JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 100, NO. B2, PAGES 2005-2012, FEBRUARY
10, 1995
Heat flow at Cajon Pass, California, revisited Arthur H. Lachenbruch, • J. H. Sass,2 Gary D. Clow, • and Ray Weldon 3
Abstract.In recentstudies of a 3Lkm borehole nearCajonPasswe showed thatthe 2 . observed high heat flow and its sharp decrease with depth are predictable effects of independentlydetermined erosion history, topography, and radioactivity, leaving little room for the large contribution from frictional heat required by conventional faulting models for the nearby San Andreas fault. We have since discovered an error in our
analysis thatlowersthepredicted surface heatflowfromtheupperend(-100 mW/m2) to thelowerend(-90 mW/m2) of therangeof measurement uncertainty at this complex site;it permits,butdoesnotrequire,a source increment of upto 10mW/m2 not accounted for in the prediction. Better agreement between the prediction and observationsat depth confinesthe permissibleextra heat flow to the upper part of the hole, making it difficult to attribute it to a deep frictional source. In any case, however, such a frictional source would be too small to attribute to conventional high-strength faulting models, and the basic conclusion of the original study is unchanged. The most likely cause of the relatively small discrepancy between predicted and observed heat flow (if it exists) is preferential three-dimensionalflow into the higher-thermal conductivity rock that occupies the upper part of the borehole. Introduction
In recent papers [Sass et al., 1992;Lachenbruch and Sass, 1992, 1988], we summarized our thermal measurements from
a 3•-kmresearch well at CajonPass,California, andesti-
possibly a uniform increment throughout the hole of about half that amount. Although this departure between predicted and measured heat flow lies in the fringe of measurement uncertainty at this complex site, it is worth investigating its possible sourcesand their implications, if any, for a permis-
mated the local heat flow and its likely perturbations from
sible
anomalous
largely superficial discrepancy is generally incompatible with a deep frictional source (it probably represents thermal refraction), but that even if it were not, such a source would generally be too small to attribute to conventional highfriction faulting models.
site conditions.
We found that the heat flow was
anomalously high(--•90-100 mW/m2)at thesurface, butthat it decreased downward to about --•75 mW/m 2 near the bottom of the hole. Although the high surface value was consistent with predictions of frictional heating from conventional strong-fault models, the rapid decrease with depth was not. We suggestedthat the observed heat flow behavior was readily explained, and to some extent anticipated [Lachenbruch et al., 1986; Lachenbruch and Sass, 1988], from independent estimates of erosion history provided by Weldon [1986], and that the heat flow at Cajon Pass did not support the high-strength model. We have since noticed that the mathematical approximation
used
to
estimate
the
sedimentation/erosion
effects
[Lachenbruch and Sass, 1992] contains an unacceptably large error (Appendix A). In this paper we show that although correcting the error makes the fit of the prediction to the data less impressive, the revised estimate still provides a good account of the large measured heat flow, and the major conclusions are unchanged. The effect of the correction is to decrease the predicted surface heat flow and its rate of falloff with depth; leaving the possibility of an unidentified anomalous heat flow increment as large as -10
mW/m2 confined largelyto the upperpart of the hole,or 1U.S.Geological Survey,MenloPark,California. 2U.S.Geological Survey,Flagstaff, Arizona. 3Department of Geological Sciences, Universityof Oregon,Eugene.
Copyright 1995 by the American Geophysical Union. Paper number 94JB02872. 0148-0227/95/94JB-02872505.00
increment
Predicted
Heat
of fault
friction.
