In uence of Surfactants on Spreading ofContaminants and Soil RemediationP. Knabner, S. Bitterlich, R. Iza Teran, A. Prechtel, and E. SchneidInstitute for Applied Mathematics, University of Erlangen{NurembergMartensstrasse 3, 91058 Erlangenemail: knabner@am.uni-erlangen.de, URL: http://www.am.uni-erlangen.de/am1Abstract. Surfactants occur already in undisturbed biological processes in soilsbut within the development of remediation techniques these substances are of largeinterest because of their interaction with hydrophobic substances. The standardmodels for (un-)saturated water ow and solute transport are extended to includethe in uence of the surfactant transport on the ow of the water phase. Two e�ectsof surfactants on (un-)saturated water ow are included: the modi�cation of theinterfacial tension between water and air and the swelling of clay minerals dueto sorption of surfactants. Simulations are presented, which exhibit the feedbackof surfactant transport on water ow and content. An identi�cation algorithm forhydraulic soil properties has been developed that supplements the simulation tool.1 IntroductionThe numerical modelling of water ow and solute transport has become astandard tool to complement �eld and laboratory studies in evaluating andforecasting the fate of pollutants. Furthermore advanced simulation is usefulto predict the success of site remediation strategies like the treatment of apolluted soil with surfactants (SEAR: surfactant enhanced aquifer remedia-tion).In the last decade, the modelling of the in uence of surfactants on ground-water ow and transport has considerably advanced and can now be incor-porated in ow and transport settings. The presence of surfactants in thepore water has an impact on the ow regime and vice versa. Surface activeagents enhance aquifer remediation by increasing the solubility of organicsubstances within the aqueous phase and the mobility of non aqueous phaseliquids. On the other hand the e�ective permeability of the porous mediummay vary as a consequence of the aggregation of the surfactant onto the clayfraction of the soil. These aspects have to be taken into account when therisk of groundwater pollution is evaluated and the fate of contaminants ispredicted.To perform practical simulations along with laboratory experiments theidenti�cation of hydraulic parameters is essential. In order to describe thewater ow in an appropriate manner, the parameters of the water retentioncurve and the unsaturated hydraulic conductivity have to be speci�ed.

2 P. Knabner et.al.2 Coupled Water{Surfactant Transport2.1 ModelTo describe the migration of a surfactant in a porous medium we have toestablish a description of the ow regime as well as a transport model whichshould include advection, dispersion, di�usion and sorption. Standard formu-lations exist in form of partial di�erential equations obeying the principle ofconservation of mass. This leads us �rst to the Richards equation as a modelfor ow in the saturated and vadose zone@@t�( ) +r � q = 0 q = �Kskr( )r( + z) (1)with parametrizations of the soil water characteristic (water retention curve)and of the unsaturated hydraulic conductiviy. Here t denotes the time, � thevolumetric water content, the pressure head, q the Darcy ux, Ks and krare the saturated and relative hydraulic conductivity and z is the elevationhead. The solute transport is described by the following partial di�erentialequation: @@t (�c) + �b @@t (�(c)) +r �w = 0 w = �Drc+ qc (2)where c is the dissolved concentration in x at time t, �b the bulk density, � theequilibrium sorption isotherm,w the mass ux andD the di�usion-dispersiontensor.These standard equations are now coupled in the following way to takethe interaction of uid ow and surfactant transport into account. Accordingto the work of [5] the surface active agent decreases the surface tension of the uid with increasing concentrations and thus a�ects the capillary pressureof an unsaturated soil. Consequently a scaling factor is introduced in thepressure-saturation relation:�( )! �(�0� ) =: �( ; c) �0� = 11� b ln(c=a+ 1) (3)with empirical, surfactant dependent parameters a and b. In a recent study[6], this model has proven its applicability in accordance with experimentallaboratory results. The variability of the hydraulic conductivity due to thesorption of the surfactant is incorporated in the model according to [4]:Ks ! Ke�(c) = K1��coarseK�clay � = �clay + �b�s�(c) (4)Ke�(c)kr( ) =: K( ; c)Ke� represents the e�ective hydraulic conductivity, Kcoarse and Kclay thesaturated hydraulic conductivities of the coarse respectively the clay fraction

