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Simulation of air and water flow in

deformable porous media and comparison

with experimental data

Z. Xiaoyong", L. Dai#

Li Danli

(Guangzhou Normal Institute, China)

ABSTRACT

The behaviour of multiphase liquid flow in deformable soil is described by acoupled set of PDEs, including equilibrium equation for soil phase andcontinuity equations for wetting and non-wetting fluids respectively. Thegoverning equations are solved by finite element method to give a description ofair and water flow in a laboratory column with porous medium deformation.Comparison with experimental data is shown with well agreement.

INTRODUCTION

Traditional descriptions of water movement in unsaturated porous media arebased on single phase fluid flow in rigid soil skeleton. Numerical simulationfollowing this idea has produced significant scientific contribution to theapplication fields of water resources, in addition to the experimental works inthe laboratory and field. To have a better understanding of the complex naturalsoil system, numerical models based on the idea of multiphase flow and soildeformation are becoming more important. In this paper, a fully coupled modelto simulate the complex behaviour of consolidation and pressures of water andair in porous media is presented. The governing equations describingdisplacement of soil and movements of water and air in saturated-unsaturateddeformable porous media are coupled non-linear partial differential equationsand are solved by finite element method. A numerical simulation on the drainageof water from and on the infiltration of air to a vertical column of sand saturated

Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

28 Computational Methods and Experimental Measurements

at initial time is shown. It is the main purpose of the paper to give a betterphysical understanding and description of two-phase fluids flow problems indeformable soil.

GOVERNING EQUATIONS

By combining Darcy's law, which is assumed for the transport of both air andwater, with the mass continuity equation, the continuity equations for air andwater are expressed by the contributions of fluid accumulation rate respectivelyin the following.

+ H^ + R r^u nT-^±L^+B B 3K. 3t 3K K. (3K./

where, by using subscript 1 to represent the phase (g for air and w for water), p]is the fluid density (M/L? ); k is the intrinsic permeability tenser (L% ); k isthe relative permeability with respect to the particular fluid phase; ji} is the fluiddynamic viscosity (M/LT); BJ is the fluid formation volume factor; pj is the

fluid pressure (M/LT% ); g is gravitational constant (L/T ); h is the elevationmeasured from a reference datum (L); (j) is the porosity of the medium; S% is thevolumetric saturation of the phase or the volumetric fraction of the total porespace occupied by the phase. For the phase water, its saturation S\y is simplerelated to the volumetric moisture content or the volumetric fraction of the bulksoil volume occupied by water by the equation 6=<S>S /. Ry is volatility of waterin air and R% is solubility of air in water.

The equilibrium equation, in incremented form, relating the total stress a to thebody forces b and to the boundary tractions s specified at the boundary F of thedomain Q, may be formulated by using the principle of virtual work,

- fSiTdbdQ- f8iTdsdr = 0Ja Jr

in which &T represents the total strain of the soil skeleton and u^ meansdisplacement. By introducing the concept of effective stress a' which is thestress controlling the changes in volume and the strength of the soil, we havea = a'-mp where m is equal to unity for normal stress components and zero


Computational Methods and Experimental Measurements 29

for the shear stress components and p the pressure of fluids, air and water, afterincorporating the constitutive relation, da'=D^(de-de^ -d6p -dej, in

which DT is the tangent matrix, depending on the level of effective stress O'and also on the total strain e if strain effects are to be considered; dec is thecreep strain, de, = Cdt where C means rate of creep strain and t the time; dEp isthe overall volumetric strain caused by uniform compression of the particle dueto the pressure of fluids,

dp

where KS is the bulk modules of solid or soil phase; dEo means initial strain orall other strains not directly associated with stress changes such as swelling,thermal, chemical, and so on; the general equilibrium equations can be writtenin term of the unknowns displacement vector u and pressure of fluids p

The continuity equations are subjected to the condition that Syy+Sg=l. Theequations for the two phases are linked by the capillary pressures existing atinterface between phases, pg-pw=Pc( w g)=Pc( w " w)=Pc( w)) where pcis the capillary pressure or the pressure difference between the nonwetting (air)and the wetting (water) fluid phases. Here, it is hypothesized that p% is relatedto 8^ alone. For the state equation, p=p(p,T). By considering isothermal

problem, it is simplified to p=p(p) or p\v=Pw(Pw)» Pg=Pg(Pg) nd

3t " dp* 3t 9t dpg 3t

for water and air, respectively.

