contaminant plume in many case studies (consult e.g. Spitz and Moreno,. 1996). At the heart of any attempt to model the evolution of contaminant migration.
A comprehensive tool for the simulation of complex reactive transport and ow in soils Eckhard Schneid, Alexander Prechtel and Peter Knabner Institute for Applied Mathematics, University of Erlangen-Nuremberg Martensstr. 3, D-91058 Erlangen, Germany
Abstract
The application of natural attenuation as a site remediation strategy depends essentially on the reliable prediction of the migration of the contaminant plume. We present a 1D simulation tool that is capable of handling a variety of complex scenarios predicted to be of interest in site remediation problems. The implementation of the di erent components is organized in a modular structure that facilitates arbitrary extensions of the incorporated models and enables the combination of the model components. Ecient, robust numerical techniques (e.g. hybrid mixed nite elements) are embedded in a menu driven, user friendly environment to serve hydrogeologists or engineers without profound knowledge of the mathematical theory. The software is suitable for unix workstations as well as inexpensive personal computers. The model components include reactive solute transport (with di usion, dispersion, advection and sorption) and single as well as two phase ow in the saturated and the vadose zone. The underlying models contain nonstandard e ects that enable the simulation of a large variety of relevant support strategies for natural attenuation. We present examples for the interaction of the transport of surface active agents and the water ow or the in uence of carriers, which can change predicted residence and travel times of strongly sorbing contaminants by several orders of magnitude. Keywords: Simulation Software, Numerical Flow and Transport Model, Finite Elements, Carrier facilitation, Surfactants Mathematical subject classi cation: 65M60, 76S05, 76T05, 86A05 This paper was presented in the session 'Contaminated sites: risk assessment, natural attenuation, in-situ remediation' at the EGS 2000 General Assembly, 25-29 April 2000, in Nice, France
1
1 Introduction When earth scientists address a site remediation problem, they have to gain a precise idea of the fate of the contaminant. Numerical models have proven their value along with laboratory and eld experiments in processing the available site information and predicting the migration and extent of the contaminant plume in many case studies (consult e.g. Spitz and Moreno, 1996 ). At the heart of any attempt to model the evolution of contaminant migration lie the description of groundwater movement and the standard formulation of reactive solute transport, which we extend in the sequel. As research about contaminant migration advanced, nonstandard e ects like carrier facilitated transport or the in uence of surfactants on contaminant spreading could be expressed in a mathematical framework and incorporated in model formulations. When mathematical models take these non-standard aspects into account, they can play an essential role in the design of site remediation strategies. We present such a mathematical software tool that has been developed with the scope of incorporating important, non-standard e ects as the aforementionend, but also with the aim of applying advanced mathematical techniques like a mixed hybrid nite element discretization or grid adaption to solve the problem eciently and accurately on a computer. Nevertheless the software named Richy (Schneid, 2000a ) is not designed as a pure research tool but it is embedded in a menu driven, user friendly environment and thus intended to serve scientists without an advanced mathematical background as well. We want to depict brie y the already implemented model components and focus on the modular structure of the software. This modular organization guarantees a exible handling, and permits arbitrary extensions of model components as well as the combination of existing modules. Two applications of the model are presented. We establish a set up for carrier facilitated transport in a multilayer soil column and demonstrate the permeabilty reduction of a porous medium by the transport of surfactants.
2 The Model Components First we want to line out the basic mathematical model components (modules) that are included in the simulation tool. Partial di erential equations (PDE) constitute the basis of the description of all the presented processes (see standard textbooks like de Marsily, 1986 ). They have to be completed by appropriate initial and boundary conditions to provide a solution of the 2
problem. These components are organized in a modular way in order to easily extend the application and facilitate its handling. Fig. 1 gives an overview of the di erent types of modules already implemented. All the PDE bear problem speci c coecients and coecient functions. We cite just a few of the incorporated functional forms, keeping in mind that we are not restricted to any of these. In a typical application, components will be combined and con gurated with the speci c properties of the occuring substances. For instance ground water will be simulated by the module '(un-)saturated ow' and a dissolved heavy metal by the 'solute transport' module. The component 'heat conduction' will not be treated in the following because it is of minor interest in site remediation problems. The combination of components is explained in section 3.1. The following assumptions underly the presented models: The porous medium is rigid, the ow regime laminar and the phases are incompressible. It is not intended to depict the complete set of the mathematical model formulations but to describe brie y the capacity of the components with references for further reading.
