Semianalytical solution for dual-probe heat-pulse applications that accounts for probe radius and heat capacity

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1 This is he uhor s finl, eer-reviewed mnuscri s cceed for ublicion. The ublisher-formed version my be vilble hrough he ublisher s web sie or your insiuion s librry. Seminlyicl soluion for dul-robe he-ulse licions h ccouns for robe rdius nd he cciy John H. Knigh, Gerrd J. Kluienberg, Tmir Kmi, nd Jn W. Homns How o cie his mnuscri If you mke reference o his version of he mnuscri, use he following informion: Knigh, J. H., Kluienberg, G. J., Kmi, T., & Homns, J. W.. Seminlyicl soluion for dul-robe he-ulse licions h ccouns for robe rdius nd he cciy. Rerieved from h://krex.ksu.edu Published Version Informion Ciion: Knigh, J. H., Kluienberg, G. J., Kmi, T., & Homns, J. W.. Seminlyicl soluion for dul-robe he-ulse licions h ccouns for robe rdius nd he cciy. Vdose Zone Journl,. Coyrigh: Soil Science Sociey of Americ Digil Objec Idenifier DOI: doi:./vzj. Publisher s Link: hs:// This iem ws rerieved from he K-Se Reserch Exchnge K-REx, he insiuionl reosiory of Knss Se Universiy. K-REx is vilble h://krex.ksu.edu

2 Seminlyicl Soluion for Dul-Probe He-Pulse Alicions h Accouns for Probe Rdius nd He Cciy John H. Knigh, Gerrd J. Kluienberg*, Tmir Kmi, nd Jn W. Homns Acceed for Publicion in Vdose Zone Journl J. H. Knigh, Fculy of Science nd Technology, Queenslnd Universiy of Technology, Brisbne, QLD, Ausrli; G. J. Kluienberg, Dermen of Agronomy, Knss Se Universiy, Mnhn, KS ; nd T. Kmi nd J. W. Homns, Dermen of Lnd, Air nd Wer Resources, Universiy of Cliforni, Dvis, CA. Conribuion no. --J from he Knss Agric. Ex. Sn., Mnhn, KS. *Corresonding uhor gjk@ksu.edu.

3 Seminlyicl Soluion for Dul-Probe He-Pulse Alicions h Accouns for Probe Rdius nd He Cciy ABSTRACT The dul-robe he-ulse DPHP mehod is useful for mesuring soil herml roeries. Mesuremens re mde wih sensor h hs wo rllel cylindricl robes: one for inroducing ulse of he ino he soil heer robe nd one for mesuring chnge in emerure emerure robe. We resen seminlyicl soluion h ccouns for he finie rdius nd finie he cciy of he heer nd emerure robes. A closed-form exression for he Llce rnsform of he soluion is obined by considering he robes o be cylindricl erfec conducors. The Llce-domin soluion is invered numericlly. For he cse where boh robes hve he sme rdius nd he cciy, we show h heir finie roeries hve equl influence on he he-ulse signl received by he emerure robe. The finie rdius of he robes cuses he he-ulse signl o rrive erlier in ime. This ime-shif increses in mgniude s robe rdius increses. The effec of he finie he cciy of he robes deends on he rio of he he cciy of he robes C nd he he cciy of he soil C. Comred o he cse where C C, he mgniude of he he-ulse signl decreses i.e., = smller chnge in emerure nd he mximum emerure rise occurs ler when C C. > When C C, he mgniude of he signl increses nd he mximum emerure rise occurs < erlier. The seminlyicl soluion is rorie for use in DPHP licions where he rio of robe rdius nd robe scing L sisfies he condiion h L..

4 INTRODUCTION The dul-robe he-ulse DPHP mehod is widely used for mesuring soil herml roeries nd volumeric wer conen. Mesuremens re mde wih sensor h consiss of wo rllel cylindricl robes. The heer robe conins elecricl resisnce wire used o inroduce he consn re during finie ime inervl yiclly seconds, nd he emerure robe conins hermisor or hermocoule o mesure chnge in emerure s funcion of ime. If he disnce beween he robes nd he re nd durion of heing re known, soil herml roeries i.e., volumeric he cciy, herml conduciviy, nd herml diffusiviy cn be esimed from he rnsien emerure resonse by using n rorie soluion of he he conducion equion. Volumeric wer conen cn be deermined from he volumeric he cciy if he soil bulk densiy nd he secific he of he soil solid consiuens re known. The soluions currenly used in conjuncion wih he DPHP mehod re he heer robe s n infinie line source from which he is relesed insnneously Cmbell e l., or during finie ime inervl Brisow e l.,. These soluions re generlly regrded s rorie for deermining herml roeries nd wer conen, bu hey do no ccoun for he finie rdius nd finie herml roeries of he robes. By using numericl models, Gurgli nd Pous nd Homns e l. demonsred h herml roery esimes my exhibi significn bis if he finie rdius nd finie herml roeries of he heer robe re no ken ino ccoun. A growing body of exerimenl evidence lso suggess h he DPHP mehod my resul in bised esimes of herml roeries nd wer conen Trr nd Hm, ; Song e l., ; Brisow e l., ; Bsinger e. l., ; Ochsner e l., ; Hm nd Benson,. To our knowledge, no em hs been mde o invesige he effecs of he

5 finie rdius nd finie herml roeries of he emerure robe. There is lso need for more comlee remen of he effecs cused by he finie roeries of he heer robe. Qunifying hese effecs is riculrly relevn in ligh of he shif owrd sensors wih robes of lrger rdius shif driven rimrily by he need o mke he sensor more robus i.e., o minimize deflecion of he robes when insered ino soil. Cmbell e l. used sensor wih robe rdius of. mm. In ler work, sensors wih robe rdius of. mm hve been used exensively e.g., Trr nd Hm, ; Ren e l., ; Bsinger e l.,, nd resuls hve been reored for sensors wih robe rdius s lrge s. mm Nusier nd Abu-Hmdeh,. We resen seminlyicl soluion h ccouns for he finie rdius nd finie he cciy of he robes of DPHP sensor. A closed-form exression for he Llce rnsform of he soluion is obined by considering he robes o be cylindricl erfec conducors of infinie lengh. The lgorihm of Sehfes,b is used o inver he Llce-domin soluion numericlly. Preceden for use of he erfec conducor ssumion cn be found in he work of Blckwell nd Jeger, who used i o derive he soluion for rdil conducion from single heed robe in conc wih medium of infinie exen. This ssumion seems resonble in he conex of he DPHP mehod insofr s he robes of DPHP sensors yiclly hve herml conduciviy much greer hn h of soil. The soluion resened herein is similr o soluions in he groundwer lierure h ke ino ccoun he finie rdius nd finie sorge cciy of ired uming nd observion wells Tongenyi nd Rghvn, ; Ogbe nd Brighm, ; Novkowski,, bu i voids he need for numericl evluion of inegrls in he Llce domin.

6 We begin by deriving generl form of he seminlyicl soluion, nd hen resen secil cse of he soluion useful for DPHP licions. The secil cse of he soluion is used o exmine he effecs of he finie roeries of he robes. Secificlly, we resen resuls h show how he he-ulse signl received by he emerure robe is modified by he finie rdius nd he finie he cciy of he robes. We hen use resuls from finie-elemen model o check he vlidiy of he seminlyicl soluion nd o ssess wheher i is rorie o re he robes s erfec conducors. We conclude wih discussion of ublished d indicing h herml roery nd wer conen esimes my be bised if he DPHP mehod is imlemened wihou ccouning for he finie roeries of he robes. We lso briefly discuss oenil limiions of he seminlyicl soluion. GENERAL SOLUTION Consider DPHP sensor wih infiniely long heer nd emerure robes of finie rdius, finie he cciy, nd infinie herml conduciviy. The heer robe hs rdius nd volumeric he cciy C, nd he emerure robe hs rdius nd volumeric he cciy C. The robes, wih cenerlines disnce L r nd wih + < L, re surrounded by soil wih volumeric he cciy C nd herml conduciviy λ. We ssume he soil o be homogeneous nd isoroic nd h C nd λ re no funcions of emerure. We lso ssume h he soil nd robes re in erfec herml conc. Le V nd V be he emerures of he heer nd emerure robes, resecively, where is ime. We derive n nlyicl soluion in he Llce domin for V, given n rbirry heing funcion φ for he heer robe. The funcion φ gives he re er uni lengh which he is relesed from he heer robe.