Fluid formation volume factors, volatility of water and solubility of air areassumed pressure dependent,

and Rg =

The pressure p is expressed by p=S^ Pw+Sg Pg and its derivative with time is



From the relation of Pc=Pc(S\v)- have

at dp, at dp,

3p=s dp* i dS, 3p,at at ™ dp_ at

3t

dS, 8p,

•dp, at

" at dp, at at at dp, at at

Now substitution of the above relations into the governing equations. Thecontinuity equation for air is written as

-V

+ E&+RB. '' B.

at B, dp, at

3,lae mnV-

3K at 3K

1 - (j)

K.

and the continuity equation for water becomes

-V R,-£- V(p. + p.gh)

dp, at at at B. dp, t, at at

B at B, dp, at

3e rn'O C

3K, at 3K.

!-()) m

K, (3K.)'



Finally, the equilibrium equation is

-nvHS^+ p -

To solve the governing equations for the unknown u, p\y, and pg, appropriateinitial and boundary conditions are required. For the initial conditions, it iscommon to assume a static state of equilibrium at zero time of computation. Theequilibrium equation has already the boundary conditions incorporated and thecontinuity equations have two types of boundary conditions, one is theprescribed pressure alone or Dirichlet boundary condition and the other one isthe prescribed flux normal to the boundary or Neumann boundary condition,

that is, pi=Pi,kk o— —

M-i+ p,gh)n = -q

where Pj and q% are prescribed functions which might be dependent on time.This group of equations describe the isothermal coupled air and water flows indeformable porous media.

FINITE ELEMENT SOLUTION

Because the governing equations are strongly nonlinear and coupled, noanalytical solution can be derived. Therefore, their solution requires the use ofnumerical technique. Here, Galerkin method on finite elements is applied. Thedependent variables and their derivatives in the governing equations areapproximated in an element by

where N is the number of nodes in the finite element grid and T is the variablesor their derivatives. In term of the basic functions N and the variables at nodesT, the approximated variables and their derivatives of the governing equationsare listed in the following



u = = Slf N

Uc

3e _ dU I c*xi— = B— e= e3t dt

yyl-xyl

-xx2"yy2-xy2

3N,

3x

0

3N,

3N,

3x

0

3N,

3y

dy

9N

^1 03x

0 *±3y

3N, 3Ng

3y 3x

Pw=Kl,Pw

3N,

3p,3t

5.dt

P - fP P^ - l ' ^

After the modelled domain is divided up into a number of subdomains orelements, Lagrangian quadratic isoparametric elements which have nodes ateach of the corners, a node along each side of the element, and additionally anode in the centre of the element are employed to represent the arbitrary lateraltopography of the domain. The basic functions which interpolate the dependentvariables and parameters over the elements are chosen as Gray (1977). Thegoverning equations are transformed into a group of semi-discrete equations bysubstitution of formulations above and incorporating the boundary conditions.The equilibrium equation is formulated as

. dU

in which A,, = jVo.,.BdQ = Aj',

r fnTn 1 TA,, = B, DT m-B^r•1 '3K,

A,3 = \ B 'DT m-B^rJn * ii<r



DT =Ql

r2233.

dt ^ dt

E1-"

where E is the Young modules (N/m ) and \) Poisson ratio. The continuityequation for water becomes

. dU _ dP.A™

dP

in which A^

,^ dp, Bgdp, B^dp

S - " 1 r

, dp.

1-tgBggh)dQ- f N[ P' ^ 3K,

The continuity equation for air is written as

dU

dt

dP,

' dtL.

' dt



Combining the proceeding equations coupling for the fluid pressures anddisplacement in a suitable order, the following system of first order non-lineardifferential equations written in matrix form is formed

22 A 23

dU

dt

dt

5.dt

or for matrix of element,

w = (u, % ... u,

U

dW

dt1- G- W = f

dVand for matrix of total domain, GS -- + GT « V = F

dtV = fu V P P ... n v P P ^^ V^l ^1 ^wl Igl ^ nodes Anodes ^wnodes ^ gnodes )

in which nodes being the total number of nodes of the discretized domain.

