Hierarchical System of Coupled Soil Porous Medium and Mass Transfer Models Système Hiérarchique Des Modèles Couplés, Porosité Du Sol, Transferts De Masse

Scientific registration n° : 652 Symposium n° : 3 Presentation : poster Hierarchical System of Coupled Soil Porous Medium and Mass Transfer Models Système hiérarchique des modèles couplés, porosité du sol, transferts de masse

ZEILIGUER Anatole M.

Moscow State University of Environmental Engineering, Prjanishnikov Street, 19, Mos- cow, 127550 Russia

Abstract The dependence of soil properties on main-made factors affecting the soil requires special model elaboration. The complex of water fluxes and unsaturated soil hydraulic properties are described by the system of coupled models. The conventional names of proposed models (textural, structural, aggregated, swelling, alkali, gypso-calcaireous) reflect the specificity of principal structural elements of the soil. The unsaturated hydraulic properties are available from models operated by a set of the traditionally determined soil physico-chemical properties such as: bulk density of soil and of soil structural elements and soil structural element size-distributions. This paper presents a compilation of elabo- rated theoretical systems of soil medium and water flow models. Introduction The soil medium provides a very complex unstable modelling system for the simulation of water flow. The mathematical simulations of water flow is based on the resolution of the differential equations describing the water movement into and through the variably saturated porous medium. Numerous models are available for performing simulations related to the movement of water. In recent years, the developing and using of soil models has increased for the purpose of estimating contaminant levels in different type of soils. The structure of soil medium is generally the most important part required in such models. Many soils change own porous medium properties during the drain- age/imbibition cycles, clay particles lessivage, soil alkalinisation due to sodium absorba- tion, gyps and calcium crystallisation/dissolution and others. Often soil models use an over-simplified presentation of the porous structure, having a little basis in real soil con- ditions. Several soil medium models have been proposed applicable to any type of soils and have met with reasonable good success in practice. Progress continues to be made in this area, for example the recent researches stressed the limitation of existing soil medium models for the simulation of water flow in real soil with complex structure [Dyk- huizen, 1987], [Fred, and al., 1990], [Nitao, and al., 1991], [Kutilek, and al., 1992], [Perrier, and al., 1992], [Gerke, and al., 1992]. Therefore the accurate water flow model application requires the development of a special model describing the peculiari- ties of water flow in the soil medium with additional techniques providing support to the hydraulic parameters estimation and correction in function of the soil chemical and physical properties changing. Concept In our concept the soil macroscopic structure is presented by a soil profile divided into horizons described by a set of physicochemical soil properties (soil profile model). Each horizon is evaluated on a macroscopic scale as homogeneous containing on a microscopic scale different types of heterogeneity of the soil medium. The nature of heterogeneity depends on physicochemical properties of structural soil elements which forms a hierarchical pore system dividing the heterogeneous soil porous medium into distinct but homogeneous pore systems. It is postulated that all of the subordinated and intercon- nected pores defined within the structural soil elements constitute on the horizon scale a hierarchically organised soil system. The first level of this soil system is created by the soil elementary particles, the second by the microaggregates and non aggregated soil elementary particles, and the third either by the macroaggregates or by the peds. The ad- ditivity hypothesis of physico-chemical and hydraulic properties of soil system levels is accepted for building on a microscopic scale the system of soil medium and appropriate water flow model. Soil Porous Medium Modelling In our approach we use deterministic soil parameters of different soil medium models corresponding to various types of soil aggregation for estimating representative soil property values. By applying a hierarchical method to the soil media, we intend to build soil medium models which take into account both solid and void phases at each microscopic level of the soil system. The void ratio of soil containing K levels of aggregation is the weighted sum of structural void ratios of each of these levels related by K ) p s e(f) = V (f) M (f) = å Fk (f)ek (f) , (1) k =1 where V p - is the volume of pore space of soil medium containing K levels including structural and textural pores, M s - is the solid mass of soil with all K levels f - is the pa- rameter) describing) the influence of some processes to the change of soil physic prope) r- p s th p ties, ek = Vk (f) M k (f) - is the void ration of structural pores of k soil level, V k - is th s the volume of structural pores at k soil level, M k is the solid mass of k level, s s Fk (f) = M k (f) M (f) - is the relative mass of soil medium containing k level.