We
conclude
that
this
Flow
The elements of the heat flow discussionare reproduced in modified form (from Figures 3, 4, and 5 of Lachenbruch and Sass [1992]) in Figures 1, 2, and 3. Figure 2 illustrates the three effects for which heat flow adjustments were calculated from independent evidence at the site. The straight line on the left is the expected decrease in heat flow with depth due to the measured rock radioactivity. Effects of changing elevation at Earth's surface are broken into two parts; the two curves in the middle represent the effects of evolution of three-dimensional topography relative to a horizontal reference plane through the present surface at the borehole and were calculated by the method of Birch [1950] assuming the evolution took place over the last 1 or 10 m.y. The second part of the surface correction is the (one-dimensional) effect of movement of the reference plane itself, i.e., of the history of sedimentation and erosion at the site, an estimate of which is shown in Figure 1. The preferred history, based on detailed field mapping and magnetostratigraphic,fossil, and radiometric age dating [Weldon, 1986; Meisling and Weldon, 1989; Weldon et al., 1993] is shown for the past 5 m.y. as curve b, Figure 1 (effects of earlier events would be uniform throughout the hole; they are treated as a contribution to "background"). The "ramps" (curves a and c) were inserted for convenient reference; they represent conservative
2005
2006
LACHENBRUCH
ET AL.:
HEAT
2.0
FLOW
AT CAJON
PASS
Figure 3a compares estimates of the background heat flow, predicted heat flow, and observed heat flow at the Cajon Pass hole (the erroneous "old" prediction from Lachenbruch and Sass [1992, Figure 5c] is dashed). The
b_a
"background" is takenasthemean(69mW/m2, Table1) of
1.0
18 surface heat flows obtained within 100 km of the site [Sass
0.5 0 0
1
2
3
I
i
4
5
Time B.P., million years
Figure 1. History of the depth of burial of present surface at Cajon Pass drilling site with observational control (dots and error bars) by R. J. Weldon [Lachenbruch and Sass, 1992]. Ramp-shaped curves (curves a and c) are simple bracketing reference models; curve b is preferred model.
upper and lower bracketing conditions of uniform erosion of 1.6 km over the past 1 m.y., and 1 km over the past 1.67 m.y., respectively. The advantage of these simple "ramp" histories is that their effects on heat flow (curves a and c, Figure 2) can be computed readily from well-known analytical results [Carslaw and Jaeger, 1959]. More refined estimates based on the preferred curve (curve b, Figure 1) require approximate methods because the velocity of the surface varies with time. As explained in Appendix A, the preferred curve can be approximated by superimposing constant-velocity ramps like curves a and c. The combined thermal effect can then be approximated analytically by adding exact results for the ramps, for which the nonlinear moving boundary condition results in a superpositionerror, or by linearizing the boundary condition and adding the approximate thermal contributions so obtained. In our previous discussion [Lachenbruch and Sass, 1992], we used the first of these methods which we later found to be unsatisfac-
tory because of larger than expected superposition errors (Appendix A). The incorrect result from this "old" calculation (shown in Figure 4 of Lachenbruch and Sass [1992]) is reproduced as the dashed curve b' in Figure 2. The curve labeled b in Figure 2 represents the revised (i.e., corrected) result. It was obtained from a numerical model originally devised (by Gary Clow) to accommodatethe corresponding thermal effects of accumulation and ablation in glaciers; it is described in Appendix B. The bracketing ramp models (a and c, Figures 1 and 2)
et al., 1992, Figure 1; 1986] reduced at depth by the effect of measuredlocal rock radioactivity. Adding the independently calculated anomaliesfor topography and the preferred erosion history (curve b, Figure 1) to the backgroundyields the predicted heat flow (solid curve, Figure 3a). Shown also are interval averages (solid dots) and error bars for heat flow measurements in the crystalline rocks [from Sass et al., 1992]and open symbolsand error bars (dashed)representing two different interpretations of heat flow in the overlying sediments (to be discussed further below). It is clear from Figure 3a (dashed and solid curves) that correction of the error substantially degrades the fit of the prediction to the observed heat flow. This is seen also in Table 1, where we include estimates of likely ranges for these quantities. The 18 nearby heat flows (ranging from 63
to 78mW/m2 witha meanof 69mW/m2) shownorelationto their distance from the fault, and we take their standard
deviation(-4 mW/m2, shadedregionsin Figure3b) as a measure of uncertainty in selecting the correct local background value. The same uncertainty is reflected in the predicted value (Figure 3b and Table 1); we ignore uncertainty in the difference (from Figure 2) between background and prediction. The range of observed heat flow is taken from the simple least squaresrepresentationof Figure 4. The revised prediction seems to agree rather well with observations at the bottom of the hole (Figures 3a and 3b). However, at the top of the hole the revised prediction leaves room for
an unaccounted for -10 mW/m2 increment, although both the revised and old predicted values probably fall within the combined uncertainty of estimating background and measuring heat flow (Figure 3b and bottom two rows of Table 1).