7 We begin by deriving n nlyicl soluion in he Llce domin h ccouns only for he finie rdius nd finie he cciy of he heer robe; h is, we ignore he resence of he emerure robe nd consider heer robe surrounded only by n infinie domin of soil. This soluion yields n exression for he rnsform of he emerure of he soil locion h coincides wih he cenerline of he emerure robe. Th exression is hen used o obin soluion for V in he Llce domin h ccouns for he finie rdius nd finie he cciy of he emerure robe. Heer Probe wih Finie Proeries The soluion h ccouns for only he heer robe is derived by using coordine sysem Fig. wih rdil coordine r h is cenered on he heer robe x, y =,. The emerure robe is loced x, y = L,. The disnce from he cenerline of he heer robe o n rbirry oin in he x-y domin is r, where r = +. The emerure of he x y soil, υ r,, sisfies he he equion Crslw nd Jeger, υ + r r υ r κ υ = ; r <, > [] where κ = λ C is he herml diffusiviy. The boundry nd iniil condiions o be sisfied re υ r, ; r, [] > υ r, = ; r <, [] = υ r, = V ; r =, [] > υ dv π λ = φ π C ; r =, > [] r d V = ; [] =

8 Equion [], he boundry condiion for medium in conc wih erfec conducor Crslw nd Jeger,,., is obined by wriing n energy blnce for he heer robe. The lefhnd side of Eq. [] is he ol flux of he er uni lengh from he heer robe ino he soil r =. This roblem is similr o one ddressed in. II of Crslw nd Jeger, bu considers he more generl cse of n rbirry heing funcion. The Llce domin soluion for υ r, is obined by mking use of he definiion ˆ υˆ r, υ r, ex d [] where υ r, is he Llce rnsform of υ r, nd is he rnsform vrible. Uon king he rnsforms of Eqs. [] [], we obin he subsidiry equion d υˆ + dr r dυˆ υ ˆ = ; dr r < [] nd he boundry condiions ˆ r, r [] υ ; υ ˆ r ˆ r = [], = V ; λ = φ π CV ; dr dυˆ π ˆ ˆ r = [] where = κ [] The rdilly symmeric soluion of Eq. [] h sisfies Eqs. [] nd [] is ˆ K r υ ˆ r, = V [] K

9 where K n denoes he modified Bessel funcion of he second kind of order n. The remining boundry condiion is sisfied by differeniing his exression wih resec o r nd subsiuing he resul ino Eq. [] o give Rerrngemen of his exression yields ˆ ˆ K π CV = φˆ πλv [] K ˆ φˆ K V = [] πλ [ K + β K ] where β = C C. Now h we hve n exression for he emerure of he heer robe in he rnsform domin, i cn be subsiued ino Eq. [] o give φˆ K r υ ˆ r, = r [] ; πλ[ K + β K ] which is he rnsform of he emerure field in he soil disnce r from he cener of he heer robe. This soluion is he generl cse i.e., he cse for n rbirry heing re of he soluion for coninuous heing given in. II of Crslw nd Jeger. Noe h Eq. [] is he roduc of he rnsfer funcion υ ˆ f,, β = [] [ K + β K ] nd Eq. [A], which is he rnsform of generl line-source soluion. Equion [] cn herefore be wrien in he form φˆ K r υ ˆ r, = υˆ f,, β ; r [] πλ ˆ I is of ineres h he rnsfer funcion υ f,, β does no deend on he funcionl form of φ. The Llce rnsform of he soluion υ r, disnce r = L is

10 φˆ K L υˆ L, = υˆ f,, β [] πλ Heer nd Temerure Probes wih Finie Proeries We now use Eq. [] o derive Llce-domin soluion for V, he emerure of he emerure robe. The soluion is obined by using n ddiion heorem for Bessel funcion soluions of he he equion. In imlemening his roch, we mke he simlifying ssumion h V is influenced by he heer robe, bu h V is no influenced by he emerure robe. In oher words, we ssume h he emerure robe does no ler he rdil symmery of he emerure disribuion round he heer robe. This ssumion is rorie if he rdius of he robes is smll relive o heir disnce of serion. We exmine is vlidiy ler by using resuls from numericl model h does no imose rdilly-symmeric emerure disribuion in he viciniy of he heer robe. The soluion is obined by using second coordine sysem Fig. wih rdil coordine r h is cenered on he emerure robe x, y = L,. The disnce from he cenerline of he emerure robe o n rbirry oin in he x-y domin is r, where erms of he new coordines, he he equion kes he form r + = x L y. In υ + r r υ + r r υ υ = ; θ κ r <, θ < π, > [] where he ngle θ is such h x L, y = r cosθ, r sin. The boundry nd iniil condiions o be sisfied re θ υ r, θ, = V ; r =, θ < π, [] > π υ dv λ dθ = πc ; r =, > [] r d

11 V = ; [] = Tking he Llce rnsform of hese exressions yields he subsidiry equion d υˆ + dr r dυˆ + dr r d υˆ υ ˆ = ; dθ r <, θ < π [] nd he boundry condiions υ r, θ, = Vˆ ; r =, θ < π [] ˆ π dυˆ ˆ dθ = πcv ; dr λ r = [] To obin he desired soluion, we consider he sil deendence K of he rnsform r of he he emied by he heer robe nd use n ddiion heorem for Bessel funcion soluions of Eq. [] Crslw nd Jeger,,., Eq. [] o wrie K in erms of soluions r I m r cos mθ of Eq. [] cenered he emerure robe, where I m denoes he modified Bessel funcion of he firs kind of order m. For m hese soluions re no rdilly symmeric. The ddiion heorem for r < L is m= m r = Km L I m r m= K cos mθ [] We look for soluion υ ˆ r, θ, of Eq. [] vlid for < r < L h sisfies υ r, θ, = Vˆ ˆ. Th is, we look for soluion of he form K r K r cos mθ [] m= m υ r θ = Vˆ ˆ,, + φ υˆ ˆ f,, β dmkm r K πλ m= wih coefficiens d m o be deermined by sisfying Eq. []. This requires h d m Km L Im = φˆ υˆ f,, β [] πλk m

12 Uon subsiuing Eqs. [] nd [] ino Eq. [], he soluion,, ˆ r θ υ kes he form = = = = θ πλ β υ φ θ πλ β υ + φ = θ υ m m m m m m m f m m m m m f K m r K I L K m r I L K K r K V r cos,, ˆ ˆ cos,, ˆ ˆ ˆ,, ˆ [] To sisfy he remining boundry condiion, we differenie Eq. [] wih resec o r nd subsiue he resul ino Eq. []. Afer evluing he inegrl, Eq. [] becomes,, ˆ ˆ,, ˆ ˆ ˆ ˆ K K I L K I L K K K V C V f f β υ φ + β υ φ + πλ = π [] Comring his exression wih Eq. [] revels h he erms eriodic in θ mke no ne conribuion o he ol he flux er uni lengh r =. Equion [] conins only he conribuion from he erms for m = in Eq. []. We now use he Wronskin relion Olver,,. K I K I = + [] o wrie Eq. [] in he form,, ˆ ˆ ˆ ˆ L K K V K C V f β υ + φ πλ = π [] Rerrngemen of his resul yields he exression ] [,, ˆ ˆ ˆ K K L K V f β + πλ β υ φ = [] where C = C β. The soluion in he rnsform domin for he emerure of he emerure robe is hen πλ φ β υ β = υ ˆ,, ˆ,, ˆ ˆ L K V f f []