Estimating of Unsaturated Soil Hydraulic Properties Soil Water Retention In water retention of soil model is assumed that the pore space volumes generated by the different size of soil structural elements fractions are considered as filled with water be-

2 ing a function of the water head h and any processes affecting pore space and describing by any parameters f presenting the affectation of water retention K ) w q (h,f) = rd (f)å Fk (f)Wk (h,f) , (2) k =1 where) rd ) - is the bulk density of soil (below is assumed rw = 1.0 ), w s th W)k (h,f) = M k (h,f) M k (f) - is the water retention of structural pore of k soil level, w th M k - is the mass of water retained in the structural pores of k soil level. The water retention of structural pore is presented that ) ) ) ) ) W (h,f) = r e (f) - q pk (f) r S w (h) + q pk (f) r , (3) k w {[ k r ) d ] ek r d }

p k where rw - is the water density, q r - is the) relative volume of soil structural pore space at kth level filled by the residual water, S w - is the water saturation of structural pore e k space parts filled by the mobile water, expressed as i= Ik ) ) n=k -1 i=In ) ) w w e k (f)S k (h)f gk (f) + e n (f)S n (h)f gn (f) ) å ei ei k ,i å å ei ei k ,i S wk (h) = i=1 n=1 i=1 , (4) e i=I k ) n=k -1i=In ) e k (f)f gk (f) + e n (f)f gn (f) å ei k ,i å å ei k ,i i=1 n=1 i=1 ) éi=I k ) i= I1 ) ù q pk (f) = r e k (f) f k (f) + e 1 (f) f 1 (f) , (5} r d êå ri k ,i å ri k ,i ú ) ) )ë i=1 i=1 û w where S n , e n , e n - are the water saturation and the void ratio of structural pore em- e i e i ri bedding the nth type of soil structural elements filled by mobile and residual water (em- s pirically known values), f gn (f) = M n (f) M s (f) - is the size distribution function of ith k ,i ki k fraction of soil structural element forming kth aggregation level, n - is the type of soil structural elements (n £ k , n =1 - soil elementary particles, n =2 - microaggregates, peds etc., n =3 - macroaggregates), M s n - is the mass of soil structural elements. k i Unsaturated Soil Hydraulic Conductivity The prediction of unsaturated hydraulic conductivity of transport pore system is realised by introducing of van Genuchten’s [van Genuchten, 1980], or/and Weibull's [Zeiliguer, 1985, 1996] multi-equation in the Burdine's and Mualem’s models presented by the following view g ¥ wk ¥ wk a é dq (h,f) dq (h,f)ù a g wk e e wk a Kr (h,f) = Se (h,f) ê b b ú = Se (h ) g (h) ,(6) k [ ] ò h ò h [ ] [ ] ëê h 0 ûú where K f is the hydraulic capillary conductivity at saturation, a , b and g are the coef-

wk wk a a ficients of Burdin’s and Mualem’s models, Se (h ) = Se [h, B (f),n (f)] - is the van na ( f ) Genuchten or Weibull equation, h = [h B a (f)] - is the formal argument, B a , n a -

3 ¥ a w b are the fitted empirical parameters, g (h) = ò dSe (h) (h) - is the function having the h following analytical views for both van Genuchten or Weibull models model van Genuchten Weibull a S w h [ b n ( f ) -1] exp - h , e ( ) [1+ h] , (7) ( ) (9) a a a [1-1 n ( f ) ] [1-b n ( f ) ] a a g (h) 1- (h) (1+ h) , (8) G[1- b n (f)]{1- P[1- b n (f) ,h]} , (10 ) where G and P are respectively complete and non-complete gamma-function. Soil Porous Medium & Water Flow Models Textural Soil The textural soil medium describes a rigid one-level porous media with constant values of bulk density rd and porosity m . It’s assumed that this soil medium is built by random

g 1 arrangement of soil elementary particle having the size-distribution function f 1,i . The unique water retention function of this soil medium model is expressed as

