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radioactive chains w ith half- li ves on the order of at least 1 billion yea rs. Elements w ith shorter half-li ves have long di sappeared, and the new rad ioactivity created by cosmic bombardment is minimal. More than half the radiogenic hea t comes from potassium-40, which occu rs primari ly in the mantl e, along w ith the radioactive seri es originated by thorium-232 (see "Signi ficant Heat-Generating Isotopes," right). Uranium, the only other significant radioelement, is found mainly in th e cru st. Although a few add iti ona l ve r y l ong -li ved isotopes exist-for example, rubidium-87, lutecium176 and rhenium-187- their concentrations and heat generati on are insignifica nt. Since the core is assumed to contain no heat sou rces, it makes no contribution to heat flow at th e Earth's surface. The fraction co ntributed by rad i oac ti v it y i s rising , although the absolute rate of heat generation by radioactivity halves about eve ry billion years. Th e current rate of rad iogen ic hea t flow is about 5 perce nt of its va l ue when the Earth was formed. Temperature readings at many points at or near the Earth 's surface yie ld an average heat flux of abou t 1.2 X 10' 6 ca l ori es per square cent im eter (em) per second, although indi vidual values may vary by up to one order of magnitude. Hea t flux from the sun is much larger, but makes no sign ifica nt contribution to the geothermal grad ient- most solar radiation is either refl ected or rad iated back into space. Furthermore, diurnal temperature variations penetrate the surface only about 1 meter [3.3 feet]; annual vari ations penetrate a few tens of meters on the continents and a few hundred meters in the oceans. Climati c variations, however, at the scale of a few thousand years, can influence measu rements hundreds of meters below the surface, and variations of 10,000year period ici ty ca n be sensed down to 2 kilometers [1 .2 milesJ.2 How Heat Reaches the Surface Conduc ti on and convection are th e two common modes of heat tran sfer in the Earth. Conduct i on a ri ses from in t e ra ct i ons between neighboring atom s, w hereas convection transports heat by fluid movement resulting from the interaction between gravity and thermally induced density cha nges. First, consider cond ucti on. In an isotropic medium, conducti ve hea t flow is q = k VT, w here q is the heat flu x (calories per square cen timeter per second), k is th erm al conduc ti vity of th e medium, Volume 1 Number 1 Significant Heat-Generating Isotopes Half-life location Potassium-40 1.3 billion years Mantle Thorium-232 14 billion years Mantle Uranium-238 4.5 billion years Crust Isotope Minerals ~Feldspars ""-· Mtcas "' • Quartz Pyroxenes and Amphiboles " Olivines, Garnet, Epidote • Water • Ice Rocks ------ /'Rock Salt - - - - - - - - - - - - - Limestone ""sandstone & Quartz - - - - - - - - Schist - - - - - - Gabbro - - - - - Tuff 0 0.05 0.1 '-.....Granite 0.1 5 0.2 Thermal Conductivity, k cal;cm s. oc 0 Ranges of thermal conductivity for minerals and rocks at standard temperature and pressure. The range of thermal conductivity of silicates (feldspars, micas, and pyroxenes), for example, is controlled by silicon-oxygen bonding: the more shared oxygen atoms, the higher the conductivity. and VT is temperature grad ient. Since co nductivity and temperature gradient can vary individually with depth, each must be specified for a given location. Near the surface of the Earth, the gradient is dT/dz (z increasing w ith depth). Thus, the rate at w hich heat is co nd ucted ou tward through th e Eart h depends in a simple way on how temperature and thermal co nductivity vary w ith depth. However, to pred ict th e variat ion of temperature w ith depth requ ires knowing th e ve rtica l d ist ribu ti o n of heat sources (rad ioact iv it y) and therm al diffusivity-a mea surement of the r elat ive ability of a mate ri al to co ndu c t and to retai n heat, k/pC. in wh ich p is density of the rock- fluid sys tem and C i s heat capac ity per unit ma ss.J Because th e verti ca l distribution of heat sources i s not routinely determined, most emphasis is given to determini ng thermal diffusivity. The wide range of thermal diffusivities in rocks and minerals results from va riation in compos ition and porosity (above). Water, w hich responds on ly sluggishl y to th ermal changes in its envi ronmen t, has a therma l d iffusivity an order of magnitude less th an that of typica l rocks and mineral s. Although there is no consensus on a mixing rule for thermal conductivity, a reason1. Howell BF )r: Introduction to Geophysics. New York: McGraw-Hill Book Co .