In AJJ stated above, the integral must be evaluated by numerical scheme. Thetwo-dimensional Simpson integration involving 9 integration points is used.



in which F is the function of unknown variables and IJI is the Jacobiandeterminant; N is the number of nodes in the finite element grid and

a=(l/9,4/9,l/9,4/9,16/9 ,4/9,1/9,4/9,1/9)

is weight factor for Simpson integration involving 9 integration points. Thismethod has an advantage. When using with explicit time different technique, noneed to assemble physically the global matrices and to solve the linear equationsat each time step should be performed. It is first proposed by Gray (1977) andthen applied to surface water problem with pollutant and heat transport withsome modifications [Zhan et ah, 1989; Chen et al., 1990; Zhan et al., 1991].

Once the set of partial differential equations governing multiphase fluid flowsand consolidation are transformed into a set of first order differential equationsthrough a finite element technique, the generalized mid-point family of method(Hughes, 1983) is employed to discretize the time derivatives.

GS • - + GT • (aV + (1 - a)V" ) = aF"+' + (1 - a)F"

(GS + aAt • GT) V"+' = (GS - (1 - a) At • GT) V" + At(aF" + (1 - a)F" )

where the superscript n denotes the quantity evaluated at time t=At and At is thetime step of computation. By changing the values of a from 0 to 1, differentmembers of this family of methods are identified, i.e.,a=0 forward Euler or forward differencea=l/2 mid-point rule or Crank-Nicolsona=2/3 Galerkina=l backward Euler or backward differenceAll, except the first forward Euler method, of the above schemes are implicit.

As far as the nonlinear iteration is concerned, the termination of the iteratingprocedure depends upon the selection of a closure criterion. There are a numberof possibilities. A common choice is the norm IIV"+l-V"lloo with its normalizedvariant ll(V"+ -V")/V I!oo. Iteration is terminated when ||Vn+l_yn|| < and||(V"+^-V")/V"+Mloo<£2, where n is iteration times. £j and e? are usuallyproblem dependent and determined heuristically through numerical experiment;Another commonly used criterion is the satisfaction of the mass or energybalance. Here the former iteration procedure is employed. Because of transientproblems, the adaptive technique of timestep size to temporal gradients ofsolution in order to reduce computer running costs is applied. This may be doneby reducing or increasing the timestep size depending upon the number ofiterations required for convergence in the previous step or based on thecomparisons of local time truncation error between a predictor Adams-Bashforth formula and a corrector Trapezoidal rule. Here the former is used.The schemes stated above are programmed and performed on the IBM Rise6000 Station/AIX and Dec station 5125/Ultrix computers.



TEST EXAMPLE

The physical experiment of Liakopoulos (1965) is chosen as a test example forthe present model. The physical scenario for the experiment is described in thefollowing. An one meter high column of perspex was packed with Del Montesand and was instrumented to measure continuously the moisture tension atvarious points within the column. Prior to the start of the experiment (t<0)uniform flow conditions (i.e. unit vertical gradient of the potential) wereestablished by supplying water inflow from the top surface and allowing freedrainage at the bottom through a filter. At t=0 the water supply from the topsurface ceased and the tensiometer readings were recorded. The sand porosity is29.75% and soil permeability K=4.5xlO~ m2. The bulk modules of soil phaseand of water phase are Kg=1.0xlO^ N/m% and KW = 1.0x1018 N/rn

respectively. Youngs modules is E=1.3xlO^ N/m^ and Poisson ratio u=0.0.The water viscosity is set at 1.0xlO~3 Ns/m^ and the air viscosity 1.0x10"5Ns/nA The model allows specification of either a van Genuchten (1980) or aBrooks-Corey (1966) type of the capillary pressure curve and the respectiveassociated relative permeability functions. For the test example performed here,the dependencies of water saturation and water permeability on water pressurewere determined by Liakopoulos as shown in Figure 1. Since here the problemis solved as a two phase flow problem, also the relative air permeability as afunction of pc is provided, based on a small range of the Brooks and Corey(1966) relationship, shown in Figure 1. For the comparison with experimentalresults and static air phase calculations the following relation is employed