13 where he rnsfer funcions corresonding o heer nd emerure robes hve excly he sme form. Noe h Eq. [] is Eq. [A], he rnsform of generl line-source soluion, mulilied by he roduc of he rnsfer funcions for he heer nd emerure robes. Wih υˆ f,, β =, Eq. [] is idenicl o Eq. [] nd gives he rnsform of he soluion for he cse where he heer robe hs finie rdius nd finie he cciy bu he emerure robe ˆ f = hs zero rdius. Wih υ,, β, Eq. [] gives he rnsform of he soluion for he cse where he emerure robe hs finie rdius nd finie he cciy bu he heer robe hs zero rdius. In deriving Eq. [], we ssumed h he soil nd boh robes hve emerure of zero ime =. Emloying his iniil condiion does no resul in loss of generliy. The rincile of suerosiion ermis licion of his soluion o cses where he soil nd boh robes hve rbirry uniform iniil emerure υ, for which we hve υ r, = V = V = υ. For such cses V reresens he emerure rise bove he iniil emerure υ. SPECIAL CASE OF SOLUTION FOR DPHP APPLICATIONS For mos licions, he robes of he DPHP sensor hve he sme rdius nd he sme he cciy. If boh hve rdius nd he cciy C, we hve funcion for he robes cn be wrien s β = C C, nd he rnsfer υ ˆ f,, β = [] [ K + β K ] This simlificion llows us o wrie Eq. [] in he form ˆ φˆ K L V = υˆ,, β f [] πλ

14 The rorie heing funcion for DPHP licions is q ; < φ = [] ; > where q is he re er uni lengh which he is relesed from he heer robe nd is he heing durion. The Llce rnsform of Eq. [] is φ ˆ = q [ ex ]. Subsiuing his resul ino Eq. [] yields q K L V ˆ P [ ex ] = υˆ f,, β [] πλ where he suerscri P indices h his exression is he Llce-domin soluion for he ˆ cse of ulsed heing. Equion [] is he roduc of υ f,, β nd he rnsform of Eq. [A], he line-source soluion used in he DPHP mehod of Brisow e l.. This soluion cn be invered numericlly o obin vlues of V P for imes of ineres, bu beer ccurcy cn be chieved by erforming he numericl inversion for he cse of coninuous heing, nd hen using he rincile of suerosiion in he ime domin o ccoun for he finie durion of heing. This requires h we use domin. Thus, Eq. [] becomes φ = q, which hs he form φ ˆ = q in he rnsform q K L V ˆ C = υˆ f,, β [] πλ for he cse of coninuous heing. Equion [] cn be invered numericlly o obin vlues of C C V nd V for riculr imes of ineres. These vlues of C C V nd V re hen used o obin he corresonding ulsed heing resuls, V P, by using he exression C V ; < P V = [b] C C V V ; >

15 Togeher, Eqs. [] nd [b] reresen seminlyicl soluion for he emerure of he emerure robe. Herefer, we refer o Eq. [] s he idenicl-cylindricl-erfec-conducors ICPC soluion. An imorn roery of he ICPC soluion is h i becomes idenicl o he line-source ˆ f = soluion of Brisow e l. when υ,, β, which corresonds o he cse where he heer robe nd emerure robe boh hve zero rdius. Thus, he effecs of he finie rdius nd finie he cciy of he robes cn be exmined by comring vlues of V P from Eq. [] wih vlues of υ L, obined by evluing Eq. [A] wih r = L. Th roch ws used o obin he resuls resened in he nex secion. The ICPC soluion nd Eq. [A] were evlued using MATLAB ver.., The MhWorks, Inc., Nick, MA. The modified Bessel funcions of he second kind nd he exonenil inegrl were evlued using he buil-in funcions BESSELK nd EXPINT, resecively. Numericl inversion of Eq. [] ws erformed by using he lgorihm of Sehfes,b wih coefficiens. Deils of he numericl inversion rocedure re given in Aendix B. A coy of he MATLAB scri used o erform hese clculions is vilble uon reques. INFLUENCE OF THE PROBES A key resul eviden from he funcionl form of he ICPC soluion is h he finie roeries of he heer robe nd he finie roeries of he emerure robe hve equl influence in modifying he he-ulse signl received by he emerure robe. This follows from he fc h boh robes hve he sme rnsfer funcion. If one of he robes hs zero rdius, Eq. [] kes he form

16 q K L V ˆ C = υˆ f,, β [] πλ regrdless of wheher h robe is he heer robe or he emerure robe. Therefore, hving heer robe of finie rdius nd emerure robe of zero rdius roduces he sme resul in he ime domin s hving heer robe of zero rdius nd emerure robe of finie rdius. This resul is consisen wih he sil sensiiviy of he DPHP mehod worked ou by Knigh e l.. They showed h he sil sensiiviy in he viciniy of he emerure robe is idenicl o h in he viciniy of he heer robe. If we consider he robes o be heerogeneiies wih he cciy slighly differen from h of he soil, he sil sensiiviy of he DPHP mehod indices h boh heerogeneiies mus hve equl influence on he he-ulse signl received by he emerure robe. Herefer, we consider only he combined effecs of he heer nd emerure robes. Desie he fc h boh robes hve equl influence, heir combined effec in he ime domin is no simly ddiive becuse i is he resul of convoluion oerions. In he reminder of his secion we use he ICPC nd line-source soluions o exmine he effec of he finie rdius of he robes nd he effec of he finie he cciy of he robes. The dimensionless quniies h rimrily deermine he exen of hese wo effecs re β. We firs resen resuls for rbirry vlues of L nd L nd β o isole he effecs of finie rdius nd finie he cciy. We hen resen resuls obined by using vlues of for yicl DPHP sensor. L nd β Finie Rdius The effec of he finie rdius of he robes ws exmined by evluing he ICPC soluion

17 wih β = nd rios L of.,., nd.. By definiion, he he cciy of he robes is idenicl o h of he soil when β =. The clculions were erformed using q = W m, = s, L =. m, C =. MJ m K, λ =. W m K, nd vlues for nd C h yielded he desired vlues for L nd β. The sme vlues for q,, L, C, nd λ were used o evlue he line-source soluion. The resuls Fig. re loed for he cse of zero iniil emerure o be consisen our mhemicl noion; however, s noed erlier, he resuls lso hold for he cse of n rbirry uniform emerure υ. In h cse, he curves in Fig. reresen he chnge in emerure from he iniil emerure υ. Comring he curves for he ICPC nd line-source soluions Fig. shows h he finie rdius of he robes cuses he he-ulse signl o rrive he emerure robe slighly erlier in ime. The mgniude of his ime shif increses s he rdius of he robes increses. In he limi s, he effec of finie robe rdius vnishes nd he ICPC soluion becomes ˆ idenicl o he line-source soluion. This is consisen wih he fc h υ f,, β s. For oher combinions of he rmeers q,, L, C, nd λ here is essenilly no chnge in he wy he effec of finie robe rdius is mnifesed in he resuls; however, for given robe rdius, he mgniude of he ime shif is influenced by he herml diffusiviy nd he robe scing. Secificlly, he ime shif decreses linerly wih κ nd increses linerly wih L. The finie rdius of he robes cuses ime shif becuse he effecive disnce rveled by he he-ulse signl is smller for he ICPC soluion hn for he line-source soluion. The line- source soluion ssumes h he he ulse origines he origin nd h he emerure rise is recorded r = L. Thus, he effecive rvel disnce is L. In conrs, he effecive rvel disnce for he ICPC soluion lies somewhere beween L nd L becuse he he ulse

