Profiles of relative permittivity and electrical conductivity from unsaturated soil water content models

Transcription

1 ANNALS OF GEOPHYSICS, 59, 3, 016, G030; doi: /ag-6990 Pofiles of elative pemittivity and electical conductivity fom unsatuated soil wate content models Robeta Poetta *, Fabio Bianchi Istituto Nazionale di Geofisica e Vulcanologia, Rome, Italy Aticle histoy Received Febuay 19, 016; accepted May 17, 016. Subject classification: Soil textue, Diffusion, Wate content, Conductivity/pemittivity pofiles. ABSTRACT A mathematical model of wate diffusion in the vadose zone has been implemented fo diffeent types of soil textues in ode to detemine the soil wate content (SWC) pofiles in dependence of depth and time. Fom these pofiles, obtained fo diffeent soils, we deived the chaacteistic electical paametes, such as elative pemittivity (f ) and electical conductivity (v), and thei vaiation in time, employing empiical elations available in the scientific liteatue. The simulation though mathematical models has been pefomed taking into account diffeent types of soils chaacteized by the pecentage composition of sand, clay and silt in the textual tiangle, which povides some physical and chemical popeties that affect the wate etention in the soil. The esulting simulated pofiles of SWC and consequently pemittivity and conductivity pofiles, span ove a cetain ange of values suggesting the best techniques and the limits in geophysical investigation. Moeove this a-pio knowledge helps in the elaboation and intepetation of pemittivity and conductivity data obtained by the measuements. Pemittivity and conductivity pofiles ae paticulaly useful in some envionmental applications when the soil textues ae (o supposed to be) known as in the typical case of landfill leachate dispesion. Since the soil textues beneath o neaby a disposal waste ae chaacteized by a SWC, the values of f and v at vaious depth can be diectly infeed. 1. Intoduction Electical conductivity (v) and elative dielectic pemittivity (f ) ae two independent physical popeties that chaacteize the behavio of soil when excited by electic cuents o electomagnetic fields [Settimi 011]. The electical conductivity and dielectic pemittivity of the constituent mineals ae the main contolling factos of the electical popeties of soils [Kiaa et al. 014]. The deivation of these quantities elated to the wate content povides useful infomation egading the pactical use of geophysical investigation (GPR, TDR, etc.), applied to unsatuated subsuface vadose zone of medium. This zone has elevant impotance in diffeent ambits and especially in geophysical pospection [Huisman et al. 003]. The unsatuated potion of soil concens the fist metes beneath the gound level down to the wate table, whee the satuated zone begins. In this section hydogeological popeties (physical and chemical) ae impotant since they dominate the electic popeties of soil, descibed by the two quantities v and f. These last quantities can be diectly measued though seveal techniques o theoetically deduced (constitutive elations) in a homogeneous medium. In this pape these paametes ae estimated though soil hydologic models [Huisman et al. 003]. In the latte case the SWC in vadose zone plays an impotant ole to detemine both elative pemittivity and electical conductivity since they ae stongly dependent on it. The diffeent soil textues can etain diffeent pecentage of wate, due to its poosity, unde the action of two agents: the gavity potential and the matix potential. The diffusive equation is diven by these quantities and cannot be solved without models. These models need in input physical paametes that quantify the textues of soil. In the past decades authos poposed diffeent numeical paametes [Mualem 1976, van Genuchten 1980] to solve the so-called Richads equation that well descibes the wate diffusion in the vadose zone. Once the input paametes fo the paticula type of soil ae established it is possible to numeically calculate the SWC, which is function of time and depth. SWC is the most impotant hydo-geological paametes that jointly to salinity and poosity affects the electical popeties in the unsatuated soils. This pape deals with simulations of soil model in ode to evaluate the SWC and consequently though Topp and Achie empiical elationships to detemine the electical paametes of soil. In this specific case, we obtained 3D pofiles that show tempoal and spatial vaiation of the electical popeties. The study focuses on soil investigation theoetical appoach, that could be a useful tool in some envionmental and geophysi- G030