Heat flow anomaly, mW mTM -10
0
10
20
30
indicatethattheerosionhistorycontributes 12-25mW/m2to the surface heat flow; the contribution is reduced by about half at the 4 km depth. Correction of the "old" estimate adjusts the surface effect of the preferred history from
almost25 mW/m2 (curveb' dashed,Figure2) to about15
mW/m 2(curve b,Figure 2),butatthe35kmdepth there is little changein its contribution (fromabout11to 9 mW/m2). The erosion effect is well constrained from below; an improbable extreme lower limit (curve c, Figures 1 and 2)
Figure 2. Contributions to predicted heat flow anomaly at wouldreducethe surfacecontribution by only3 mW/m2. In Cajon Pass. Straight line on left is decrease in heat flow with depth resultingfrom rock radioactivity. The next two curves combination with effects of radioactivity and topography to the right are bracketing effects on local heat flow from (averaged for the 1 and 10 Ma histories, Figure 2), the evolution of local three-dimensional topography in 1 or 10 preferredcurveyieldsa surfaceanomalyof-20 mW/m2 (the m.y. Right-hand curves are calculated effects on heat flow of bracketing range is17-30 mW/m 2)anda falloff overthe37 preferred erosion history (curve b) and the bracketing histokm depthof the hole of about 17 mW/m2 (rightcolumn, ries (curves a and c) in Figure 1. Curve b is the corrected Table 1). version of the previously published curve b' (dashed).
LACHENBRUCH
ET AL.' HEAT
FLOW AT CAJON PASS Heat flow, mW mTM qo 7.4 80 90
Heat flow, mW m TM 60
70
80
90
qo' 7.0
100 6O 0
'
x/, ' . ,
1t
/ 1'
I.# -c•/ /-•
/ 4
//
,
,
,
i
,, .,.'
// ,
1 oo
I
E
//
2007
,
•iBure •a, Comparison of background, •redicted and observed heat flow as a fu•ctio• of de•th at Cajo• •ass.
AveraBe backsround heatflowisthemea•(6• mW/m•)of •8 •st• •or •cts o• r•io•cti•it• •it• •t• •i•r• •). •redicted heat flow is the sum of the backsround and the •redicted e•ects of to•oBra•hy and erosion from •iBure (Solid curve ("revised •redictio•") is codected for tio• e•or; dashed curve ("old •redictio•") is •ot.)
.
." ,-"'! ./'
_
I
_: ,'" prediction
..'
.....> !..,
/
!.' •//
I / / ,/
... /r / ,...'/,/
.'-•--• revised
.. ,..'
prediction ,
Figure 3c. Adjustment of background heat flow to obtain best least squaresfit of prediction to the heat flow data (solid circles). Solid line and solid curve (representing revised
prediction)were adjustedto the right by 5 mW/m2 from Figure 3a; dashedline and dashed curve (representing "old"
erroneous prediction)were adjustedby 1 mW/m2. Dotted curves represent solid curve -+2 standarddeviations; q0 and
q• areadjusted surface heatflow(69+ 5 and69+ 1mW/m2,
circles with error bars are mea• values of observed heat flow
respectively).
i• comerruinous de•th i•terva]s i• crystaHi•e rock (see •i• 4). O•n •kd•s and s•ar•s r•r•s•nt sc•m•s •or r•d•cJn• •at •o• data Jnt• •oro•s s•dJm•nts
is 74 mW/m2 (solidline, Figure3c). In this sense,the high
•s•
measured surface heat flow is reduced by the prediction to 74
t•0.
mW/m2, a plausibleundisturbed valuewithinthe rangeof As an alternate to the forward prediction in Figure 3a, for which background surface heat flow is assumed to have its
nearby measured heat flows [Sass et al., 1992, Figure 1]. If, however, the local undisturbed background were in fact the
meanregionalvalue,69 mW/m2, an additionalunidentified mostlikelyvalue(69mW/m2),wecanadjustthebackground uniformsourceof 5 mW/m2 wouldbe permitted(as dissurface heat flow (q0, Figure 3c) to obtain the best fit
cussedbelow, however, it could account for only 10 or 15 of the 50-100 MPa of friction expected of the strong-fault models). The dashed line and dashed curve (Figure 3c) curveof Figure3a to theright5 mW/m2, i.e., by adjusting illustrate the remarkably good fortuitous fit of the erroneous theassumed background from69to 74mW/m2 (q0, Figure old prediction. 3c). Thus the revised prediction (solid curve, Figure 3c) accounts for the heat flow observations best (in a least Heat Flow squares sense) if the background surface heat flow at the site Measured It seemsclear from Figure 3 and Table 1 that predictions from independently observed site conditions can account for
between the "prediction" and observations. For the revised prediction this is achieved by slidingthe solid line and solid
Heat flow, mW mTM 60
0
70
80
90
100
Table 1. Research
Predicted and Observed Heat Flow, Cajon Pass Well Bottom
Surface (0km)
Hole
(3•km)
Downhole
Decrease
E
Backgrounda
Topography b
Figure 3b. Estimated ranges of uncertainty of background, predicted, and observed heat flow. Shaded regions show effect of a range of -+1 standarddeviation in backgroundheat flow; hatched region is uncertainty in observed heat flow (from Figure 4).
