ANALYSIS OF THERMAL MEASUREMENTS ACQUIRED IN NUNAVIK: COMPARISON OF FIELD DATA WITH NUMERICAL MODELS

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1 ANALYSIS OF THERMAL MEASUREMENTS ACQUIRED IN NUNAVIK: COMPARISON OF FIELD DATA WITH NUMERICAL MODELS K. Hadj-Rabia 1, A.M. Cames-Pintaux 1, M. Allad 2 1 LMSGC, UMR 113 L.C.P.C./C.N.R.S., Champs su Mane,Fance cames@lcpc.inets. 2 Cente d'žtudes nodiques, UnivesitŽ Laval, QuŽbec, Canada michel.allad@cen.ulaval.ca Abstact Gound themal data om the Cente d'ƒtudes Nodiques study site at Manitounuk Sound along the Hudson bay coast wee used o validation o two numeical themal models designed at Fance's C.N.R.S. The numeical esults obtained with the models show good ageement with the ield measuements. As the models ae sensitive and pecise, the esults also show that pecision in calculated esults stongly depends on detailed inputs o gound ice statigaphy. Intoduction Mathematical models have become tools that help to simulate and undestand pemaost themal behaviou in vaious geomophological, ecological and climatic contexts. Thei use is necessay in ode to develop an analytical capability and o making egional pedictions o pemaost conditions. Also compaing models with ield data is a good means to detect, though discepancies, new teain actos that ae not easily ecognized in the ield. An analysis o the themal egime in pemaost equies solution o the Stean equation, which contols heat tanse poblems with phase change. Two numeical models wee chosen to be validated by compaison with gound themal data om the Manitounuk site in nothen QuŽbec, Canada. The ist model tested was the One-dimensional axisymetical model with phase change (Cames-Pintaux et al, 1983). It detemines with pecision and minimum calculation eot the displacement o the eezing ont and soil tempeatues at each time step. It is available in a plane, cylindical o spheical domain. The second tested model was the Two-dimensionnal Stean poblem teated by a numeical method with altenate diections (also called A.D.I. model) (Cames-Pintaux et al, 1990). Using an altenate diections method, the teatment at each time step o the displacement o the eezing ont ollowing two axes was made by adapting the one-dimensional method o Goodich (1978). This A.D.I. model applies to domains composed eithe o one mateial (homogeneous) o seveal mateials (heteogeneous). The mathematical pecision o both models has poven to be vey high in past applications in engineeing. Howeve, the level o detail they allow has not peviously been tested with high quality ield measuements in tue pemaost conditions. The numeical esults obtained with these two models wee compaed with ecoded measuements at the Manitounuk site in ode to test the accuacy o the models and to identiy the gound paametes most likely to inluence model peomance The Manitounuk site At the Manitounuk site along the east coast o Hudson bay, the Cente d'etudes Nodiques measued tempeatue and caied out studies o a pemaost plateau om Apil 1988 until Decembe The pemaost plateau, typical o the discontinuous pemaost zone, is located along the shoeline and is 5.6 m high. It is about 185 m long and 150 m wide (Figue 1). At this location, the pemaost depth is about 16 m (Allad et al., 1996). The soil o the plateau consists o Quatenay clayey silt o the Tyell Sea, a highly ostsusceptible mateial. The plateau is instumented with ou themisto cables, emplaced within boeholes in August The cables ae labelled TP1-TG1, TG2, TG3 and TG4. Themisto beads wee placed at depths o 0, 1, 2, 3, 4, 5, 6, 7, 8, 12 and 16 metes along each o K. Hadj-Rabia, et al. 397

2 Figue 1. Aeial photogaph o the MANITOUNUK study site. Location o the ou themisto cables is shown and the pemaost is delineated (modiied om Allad et al., (1996). the cables. Daily tempeatue measuements wee ecoded automatically o cable TP1-TG1 om Apil 1988 until Decembe The thee othe, nonautomated cables, also povided data as they wee manually ead thee times a yea duing site visits: Mach 8, 1988, Apil 7, 1988, August 12, 1988, Apil 11, 1989, August 5, 1989 and Mach 12, In 1994, twenty-ou boeholes wee dilled down to 4 metes and coes wee descibed. One deepe boehole was also dilled down though the pemaost base at 16 m. This dilling pogam povided knowledge o the cyostatigaphy o the soil (Allad et al., 1996). Fou layes make