Pc=Pg-Pw=(Pg-PrefMp\v-pref)

where pref is reference pressure equal to 1 meter of water or 9810 N/mA

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Pc-(Sw, Krw, & Krg)

Pc-SaturationPc-KrwPc-Krg

0.2 0.4 0.6Pc(m)

0.8

Figure 1, Relations between saturation, relative permeabilities of water and airand capillary pressure



Numerically the problem was solved as a 2-D problem and for the spatialdiscretization a column of 10 equally sized elements were employed. Tosimulate conditions similar to the static air phase assumption, the lateral surfacewas first made permeable to air. Hence the boundary conditions are thefollowing,lateral surface: Pg=pref> qw=0, uh=0 where uh means horizontal displacementof soil;bottom surface: Pg=pref, Pw=Pref, uh=uy=0 where iiy means verticaldisplacement of soil;top surface: pg=pref» Qw=0;the initial conditions are

Full two-phase flow conditions are obtained by making the lateral nodesimpermeable to air, i.e. qg=0. This corresponds to the experimental conditions.All initial and boundary conditions are the same as in the static air phase caseexcepting no air flux instead of constant air pressure at lateral boundary.

The physical phenomena are displacement of water from the column by anincoming air with soil deformation. The profiles of vertical displacements,water pressure, air pressure, and water saturation for both lateral surfacepermeable and impermeable to air cases are shown in Figure 2-5. The results inFigure 2-5 by assuming lateral surface permeable to air are indicated in simonibecause it was simulated by using one phase fluid flow technique (Schrefler andSimoni, 1988), while the results by assuming lateral surface impermeable to airare denoted in liakop since it is a two phase fluids flow description ofexperiment performed by Liakopoulos (1965). Figure 2 shows the different forvertical displacement between one and two phase methods. By two phasemethod, we got larger vertical displacement at earlier time span, but smaller laterthan that by one phase method. Water pressure by two phase method agreesbetter than that by one phase method with experimental data, as shown inFigure 3. As presented in Figure 4, the change in air pressure is small by onephase method, as traditional description of unsaturated theory. However, thechange of the air pressure during the experiment inside the column is far fromnegligible and performs important role in the saturation distribution as shown inFigure 5, where saturation profile by two phase method is closer to observationphenomena.

To point out the influence of the permeability and Youngs modules, a secondsimulation is performed again with a larger permeability K=7.0xlO"^m^ andsmaller Youngs modules E=0.7xlO^ N/m^. The new profiles of verticaldisplacements, water pressure, air pressure and saturation are shown in Figure6-9. They are different from those in Figure 2-5 in that larger permeabilitydrains two phase fluids flow faster and water pressures agree well withexperimental data. Also smaller Young modules causes larger verticalconsolidation. The influence of materials on the soil deformation fromsimulation remains to be investigated. From figures presented above, it is clearthat the soil deformation and water pressure at the onset of the experiment isinfluenced by the change of lateral boundary conditions as well as the



Height(m)1

0.8

0.6

0.4

0.2

0

liakopsmon -----

5, 10, 20, 30, 60, 120 mins from right to left

-0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0Vertical displacement (m)

Figure 2, Profiles of vertical displacement (m) with lateral surface of the columnpermeable (indicated by simoni) and impermeable to air (indicated by liakop) byusing permeability 4.5x10 3 2 %nd Youngs modules L3xlO%/m-

Height(m)

1

0.8

0.6

0.4

0.2

0

computed liakopcomputed simoni •experiment liakop


-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0Water pressure (m)

Figure 3, Profiles of water pressure with lateral surface of the columnpermeable (indicated by simoni) and impermeable to air (indicated by liakop),comparing with experimental data by Liakopaulos, by using permeability4.5x10" m^ and Youngs modules 1.3x10*5




-0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0Air pressure (m)

Figure 4, Profiles of air pressure with lateral surface of the column permeable(indicated by simoni) and impermeable to air (indicated by liakop) by usingpermeability 4.5x10" m^ and Youngs modules 1.3x10^ N/m^