69 --- 4
5
63 --- 4
0 9 (11)
6
5
Erosion c
15 (24)
Predicted c'•t
89 +__ 4 (98 +--4)
72 +--4 (74 +__ 4)
17(24)
6 (13)
Observed e
97 +-- 5
75 +__5
22 +-- 5
Values are in milliwatts per square meter. a"Background" is mean (+standard deviation) of 18 surface heat flows [$ass et al., 1992, 1986] within 100 km, less contribution of rock radioactivity at depth x kilometers (69 +__ 4 - 1.8x milliwatts per square meter).
øAverage of the 1 and10Ma estimates of Figure2. CErroneous "old" parentheses.
value from Lachenbruch and Sass [1992] in
•t"Predicted" is sum of first three rows. eAs describedby linear regressionin Figure 4.
2008
LACHENBRUCH
ET AL.:
HEAT
Heat flow, mW m TM 70
80
90
lOO
0
FLOW
AT CAJON
PASS
squares by Sass et al. [1992], which contains a discussion of the difference. Since the values differ by --•10%, the choice between them is significant to the discussion;both interpretations fall approximately within the formal range of two standard deviations from the straight line which is _+5
mW/m2 (dottedlines,Figure4). Because thesolidcirclesin Figure 4 represent independent heat flow determinations
(overconterminous intervals),we adoptthe -+5mW/m2 asa
1.0
measureof uncertainty in heat flow in Table 1 and Figure 3b. The width of the comparable error interval for the corresponding curvilinear fit (dotted curves, Figure 3c) is about the same.
E
.t 3.0
4.0
Figure 4. Mean heat flow in crystalline rock (solid circles) in contiguousdepth intervals (vertical bars) with horizontal error bars and sliding average [from Sass et al., 1992]. Open circles and squares represent alternative interpretations of heat flow in the porous sedimentary rock. Shown also is the heat flow regression line for crystalline rock +-2 standard deviations (dotted lines).
Abundant evidence for lateral inhomogeneity in the well has been discussed[Sass et al., 1992]. A comparison of the lithologies of the research well and its neighbor (the original Arkoma oil test) only 50 m away indicates a fault with 150 m of vertical displacement [Silver and James, 1988], and there is a sizable discrepancy in the form of the heat flow between the two wells (cf. Sass et al., 1992, Figures 12 and 14]. Additionally, although conductivity and gradient are negatively correlated [Sass et al., 1992, Figure 14], anomalies in their product, heat flow, are dominated by anomalies in conductivity. This suggests that the gradient profile is smoothed by laterally variable conductivity, i.e., by threedimensional heat refraction. It is likely that higher-order irregularities in the heat flow profile (Figures 3 and 4) are causedby horizontalinhomogeneitiesof comparablescaleand that they are local effects that will look quite different in holes drilled only a few kilometers apart. These downhole irregularities add to the difficulty of identifying the larger-scalepredicted anomaloustrends addressedin Figures 2 and 3.
DiscrepanciesBetween Observed and Predicted Heat
much or all of the anomalous high surface heat flow and its decrease with depth (with no contribution from frictional heat). The reason we do not know whether it is "much" or
"all" is because of our limited knowledge of the heat flow anomaly itself. In addition to the ever presentuncertainty in selectingthe proper "background heat flow," the two main problems are (1) uncertainty in how to estimate conductivity in the upper 500 m (sedimentary)section,and (2) uncertainty in the local effects on heat flow from laterally variable lithology. The error bars for heat flow (Figure 3) do not accountfor uncertainties of these kinds; they are only formal representationsof the statistical scatter of conductivity and thermal gradient measurements. A formal least squaresrepresentationof the linear trend in
heat flow with depth (Figure 4) gives a perspectiveon the
Flow
Significancefor Fault Friction
Conversion of anomalous heat flow to average fault friction is generally based on simple analytical heat conduction modelsfor idealized fault configurations[e.g., Henyey, 1968; Brune et al., 1969; Henyey and Wasserburg, 1971; Lachenbruch and Sass, 1973, 1980]. For the usual geometric assumptions they generally indicate that the maximum anomaly for a mature vertical fault (•>8-10 m.y. old) is greater than one-half the average rate of frictional heating on the fault; at a distance of 4 km from the San Andreas fault, the fraction could be as low as one third if the fault were only 5 m.y. old [Lachenbruch and Sass, 1992, Figure 7a; 1980,
AppendixA]. Thusa heatflowanomalyof 5 mW/m2would implyan averagefrictionalsourceof 10-15mW/m2, which,
highe. r-order(i.e., shorterwavelength) irregularities. For for a slip velocity of 3 cm/yr, suggestsaverage fault friction consistency the analysis is based only on the heat flow estimates in crystalline rock to which we assign equal weights(solid circles, Figure 4; the irregularcurve is average heat flow in a sliding 250-m window [Sass et al., 1992]. Because core samples were obtained from only 3% of the hole, the conductivity profile is based largely on estimates from drill cuttings which pose special problems for the porous sedimentary rocks that occupy the upper 500 m of the section. The open circles and squaresin Figure 4 represent estimates based on two plausible, but different, reduction
strategiesfor the samedata; the schemeleadingto the open circles was used by Lachenbruch and Sass [1988] and the
of 10-15
MPa.