up the complete pemaost statigaphy; thei aveage thicknesses acoss the plateau ae as ollows: 1- Laye I is the active laye. Its mean depth is about 0.80 m. 2- Laye II is an ice ich laye (45 to 80 pe cent ice content) om 0.80 m to 2.30 m. 3- Laye III contains a low volume o eticulate ice (15 to 40 pe cent) om 2.30 m to 3.50 m. 4- Laye IV has a vey high ice content made o lenses that get thicke with depth (50 to 80 pe cent ice content) om 3.50 m until the pemaost base at 16 m. Themophysical chaacteistics To conduct numeical analyses, it was essential to know the themal conductivity and volumetic heat capacity o the dieent layes, as well as the latent heat involved duing phase change. Samples om coes wee analysed at the Geological Suvey o Canada (GSC) laboatoy and a ist detemination o themal conductivity was obtained on numeous samples. A compaative analysis o the themophysical chaacteistics om statigaphic hoizons o the site o Umiujaq (100 km noth o Manitounuk, also in clayey silt o Tyell Sea) and o this site was made in 1993 (Haoulani et al., 1993). Subsequent to the dilling o the boeholes in Manitounuk site, measuements o volumetic ice content wee caied out in 1994 (Allad et al., 1996). The moe pecise inomation on phase composition at vaious depths in the pemaost allowed much moe accuate detemination o the themal chaacteistics o the dieent layes. Volumetic actions o ai and oganic mateial wee judged to be non-signiicant at the Manitounuk site since the voids wee satuated and no peat laye was pesent on the suace o in the statigaphy. Accoding to the boehole data, the mean volumetic ice content allowed the calculation o the volumetic actions o solids, ice and unozen wate. The detemination o 398 The 7th Intenational Pemaost Coneence

3 vaiable unozen wate content with tempeatue was made by applying the statistical elationships established by Fotie et al. (1991) using an adiabatic caloimety method. Computations o conductivity l, volumetic heat capacity (C) and latent heat (L) could be undetaken. Fo a ange o tempeatues, the themophysical mean popeties used hee ae indicated o the active laye in Table 1, o ice-ich pemaost (70% ice content) in Table 2 and o ice-poo pemaost (30% ice content) in Table 3. Despite the maine oigin o the sediments and the poximity o the site to the seashoe, measuements on soil wate and on ice lens wate showed no salt content in the soil. Table 1. Themophysical popeties o the active laye Table 2. Themophysical popeties o ice-ich pemaost (70%) Numeical appoach The validity o the one-dimensionnal and two-dimensionnal models was tested with the help o expeimental measuements in heteogeneous soils. But as oiginally conceived, the themophysical chaacteistics in these models did not vay with tempeatue and theeoe wee not able to epoduce accuately tempeatues in pemaost. Fo this study, it was necessay to modiy the mathematical and numeical appoach o these two models to incopoate the geometical heteogenity o the cyostuctue (i.e., ice poo and ice ich layes) and allow the themophysical chaacteistics to vay vaiable with tempeatue. The equations o the omulation o the Stean poblem wee completely evised within the A.D.I. model and the axisymmetical model because a themodependent model is moe complex than a model with themophysical chaacteistics that depend only on the physical popeties o the soils unde investigation. The Stean poblem duing the inteval time o study is witten in a two-dimensional domain W, in the absence o heat souces and heat sinks. The suace S that sepaates unozen (u) and ozen () phase o domain W, is always at the eezing tempeatue q. The Stean equations ae witten (themophysical chaacteistics depending on tempeatue): q ( q, xyc, ) ( q, xyc, ) ( q, xy, ) div l q, x, y gad q t ( ( ) ( ) )= 0 in ozen zone q u( q, xyc, ) u( q, xy, ) div lu q, x, y gad q t = 0 in unozen zone æ q çlu( q, xy, ) ö è n æ ( ) q ö çl = ö ø q, xy, L( q, x, y) ds è n ø dt ø n on S u e Table 3. Themophysical popeties o ice-poo pemaost (30%) (ds/dt epesents the velocity and n the unit nomal o a point on the ont). In the axisymetical model established in 1983 (Cames-Pintaux et al., 1983) and the A.D.I. model established in 1990 (Cames-Pintaux et al., 1990), the equations (1-3) wee witten: q mcm lmdiv gad q t - ( ( ))= 0 (m is equal at in ozen zone and at u in unozen zone) and æ l q ö æ ç u - çl è nø è u q ö = L ds ö nø dt ø In the adapted models, the axisymetical tempeatue dependent model and the A.D.I. tempeatue dependent model, at each time iteation, conductivity l(q), volumetic heat capacity C(q) and latent heat L(q) ae again calculated in each element o the studied domain accoding to tempeatue in this element. So, the esults obtained with these two adapted models ae moe accuate than the esults calculated by the ist models established in 1983 and e n [1] [2] [3] K. Hadj-Rabia, et al. 399