• Inc., 1959. Anderson RN: Marine Geology. New York: John Wiley & Sons, 1986. 2. Garland GD: lntrodvcliontn GPnphy.< ics-Mantle, Core, and Crust. Philadelphia: W.IJ. Saunders Co., 1971. 3. Hearst )Rand Nelson PH : Well Logging for Physical Properties. New York: McGraw-H ill Book Co., 1985. Horai K-1: "Thermal Conductivity of Rock-Forming Minerals," journal of Geophysical Research 76, no. 5 (February 10, 19711: 1278 - 1308. 55 Descending Convection Currents Descending Convection Currents 0 The relation bet ween convection and plate tectonic movement, as proposed by A rthur Holmes in 1929 (vertical scale exaggerated). Rising con vection currents pull continents apart and create new ocean in a rift zone. M o untain ranges and deep sea trenches form where convection currents descend. (A fter Press F and Siever R: Earth. San Francisco: WH. Freeman and Co., / 978: 459.) ably succ ess fu l empiri ca l approach is to take the volume-weighted geometric mea n of the conductivities of the constituents: kmix =D i<f· in which kmix is the conductivity of the mixture, and k; and ¢1; are the conductivity and vo lume fra cti o n, respec tively, of th e i-th co nstituent. This rule applies to both nonporous and liquid-filled porous rock.4 Conductio n was in itia lly thought to be responsible for most heat flow in the Earth, but studies show that convection also transports heat efficiently and often plays a mo re important role than conduction. In relatively uncon strained systems, convection occurs by means o f convection cells, in w hi ch a rising fl uid carries heat upward and transfers it to cooler material at the top of the ce ll. As the fluid ri ses, it cools, becomes more dense and sinks to the botto m, where it is heated again . In some environments the fluid may not find a return path to the bottom. In this case th e convecti on fluid must be rep lenished from another source, for example, wa ter entry from the surface through an aquifer. In either case, the heat flux, q, car- ri ed by the rising current is Cvt!.T, i n w hich C is the fluid's heat capac ity (calori es per cubic centimeter per degree), vi s its vertica l velocity (centimeters per second), and LiT is the temperatu re difference between the top and bottom of the convecti on cell. The mantle itself is thou ght to be sufficiently fluid to parti c ipate in convec tio n . For example, a rock flow rate as low as 1 em 10.4 inch! per year across a temperature d rop of several tens of degrees produces a heat fl ux of about 10'6 ca lories per square ce ntimeter per second, th e same order of magnitude as the total o utward fl ux at the surface.2 Not o nly does this con vection in the mantle influence heat fl ow to the crust, but large-scale convection cell s d rive p late tectonic movement (above). Hydrothermal convection takes p lace at lower temperatures, where w ater can exist as a separat e pha se . Hi gh -temp erature hydroth ermal convection is responsible for loca l heat transfer to th e Earth's surface in the form o f hot spri ngs and geysers. Lowtemperature co nvecti on is probabl y an impo rtant mechanism of fl uid (inc lud ing hydrocarbon) flow in sedimentary basins. u.. 0 ~ :::J ~ • Ql Q_ E Ql f- 105 Ql 0 .r:. E 0 g CD 100 95 90 ~--------------------~ 1 2 3 tk + tlt M Measurement of H eat Flow Rate Direct measu rement of heat flow rates is difficult to achieve, even in the subsurface portion of th e Earth immediately avai labl e to us. Co nseque ntl y, heat fl ow is al most 56 120 . ------------------------ . O Bottomhole temperature recordings in a Paris Basin well from fo ur comecutive logging rum made up to 18 hours since mud circulation ceased. The measurements are extrapolated on a Homer-type plot to yield true forma tion temperature. tk is the length of time the mud was circulated; 6 t is the elapsed time between