) i=I1 ) ) i=I1 ) i=I1 ) w w p 1 1 g1 1 1 1 1 q (h) = (m -qr ) e e S e (h) f1,i e e f1,i + rd e r f1,i , (11) å i i å i å i i=1 i=1 i=1 The following equation simulating the vertical water flow in textural soil governed by the appropriated water pressure head h conceives the porous medium as a single continuum of a capillary transport) pore system ) w p dSe ¶h ¶ é æ ¶h ö ù (m -qr ) = êKç +1÷ ú , (12) dh ¶t ¶x ë è ¶x ø û Structural Soil Under the notion of the structural soil medium we assume rigid porous media consisting of two levels of aggregation with appropriate stable pore systems) made up of a rigid skeleton. The second level of porous medium with porosity m 2 is built by the random ar-

g 2 rangement of microaggregates having the size-distribution function f 2,i and non- aggregated soil elementary particles having own size-distribution function f g 1 forming 2,i ) the inter-micro aggregate pore subspace. The first level of this model porosity m 1 is

g 1 build by the aggregated elementary particles (f 1,i ) forming in the interior of the micro aggregates the intra-microaggregate pore subspace. The total water retention of structural soil medium model is expressed by the equations (2-5) for K=2. Conforming to the textural soil medium model that conceives the porous medium as consisting of two connected systems of transport and dead-end capillary pores, the vertical water flow equation governed by water pressure head h have the following view ) ) 2 ) ) ) dS w2 ¶h ) dS wd1 ¶ é æ ¶h ö ù p2 e 2 p1 e 2 (m2 - qr ) + (m1 -qr ) = êK2 ç + 1÷ ú , (13) dh ¶t dt ¶x ¶x 2 ë è ø û ) s wd where K - is the hydraulic conductivity of pore of second level, S 1 - is water satura- a2 e tion of pore space of first level formed of dead-end pores.

4 Aggregated Soil Aggregated soil medium is assumed porous media rigid in exterior and deformable in interior consisting of three levels. The third level is built by non-rigid macroaggregates forming the inter-aggregate) pore subspace which porosity is described by the shrinkage- g 3 swelling characteristic m3 (h) having the aggregate size-distribution f 3,i , the second - by rigid microaggregates (f g 2 ) forming in the interior of macroaggregates the intra- 2,i ) aggregate transport pore subspace with the porosity m2 (h) and the first by elementary soil particles (f g 1 ) forming in its turn in the interior of microaggregates the intra- 1,i ) microaggregate pore subspace with the own porosity m 1 . The water retention of aggregated soil model is presented as ) ) ) i=I 3 ) ) 1 ) ü w wk p1 w1 1 g1 q (h) = mk (h)Se (h) + (m1 - qr )Se (h) + rd e r f1,i ï å å i k =2 i=1 ï ý , (14) ) i= Ik ) ) i= Ik ) w wk k k gk k gk ï Se (h) = e S (h)f k ,i e f k ,i å ei ei å ei ï i=1 i=1 þ From water flow point of view the aggregated soil model is consisting of two interacting continuous systems of transport inter-aggregate and intra-aggregate capillary pore as well as that one discontinuous system of dead-end capillary pores connected with intra- aggregate transport pore system. For this model the vertical water flow governed by the h h water pressure heads of the third and second levels) 3 and )2 is described by the following equations taking in consideration that d m 3 dh = - d m 2 dh ) ) 2 2 w ) ) dS 3 ¶h dm ¶h ¶ é æ ¶h ö ù ü e 3 w3 2 2 3 m3 - Se = êK3 ç +1÷ ú - q ï dh3 ¶t dh2 ¶t ¶x ë è ¶x ø û ï ) ) ) w ) ) wd ï é ) dS 2 dm ) ¶S 1 ù ¶h ¶ é æ ¶h ö ù ï e w2 1 p1 e 2 2 êm2 + Se + (m1 -qr ) ú = êK2 ç +1÷ ú + qý, (15) ë dh2 dh2 ¶h2 û ¶t ¶x ë è ¶x ø û ï ï q = b 3 h - h 2 ( 3 2 ) ï ï þ 3 where K 3 , K 2 - are the hydraulic conductivity of both transport subspaces, b 2 - is the transfer coefficient of water exchange between both transport subspaces. Swelling Soil Under the notion of the swelling soil medium we assume a deformable porous medium consisting of two levels. The second level is built by non-rigid peds (f g 2 ) described by ) 2,i the shrinkage-swelling characteristic e2 (h) forming the inter-ped pore subspace and the first level is built by elementary particles (f g 1 ) and forming in the interior of peds the in- 1,i ) tra-ped pore subspace with the shrinkage-swelling characteristic e1(h). The total water retention of swelling soil is presented by the following