In uence of Surfactants on Spreading of Contaminants 3of the soil, �clay is the volume fraction of the clay component and �b and �s arethe densities of the bulk soil and the surfactant, respectively. Surfactants sorbpreferentially to the clay and therefore increase the volume fraction of theclay-surfactant conglomerate, therefore the e�ective permeability decreasesdue to the sorption of the surfactant. These mechanisms couple the surfactanttransport to the water ow. On the other hand, the actual water contentand Darcy ux in the transport equation are determined by the Richardsequation. As the surfactants have an in uence on the ow regime, they do alsoa�ect the fate of a contaminant which is transported in the subsurface. Theupstanding mutually coupled system of partial di�erential equations alongwith its de�ning coe�cient functions has to be discretized to be solved on acomputer.2.2 DiscretizationThe transport equations (1) and (2) are written in a mixed variational for-mulation. The time evolution is discretized by the backward Euler scheme.The spatial dicretization is introduced by lowest order mixed �nite elementson triangles. The water pressure and the surfactant concentration c areapproximated in P 0(T ), i.e. piecewise constant on elements. The uxes ofwater q and of surfactant w are approximated in RT 0(T ), i.e. piecewise lin-ear on elements, having continuous normal components over the edges (inthe sequel abbreviated by \continuity of uxes"). To circumvent the result-ing saddlepoint problem, we introduce hybridization of the ux ansatz space.We enlarge the discrete space of uxes to the product space QT2T RT 0(T ),neglecting continuity of uxes. Additional Lagrange{multipliers on edges (�,�) and a new constraint equation are introduced to re-establish the continuityof uxes. After this extension of the problem formulation and its variables thesystem is twice reduced by condensation. First the ux variables are explicitlyeliminated. The resulting system of equations that describes the evolution ofone timestep is then�( T ; cT ) + �jT jK( T ; cT ) XE;E0�T B�1T;EE0 ( T � �E) = �( oldT ) (5)XT�EE0�T K( T ; cT )B�1T;EE0( T � �E0 � zTE0) = 0 (6)�( T ; cT )cT + �b�(cT ) + �jT j XE;E0�T A�1T;EE0(cT � �E) (7)+cT �jT jK( T ; cT ) XE;E0�T B�1T;EE0( T � �E) = �( oldT )coldT + �b�(coldT )

4 P. Knabner et.al.XT�EE0�T �cTK( T ; cT )B�1T;EE0( T � �E0 � zTE0) (8)+A�1T;EE0(cT � �E0)� = 0where T , cT are pressure and concentration values on the elements (the de-grees of freedom of the discrete ansatz space) and �E , �E are the Lagrange{multipliers on edges. BT is the element matrix, composed of the L2-scalarproducts of the basis functions from the RT 0 Ansatz space and connectingthe edges on that element. AT is the same matrix but including the inverseof the di�usion-dipersion tensor D, which itself depends on the Darcy uxq. All variables but oldT and coldT correspond to the new time level. Thesecond step of variable elimination is done in an implicit way. The element-wise equations (5) and (7) represent the discrete evolution of element values( T , cT ) and involve only degrees of freedom from one element. These equa-tions together de�ne an implicit expression of the element values for givenLagrange{multipliers (�E , �E for E � T ). In this treatment the dependenceof AT on D�1(q) and thus on T , cT and �E for E � T because ofqT;E = XE0�T K( T ; cT )B�1T;EE0( T � �E0 � zTE0)is not stated explicitly. Within the iteration process to be used for the solutionof the evolving system (see section 2.3) the Jacobians do not include thisdependence, but the evaluation of residuals considers the correct form ofAT . The solution of equations (5) - (8) requires only evaluation of theseequations and of their derivatives for given variables. Therefore the implicitde�nition of T and cT can formally be included in the edgewise equations(6) and (8), describing the continuity of ux across an edge. Finally (6) and(8) represent the global system of equations, that has to be solved for theLagrange{multipliers in each time step.2.3 AlgorithmA damped version of Newton's Method (Armijo's rule) is used to solve thelocal ((5), (7)) and the global ((6), (8)) nonlinear equations. Due to thetreatment of the q-dependence of the dispersion tensor, actually this is amodi�ed Newton's method, ignoring the corresponding terms in the Jacobian.Within each iteration of the global solution process, the evaluation of (6)and (8) and of their derivatives with respect to the Lagrange{multipliers ispreceeded by solution of the local problems (5) and (7).The global system of linear equations is solved by a multigrid method. Thesystem matrix is the Jacobian of (6), (8), the right hand side is the residualof this set of equations and the unknowns are the updates of the Lagrange{multipliers on edges within the Newton iteration. Based on the equivalenceof nonconforming and mixed �nite elements [1] the multigrid method is built