Height(m)


i i i I 10.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

Saturation

Figure 5, Profiles of saturation with lateral surface of the column permeable(indicated by simoni) and impermeable to air (indicated by liakop) by usingpermeability 4.5x10" m and Youngs modules 1.3x10^



Height(m)1


-0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0Vertical displacement (m)

Figure 6, Profiles of vertical displacement (m) with lateral surface of the columnpermeable (indicated by simoni) and impermeable to air (indicated by liakop) byusing permeability 7.0x10" m^ and Youngs modules 0.7x10 N/m^

Height(m)

computed liakopcomputed simoniexperiment liakop


J 1 i J i i t i

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0Water pressure (m)

Figure 7, Profiles of water pressure with lateral surface of the columnpermeable (indicated by simoni) and impermeable to air (indicated by liakop),comparing with experimental data by Liakopaulos, by using permeability

3m2 and Youngs modules 0.7x10 N/m^



Height(m)


-0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0Air pressure (m)

Figure 8, Profiles of air pressure with lateral surface of the column permeable(indicated by simoni) and impermeable to air (indicated by liakop) by usingpermeability 7.0x10" m^ and Youngs modules 0.7x10 N/m%

Height(m)

0.8

0.6

0.4

0.2

00.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

Saturation

Figure 9, Profiles of saturation with lateral surface of the column permeable(indicated by simoni) and impermeable to air (indicated by liakop) by usingpermeability 7.0x10" m^ and Youngs modules 0.7x10 N/m^

naKop -simoni

5,10,20,30,60,120

i i i

mins fromi i

right to lefti i

1



saturation. The results confirm that the model proposed can reproducemultiphase flow in deformable porous media and that the main features of themodel, i.e. air flow and solid deformation have their importance.

CONCLUSIONS

From the context of the presentation, the main conclusions can now besummarized. A fully coupled model for both air and water flows in deformingporous media and its numerical solution have been presented. A test exampleswas performed with different permeabilities and Youngs moduli andcomparison with experimental data shows good agreement. It has been shown,from the results, that this approach can be used for a better physicalunderstanding and description of air and water movements in deformableporous media since from this two phase fluids flow model more physicalinfomation are provided than by traditional one phase fluid flow description..

REFERENCES

1. Brooks, R.N. and Corey, A.T., Properties of porous media affecting fluidflow, J. Irrigation and Drainage Division, Proc. Am. Ser. Civ. Eng., 92(IR2),61-68, 1966.2. Chen Chuping, Zhan Xiaoyong, and B.A. Schrefler, Two dimensional finiteelement model for tidal flow in bays, Computational Method in SurfaceHydrology (Eds. Gambolati, G. et al.), p. 85-90, CMP & Springer-Verlag,Southampton, UK, 1990.3. Gray, W.G., An efficient finite element scheme for two dimensional surfacewater computation, Finite Element in Water Resources (Ed. Gray, W.G. et al.),p. 4.33-4.49, Pentech, London, UK, 1977.4. Hughes, T.J., Analysis of transient algorithms with particular reference tostability behaviour. Computational Methods for Transient Analysis, ElsevierScience Publishers, 1983.5. Liakopoulos, A.C., Transient flow through unsaturated porous media. D.Eng. dissertation, Univ. of Calif., Berkeley, 1965.6. Schrefler, B.A., and L. Simoni, A unified approach to the analysis ofsaturated-unsaturated elastoplastic porous media, in Numerical Method inGeomechanics, (G. Swoboda ed.), Balkema, 205-212, 1988.7. van Genuchten, M.T., A closed form equation for predicting the hydraulicconductivity of unsaturated soil, Soil Sci. Soc. Am. J., 44, 892-898, 1980.8. Zhan Xiaoyong, Chen Chuping, and B.A. Schrefler, Two dimensional finiteelement model for thermal dispersion in bays, Numerical Methods in ThermalProblems (Eds. Lewis, R.W. et al.), p. 469-475, Pineridge, Swansea, UK,1989.9. Zhan Xiaoyong, B.A. Schrefler, and Li Danli, Numerical simulation of tidalflow and contaminant transport in shallow water bays, Computer Modelling inOcean Engineering 91 (Eds. Arcilla, A.S. et al.), Balkema, Rotterdam, 1991.


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