) i=I1 ) ) i=I1 ) i= I1 ) ) ì w ü w 1 1 g1 1 g1 1 g1 w1 q (h) = rd íe1(h) e e S e (h)f1,i e e (h)f1,i + e r f1,i [1- Se (h)]ý ,(16) å i i å i å i î i=1 i=1 i=1 þ

5 The swelling soil is conceived as the soil medium consisting of two interacting transport systems of transcapillary inter-ped and intra-ped capillary pores. Vertical water flow in such soil governed by pressure heads of the first h1 and second level h2 is expressed by the following equation) (neglecting) the vertical soil movement) w ) é ) dS 2 dh de ¶h ù ¶ é æ ¶h öù ü r e e 2 - S w2 2 1 = K ç 2 + 1÷ - b 2 h - h d ê 2 e ú ê 2 è øú 1 ( 2 1 ) ï ë dh2 ¶t dh1 ¶t û ¶x ë ¶x û ï ) ) ý ,(17) ) ) w1 ) é p dSe w de1 ù ¶h1 ¶ é æ ¶h1 öù 2 ï r e -q 1 + S 1 = K ç + 1÷ + b h - h d ê( 1 r ) e ú ê 1è øú 1 ( 2 1 )ï ë dh1 dh1 û ¶t ¶x ë ¶x û þ 2 whereK 2 , K 1 - are the hydraulic conductivity of both transport subspaces, b1 - is the transfer coefficient of water exchange between two transport pore systems. Alkali Soil Under the notion of the alkali soil medium we assume non-rigid porous medium consist- 1 ing of one level pore system built by elementary soil particles (f 1,i ) containing mont- 1 morillonite (f 1,M ) changing appropriate water retention characteristic in function of ab- + Na + sorbed sodium (N CEC = Na CEC , where CEC is the cation exchange capacity of Na + soil). The water retention of alkali soil depending on two arguments h and N CEC is ) ) ) ü + + + + q w h, N Na = m - q p N Na S w h, N Na + q p N Na ï ( CEC ) [ r ( CEC )] e ( CEC ) r ( CEC ) ï i= I1 ) ) ) ) ï w + w 1 1 g1 1 Na 1 g1 ï ) åe e S e (h) f1,i + e e N CEC S e (h) f1, M + i i M ( ) M ï S w h, N Na = i=1 ý , (18) e ( CEC ) i=I1 ) ) + e 1 f g1 + e 1 N Na f g1 ï å ei 1,i eM ( CEC ) 1, M i=1 ï ï ) i= I1 ) ) + é + ù ï q p N Na = r e 1 f g1 + e 1 N Na f g1 r ( CEC ) d êå ri 1,i rM ( CEC ) 1, M ú ï ë i=1 û þ The vertical water) flow in alkali) soil is described by the following equation w é w p ù Na+ p ¶Se ¶h ¶Se ¶qr ¶N CEC ¶ é æ ¶h ö ù (m -qr ) + ê + + + ú = êKç +1÷ ú , (19) ¶h ¶t Na Na ¶t ¶x è ¶x ø ëê¶N CEC ¶N CEC ûú ë û Gypso- Calcareous Soil The gypso-calcareous soil is presented as metamorphic porous medium consisting two

g 1 level subspaces. The first is built by soil elementary particles (f 1,i ) and calcium crystals (f C 1 ) forming in the interior of aggregates the intra-aggregate subspace with the porosity )1,i ) m1(C) . The second level forming the inter-aggregate subspace with the porosity m2 (G)

g 2 G 1 is built by aggregates (f 2,i ) and gyps crystals (f 2,i ).The total water retention of soil depending on three arguments h , G (mass of gyps crystal) and C (mass of calcium crystal) is expressed as