18 origines r = nd he signl is received r =. The effecive rvel disnce clerly decreses wih incresing robe rdius. Alhough he ICPC soluion ccouns for he finie rdius of he heer nd emerure robes, i does so only roximely becuse he robes re considered o be erfec conducors. By no ccouning for he finie conduciviy of he robes, he ICPC soluion overesimes he ime-shif cused by he finie rdius of he robes. I follows h his soluion will be useful for DPHP licions only if he overesimion of his ime-shif is miniml. In oher words, he ICPC soluion will be useful only if i llows he effec of finie robe rdius o be roximed wih sufficien ccurcy. This will be he cse if he herml conduciviy of he robes is sufficienly lrge relive o he conduciviy of he soil. We exmine his issue in deil ler by using resuls from numericl model h ccouns for he finie conduciviy s well s he finie he cciy of he robes. Finie He Cciy The effec of he finie he cciy of he robes ws exmined by evluing he ICPC soluion wih L =. nd β vlues of.,, nd. Clculions were erformed using he sme rmeer vlues h were used in he revious secion. The influence of he finie he cciy of he robes cn be undersood by comring he curves for β =. nd β = wih he curve for β = Fig.. Less energy is needed o rise he emerure of he robes when β <. This resuls in n increse in he mgniude of he he-ulse signl, nd he signl is osiively skewed so he mximum emerure rise occurs erlier. Conversely, when β >, more energy is needed o rise he emerure of he robes hn for he cse where β =. As

19 resul, he mgniude of he he-ulse signl decreses nd he signl is negively skewed so he mximum emerure rise occurs ler. To undersnd why β influences he skewness of he he-ulse signl, i is useful o consider he ime re of chnge of he he flux he inerfce beween he heer robe nd he soil. The he flux r = increses ridly wih ime following he onse of heing. Becuse less energy is needed o rise he emerure of he heer robe when β <, he he flux r = increses more shrly wih ime hn for he cse where β =. This roduces signl h is osiively skewed relive o he signl for he cse where β =. The negively skewed signl for he cse of β > is cused by similr bu oosie effec. As noed erlier, he ICPC soluion only roximely ccouns for he effec of finie robe rdius becuse i does no ccoun for he finie conduciviy of he robes; however, i ccouns for he finie he cciy of he robes in wy h is hysiclly correc. Thus, he oenil uiliy of he ICPC soluion for DPHP licions deends on wheher he benefi gined by ccouning for he finie he cciy of he robes offses he loss of ccurcy cused by roximing he effec of finie robe rdius. Resuls for Tyicl DPHP Sensor Here we resen resuls obined by evluing he ICPC nd line-source soluions using vlues of nd C for he DPHP sensor of Bsinger e l.. Figure shows cross-secion of he robes of heir sensor. The robes were fbriced from ye sinless-seel ubing.-mm i.d.,.-mm o.d. h ws filled wih hermlly conducive eoxy Omegbond, Omeg Engineering, Inc., Smford, CT. The volumeric he cciy of he robes cn be roximed s he weighed verge

20 C ] = e Ce + [ e Css [] where e is he rdius of he eoxy-filled region of he heer nd emerure robes, C e is he volumeric he cciy of he hermlly conducive eoxy, nd C ss is he volumeric he cciy of he sinless seel. This exression yields C =. MJ m K for he rmeer vlues in Tbles nd. Resuls for he sensor of Bsinger e l. were obined for ir-dried snd, wersured snd, nd wer. The dry nd we snds were chosen becuse heir conduciviies bound he rnge of conduciviies yiclly encounered in minerl soils. Wer ws chosen becuse sensor clibrion i.e., deerminion of ren robe scing yiclly involves mesuremen in wer immobilized wih gr Cmbell e l.,. The volumeric he cciies of hese medi Tble give β vlues of.,., nd. for he dry snd, we snd, nd wer, resecively. Vlues for he rmeers h remined fixed for hese clculions Tble yield rdius/scing rio of L.. Consider firs he resuls for we snd Fig., where he he cciy of he robes is only slighly smller hn h of he soil β =.. Becuse β, he finie he cciy of he robes hs miniml effec. The difference beween he curves for he ICPC nd line-source soluions is rimrily due o he finie rdius of he robes. Accouning for he finie rdius of he robes roduces he-ulse signl h rrives roximely. s erlier. Alhough he effec of finie robe rdius for he dry snd β =. nd wer β =. cnno be deermined by insecing resuls in Fig., i cn be esimed by considering he diffusiviies of he hree medi. The ime shif for he dry snd is roximely. s becuse he diffusiviy of he dry snd is smller hn h of he we snd by fcor of bou.. Similrly, he ime shif for wer is roximely. s becuse he diffusiviy of wer is bou. imes smller hn h of he

21 we snd. For he dry snd, he lefwrd shif cused by he finie rdius of he robes is offse by subsnil effec due o heir finie he cciy Fig.. Combined, he wo effecs reduce he mximum emerure rise by bou. K nd dely he rrivl ime of he mximum emerure rise by roximely s. In conrs, for wer, he wo effecs combine o increse he mximum emerure rise by bou. K, nd hey cuse he mximum emerure rise o er roximely s erlier. The resuls resened in Fig. clerly indice h he DPHP mehod will yield bised esimes of herml roeries if he line-source soluion is used for herml roery esimion. To qunify his bis, we deermined he herml roeries required o force he line-source soluion ino greemen wih he curves in Fig. h were genered wih he ICPC soluion. This ws ccomlished by deermining he mximum emerure rise, V m, nd he ime which i occurred, m, for ech of he hree curves genered wih he ICPC soluion. These irs of V m nd m vlues were hen subsiued ino Eq. [] [] of Brisow e l. o deermine herml roeries using line-source heory. The resuls for dry snd Tble show h, if he finie rdius nd finie he cciy of he robes re no ken ino ccoun, he cciy is overesimed by.%, nd he conduciviy nd diffusiviy re underesimed by.% nd.%, resecively. Ignoring he finie roeries of he robes hs less of n effec on he herml roeries of we snd nd wer Tble, bu he errors in he conduciviy nd diffusiviy of wer re sufficienly lrge o be of concern. Clerly, if he ICPC soluion rovides resonble roximion of reliy, he resuls resened here for yicl DPHP sensor show h he effecs of he finie rdius nd finie he cciy of he robes my be subsnil, esecilly for he cse where here is lrge conrs beween he he cciy of he robes nd h of he soil.

22 VALIDITY OF THE SOLUTION In his secion we use resuls from numericl model o check he vlidiy of he ICPC soluion nd o ssess wheher i is rorie o re he robes s erfec conducors. We begin wih descriion of he numericl model. Numericl Model In he numericl model, we relx he consrin on herml conduciviy nd consider robes of finie rdius h hve finie conduciviy s well s finie he cciy. We lso ccoun for he fc h he heer nd emerure robes re comosie solids h consis of sinless seel nd hermlly conducive eoxy Fig.. As in he derivion of he seminlyicl soluion, we consider boh robes o be infinie in lengh nd we model he rnsfer in lne norml o he xis of he robes h coincides wih he locion of he hermisor. The heer robe is cenered he origin nd he emerure robe is cenered x, y = L,, s in Fig.. Boh robes hve rdius nd eoxy-filled regions of rdius e. We do no ccoun for he finie herml roeries of he hermisor nd elecricl resisnce wire, effecively ssuming h hey hve herml roeries idenicl o hose of he eoxy. The roblem domin is he semicirculr region defined by r nd θ π. We ssume no conc resisnce he meril inerfces b wihin his domin, nd h he herml roeries C nd λ wihin ech orion of he domin re homogeneous, isoroic, nd indeenden of emerure. The emerure T x, y, in he roblem domin sisfies he he equion T C = λ T + g ; > []

23 where g x, y, is he re of he generion er uni volume nd r = +. The he x y equion ws solved wih zero iniil condiion nd wih emerure T = on he exernl boundry r = b. The rdius b ws mde sufficienly lrge h he soluion is effecively he soluion for boundry condiion of T s r. Owing o symmery considerions, zeroflux condiion ws imosed long he exernl boundry coinciden wih he x xis. The soluion of Eq. [] ws lso subjec o emerure coninuiy nd norml he flux coninuiy condiions ll inernl boundries i.e., meril inerfces. Becuse he heer robe is usully modeled s line source of infinie lengh, i is cusomry o qunify he he generion for DPHP mesuremen in erms of q, he re of energy relesed er uni lengh of line source. Here we ssume h he he relesed by he four srnds of resisnce wire Fig is uniformly disribued over he cross-secion of he eoxy in he heer robe. Thus, he re of he generion is reresened by q πe ; r < e, θ < π, < g x, y, = ; r < e, θ < π, > [] ; r > e, θ < π, > The boundry vlue roblem described bove ws solved wih COMSOL Mulihysics finie-elemen sofwre Version., COMSOLAB, Sockholm, Sweden. All simulions were erformed wih domin rdius of b =. m nd mesh h consised of, ringulr elemens. Elemen size rnged from roximely. m he erimeer of he domin o roximely. m long he inerfces beween he hermlly conducive eoxy nd he sinless seel ubing Fig.. The emerure T L,, x, y = L, ws ken o reresen he emerure rise recorded by hermisor.