To place our overall discussion in context, it will be recalled that early heat flow measurements near the San Andreas fault at Cajon Pass (in the Arkoma well [Lachen-
bruch et al., 1986] and phase 1 of the Cajon Pass research well [Lachenbruch and Sass, 1988]) indicated that the near-
surfaceheat flow was 20-30 mW/m2 abovethe regional background, as might be expected from frictional heating if the conventional high-strength(--•50-100 MPa average) frictional model applied to the San Andreas fault (i.e., MohrCoulomb frictional failure assuming Byerlee's law, hydrostatic fluid pressure, and failure angles between optimum
LACHENBRUCH
ET AL.:
HEAT
and 45ø [e.g., Lachenbruch and Sass, 1992, Figure 8]. The purpose of the related discussionby Lachenbruch and Sass [1992] was to show that the surface anomaly and its decrease with depth (documented in phase 2 drilling) had a different explanation (principally rapid local erosion) and consequently that the observations did not support the highstrength frictional model. The remarkable agreement (dashed curve, Figures 3a and 3c) between the heat flow predicted (incorrectly because of an error of calculation) and the observations left little room for any significant frictional heating, let alone the 20-30
FLOW
AT CAJON
PASS
2009
shown that significant vertical and lateral variation occurs on many scales (see section on lateral variability in the work by $ass et al. [1992, p. 2025]), and it is likely that a horizontal hole in the crystalline rock would expose variations in conductivity comparable to those observed in a vertical drill
hole. Averageconductivity in the upper15 km of the research well is -15% greater than in the lower 2 km [Sass et al., 1992, Table 7]. If the upper "layer" had a lateral
• km), thermal extentno greaterthanits depth(i.e., 0
T(y, t) = 0
t- 0
y = -H(t)
(A5) (A6)
where H(t) is a specifiedhistory of the height of the surface; it representsa specifiedhistory of deposition and erosion at
the rate j_/r(t)-- u(t). At the upper boundary, (A6) imposes the simultaneous conditions
mW/m2) can reasonably be ruledout. It shouldbe noted, however, that a refined upper limit on the permissible frictional contribution ("the heat flow constraint") is not easily established from measurements in a single borehole (even a deep one) because of the inherent uncertainty in establishingmagnitudesof both the present heat flow and the appropriate "background" value. Although deep boreholes provide unique information on the form of heat flow variation with depth, permitting discrimination among anomalous sources, the best source of information on the heat flow constraint remains the (negative) statistical results on variation in heat flow with distance from the fault at many sites [e.g., Brune et al., 1969; Henyey and Wasserburg, 1971; Lachenbruch and $ass, 1973, 1980, 1981].