2 PORRETTA AND BIANCHI in [cm/day] and i is the soil wate content [cm 3 /cm 3 ]. It descibes the movement of the fluid (fluid motion) in a non-satuated and poous medium. It is a non linea patial diffeential equation and, as such, it has no analytical closed fom solution and must be integated numeically. i t z K = " W Q i VS z + 1 X% (3) Figue 1. Wate etention cuve that links the two vaiables i o } depending on the type of soil [Tulle and O 005]. cal applications. Futhemoe it could epesent a peliminay phase which pecedes the measuements and facilitates the intepetation of the esults (GPR, TDR, ERT and othe simila geophysical investigation techniques e.g. afte aining suvey o in landfill contol).. Wate content models Wate flows in satuated poous medium and wate content etention in diffeent kind of soils has been subject of many studies. To descibe this phenomenon in the vadose zone Richads poposed an equation which is a combination between the flux equation and mass consevation, and it chaacteizes the SWC tend ove time [Richads 1931]. The softwae developed to deivate SWC pesented in this pape it s based on well-known liteatue models such as van Genuchten, Mualem, Books and Coey. The intoduction of these models was necessay, due to the impossibility to analytically solve the Richads equation. The wate diffusion equation into unsatuated soil in one-dimensional tems can assume the following fom: i t = z SK H z X (1) Futhemoe, K(i) is not a constant in the stict sense, as it depends on the same i, hence the poblem need an appopiate model to be teated. Hee it is pesented in one-dimensional tem along the vetical z. Equations () and (3) have a two dependent vaiables i and H (o }), so the esolution of the two foms equies the definition of a constitutive equation that binds i and } as in Figue 1 [Mualem 1976]. Fom USDA (United States Depatement of Agicultue) tiangle textues shown in Figue typical van Genuchten paametes ae epoted in Table 1 [Leij et al. 1996]. As mentioned ealie K is not a constant, because of its dependence on i and fo this easons, a model fo each diffeent soil textues is equied. This demands a constitutive equation that takes into account the wate etention cuve. Such a cuve depends on the soil pososity, o, even bette depends on the effective satuation Se [Leij et al. 1997]. The latte can be expessed as: Se = Qi- iv Qis - iv whee: is is the wate content in satuated conditions and i is the esidual wate content. It means that if Se=0 the fluid can be found only in gaseous phase, (4) whee i is the SWC [m 3 /m 3 ], H epesents the hydaulic head: H = } + z, being } the matix potential and z the elevation above a efeence plan (level). K is hydaulic conductivity in (m/s). The above is the socalled Richad s equation that has no closed fom solution because K is depend dependent on i. The substitution of the hydaulic head H with }+ z, leads to the so-called Richad s mixed fom: i t = z " KS W z + z z X% = z " K S W z + 1X% () Hee please note that K is the hydaulic conductivity Figue. USDA (United States Depatement of Agicultue) textues tiangle.