24 Vlidiy of he Soluion In deriving he ICPC soluion, we ssumed h he emerure disribuion in he viciniy of he heer robe is rdilly symmeric. For his ssumion o hold, he rio L mus be sufficienly smll h he emerure robe hs negligible effec on he emerure disribuion ner he heer robe. Obviously, his condiion mus be sisfied if he ICPC soluion is o rovide good roximion of he emerure robe emerure for he cse where he boh robes hve infinie conduciviy. In forhcoming er we will exmine he vlidiy of he soluion in greer deil. Here we ddress he vlidiy of he soluion only for he DPHP sensor of Bsinger e l., for which L.. Evluing he vlidiy of he ICPC soluion requires model h res he robes s erfec conducors nd llows for emerure disribuion in he viciniy of he heer robe h is no rdilly symmeric. This ws chieved wih our numericl model by ssigning n rbirrily lrge vlue of conduciviy for he orions of he domin conining eoxy or sinless seel. Resuls were obined for conduciviy of. W m K, which is more hn, imes greer hn he conduciviy of he sinless seel. All simulions were erformed using vlues for C e nd C ss from Tble nd he vlues for e,, q, nd L given in Tble. Resuls for dry snd, we snd, nd wer were obined by using he soil herml roeries given Tble. For ll hree medi, he numericlly simuled emerure rise curves, T L,,, re nerly idenicl o he curves for he ICPC soluion shown in Fig.. We comre resuls from he wo soluions by exmining he difference V P T L,, s funcion of ime Fig.. Agreemen beween he nlyicl nd numericl resuls ws bes for he we snd nd wer Fig., wih vlues for V P T L,, no greer hn. K nd. K for he we snd nd

25 wer, resecively. Thus, for he we snd nd wer, i ers h L. is sufficienly smll h he ICPC soluion rovides n excellen roximion of he emerure of he emerure robe for he cse where boh robes re erfec conducors. Agreemen beween he nlyicl nd numericl soluions ws no s good for he dry snd Fig.. The ssumion of rdilly symmeric emerure disribuion ner he heer robe ws no sisfied o he sme exen for dry snd s for we snd nd wer. Neverheless, he mgniude of he difference beween he nlyicl nd numericl resuls no greer hn. K suggess h he ICPC soluion roximes he emerure of he emerure robe wih sufficien ccurcy o be of use for DPHP licions where L.. Arorieness of Soluion for DPHP Alicions Alhough he ICPC soluion closely roximes he emerure of he emerure robe for he cse where boh robes re erfec conducors, i will be rorie for DPHP licions only if i is rorie o re he robes s erfec conducors. Wheres he ICPC soluion correcly ccouns for he finie he cciy of he robes, i only roximely ccouns for heir finie rdius becuse i does no ccoun for heir finie conduciviy. Thus, is rorieness for DPHP licions deends on wheher he benefi gined by ccouning for finie he cciy offses he loss of ccurcy cused by roximing he effec of finie rdius. To exmine his issue, we erformed simulions for he sme hree cses described in he revious secion, bu wih version of he numericl model in which he robe merils hd finie conduciviy s well s finie he cciy. Simulions were erformed using he conduciviies for eoxy nd sinless seel given in Tble. We comred hese resuls o resuls obined wih boh he ICPC soluion nd he line-source soluion. The rorieness of

26 he ICPC soluion is ddressed by evluing is erformnce relive o h of he line-source soluion, which does no ccoun for he finie roeries of he robes. For ll hree medi, he numericlly simuled curves, T L,,, re nerly idenicl o he curves for he ICPC soluion shown in Fig.. The level of greemen cn be seen by exmining he difference V P T L,, s funcion of ime Fig.. The difference curves for ll hree medi show h he ICPC soluion overesimes he emerure of he emerure robe relively erly imes nd underesimes i relively le imes. The reson for his rend is ddressed ler in his secion. Here we drw enion o he fc h he numericlly simuled vlues gree quie well wih he vlues of V P from he ICPC soluion. The mgniude of he difference V P T L,, is no greer hn. K for he dry snd, no greer hn. K for he we snd, nd no greer hn. K for wer. In conrs, he level of greemen beween he numericlly simuled curves nd he linesource soluion is no nerly s good. This cn be seen by exmining he difference υ d, T L,, s funcion of ime Fig.. The line-source soluion overesimes he emerure of he emerure robe by s much s. K for he dry snd, nd underesimes i by s much s. K nd. K for he we snd nd wer, resecively. Clerly, he ICPC soluion offers significn imrovemen over he line-source soluion in chrcerizing he emerure of he emerure robe. We herefore conclude h i is rorie o re he robes of DPHP sensor s erfec conducors for licions where L.. Furher insigh regrding he rorieness of he ICPC soluion cn be obined by using resuls from he numericl simulion o exmine he sil disribuion of emerure rdilly ouwrd from he cenerline of he heer robe ino he medium surrounding i. We show resuls for he cse where he medium is wer Fig.. The lbels on he curves denoe he ime in

27 seconds from he onse of heing. The emerure of he heer robe increses ridly wih ime during he inervl <, hen grdully decreses fer he curren o he heer robe is swiched off. Of ineres here is he fc h he orion of he heer robe mde of sinless seel remins nerly isoherml for ll imes. This orion of he heer robe herefore behves very much like erfec conducor. The eoxy-filled orion of he heer robe lso behves like erfec conducor for imes greer hn. Only during he heing inervl does he eoxy-filled orion of he robe exhibi behvior inconsisen wih h of erfec conducor. Clerly, here ers o be good reson for he fc h he ICPC soluion rovides n excellen descriion of he emerure of he emerure robe. Reurning o he resuls of Fig., we now reconsider he fc h he ICPC soluion overesimes he emerure robe emerure relively erly imes nd underesimes i relively le imes. Treing he robes s erfec conducors roduces his resul becuse he effec of he finie rdius of he robes is overesimed, resuling in underesimion of he effecive disnce rveled by he he-ulse signl. This cuses lefwrd shif of he signl received by he emerure robe. The fc h he ICPC soluion iniilly overesimes he emerure nd hen ler underesimes i is direc consequence of his lefwrd shif. Desie he fc h β for hese resuls Fig., he finie he cciy of he robes hs no effec on he differences V P T L,, becuse he ICPC soluion nd he numericl model boh ccoun for he finie he cciy of he robes. Ineresingly, he lefwrd shif is smll nd relively similr for ll hree medi, flling somewhere beween. nd. s, which suggess he ossibly of inroducing ime offse correcion o furher imrove he ccurcy of he ICPC soluion for DPHP licions.