Appendix A In this section we describethe problem of constructingan approximateanalytic solutionfor effectsof erosionand sedimentationon heat flow. In doing so, we presentgeneralgraphs for rapid estimationof the effectsand identify the natureof our error in the work by Lachenbruch and $ass [1992]. To find the one-dimensional
thermal
effect of erosion and
depositionat rate u(t), we considerthe followingproblem:The region x > 0 moves along the x axis with velocity u(t) (x is positive downward and u is positivefor deposition).The initial temperatureis ax, and the surfacex - 0 is maintainedat zero temperature. The equationsto be solved are
02T u(t) OT 10T Ox2
k
Ox
=
(A1)
k Ot x
T( x, O) = ax T(0, t)=0
x>0 x=0
t=0 t>0
(A2) (A3)
where k is thermal diffusivity. For the "ramp" histories of Figure 1, we have used the analytical solution to this problem given by Carslaw and Jaeger [1959, p. 388] for the special case in which u is
OT
dT= •
oy
OT
dy +
dy dt
ot
dt = 0
- J-/(t)
(A7a)
(A7b)
leading to the implicit relation OT/Ot
J-/(t)=
OT/Oy
(A8)
Thus when we specify the rate of erosion or deposition, we are specifyingthe ratio of the time and spacederivatives of temperaturenear the surface. As this conditionis nonlinear, if T1(Hi) and T2(H2) are temperaturescompatiblewith erosion/depositionhistories Hi(t) and H2(t), respectively, and if Ti2(Hi + H2) is the result for the combinedhistory Hi(t) + H2(t), then in general
Tl(H•) + T2(H2)y•T•2(H • + H2)
(A9)
and we must be careful in approximating effects of complex erosion histories by adding exact results for elementary forms, e.g., ramps. According to (A8), at a fixed point y* close to the surface OT
OT
• •.--
Ot y,
(AlO)
Oy
If thevelocityJ-/issoslowthattemperatures nearthesurface (at y = y*) adjustto the changingoverburdenwith little change in gradient,we can replaceOT/Oyin (A10) by the undisturbed gradienta. Under this condition,integrationof (A10) yields r = all(t)
y = y*
(A11)
which provides a new (and linear) boundary condition at a point y*, convenientlychosento be Earth's surface at the conclusion of the event H(t).
LACHENBRUCH
ET AL.'
HEAT
FLOW
0
would be an equalityif T were replacedby Tc); however, the resultingapproximation may or may not be a good one. For uniform erosion in the amount H at constantrate J/= H/t over a period of duration t (the "ramp" condition), the nonlinear boundary condition (A6) may be replaced with (A11) if the Peclet number for this problem
is sufficiently small. Figure A1 compares the anomalous surface gradient (or heat flow) normalized by its undisturbed value, for the ramp
G and its climatic approximationG c. Dimensionalresults (upper axis) show how the anomaly varies with H for events of 1 m.y. duration. (For example, uniform erosion of 1.5 km over 1 m.y. increases the surface heat flow by 34%; deposition of 1.5 km in 1 m.y. decreases it by 27%. The correspondingclimatic approximationsare both 30%.) The dimensionless lower axis shows the general relation between the two representations(G and Gc) of the heat flow anomaly and p (equation (A12)). The climatic approximation varies linearly with H (or p); the exact result does not. The approximation, Gc, underestimatessurfaceheat flow, G, for both erosion and deposition. For events lasting 1 m.y., the discrepancy is not large if erosion or deposition (H) is less than ---0.5 km. If t = 4 m.y., the correspondingdiscrepancy occurs when H • 1 km; more generally, it occurs when p •
H, km, for t = 1my
•-
0
1
2
0.4
o
0.2
o•(.D -a
0
•
4
8
12
16
18
-0.3 • .................................... 0
0.5
1.0
1.5
z=x/• .
0.1. The upper and lower axes of Figure A1 illustrate that the scale of the major event in the erosion history at Cajon Pass is roughly p --- 0.2 (where the discrepancy between G and G c is startingto become significant). Figure A2 shows correspondingresults for anomalous heat flow as a function of depth, dimensionless for the cases p = -0.1 and _+0.25read from the lower axis, or dimensional for events lasting 1 m.y. with H = _0.56 km and _+1.4 km, where depth in kilometers is read from the upper axis. (Ther-
maldiffusivity k is generally takento be 10-6 m2/s.) To approximate an irregular erosion/deposition history, one might consider trying the approximate addition of nonlinear exact solutions G, or the exact addition of linear
approximate solutionsG c. Noting that the climate approximation underestimated heat flow in both erosion and deposition (see Figure A1), we opted for superposition of exact results in the work by Lachenbruch and Sass [1992]. As this involved adding large erosion and deposition effects of opposite sign and large p, unacceptably large errors accumulated. After investigating this problem with the numerical model (Appendix B), it is clear that a combination of these schemeswould have given very good results, namely, using the exact solution for a ramp that fits well near the time origin and correcting it for the less important earlier events with a superimposed climatic approximation.
erosion'• •,,' Appendix B In this section
-(].2
N
Z
2011
Figure A2. Conditions of Figure A1 with disturbance shown as a function of dimensionlessdepth z (lower axis) for (A12) p = • 0.1 and •0.25, or depth x kilometers (upper axis) for H = •0.56 km and • 1.4 km for events of 1 m.y. duration.