In uence of Surfactants on Spreading of Contaminants 5

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Fig. 1. Water{surfactant transport with (middle and right column) and without(left column) permeability reduction: contour plots of surfactant concentration andwater content for successive timepoints t 2 f5; 10; 30g.from grid transfer operators, derived for the Crouzeix-Raviart element [2].The coarse grid matrices are de�ned by Galerkin approximation. The linearproblem on the base level of the grid hierarchy is solved by LU decomposition.On �ne grid levels smoothers like Gauss-Seidel or ILU are used.2.4 Simulation ResultsThe introduced model is used to demonstrate the e�ects of varying surfacetension and permeability in a two dimensional simulation with an arti�cialdata set. All the results are compared to the corresponding ow and transportscenario with constant surface tension and without permeability reduction(labelled 'Reference' in Fig. 1 and Fig. 2).The Richards equation is complemented by exponential parametrizations of

6 P. Knabner et.al.the water retention curve and the unsaturated hydraulic conductivity. Weimpose a stationary unsaturated ow regime from the northern to the south-ern boundary of a square domain by de�ning the ux and the correspondinginitial pressure pro�le. Eastern and western boundaries are impermeable (no ow boundaries). Thus when assuming no impact of the surfactant transporton the water ow, the water content � = 0:295 (color-coded in Fig. 1 andFig. 2 with green), the pressure and the ux remain constant throughout theentire time interval.For the transport of the surfactant, we included the e�ects of advection, dif-fusion, longitudinal and transversal dispersion as well as linear equilibriumsorption. The domain is initially void of surfactant. The substance is injectedat the western half of the northern boundary within the time interval [0; 6],after which no more surfactant enters the area. At the southern boundary afree outlet is given (homogeneous Neumann condition). See the migration ofthe contaminant plume at di�erent time steps on the left hand side of Fig. 1and Fig. 2.In a �rst step we disregard pressure scaling and respect only the perme-ability reduction through the sorptive capacity of the surfactant as proposedin model equation (4). As the sorbed mass of surfactant per mass of soil �(c)is increasing, the e�ective hydraulic conductivity Ke� will be dominated bythe low conductivity of the clay fraction Kclay and thus decrease. As we im-pose the same water ux at the boundaries as in our reference case, this leadsto a retarded water transport in the area of low permeability, thus an increasein the water content (see right column of Fig. 1). As water ow is hindered bythe surfactant plume, water content in front of the plume now decreases. Thepermeability reduction induces the retardation of the surfactant migration,as can be seen in the middle column of Fig. 1.In a second simulation, we disregard the permeability reduction, but en-able the surface active agents to change the surface tension and thus a�ectthe hydraulic pressure head as described by model equation (3). Increasingsurfactant concentration c implies lower surface tensions and consequentlyhigher pressure heads. As a consequence of the generated hydraulic gradi-ent, water content decreases (right column of Fig. 2) with the solute front. Ifsurface tension is assumed constant, increases in pressure head indicate in-creased water contents, i.e. a wetting front. But as demonstrated in a recentexperimental study by [6] pressure heads may increase substantially while thewater content decreases in the presence of surface active agents. This observa-tion can be clearly reproduced by this two dimensional numerical simulation.Another remarkable e�ect is the change of the water ow in the eastern halfof the domain. As water content decreases where the surfactant concentra-tion is high, it increases in the eastern half, i.e. the ow is deviated towardsthe east. The constant water content in the reference case is � = 0:295. Thisexplains the extended spreading of the surfactant plume (middle column of