28 DISCUSSION Hving esblished he vlidiy of he ICPC soluion, nd hving demonsred is usefulness for DPHP licions, we now briefly consider exerimenl evidence suggesing h he DPHP mehod my resul in bised herml roery nd wer conen esimes if he finie roeries of he robes re no ken ino ccoun. We lso exmine oenil limiions of he ICPC soluion for siuions where L >.. Exerimenl Evidence for Probe Effecs Perhs he sronges evidence for he foremenioned robe effecs is found in he resuls of Hm nd Benson, who clibred DPHP sensors i.e., mesured ren robe scing in medi wih rnge of known volumeric he cciies. Aren scing ws deermined from mesured vlues of mximum emerure rise using infinie line source heory. Their resuls showed h ren scing incresed significnly s he he cciy of he clibrion medi decresed. Furhermore, hey showed h ren scing ws greer hn he hysicl robe scing for clibrion in dry glss beds wheres ren scing ws smller hn he hysicl scing for clibrion in sured glss beds nd gr-immobilized wer. These resuls re enirely consisen wih he differences beween he mximum emerure rise of he ICPC nd line-source soluions Fig.. The resuls in Fig. show h ren scing will be overesimed when β > s for clibrion in dry glss beds nd underesimed when β < s for clibrion in sured glss beds nd gr-immobilized wer. Alhough he resuls of Hm nd Benson cnno be exlined enirely by he finie rdius nd finie he cciy of he robes, hese robe effecs likely conribued o he observed rends. Clerly, here is need for exerimenl work o deermine if use of he ICPC

29 soluion migh minimize he deendence of ren scing on he he cciy of he clibrion medium, or erhs even elimine he need o deermine ren scing. Furher evidence for robe effecs cn be found in he fc h mny hve reored surion-deenden bis in wer conen esimes derived from he cciy esimes when infinie line source heory is used for rmeer esimion Trr nd Hm, ; Song e l., ; Brisow e l., ; Bsinger e. l., ; Ochsner e l.,. The bis reored in hese invesigions is consisen wih he resuls of Fig. in h overesimion of wer conen ws grees smller wer conens i.e. for smller vlues of he cciy when he conrs beween he he cciy of he robes nd h of he soil would be grees. Clerly, here is lso scoe for exerimenl work o deermine if use of he ICPC soluion migh reduce surion-deenden bis in he cciy nd wer conen esimes. Poenil Limiions of he Soluion The vlidiy of he ICPC soluion nd is rorieness for DPHP licions were esblished using resuls for he cse where L.. I follows, of course, h he soluion will be vlid nd rorie for licions where L <., bu here re oenil limiions for licions where L >.. Addiionl invesigion is required o deermine he exen o which L cn be incresed before he ICPC soluion fils o rovide good roximion of he emerure robe emerure for he cse where boh robes hve infinie conduciviy. Also required is nlysis o deermine he exen o which before he rorieness of he erfec conducor ssumion begins o fil. L cn be incresed Nowihsnding hese oenil limiions of he heory, i is nurl o consider wheher use of he ICPC soluion migh mke i fesible o imlemen he DPHP mehod wih sensors h

30 hve robes of lrger rdius. This ossibiliy cerinly meris considerion, bu words of cuion re in order. The ICPC soluion ccouns for he finie rdius of he robes, bu i does no ccoun for heir finie lengh or for he ossibiliy of xil conducion in he robes. These issues re no generlly regrded o be of mjor concern for sensors wih robe scing nd robe geomery i.e., rdius nd lengh similr o h of he Cmbell e l. nd Bsinger e l. sensors, bu hey my become significn if robe rdius is incresed wihou corresonding increse in robe lengh. Thus, i is no given h he rdius of he robes cn be incresed simly by using heory h ccouns for heir finie rdius nd finie he cciy. Exerimenl work s well s ddiionl nlysis will be required. Finlly, we emhsize h he ICPC soluion does no llow for he ossibiliy of imerfec conc beween he soil nd he robes. In his regrd, he ICPC soluion is no beer hn he line source soluion. The imornce of conc resisnce ws no ddressed in his invesigion, bu i cerinly meris furher considerion. I hs been shown conc resisnce cuses miniml error in he cciy esimes Noborio e l., ; Liu nd Si,, bu Noborio e l. lso showed h.-mm ir g beween he soil nd he robes my cuse error s lrge s % in esimes of herml conduciviy. SUMMARY AND CONCLUSIONS We hve derived seminlyicl soluion h ccouns for he finie rdius nd finie he cciy of he robes of DPHP sensor. The soluion consiss of closed-form exression in he Llce domin h is invered numericlly using he Sehfes lgorihm. Equion [] is he Llce-domin exression for he emerure of he emerure robe for he generl cse where he heer nd emerure robes hve differen rdius nd he cciy. This resul ws

31 used o derive Eq. [], which gives he emerure of he emerure robe for he secil cse where boh robes of he DPHP sensor hve he sme rdius nd he sme he cciy. Equion [], which we refer o s he ICPC soluion, ws used o invesige he effecs of he finie rdius nd finie he cciy of he robes. An imorn resul reveled by he funcionl form of Eq. [] is h he finie roeries of he heer robe nd he finie roeries of he emerure robe hve equl effecs on he emerure of he emerure robe. We lso hve shown h he finie rdius of he robes cuses he he-ulse signl o rrive he emerure robe slighly erlier in ime. The mgniude of his ime shif deends rimrily on he mgniude of he rio L. For sensor wih fixed robe scing L, he mgniude of he ime shif increses s he rdius of he robes increses. The effec of he finie he cciy of he robes deends on he rmeer β, which is he rio of he he cciy of he robes nd he he cciy of he soil. Relive o he cse where β =, less energy is needed o rise he emerure of he robes when β <. This cuses n increse in he mgniude of he he-ulse signl, nd he signl is osiively skewed so he mximum emerure rise occurs erlier in ime. When β >, he mgniude of he he-ulse signl is decresed nd he signl is negively skewed so he mximum emerure rise occurs ler. Resuls obined wih he ICPC soluion for yicl DPHP sensor showed h he effecs of he finie rdius nd finie he cciy of he robes re no insignificn. This is riculrly rue when β is lrge, which occurs when mesuremens re mde in relively dry soil. The lrge conrs beween he he cciy of he robes nd h of he soil hs subsnil effec on he he-ulse signl received by he emerure robe. Our resuls for yicl sensor re consisen wih ublished exerimenl d suggesing h he DPHP mehod my resul in

32 bised herml roery nd wer conen esimes if he finie roeries of he robes re no ken ino ccoun. Severl simlifying ssumions were mde in deriving he seminlyicl soluion: i h he robes re erfec conducors, ii h he emerure robe does no ler he rdil symmery of he emerure disribuion round he heer robe, nd iii h here is no conc resisnce he soil-robe inerfces. The vlidiy of he firs wo ssumions ws invesiged wih finie-elemen model h ccouned for he finie conduciviy s well s he finie he cciy of he robes. The resuls showed h boh of hese ssumions re vlid for DPHP licions where he robes sisfy he condiion h L.. Addiionl invesigion is required o deermine he exen o which L cn be incresed before hese ssumions become inrorie. If he ICPC soluion remins vlid for lrger vlues of L, i my be fesible o imlemen he DPHP mehod wih sensors h hve robes of lrger rdius. This would be desirble from he sndoin of incresing he rigidiy of he robes, bu i is no given h he rdius of he robes cn be incresed by simly using heory h ccouns for heir finie rdius nd finie he cciy. Oher effecs e.g., xil conducion my become significn s robe rdius is incresed. Exerimenl work s well s ddiionl nlysis will be required o deermine wheher oher effecs my offse he benefi gined by ccouning for he finie rdius nd finie he cciy of he robes. NOMENCLATURE rdius of he robes for he cse where hey boh hve he sme rdius m rdius of he heer robe m rdius of he emerure robe m

