. Y. Ho, 503-548, McGraw-Hill, New York, 1981. Trimmer, D., B. Bonner, H. C. Heard, and A. Duba, Effect of pressure and stress on water transport in intact and fractured gabbro and granite, J. Geophys. Res., 85, 7059-7071, 1980. Viilinger, H., and E. E. Davis, A new reduction algorithm for marine heat flow measurements,J. Geophys. Res., 92, 12,846-12, 856, 1987.
C. E. Neuzil, Water ResourcesDivision, U.S. Geological Survey, 431 National Center, Reston, VA 22092. S. E. Silliman, Department of Civil Engineering, University of Notre Dame, Notre Dame, IN 46556.
(Received July 18, 1988; revised January 9, 1990; accepted January 10, 1990.)