3 ELECTRICAL PROFILES FROM SWC MODELS Textual class N i [cm 3 /cm 3 ] is [cm 3 /cm 3 ] a [1/cm] n Sand Loamy sand Sandy loam Loam Silt Silt loam Sandy clay loam Clay loam Silty clay loam Silty clay Clay Table 1. Typical van Genuchten model with paametes (a, n) including esidual (i) and satuated (is). while if Se =1 we ae in satuated conditions. Because the hydaulic conductivity K depends on Se, theefoe the intoduction of models [Books and Coey 1964, Mualem 1976] that take into account such a dependence K K(Se) is necessay. In this wok the following model poposed by van Genuchten has been adopted: KS Q V= K S" 1 - RS e Such a model [van Genuchten 1980] is widely employed in this field. In this wok was also intoduced the dependence on the matix potential like: Se = 1 Q1+ with tabulated paametes a, n =1 1/m Q n a} V m - Vnm Q- hv n m+ 1 Q1 V s i = i i a a W + a} e 1/ m m e W % n- 1 Equation (3) afte some few aangements becomes i W t z K W = " Qi VS z + 1X% (8) W (5) (6) (7) Equation (8) is numeically solved in Matlab pogam though the van Genuchten s model, given in elations (5), (6) and (7). This allows to obtain both i (z,t) and } (z,t) vaiables in function of time and depth. Since the implemented model it s based on equations having validity in vadose zone the simulated depth can only each the wate table. Limits on the depth paamete ae intinsic in the matix potential that cannot act on moe than 10 m above the wate table in ideal conditions, which eflects in a -5 m action in eal conditions depending on soil textue. Also the heteogeneity of the soil has been stongly simplified and the missing data intepolated with a statistical algoithm. 3. Empiical elations to deive elative pemittivity of the soil Many physical and empiical models have been suggested fo the evaluation of the i f elationship in the liteatue. The elative pemittivity is the main electic quantity employed to define SWC in the soil since it can be easily measued though vaious techniques. The elation that ties these two quantities was empiically established [Topp et al. 1980]. This equation is valid fo a wide ange of mineal soils and independent fom soil bulk density, ambient tempeatue, and salt content. This led many authos to the use of the tem univesal fo this equation with appopiate caveat that in oganic soils o heavy clay soils poblems aise which may equie site-specific calibation [Cosenza et al. 003, and the efeence theein]. In this specific study the Topp s fomula [Topp et al. 1980] has been used. Fom well-known wate content pofiles, obtained by a modeling softwae [Bianchi et al. 015], Topp s elation (9) allows to deivate the elative pemittivity values of soil fom the SWC. This empiical model was geneated using time-domain eflectomety (TDR) at a fequency between 1 MHz and 1 GHz to measue f fo seveal mineal soils. The estimated eo in this model is Geneal Topp s model can be expessed: = f i i i whee f is the elative pemittivity and i is the soil volumetic wate content. They also povided anothe invese elation as follows: (9) 3

4 PORRETTA AND BIANCHI - - wq V=- 53. # # # 10 f # 10 f i f f (10) A specific calibation is needed fo soils with highe wate content o oganic matte: whee b 0 and b 1 ae two empiical paametes depending on soil type. In the same pape Ledieu et al. [1986] found the following elationship: i = f (14) fo oganic soil; = f i i i = f i i i (11) (1) This elationship appeas to wok bette fo most mineal non-magnetic soils ove a ange of fequencies between 1 MHz and 10 GHz [Hamed et al. 006]. Roth et al. [1990] fo mineal soil poposed this empiical elationship: fo 450 µm glass beds [Mukhlisin and Saputa 013]. The model elationships woks bette fo fequencies aound 100 MHz. At highe fequencies and moistue contents nea to satuation (i~0.4) the Topp-model ove-pedicts elative pemittivity by up to 0%. At vey low wate contents the Topp-model doesn t pefom well, paticulaly fo soils with a lage clay content [van Dam et al. 005]. Howeve, late studies have shown the dependency of the i f elationship on clay content [Pesson et al. 000, Bouksila et al. 008] and minealogy [Cosenza and Tabbagh 004], oganic matte and poosity o soil density [Malicki et al and Pesson et al. 00], and soluble salt content [Dalton 199, Nadle et al. 1999, Pesson et al. 000]. Ledieu et al. [1986] have shown anothe fom of empiical equations to descibe the i f elationship, that can be used to expand the Topp model fo highe wate content [van Dam et al. 005] of the fom: f = b + b i 0 1 (13) i = f f f 3 While fo oganic soil and mateial is: i = f f f 3 (15) (16) the eo estimations of these equations fo mineal soil and oganic soil ae and cm 3 /cm 3, espectively [Mukhlisin and Saputa 013]. Although a compaison of all the above elationships showed a simila tend (Figue 3), Topp s fomula (blue cuve) esulted the elationship that best fitted the input data, since is calibated fo soils that can hold a wate content up to 50%. Indeed in unsatuated zone the poes in the soil can be filled not only by wate but also by ai. 4. Empiical elations to deive conductivity of the soil Anothe majo contolling facto, of the soil electical popieties estimated in this study, is electical Figue 3. Empiical elations compaison. 4