33 e rdius of he eoxy-filled region in he robes m b rdius of he semicirculr domin for numericl model m g re of he generion er uni volume for numericl model W m Llce rnsform vrible s q re of he er uni lengh relesed from he heer robe or line he source W m r rdil coordine for coordine sysem cenered on he heer robe m r rdil coordine for coordine sysem cenered on emerure robe m ime s heing durion s x Cresin coordine m y Cresin coordine m C volumeric he cciy of he soil J m K C volumeric he cciy of he robes for he cse where hey boh hve he sme he cciy J m K C volumeric he cciy of he heer robe J m K C volumeric he cciy of he emerure robe J m K C e volumeric he cciy of he hermlly-conduciviy eoxy J m K C ss volumeric he cciy of he sinless-seel ubing J m K I m modified Bessel funcion of he firs kind of order m K n modified Bessel funcion of he second kind of order n L disnce beween he cenerlines of he heer nd emerure robes m T emerure of he soil nd robes in he numericl simulions K V emerure of he heer robe in he seminlyicl soluion K

34 V emerure of he emerure robe in he seminlyicl soluion K β rio of he volumeric he cciy of he robes nd he volumeric he cciy of he soil for he cse where boh robes hve he sme he cciy dimensionless β rio of he volumeric he cciy of he heer robe nd he volumeric he cciy of he soil dimensionless β rio of he volumeric he cciy of he emerure robe nd he volumeric he cciy of he soil dimensionless θ ngulr coordine for coordine sysem cenered on emerure robe rdins φ rbirry heing funcion κ herml diffusiviy of he soil m s λ herml conduciviy of he soil W m K υ emerure of he soil in he seminlyicl soluion K APPENDIX A The line-source soluion of Brisow e l. is widely used for DPHP licions. Here we show h heir soluion is secil cse of generl line-source soluion. We lso give he Llce rnsform of h soluion. The rnsform of he generl line-source soluion is of ineres becuse i ers in he derivion of he seminlyicl soluion. Consider line source of infinie lengh h releses he ino n infinie medium iniilly zero emerure. The line source is loced x, y =,, so he rdil disnce from he line source in he x-y lne is r, where r = +. If he is relesed consn re during he x y ime inervl <, he soluion of he he equion is de Vries,

35 ; Ei Ei, ; Ei, r r q r r q r > κ κ πλ = υ < κ πλ = υ [A] where q is he re er uni lengh which he is relesed, is he heing durion, nd Ei x is he exonenil inegrl of rgumen x. This is he soluion emloyed in he DPHP mehod of Brisow e l.. I is esily shown h Eq. [A] is secil cse of he generl soluion Crslw nd Jeger,,. d r r κ φ πλ = υ ex, [A] For ulsed heing, he generl line-source soluion yields Eq. [A] by using he form for φ given in Eq. []. By mking use of Duhmel s heorem Crslw nd Jeger,,., i cn be shown h he Llce rnsform of he generl line-source soluion is ˆ, ˆ r K r πλ φ = υ [A] In he body of he ex, we refer o Eq. [A] in discussing he form of Eq. []. We lso use Eq. [A] wih r = L in discussing he form of Eq. []. APPENDIX B The Sehfes lgorihm for invering Eq. [] cn be wrien in he form = β + ω πλ N i i i i i i i K K L K i q V C ]} [ { [B] where

36 ln i = i ; i =,, K, N [B] κ All inversion clculions were erformed by evluing Eq. [B] wih N = nd he Sehfes weighing coefficiens, ω i, given in Tble B. Uon relcing wih in Eq. [B] nd [B], C he sme lgorihm cn be used o obin vlues of V for imes greer hn. Insmuch s Eq. [B] is funcion of ime, i mus be evlued ech ime for which vlue of V C is C desired. I mus lso be evlued ech ime for which vlue of V is desired. The vlues of C C V nd V obined in his wy were used o evlue Eq. [b]. We used N = for our clculions becuse i MATLAB s double recision floing-oin rihmeic yields resuls wih roximely significn deciml digis, nd ii he funcion BESSELK evlues he Bessel funcions K nd K o n ccurcy of bou deciml lces. Oiml resuls re generlly obined when N is bou equl o he number of significn deciml digis used in he clculions Knigh nd Riche,. ACKNOWLEDGMENTS This meril is bsed on work suored by he Nionl Science Foundion under Grn No. ECS-. REFERENCES Bsinger, J.M., G.J. Kluienberg, J.M. Hm, J.M. Frnk, P.L. Brnes, nd M.B. Kirkhm.. Lborory evluion of he dul-robe he-ulse mehod for mesuring soil wer conen. Vdose Zone J. :.

37 Blckwell, J.H.. A rnsien-flow mehod for deerminion of herml consns of insuling merils in bulk: Pr I. Theory. J. Al. Phys. :. Brisow, K.L., G.J. Kluienberg, C.J. Goding, nd T.S. Fizgerld.. A smll muli-needle robe for mesuring soil herml roeries, wer conen nd elecricl conduciviy. Comu. Elecron. Agric. :. Brisow, K.L., G.J. Kluienberg, nd R. Horon.. Mesuremen of soil herml roeries wih dul-robe he-ulse echnique. Soil Sci. Soc. Am. J. :. Cmbell, G.S., C. Clissendorff, nd J.H. Willims.. Probe for mesuring soil secific he using he-ulse mehod. Soil Sci. Soc. Am. J. :. Crslw, H.S., nd J.C. Jeger.. Conducion of he in solids. nd ed. Clrendon Press, Oxford, UK. de Vries, D.A.. A nonsionry mehod for deermining herml conduciviy of soil in siu. Soil Sci. :. Gurgli, D.O., nd J.L. Pous.. An elecricl model of he flow in soil. Soil Sci. Soc. Am. J. :. Hm, J.M., nd E.J. Benson.. On he consrucion nd clibrion of dul-robe he cciy sensors. Soil Sci. Soc. Am. J. :. Homns, J.W., J. Simunek, nd K.L. Brisow.. Indirec esimion of soil herml roeries nd wer flux using he ulse robe mesuremens: Geomery nd disersion effecs. Wer Resour. Res., doi:./wr. Incroer, F.P., nd D.P. De Wi.. Fundmenls of he nd mss rnsfer. h ed. John Wiley & Sons, New York.

38 Jeger, J.C.. Conducion of he in n infinie region bounded inernlly by circulr cylinder of erfec conducor. Aus. J. Phys. :. Kmi, T., G.J. Kluienberg, nd J.W. Homns.. Design nd numericl nlysis of buon he ulse robe for soil wer conen mesuremen. Vdose Zone J. :. Knigh, J.H., W. Jin, nd G.J. Kluienberg.. Sensiiviy of he dul-robe he-ulse mehod o sil vriions in he cciy nd wer conen. Vdose Zone J. :. Knigh, J.H., nd A.P. Riche.. Trnsien elecromgneic clculions using he Gver- Sehfes inverse Llce rnsform mehod. Geohysics :. Liu, G., nd B.C. Si.. Errors nlysis of he ulse robe mehods: Exerimens nd simulions. Soil Sci. Soc. Am. J. :. Noborio, K., K.J. McInnes, nd J.L. Heilmn.. Mesuremens of soil wer conen, he cciy, nd herml conduciviy wih single TDR robe. Soil Sci. :. Novkowski, K.S.. Anlysis of ulse inerference ess. Wer Resour. Res. :. Nusier, O.K., nd N.H. Abu-Hmdeh.. Lborory echniques o evlue herml conduciviy for some soils. He Mss Trnsfer :. Ochsner, T.E., R. Horon, nd T. Ren.. Use of he dul-robe he-ulse echnique o monior soil wer conen in he vdose zone. Vdose Zone J. :. Ogbe, D.O., nd W.E. Brighm.. A model for inerference esing wih wellbore nd skin effecs boh wells. Per SPE resened he h Annul Technicl Conference nd Exhibiion, Houson, TX, Se. -,. Sociey of Peroleum Engineers. Olver, F.W.J.. Bessel funcions of ineger order... In M. Abrmowiz nd I.A. Segun ed. Hndbook of mhemicl funcions wih formuls, grhs, nd mhemicl bles. Dover, New York.