p 2 = H2/kt
-1
PASS
x, km, for t = 1my
Replacing the boundary condition (A6) with (A11) reduces the problem of a half-space whose surface is eroding or accreting by H(t) to one undergoing a "climatic change" in surface temperature by all(t) (discussed by Carslaw and Jaeger [1959, p. 63]). We shall refer to this as the climatic approximation and denote it by the subscriptc. This type of approximation (replacing changing surface elevation by a temperature equivalent on a reference plane) is the basis for most [e.g., Jeffreys, 1938;Bullard, 1938;Birch, 1950] but not all [e.g., Jaeger, 1965;Jaeger and Sass, 1963; Lachenbruch, 1968] schemes for terrain corrections to heat flow. In the climaticapproximationthe thermal effectsof elementaryforms of H(t) may be added to represent complex ones (i.e., (A9)
-2
AT CAJON
calculate
deposition• ••-'•,,
-0.4 i
0
i
i
i
•
012
i
i
i
0.4
p = H/v•
Figure AI. Fractional disturbance to surface heat flow for uniform ("ramp") erosion or deposition in the amount H in time t shown as a function of p (on the lower axis) or H kilometers for t = 1 m.y. (on the upper axis). Solid curves represent exact result G; dashed are climatic approximation Gcß
we outline
the numerical
the effects of erosion
method
and sedimentation
used to
on heat
flow. It is applied to the solution of the one-dimensionalheat flow equation,
OT0 (KOTi uOT .c. =ZzzVzz/.c. Vzz +s wherep is density,Cp is the specificheat, K is thermal conductivity, T is temperature, t is time, u is the vertical velocity, and $ is the heat production rate per unit volume. Equation (B1) is solved using the well-known Crank-
2012
LACHENBRUCH
ET AL.:
HEAT
Nicolson scheme [see Richtmyer and Morton, 1967] modified to include the advective term. The algorithm consistsof solving a tridiagonal system of equations, Tn+l -Ai•i_ 1 + BiT•+1 - CiT•++l 1= Di
(B2)
where the subscripti and superscriptn refer to the depth and time nodes, respectively, in the space-timegrid. The coefficients are defined by
A i = Ai[1 - Ti(AZ/2)]
Bi=2(1+Ai)
FLOW
AT CAJON
PASS
Henyey, T. L., and G. J. Wasserburg,Heat flow near major strike-slip faults in California, J. Geophys. Res., 76, 7924-7946, 1971. Jaeger, J. C., Application of the theory of heat conduction to geothermal measurements, in Terrestrial Heat Flow, Geophys. Monogr. Ser., vol. 8, edited by W. H. K. Lee, pp. 7-23, AGU, Washington, D. C., 1965. Jaeger, J. C., and J. H. Sass, Lee's topographic correction in heat flow and the geothermal flux in Tasmania, Geofis. Pura Appl., 54, 53-63, 1963.
Jeffreys, H., The disturbance of the temperature gradient in the Earth' s crust by inequalities of height, Mort. Not. R. Astron. Soc., Geophys. Suppl., 4, 309-312, 1938. Lachenbruch, A. H., Rapid estimation of the topographic disturbance to superficialthermal gradients,Rev. Geophys., 6,365-400,
(B3)Lachenbruch, A. 1968.
Ci = Ai[1 + ¾i(Az/2)]
D i = AiTin_•+ 2(1 - Ai)T• + CiTi• + [2SiAt/(pCp)i] Where
Ai: kiAt/(Az)2
Ti = -u/ki
and k is thermal diffusivity. In the present implementation, the thermal conductivity is assumed to be constant throughout the problem domain. The temperature at the surface of Earth establishes the upper boundary condition, while the heat flux is assumed to be constant at the lower boundary. This scheme is numerically stable as long as the absolute value of the "grid" Peclet number, defined by
P e : uAz/k
(B4)
remains less than 1 [Patankar, 1980]. In a verification test, the numerical scheme reproduced the analytic solution for the case of erosion (or sedimentation) at a fixed rate with a
relative error less than 10-4 throughoutthe upper few
H., and B. V. Marshall, Heat flow through the Arctic Ocean floor: The Canada Basin-Alpha Rise boundary, J. Geophys. Res., 71, 1223-1248, 1966. Lachenbruch, A. H., and J. H. Sass, Thermomechanical aspectsof the San Andreas fault system, in Proceedingsof the Conference on the Tectonic Problems of the San Andreas Fault System, edited by R. L. Kovach and A. Nur, Stanford Univ. Publ. Geol. Sci., 13, 192-205, 1973. Lachenbruch, A. H., and J. H. Sass, Heat flow and energeticsof the San Andreas fault zone, J. Geophys. Res., 85, 6185-6223, 1980. Lachenbruch, A. H., and J. H. Sass, Corrections to "Heat flow and energeticsof the San Andreas fault zone" and some additional comments on the relation between fault friction
and observed heat
flow, J. Geophys. Res., 86, 7171-7172, 1981. Lachenbruch, A. H., and J. H. Sass, The stress heat-flow paradox and thermal results from Cajon Pass, Geophys. Res. Lett., 15, 981-984, 1988. Lachenbruch, A. H., and J. H. Sass, Heat flow from Cajon Pass, fault strength, and tectonic implications, J. Geophys. Res., 97, 4995-5015, 1992. (Correction, J. Geophys. Res., 97, 17,711, 1992.) Lachenbruch, A. H., J. H. Sass, T. H. Moses Jr., and S. P. Galahis Jr., Thermal considerations and the Cajon Pass borehole, U.S. Geol. Surv. Open File Rep., 86-469, 1986.