5 ELECTRICAL PROFILES FROM SWC MODELS conductivity. Although usually geoelectic investigations ae focused on the measuement of soil esistivity, in this pape conductivity is pefeed to esistivity (i.e. the invese quantity), since pevalently we efe to ain wate data given in tems of conductivity enteing in the following equations. The empiical elation to estimate the electical conductivity both fo satuated and unsatuated soil is the Achie s law [Shah and Singh 005] also spead in the liteatue. This last one connect electical conductivity of satuated ocks (v o ) to the conductivity of the electolyte poe of soils (v w ), it can be expessed as: whee v qo is the conductivity contibution which chaacteizes high clay pecentage soils. It can only be neglected fo high saline wate content filling high poosity soils. F is the fomation facto which is equal to: (18) The paamete z is the poosity, and m is the cementation exponent which inceases with compaction, cementation and consolidation; it vaies between 1.3 and.5. Unconsolidated sands have values in the ange between 1.3 and 1.5 [Dannowski and Yaamanci 1999, and the efeence theein]. Geneally, the esistivity value is geatly influenced by basic soil chaacteistics vaiation such as faction of solid, ai and wate. Accoding to [Giffiths and King 1981, Telfod et al. 1990], esistivity value was highly influenced by poe fluid and gain matix of geomateials. In compact condition, it was found that at high compaction values coesponds lowe esistivity value (o highe electical conductivity). The volumes of poe in compact condition wee educed and cause the cuent to easily popagate especially duing the existing of wate [Abidin et al. 013]. Anothe fom has been poposed, known as Achie s second law: v v o 1 = v F w+ v qo F m = z F I o = v w + vqo (17) (19) which has been developed as an extension fo unsatuated ocks and soils including the I facto - I = Sw n (0) The satuation index I depends on the degee of satuation Sw and the satuation exponent n. This latte was obseved to be about fo consolidated ocks and to ange fom 1.3 to fo unconsolidated sands [Lesmes 005, and the efeence theein]. Achie s second law (19) has been chosen and implemented in the modeling softwae to obtain pofiles of electical conductibility tend ove time which chaacteizes the vadose zone. This physical quantity depends on a numbe of paametes as soil textue, wate holding capacity, oganic matte, salinity and ions exchange capacity. The conductivities of wate filling poes is an impotant facto in the pocess of electic cuent flow though the soil; especially when a quantity of salts ae dissolved in it. 5. Simulations and esults The softwae estimation of the effective wate content, in a poous soil is detemined by Richad s equation (Equation 3). The latte is a patial diffeential equation that can only be solved with the implementation of the van Genuchten soil models. Hence wate content is stongly affected by physical popety of soil mateial (poosity, capillaity, etc.) and fluid popeties (viscosity). Indeed at equal initial conditions, etention wate content only depends on matix potential in opposition to the effect of gavity, peventing leaching though the gound till the wate table. In the pefomed softwae simulation, specific initial condition (infiltation velocity), bounday conditions (wate table depth, gound level and satuated zone hydaulic head) and simulation paametes as time and depth have been taken into account. Infiltation is the volume of wate passing into the soil pe unit of aea pe unit of time. Simplifying it has the dimensions of velocity [m 3 m - s -1 ]. Depth is intended not only as the depth of the wate table, o line of satuation, but also the points in the modeled space Soil type Poosity [cm³/cm³] Infiltation velocity [cm/day] vw [ms/m] Cementation exponent Satuation exponent Wate table depth [cm] Time [day] Sand Silt Clay Table. Simulation paametes. 5