4 a) Figue 2. Deinition o the two-dimensional domain. Compaison o values obtained with the help o numeical models with in situ measuements Fo the Manitounuk site, the axisymetical tempeatue dependent model was used to calculate the behaviou o the soil along the cable TP1-TG1 and the A.D.I. tempeatue dependent model was used o the twodimensional behaviou o the soil between two themisto cables. b) The two-dimensional behaviou was calculated with the help o a ectangula domain W (Hadj-Rabia, 1997). The ectangula numeical gid was 80 m in length (x) and 20 m in depth (y) inside a tansvese vetical section (Figue 2). In this section, the cables TP1-TG1 (x tg1 ) and TG2 (x tg2 ) ae 26 m apat. The domain is coveed by a egula mesh o 50x50 cm squaes. Bounday conditions o the calculations ae: - tempeatue is imposed at 20 m depth (y = 0 m): = 0.01 C. - on vetical boundaies (x = 0 m and x = 80 m), heat luxes ae equal to 0. -T, (= q ), the tempeatue o eezing ont, is equal to 0 C. -the initial tempeatue distibution is q(x,y,t 0 (measuements o C.E.N o t 0 = Apil 8, 1988 ). Fo 0 m <= x <= (x tg2 +xt g1 )/2, q 0 (x,y) = q 0 (x tg2,y) and q 0 (x,y) = 0(x tg1,y) o (x tg2 + x tg1 )/2 <= x <= 80 m, with 0 m<= y <= 20 m. - suace tempeatue is vaiable: measuements o C.E.N. q(x) = (x,t) with t Î [apil 8,1988, apil 7,1991] om themisto bead at 0 m depth. Figue 3. a) Tempeatue poiles at 0.5 m along cable TP1-TG1, Januay- Decembe b) Tempeatue poiles at 2 m along cable TP1-TG1, Januay-Decembe Fo 0m<= x <=x tg2, q(x,t) = q(x tg2,t), o x tg2 <= x <= x tg1, q(x,t)= q(x,t) and o x tg1 <= x <=80m, q(x,y)= (x tg1,t). Fo x tg2 <= x <= x tg1, the values o q(x,t) ae obtained by intepolation between q(x tg2,t) and q(x tg1,t) with q(x tg1,t), the tempeatue measued in situ, and q(x tg2,t), the tempeatue valued om manually ead data duing site visits. In the one-dimensional model, the space-steps ae 25 cm along the vetical axis. The bounday conditions ae the same at the suace o soil and at 20 m depth. This model allows a check to see i calculated measuements ae in good ageement with themal measuements ecoded evey day along cable TP1-TG1 beoe using the moe complex two-dimensional model. 400 The 7th Intenational Pemaost Coneence

5 Figue 4. Tempeatue poiles at dieent depths between cables TG2 and TG1, Septembe 11, 1988 (70%-30% and 70%-30%). Soil conditions and paametes o model application used the themal mean popeties shown in Table 1 om 0 m to 0.8 m, and om 16 m to 20 m, in Table 2 om 0.8 m to 2.3 m, and om 3.5 m to 16 m, and in Table 3 om 2.3 m to 3.5 m. Numeical values o the tempeatue ield in the pemaost plateau wee calculated with the help o the axisymetical tempeatue dependent model, om 1988 until 1991 (measuements ae absent between Apil 7, 1991 and August 29, 1992) evey day along the cable TP1-TG1. They ae compaed with ield data. Figues 3a and 3b indicate tempeatue changes at 0.5 m and at 2 m depths along cable TP1 om Januay 1989 until Decembe Numeical values obtained by the numeical models ae in vey good ageement with the ield data at 0.5 m. The geatest depatue between the ield data and the numeical esults is o 0.65 C. With a domain coveed by a ine numeical gid, the numeical esults would be still Figue 6. Tempeatue poiles at dieent depths between cables TG2 and TG1, Septembe 11, 1988 (70% -30% and 45%-15%). bette, but the calculation time would be too expensive. The ageement is less good at 2 m in Mach and Apil. Indeed, ice and wate content in the dieent layes o pemaost change ollowing seasonal tempeatue vaiations along the poile, especially in the 2-3 metes below soil suace. Paticulaly, the depth o 2 m is nea the base o ice ich laye II which undegoes complex intenal vaiations in ice and wate content. Since in these ine soils the tempeatue o phase change may be < 0 C, the calculations wee