cessation of circulation and a given logging run . The extrapolated value of 116°F was confirmed by subsequent continuous temperature logging. (After Dowdle and Cobb, reference 5.) Oilfield Review J alw ays determ ined by independent in situ m easu rement o f the geo th erm al gradien t and of the thermal conductivity of rock samp les in the laboratory. Very recently, however, a w ireline method for thermal conductiv- · ity determination was successfull y tested in several we ll s.4 Th e geoth erm al grad ient is most c om monly acquired by w irel ine measurement of tempera ture in boreho les, alt hough measurements have been made i n ocean sed iments by implan ti ng a spear w ith vertically spaced temperature sensors (high heat flow in ocean sediments correlates w ith hyd roca rbon deposits). M ost w ells are logged and the logging tool invariably records a temperature maxim um, usuall y bottomho le temperature, that depends on the elapsed ti me since m ud ci rcu latio n ceased. A Horn ertype plot has been found empirically to convert temperatu res recorded on consecutive logging runs to the true fo rmatio n temperatu re (below left ).5 Bottomhole readings, unfo rtunately, are not continuou s measurements. At best they yi eld onl y a grad ient average in a sma ll area, w hen one bottom hole measurement is combi ned with another at a different depth in a nearby w ell . Co ntinu o u s l o gs o f t emper atur e a re recorded w ith platin um resistance th ermometers or therm istors, usuall y located at the bottom of the logging string to min imize infl uenc e of the sonde body o n the measu r e ment. I f th e dri ll i n g mud h as not reached ther mal equi librium w ith th e fo rm ati o ns, cor rec ti o ns mu st b e app lied at each depth . A con ti nuous temperature log pro vides T(z), from which the local gradient dT/ dz may be determin ed directly {right; next page, left, top and bottom ). Sometimes the gradient at a parti cular depth in a given w ell does not represent the regional geo thermal gradient. Local departures can result fro m stratigraphi c conducti v ity c hange s, water flow {convecti ve or otherw ise), expand ing (coo ling) gas enteri ng th e ho le, c ement top s not yet i n th ermal equilibrium, and recent fluid injection.J Rock thermal conducti vity is determined preferab ly w ith the so-ca lled d ivided-bar method.6 A disk cut from a core is held firmly between two materials o f the sa me di a- Temperature OF 250 150 Tension o 150 .c 1,000 lbs ------ ------------ Temperature OF a. <ll 160 0 Casing Collar Locator Temperature Gradient - 0.5 0.5 ---------------- ----------------- ---- ,_ 11-: l= i 1- ' I= ; h - t-1..-.::-J I= ·. ~- t-H::. !: HL~-1- - I= ~ 1- t--;:_;_ Jd :: ti8-t-= 1--Ef ~- 1- ; ~ r-::t= 1- : c-~ : -~=; = = ~ ~nf = l 1-1-1-1-· · - - -- 1 8 i~ ~ CO -L _. ~: l - L-1-• ·~ ~ 0 : ~ . l . ! ·: 1- ~~ 1- H :=== 't, 1-l I-!== I= 1=11="~ 1- ; ! f- 1- 8C\J co ·- D Use of a temperature log to identify the source of water production in cased hole, indicated by heating anomalies. Based on production in an offset well, and a sonic log, the operator perforated this well from 8118 to 8133 feet. But the well produced water and the shut-in pressure was considerably higher than expected for a pressure-depleted gas sand. The temperature log shows that very little flow is coming from the perforations, and most is from an interval belo w, around 8136 feet. The heating anomaly from 8 176 feet upward indicates that water production is coming from at least that deep and channeling up between the formation and casing. The final diagnosis: the perforation job ruptured the casing, producing a vertical gash that permitted water to migrate upward, possibly robbing production from the perforated zone. 4. Dove RE and W illiams C: " Thermal Conductivity Estimated From Elemental Concen tration Logs," Nuclear Geophysics 3, 1989 (in press). 5. Dowdle WL and Cobb WM: "Static Formation Temperature from Well Logs-An Empirical Method," journal of Petroleum Technology, 28, no. 35 (November 1975): 98-10 5. 6. Costa in JK and W right PM: "Heat Flow and Precision Temperature Measurements in Boreholes," Transactions of the SPWLA lOth Annual Logging Symposium, Houston, May 25-28, 1969, paper ). Vo lume 1 Number 1 57