6 PORRETTA AND BIANCHI whee the solution is calculated. Hydaulic head is an equivalent measue of the pessue expessed in height of wate column, and it is the sum of pessue head (fluid intenal pessue) and elevation head (pessue due to gavity). Following a summaizing Table of the main simulation coefficients, paametes and physical quantities that wee used to deteminate the electical conductivity, pemittivity and wate content elative to the soil textues object of this pape. The simulation though mathematical models of the WC pofiles has been epoted, espectively fo the most common textue soils types (Figues 4, 5, 6). The pictue obtained by the modeling softwae, shows ange of wate content up to 50%, and the elative pemittivity and electical conductivity values linked to it, plotted in a 3D space. A time of fou days has been chosen because sand soils afte that amount of time lose all the wate etained in it, and esulted completely dy. Due to this quite fast change in SWC, the electical popeties ae also subject to a athe emakable vaiation. So the esults of the simulation show how taking a measue on a cetain soil textue (afte-ain suvey) could be misleading if the soil etention is not taken in consideation. Fom Figue 4 it is possible to see how, in a typical clay soil, wate content in the topsoil laye changes vey little due to its high degee of etention. In the othe hand, sand soil (Figue 5) with mediumsize textue have a low moistue holding capacity, which esults in a seep though the gound downwads. Silt take up an intemediate behavio between sand and clay soils (Figue 6). Topp s deivated elative pemittivity as shown below fo clay, sand and silt (Figues 7, 8, 9) and Table 3 shown an aveage of elative pemittivity values (f ). Figue 4. Wate content section in clay soil textues. Figue 5. Wate content section in sand soil textues. 6

7 ELECTRICAL PROFILES FROM SWC MODELS Figue 6. Wate content section in silt soil textues. Figue 7. Relative pemittivity section in clay soil textues. Figue 8. Relative pemittivity section in sand soil textues. 7

8 PORRETTA AND BIANCHI Figue 9. Relative pemittivity section in silt soil textues. i [cm 3 /cm 3 ] Table 3. Relative Pemittivity and coesponding SWC values accoding to Topp elation. f Since elative pemittivity is an electical physical chaacteistic which exclusively depends on inne wate content, fo high degee of moistue content, elative pemittivity in simulation test incease. Indeed, clay soils textue ae chaacteized by highe elative pemittivity values than sand soils, which geneally shows much lowe elative pemittivity. Silt soils, show a ange of values between 0 and 4. Pofiles fo the electical conductivity have been pesent in accoding to Achie s second law (Equation 19). Also in this case, the pofiles wee obtained simulating a time of fou days as shown in Figues 10, 11 and 1. Since electical conductivity is linked to total dissolved solids (TDS) and incease consideably in function of this quantity. To simplify the model, a fixed total solute concentation has been consideed, which esults Figue 10. Electical conductivity section in clay soil textues. 8

9 ELECTRICAL PROFILES FROM SWC MODELS Figue 11. Electical conductivity section in sand soil textues. Figue 1. Electical conductivity section in silt soil textues. in a typical ain wate electical conductivity value of about 5 ms/m. Futhemoe, in the specific case of clay model, the contibution of the suface gains o paticle conductivity (of about ms/m) has been taken into consideation. Clay suface s mineal paticles influence consideably the electical conductivity of this specific textue which esults in an incease of the total cuent flow though soil as shown in Figue 10 and Table Conclusions This pape dealt with the estimation of elative pemittivity and electical conductivity pofiles infeed fom SWC models fo diffeent soil textues. As well known the diffeent pecentages of the soil components, namely sand, clay and silt, descibed tough a numeical model allows to solve the Richads diffusion equation and detemine the SWC in dependence of depth and time. Such quantity is mainly esponsible fo the values assumed by the elative pemittivity and conductivity along depth and time. Because of seldom geophysicists have a pio knowledge of the subsoil electical paametes even in the case of homogeneous soil s composition these simulation yields of SWC along the vadose zone once ecognized some chaacteistic popeties of soil textues. Convesely, moe commonly, this infomation can be used in the opposite sense infeing the SWC fom the elative pemittivity and conductivity. The pio knowledge of the pemittivity and conductivity modelled pofiles could help geophysicists and opeatos to bette intepet the field measuement s esults as in application of landfill monitoing whee the soil textues ae known. The modelled SWC and coesponding electic paametes ae useful to choice the suitable tech- 9