also made with T = -0.3 C. With the help o A.D.I. tempeatue dependent model, numeical values wee calculated inside a plane domain passing though cables TP1-TG1 and TG2. Figues 4 and 5 show the tempeatues between cables TP1-TG1 and TG2 o speciic days: Septembe 11, 1988 and June 7, The in situ measuements ae indicated only o cable TP1-TG1 because the tempeatues at cable TG2 wee not ecoded o these dates. The mean ice volumetic contents o the pemaost layes om boeholes nea cable TP2 ae taken to be simila to those o cable TP1 (70% o the ice-ich laye and 30% o the ice-poo laye). In act, the pemaost layes within boeholes nea the two cables ae dieent (Allad et al., 1996). Pemaost layes om boeholes nea TG2 have a mean ice volumic content o 45% o the ice-ich laye and 15% o the ice-poo laye. Figue 6 shows isothems between TP1-TG1 and TG2 o Septembe 11, 1988 with the themophysical chaacteistics o elements nea the cable TP1-TG1 dieent to those nea cable TG2. The numeical measuements ae now in bette ageement with the ield data nea cable TP1. Figue 5. Tempeatue poiles at dieent depths between cables TG2 and TG1, June 7, 1989 (70%-30% and 70%-30%). The two numeical models calculate the location o the phase change inteace by solving the moving K. Hadj-Rabia, et al. 401

6 inteace equation at each time step. This allows an accuate pediction o the isothem's location inside the active laye. But, tempeatues collected on site (measuements ae made at 0.50 m and 1 m only) do not allow a good compaison. Conclusion Two numeical models using tempeatue dependent themophysical chaacteistics and pemaost cyostatigaphy wee used to calculate the tempeatue ield in a pemaost plateau at Manitounuk in Nunavik (Nothen QuŽbec). The pemaost at this site epesents a elatively simple case as the soil is homogenous. Howeve, its ice content is vaiable with depth and unozen wate content vaies in time with seasonal tempeatue vaiations. These vaiations ae complex in time and space which indicates that thee is oom o impoving ield measuements, o example by having themistos moe closely spaced, paticulaly in the active laye and the uppe pemaost layes. Themophysical chaacteistics o dieent statigaphic hoizons o pemaost wee measued o calculated om samples om the ield site. These values wee intoduced in the two numeical models with geometies o the studied domain. These models can be used to develop an analytic capability and o making egional pedictions o pemaost conditions. Acknowledgments Many pesons povided help and assistance ove the seveal yeas it took o instument installation and data gatheing: Richad Fotie and Janusz Fydecki. Comments om the anonymous eviewes vee vey helpul in impoving this manuscipt. The poject was made with the help o Minist e de la Recheche et de la Technologie en Fance. Reeences Allad, M., Caon, S. and Begin, Y. (1996). Climatic and ecological contols on ice segegation and themokast: The histoy o pemaost plateau in Nothen Quebec. Pemaost and Peiglacial Pocesses, 7, Cames-Pintaux, A.M., Giat, M. and Aguie-Puente, J. (1983). Intoduction des nouvelles gžomžties et conditions aux limites dans la mžthode de Goodich pou le changement de phase. In Poceedings 16th Cong s Intenational du Foid, II, Cames-Pintaux, A.M., Goodich, L.E. and Aguie-Puente, J. (1990). RŽsolution du pobl me bidimensionnel de Stean pa une mžthode implicite de diections altenžes. In Poceedings 5i me ConŽence Canadienne su le PegŽlisol, Quebec. NORDICANA, 54, Fotie, R., LŽvesque, R., Allad, M. and Seguin, M.K. (1991). CaactŽisation du pegžlisol de buttes cyog nes ˆ l'aide de diagaphies Žlectiques au NUNAVIK, QuŽbec. Pemaost and Peiglacial Pocesses. 2, Goodich, L.E. (1978). Eicient numeical o one dimensional themal poblems with phase change. Intenational Jounal Heat Tanse, 21, Haoulani, H., Bouchad, C., Roche, Y., Cames-Pintaux, A.M. and Allad, M. (1993). Compaison o numeical modelling with measuements o pemaost themal egime in a discontinuous pemaost site in nothen Quebec. In Poceedings, 4th Intenational Symposium on Themal Engineeing and Science o Cold Regions, Hanove, pp Hadj-Rabia, K. (1997) Analyse, taitement des mesues themiques acquises dans le pegžlisol du Nod QuŽbec: conontation avec des mod les numžiques. Th se d'univesitž, UnivesitŽ de Mane La VallŽe, Mane La VallŽe (146 pp.) 402 The 7th Intenational Pemaost Coneence