6 ) ) ) ) p2 p2 w2 p2 q(h,G,C) = [q (G) - qr (G)]Se (G,h) + qr (G) +

g1 C1 ) ) ) ) , (20) f1,i + f1,i (C) p + q 1 (C) - q p1 (C) S w1 (C,h) + q p1 (C) g1 C1 G1 {[ r ] e r } f1,i + f 1,i + f 2,i (G) where

i = I 2 ) ) i= I1 ) ) w w ü e a2 S 2 (h) f g2 + e a1 S 1 (h) f G1 (G) ) å i ei 2,i å i ei 2,i ï i =1 i=1 S w2 (h,G) = ï e i= I 2 ) i =I1 ) ï a2 g2 a1 G1 å e i f 2,i + å e i f 2,i (G) ï i =1 i=1 ï ) i= I1 ) ) i = I1 ) ï w a w1 g C a g C ï S 1 (h,C) = e 1 S (h) f 1 + f 1 (C) e 1 f 1 + f 1 (C) e å i ei [ 1,i 1,i ] å i [ 1,i 1,i ]ý , i=1 i =1 ï ) i = I i= I é 2 ) 1 ) ù ï p2 r2 g2 r1 G1 q r (G,C) = rd (G,C)êå e i f 2,i + å e i f 2,i (G)ú ï ë i=1 i=1 û ï

) i= I1 ) ï p1 r1 g1 C1 ( ) ï q r (G,C) = rd (G,C)å e i [ f 1,i + f 1,i C ] i=1 þï (21) The vertical water flow in gypso-calcareous soil is described by the following equation ) ) ) ) p ) ) w2 w2 ) 2 ü æ ¶S ¶h ¶S ¶Gö ¶(m2 - qr ) ¶ é æ ¶h ö ù p2 e 2 e w2 2 (m2 - qr )ç + ÷ + Se = êK2 ç + 1÷ ú - qï è ¶h2 ¶t ¶G ¶t ø ¶t ¶x ëê è ¶x ø ûú ï ) ) ) ) p ï ) ) w1 w1 ) 1 æ ¶S ¶h ¶S ¶Cö ¶(m1 - qr ) ¶ é æ ¶h ö ù ï , (22) p1 e 1 e w1 1 (m1 - qr )ç + ÷ + Se = êK1 ç + 1÷ ú + q ý è ¶h1 ¶t ¶C ¶t ø ¶t ¶x ëê è ¶x ø ûú ï 2 ï q = b1 (h2 - h1 ) ï ï þ Conclusion In this paper simulation principles of six coupled models of soil medium and water flow are presented. The choice of various initial distributions of structural elements, and the multiple ways to aggregate them can represent a wide range of real soils. References Dykhuizen, R. C. Transport of solutes through unsaturated fractured media. Water Res. Res. 24, 8: 1225-1236, 1987. Fred, K. F., and L. A. Mulkey. Solute transport in Aggregated Media: Aggregate Size Distribution and Mean Radii. Water Res. Res. 26, 6, 1291-1303, 1990. Garnier, P., E. Perrier, R. Angulo-Jaramillo, and P. Baveye. Numerical Model of 3-Dimensional Aniso- tropic Deformation and 1-Dimensional Water Flow in Swelling Soils. Soil Sc., 162, 2, 410-420,1997. Gerke, H. H., and M. T. van Genuchten. A Dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Res. Res. 29, 2: 305-319, 1993. Kutilek, M., R. Rosselerova, and H. Othmer. Models of porous systems for un-saturated flow. In: Report at Colloquium Porous or Fractured Unsaturated Media: Transport and behaviour. Monte Verita. 182- 205, 1992. Nitao, J. J., & T. A. Buscheck. Infiltration of a liquid front in an unsaturated, fractured porous medium. Water Res. Res. 27, 8, 2099-2112, 1991.

7 van Genuchten, M. T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J., 40, 892-898, 1980. Zeiliguer, A. M. Comparison of different hydraulic function models fitted by experience data. In: Opti- mizatsia prossesov kompleksnogo meliorativnogo regulirovaniya. Ed. Moscow Hydromeliorative Inst. Moscow, 61-75, 1985, (in Russian). Zeiliguer, A. M. Coupled Models for Estimating Hydraulic Properties and Simulating Water Flow into & through Soil Medium. In: A. Müller (ed) Hydroinformatics-96. A. A. Balkema/Rotterdam/Brookfield, ETH, Zurich. 581-589, 1996.

Keywords : hierarchical system, coupled models, soil porous medium, mass transfert Mots clés : système hiérarchique, modèle couplé, porosité du milieu sol, transfert de masse