Meisling, K. E., and R. J. Weldon, Late Cenozoic tectonicsof the northwestern San Bernardino Mountains, southern California, Geol. Soc. Am. Bull., 101, 106-128, 1989. Pantankar, S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980.
kilometers of the grid. For Cajon Pass the numerical method was used to calculate the effect of the erosion/sedimentation history on the Richtmyer,R. D., and K. W. Morton, DifferenceMethodsfor InitialValue Problems, 2nd ed., Wiley Interscience,New York, 1967. heat flow profile. Changes in surface temperature were not considered; they are discussed separately in the text. The Sass,J. H., A. H. Lachenbruch, S. P. Galahis Jr., R. J. Munroe, and T. H. Moses Jr., An analysisof thermal data from the vicinity of total depth of the grid was set at 40 kin, and a grid spacing Cajon Pass, California, U.S. Geol. Surv. Open File Rep., 86-468, dz = 100 m was used. Given the likely erosion/sedimenta1986. tion velocities at Cajon Pass, the grid Peclet number re- Sass, J. H., A. H. Lachenbruch, T. H. Moses Jr., and P. Morgan, mained
less than 0.01 for all simulations.
Acknowledgments. We are grateful to our colleagues,Joe Andrews, John Bredehoeft, Peter Galahis, Tom Hanks, Yousef Kharaka, Manuel Nathenson, Jack Healy, Steve Hickman, Jim Savage, Colin Williams, and Mark Zoback for helpful discussion,and for constructive commentsfrom David Deming, Tom Henyey, and Paul Morgan during the Journal of GeophysicalResearch review processes.
Heat flow from a scientificresearchwell at Cajon Pass, California, J. Geophys. Res., 97, 5017-5030, 1992. Silver, L. T., and E. W. James, Lithologic column of the "Arkoma" drillhole and its relation to the Cajon Pass drillhole, Geophys. Res. Lett., 15,945-948,
1988.
Weldon, R. J., Late Cenozoic geology of Cajon Pass: Implications for tectonics and sedimentation along the San Andreas fault, Ph.D. thesis, 400 pp., Calif. Inst. of Technol., Pasadena, 1986. Weldon, R. J., K. E. Meisling, and J. Alexander, A speculative history of the San Andreas fault in the central Transverse Ranges, California, Mem. Geol. Soc. Am., 178, 161-198, 1993.
References G. D. Clow and A. H. Lachenbruch, U.S. Geological Survey,
Birch, F., Flow of heat in the Front Range, Colorado, Geol. Soc. Am. Bull., 61,567-630, 1950. Brune, J. N., T. L. Henyey, and R. F. Roy, Heat flow, stress, and rate of slip along the San Andreas fault, California, J. Geophys. Res., 74, 3821-3827, 1969. Bullard, E. C., The disturbance of the temperature gradient in the Earth's crust by inequalities of height, Mort. Not. R. Astron. Soc., Geophys. Suppl., 4, 360-362, 1938. Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, 510 pp., Oxford University Press, New York, 1959. Henyey, T. L., Heat flow near major strike-slip faults in California, Ph.D. thesis, Calif. Inst. of Technol., Pasadena, 1968.
Mail Stop 977, 345 Middlefield Road, Menlo Park, CA 94025. (e-mail:
[email protected]; alachenbruch@isdmnl. wr.usgs.gov) J. H. Sass, U.S. Geological Survey, 2255 North Gemini Drive, Flagstaff, AZ 86001. (e-mail:
[email protected]) R. J. Weldon, Department of Geological Sciences,University of Oregon, Eugene, OR 97403. (e-mail:
[email protected])
(Received November 10, 1993; revised September 16, 1994; accepted November 2, 1994.)