In uence of Surfactants on Spreading of Contaminants 7

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Concentration(Reference) Concentration Water Content

Fig. 2. Water{surfactant transport with (middle and right column) and without(left column) pressure scaling: contour plots of surfactant concentration and watercontent for successive timepoints t 2 f5; 10; 30g.Fig. 2) in the transversal direction. Regarding the travel times, the surfactantmigration is not retarded in this case.A subsequent simulation of contaminant transport strongly depends onthe water content and ux distributions that result from coupled water-surfactant transport. For an inert contaminant, i.e. neglecting sorption ef-fects, the migration velocity is given by q=�. Not only the direction of the ux is altered but also the modi�cation of water content will change this mi-gration velocity. In addition the micellar pseudophase occuring at surfactantconcentrations above a critical value acts as a solubilizer for organic solutes.Taking this solubilization into account, e.g. using the model of carrier facil-itation [3], is a next step for the model development. Further in uences ofsurfactants on spreading of contaminants in soils are related to the modi�ca-tion of interfacial properties that govern sorption processes.

8 P. Knabner et.al.3 Identi�cation of Hydraulic PropertiesThe numerical simulation of (un-)saturated water ow critically depends onthe knowledge of the hydraulic soil properties, expressed by the characteris-tics of water retention � and unsaturated hydraulic conductivity K. Thesehydraulic properties can be identi�ed by measurements obtained from soilcolumn out ow experiments.The design of a suitable experiment for this purpose is as follows. Avertically oriented soil column with a known initial pressure head equilibriumdistribution ( � 0, q = 0) is drained by slowly decreasing the pressure headat the outlet (lower boundary, x = 0) and preventing any ow at the top ofthe column (upper boundary, x = L). Mathematically this is modelled bya Dirichlet boundary condition (0; t) = h(t) at the lower boundary and ahomogeneous ux boundary condition (q(L; t) = 0) at the upper boundary.The experimental setup is complemented by two measurements, f and g, overtime t: The cumulative out ow f(t) := R t0 q(0; �) d� is measured at the outletof the soil column and the pressure head g(t) := (L; t) is measured at thetop of the column.We consider the model of the Richards equation in pressure head form(1) in one spatial dimension. Solving this model equation for given hydraulicfunctions � and K and assigning the measurements f(t) and g(t) to thehydraulic functions � and K characterizes the direct problem (DP: (�;K) 7!(f; g)). The inverse problem (IP: (f; g) 7! (�;K)) consists of determining thehydraulic functions � and K from given measurements f and g .3.1 Identi�ability and StabilityIf the applied suction h(t) at the lower boundary is assumed to be smoothand monotone decreasing in time, the mapping DP is injective for su�cientlysmooth hydraulic functions � and K. Therefore it is meaningful to assumethat the inverse problem IP is uniquely solvable. Then the hydraulic functionsare identi�able from out ow experiments.The inverse problem is stabilized by parametrizing the hydraulic func-tions. We are looking for an unbiased parametrization, which does not takeany a-priori shape information like the van Genuchten-Mualem parametriza-tion into account. A general approach uses splines to parametrize these func-tions. In this way we obtain a piecewise linear or a piecewise quadraticparametrization. The unknown hydraulic functions are de�ned by parametervectors p� and pK of dimension r. A special aspect of such parametriza-tions is the fact that the water retention and the hydraulic conductivity arenot coupled like in the van Genuchten-Mualem model. The low smoothness ofpiecewise linear functions applied for � and K leads to low smoothness of theobservations f(t) and g(t), corresponding to the measurements. Therefore anapproach with quadratic B-splines is pro�table. Because of the ill-posednessof the inverse problem it exists a threshold rmax depending on the error of the