39 Ren, T., G.J. Kluienberg, nd R. Horon.. Deermining soil wer flux nd ore wer velociy by he ulse echnique. Soil Sci. Soc. Am. J. :. Ren, T., K. Noborio, nd R. Horon.. Mesuring soil wer conen, elecricl conduciviy, nd herml roeries wih hermo-ime domin reflecomery robe. Soil Sci. Soc. Am. J. :. Song, Y., J.M. Hm, M.B. Kirkhm, nd G.J. Kluienberg.. Mesuring soil wer conen under urfgrss using he dul-robe he-ulse echnique. J. Am. Soc. Horic. Sci. :. Sehfes, H.. Algorihm : Numericl inversion of Llce rnsforms [D]. Commun. ACM :. Sehfes, H. b. Remrk on Algorihm [D]: Numericl inversion of he Llce rnsforms. Commun. ACM :. Trr, J.M., nd J.M. Hm.. Mesuring soil wer conen in he lborory nd field wih dul-robe he-cciy sensors. Agron. J. :. Tongenyi. Y., nd R. Rghvn.. The effec of wellbore sorge nd skin on inerference es d. J. Pe. Technol. :. FIGURE CAPTIONS Fig.. Coordine sysems used in deriving he seminlyicl soluion. The heer robe hs rdius, he emerure robe hs rdius, nd he cenerlines of he robes re disnce L r. The olr coordine sysem cenered on he heer robe x, y =, hs rdil coordine r, where r = +. The olr coordine sysem cenered on he emerure x y

40 robe x, y = L, hs coordines r nd θ h sisfy he condiions nd x L, y = r cosθ, r sin. θ r x L + y = Fig.. Temerure or chnge in emerure of he emerure robe s funcion of ime. Resuls for he ICPC soluion re from Eq. [] wih β = nd dimensionless robe sizes L of.,., nd.. Resuls for he line-source soluion re from Eq. [A] wih r = L. The ICPC soluion is idenicl o he line-source soluion for he limiing cse where. Boh soluions were evlued using q = W m, = s, L =. m, C =. MJ m K, nd λ =. W m K. Fig.. Temerure or chnge in emerure of he emerure robe s funcion of ime. Resuls for he ICPC soluion re from Eq. [] wih L =. nd β vlues of.,., nd.. Resuls for he line-source soluion re from Eq. [A] wih r = L. The ICPC soluion is idenicl o he line-source soluion for he limiing cse where. Boh soluions were evlued using q = W m, = s, L =. m, C =. MJ m K, nd λ =. W m K. Fig.. Cross-secion of he heer robe lef nd emerure robe righ of he dul-robe he-ulse sensor of Bsinger e l.. The cross-secion is for lne norml o he xes of he robes h coincides wih he locion of he hermisor. The heing elemen in he heer robe consiss of four srnds of resisnce wire. The orions of he robes filled wih hermlly conducive eoxy re shown in whie. Fig.. Temerure or chnge in emerure of he emerure robe s funcion of ime for dry snd β =., we snd β =., nd wer β =.. Resuls for he ICPC soluion re from Eq. [] wih L.. Resuls for he line-source soluion re from Eq.

41 [A] wih r = L. Boh soluions were evlued using medi herml roeries from Tble nd he rmeer vlues given in Tble. Fig.. Mesh of ringulr elemens used for he numericl model. A Mesh for he heer nd emerure robes nd orion of he soil surrounding hem. The x xis coincides wih he horizonl line h sses hrough he cener of he robes. B Exnded view of he mesh for he emerure robe nd orion of he soil surrounding i. The smll circle idenifies he node x, y = L,. The emerure his node ws ken o reresen he emerure recorded by he hermisor. Fig.. Difference beween he emerure of he emerure robe deermined using he ICPC soluion, V P, nd he emerure of he emerure robe deermined using he numericl model, T L,,. The difference V P T L,, is shown s funcion of ime for dry snd, we snd, nd wer. Resuls for he ICPC soluion re from Eq. []. Resuls from he numericl model re for he cse where boh robes were ssigned n rbirrily lrge vlue of conduciviy so hey were effecively erfec conducors. The ICPC soluion nd he numericl model were boh evlued using medi herml roeries from Tble nd he rmeer vlues given in Tble. Vlues for C e nd C ss in he numericl model were from Tble. Fig.. Difference beween he emerure of he emerure robe deermined using he ICPC soluion, V P, nd he emerure of he emerure robe deermined using he numericl model, T L,,. The difference V P T L,, is shown s funcion of ime for dry snd, we snd, nd wer. Resuls for he ICPC soluion re from Eq. []. Resuls from he numericl model re for he cse where he robes hd finie conduciviy s well s finie he cciy. Boh soluions were evlued using medi herml roeries from

42 Tble nd he rmeer vlues given in Tble. For he numericl model, vlues from Tble were used for he herml roeries of he robe merils. Fig.. Difference beween he emerure of he emerure robe deermined using he linesource soluion, υ L,, nd he emerure of he emerure robe deermined using he numericl model, T L,,. The difference υ L, T L,, is shown s funcion of ime for dry snd, we snd, nd wer. Resuls for he line-source soluion re from Eq. [A] wih r = L. Resuls from he numericl model re for he cse where he robes hd finie conduciviy s well s finie he cciy. Boh soluions were evlued using medi herml roeries from Tble nd he rmeer vlues given in Tble. For he numericl model, vlues from Tble were used for he herml roeries of he robe merils. Fig.. Sil disribuion of emerure rdilly ouwrds from he cenerline of he heer robe ino wer. The lbels on he curves denoe he ime in seconds from he onse of heing. The oen ringle shows he locion of he inerfce beween he eoxy nd sinless seel r = e. The filled ringle shows he locion of he inerfce beween he sinless seel nd wer r =. Resuls from he numericl model re for he cse where he robes hd finie conduciviy s well s finie he cciy. The model ws evlued using medi herml roeries from Tble nd he rmeer vlues given in Tble. Vlues from Tble were used for he herml roeries of he robe merils.

43 Tble. Volumeric he cciy C, herml conduciviy λ, nd herml diffusiviy κ of he merils used in his invesigion. The relionshi κ = λ C ws used o clcule diffusiviy vlues. Meril C λ κ MJ m K W m K m s Tye sinless seel... Omegbond eoxy... Air-dried Clyon snd... Sured Hnlon snd... Wer... Vlues from Tble A. of Incroer nd De Wi. Vlues from Tble of Kmi e l.. Vlues ded from Tble of Brisow e l.. Vlues from Tble of Ren e l.. Tble. Prmeer vlues corresonding o he DPHP sensor of Bsinger e l.. Prmeer Vlue Unis e. m. m L. m C. MJ m K q W m s

44 Tble. Esimes of volumeric he cciy C, herml conduciviy λ, nd herml diffusiviy κ deermined by using he mehod of Brisow e l. o fi he line-source soluion o he curves for he ICPC soluion in Fig.. The vlues in renheses give he relive errors % in he esimed herml roeries i.e. he errors in esimed herml roeries exressed s ercen of he originl vlues in Tble. Meril C λ κ MJ m K W m K m s Air-dried Clyon snd Sured Hnlon snd Wer Tble B. Weighing coefficiens ω i for he lgorihm of Sehfes,b for he cse where N =. i ω i

45 Figure y x, y r r Temerure robe θ Heer robe L x

46 Figure. β = Temerure K.. ICPC soluion, / L =. ICPC soluion, / L =. ICPC soluion, / L =. Line-source soluion. Time s

47 Figure. / L =. Temerure K... ICPC soluion, β =. ICPC soluion, β =. ICPC soluion, β =. Line-source soluion Time s

48 Figure Sinless seel ubing Eoxy Thermisor Eoxy Resisnce wires Sinless seel ubing

49 . Temerure K.... Dry snd β =. ICPC soluion Line-source soluion Figure... We snd β =. Temerure K... Temerure K..... Wer β =.. Time s

50 Figure A Soil Heer robe Temerure robe Figure b B Soil Sinless seel ubing Eoxy