10 PORRETTA AND BIANCHI i [cm 3 /cm 3 ] v Sand [ms/m] v Silt [ms/m] v Clay [ms/m] E E E E E E E E E E E E E E E E E E E E E E E E E E E+01 Table 4. Electical conductivity and coesponding SWC values accoding to Achie elation. niques to be employed and the optimal mathematical invesion algoithm to etieve the electic pofiles. The esults of these simulations as epoted in Tables 3 and 4, and Figues 7, 8 and 9 show the stong dependence of the pemittivity fom the SWC. Othe soil paametes have negligible influence on this quantity. Diffeent is the case in which we conside the conductivity. In fact SWC dependence is impotant even if salinity, tempeatue, etc. ae less elevant but still impotant fo soil conductivity. The estimation of this two pofiles have elevance especially when the soil textues ae well chaacteised. In such a case this allows to simulate SWC though the above discussed model and consequently estimate these two electic quantities in function of depth and time in unsatuated soil. A compaison with othes studies caied out peviously [Zhou et al. 001, Michot et al. 003, Schwatz et al. 008], showed the validity of the invese method to convet soil moistue content to electical esistivity using ERT with a modified fom of the Achie s law. Stating fom a given wate content, electical esistivity tend in time can be taced. Application in landfill monitoing is one the fuitful application since leachate dispesion can be pofitably infeed though the measuements of the soil electic paametes f and v. Refeences Abidin, M.H.Z., F. Ahmad., D.C. Wijeyesekea., R. Saad and M.F.T. Bahauddin (013). Soil Resistivity Measuements to Pedict Moistue Content and Density in Loose and Dense Soil, Applied Mechanics and Mateials, 353/356, Bianchi, F., M. Chiappini and R. Giodano (015). Fomazione, diffusione e distibuzione del pecolato: modelli matematici integati del pogetto SIGLOD (Fomation and distibution of leachate: integated mathematical models of SIGLOD poject), Quadeni di Geofisica, 18, 5 p., ISSN , (in Italian). Bouksila, F., M. Pesson, M.R. Bendtsson and A. Bahi (008). Soil wate content and salinity detemination using diffeent dielectic methods in saline gypsifeous soil, Hydological Sciences Jounal, 53 (1), Books, R.H., and A.T. Coey (1964). Hydaulic popeties of poous media, Hydology Papes, 3, Coloado State Univesity, Fot Collins, Coloado. Cosenza, P., C. Camelynck and A. Tabbagh (003). Diffeential effective medium schemes fo investigating the elationship between high-fequency elative dielectic pemittivity and wate content of soils, Wate Resouces Reseach, 39 (9). Cosenza, P., and A. Tabbagh (004). Electomagnetic detemination of clay wate content: ole of the micopoosity, Applied Clay Science, 6, Dalton, F.N. (199). Development of time-domain eflectomety fo measuing soil wate content and bulk soil electical conductivity, In: G.C. Topp and W.D. Reynolds (eds.), Advances in Measuement of Soil Physical Popeties: Binging Theoy into Pactice, Soil Science Society of Ameica Spec. Publ. no. 30, Madison, Wisconsin, USA, Dannowski, G., and U. Yaamanci (1999). Estimation of wate content and poosity using combined ada and geoelectical measuements, Technical Univesity of Belin, Dept. of Applied Geophysics, Ackest , D Belin, Gemany. Giffiths, D.H., and R.F. King (1981). Applied Geophysics fo Geologist and Enginees, The Element of Geophysical Pospecting, Pegamon Pess, Oxfod. Hamed, Y., S. Ghada and M. Pesson (006). Evaluation of the WET senso compaed to time domain Reflectomety, Hydological Sciences Jounal, 51 (4), Huisman J.A., S.S. Hubbad., J.D. Redman and A.P. 10

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