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� K Fig. 3. Hydraulic properties (dashed) and identi�ed hydraulic properties (piecewisequadratic) for 9 degrees of freedom (solid)discretization of the direct problem, measurement errors and the type of theparametrization, such that a parametrization with more than rmax degreesof freedom leads to instabilities.3.2 Numerical Identi�cation by Output Least SquaresThe solution of the inverse problem is based on the minimization of an errorfunctional MXi=1 �i �f(t̂i)� f̂ i�2 + NXi=1 �i �g(~ti)� ~gi�2with positive weighting factors �i, �i and measured values f̂ i and ~gi at timepoints t̂i and ~ti, respectively. E�cient optimization algorithms (like quasi-Newton methods) need the value of the error functional and its gradientin every step of the optimization. The computation of the Hessian is notnecessary. The gradient of the error functional can be evaluated in two ways.{ Finite Di�erence Method: We have to solve the direct problem 2r timesfor the one-sided and 4r times for the central di�erence ratio to computean approximation of the gradient.{ Direct Method: Di�erentiation of the direct problem with respect to thedegrees of freedom leads to a system of linear equations. The combinedsolution process (direct problem and gradient) needs about twice theCPU time of the solution of the direct problem DP.An appropriate choice of initial values accelerates the minimization of theerror functional and avoids an early abort of the optimization procedure. Weobtain appropriate start values for our parametrization of the hydraulic prop-erties �n and Kn with rn degrees of freedom by interpolating the functions�m and Km, which minimize the error functional for rm degrees of freedom(rm < rn). The values of the hydraulic functions � and K at saturation

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g ft tFig. 4. \Measurements": original-undisturbed (dashed), disturbed (points), recon-structed (solid)( � 0) are often known from independent experiments and can be usedto determine initial values for the parametrization with the least possiblenumber of degrees of freedom.We simulate an experiment for a column with a van Genuchten-Mualemparametrization of the hydraulic functions and disturbe the \simulated mea-surements" by a gaussian distributed noise (5% of the maximum value). Theidenti�cation results are shown in Figure 3 and Figure 4.References1. D. N. Arnold and F. Brezzi. Mixed and nonconforming �nite element methods:Implementation, postprocessing and error estimates. RAIRO Model. Math. Anal.Numer., 19(1):7{32, 1985.2. D. Braess and R. Verf�urth. Multigrid methods for nonconforming �nite elementmethods. SIAM J. Numer. Anal., 27(4):979{986, August 1990.3. P. Knabner, K.U. Totsche, and I. K�ogel-Knabner. The modeling of reactivesolute transport with sorption to mobile and immobile sorbents. Part 1: Exper-imental evidence and model development. Water Resources Research, 32:1611{1622, 1996.4. C. E. Renshaw, G. D. Zynda, and J. C. Fountain. Permeability reductionsinduced by sorption of surfactant. Water Resour. Res., 33(3):371{378, March1997.5. J. E. Smith and R. W. Gillham. The e�ect of concentration dependent surfacetension on the ow of water and transport of dissolved organic compounds: Apressure head-based formulation and numerical model. Water Resour. Res.,30(2):343{354, 1994.6. J. E. Smith and R. W. Gillham. E�ects of solute concentration{dependent sur-face tension on unsaturated ow: Laboratory sand column experiments. WaterResour. Res., 35(4):973{982, 1999.