[…]
[…]
(un-) saturated flow
heat conduction
[…]
[…]
multiphase flow
solute transport
[…] (un-) saturated flow « surfactant transport
Fig. 1: Fundamental model components of Richy. In an application, the modules will represent special substances like [water] in the '(un-)saturated
ow' module or [phenanthrene] in the 'solute transport' module, by providing their speci c parameters. 3
2.1 Variably saturated ow
When transport processes in the subsurface are to be considered, water movement has a major impact on the spreading of the solutes. Therefore we start the description of the di erent model components with the characterization of uid ow in the saturated and the vadose zone, based on the conservation of mass and Darcy's law. We establish the well known Richards equation for
uid ow in one spatial dimension: @
(p) + r ~q = 0 @t
~ q=
? k kr (p)r(p + gz)
(1)
Here t denotes the time, the volumetric water content, p is the pressure head, ~q the Darcy ux, k is the intrinsic permeabilty of the porous medium, the viscosity, kr is the relative hydraulic conductivity, the density of the
uid, g the acceleration due to gravity and z is the elevation head. This model bears the following coecient functions: As indicated, the water content is a function of the pressure head p, the so called water retention curve, and the relative hydraulic conductivity kr depends as well on p in a nonlinear form. For these functional relationships di erent parametrizations exist in the literature, and can be incorporated in the model. We added e.g. the van Genuchten - Mualem model (van Genuchten, 1980 ) or a form-free ansatz based on linear or quadratic spline interpolation. The identi cation of the model parameters is subject of section 3.1.
2.2 Multiphase ow
In many contaminated sites we encounter a second liquid phase that does not mix with water and thus has to be treated seperately (non aqueous phase liquids, NAPL). To simulate the ow of two immiscible liquids (indicated by the subscripts o and w) and a gazeous phase (assumed to be at constant pressure) we have to modify the Richards equation for every liquid phase i 2 o; w: @ @t
(si(po; pw )!) + r q~i = 0
q~i
= ? k kr (po ; pw )r(pi + igz) (2)
i
with si being the saturation of the phase i and the total porosity !. The coecient functions to specify are now for every liquid again kr and the saturation function si. These relationships can be obtained in the laboratory and we included two common models to describe them: the Brooks-Corey model and the parametrization of Parker et al. (see e.g. Helmig, 1997 ). i
4
2.3 Solute transport
The equation that describes the reactive transport of a solute in a porous medium is also based on the principle of mass conservation. We include the phenomena of advection, di usion, dispersion and equilibrium sorption as well as reaction kinetics in the problem formulation and obtain a PDE of the following form: (c) + r w~ = ?b @ ( (c) + s) @t @t w ~ = ?D rc + ~ qc @t s = r ((c) ? s) @
(3)
where c is the dissolved concentration, b the bulk density, w~ the mass ux, and are the equilibrium and nonequilibrium sorption isotherms, s is the sorbed concentration, D the di usion-dispersion tensor and r is the rate parameter for sorption kinetics. The coecient functions to be speci ed here are the sorption isotherms which can be determined in column experiments and may be of general shape (e.g. linear, Freundlich or Langmuir type). They are not restricted to a special functional form, a form free ansatz based on spline interpolation of discrete data points is available, too.
2.4 Coupled (un-)saturated ow and surfactant transport
We present this problem class as a single component, because the coupling of ow and transport is a mutual one and thus di ers from the unidirectional combination described in section 3.1. In the last decade surfactant enhanced aquifer remediation (SEAR) became an attractive alternative to conventional pump-and-treat remediation techniques. Surfactants alter the surface tension of the liquid phase, may mobilize entrapped NAPL or reduce the permeability of the porous medium by sorption. As research in this area advanced, mathematical formulations of the interaction of surface active agents and water ow can now be incorporated in numerical simulation tools. We extend the standard formulations of uid ow (1) and solute transport (3) by taking two additional aspects into account: Firstly, the variation of the surface tension of the uid by the surfactant, and secondly, the impact on the hydraulic conductivity due to the preferential sorption of surfactants to the clay fraction of the soil. According to the work of Smith and Gillham (1994) surface active agents decrease the surface tension of the uid with increasing concentrations and 5
thus a ect the capillary pressure in the porous medium. Consequently a scaling factor is introduced in the pressure-saturation relation: 0 1 (p) ! ( 0 p) =: (p; c) = (4) 1 ? b ln(c=a + 1) with empirical, surfactant dependent parameters a and b. In a recent study (Smith and Gillham, 1999 ) this model has proven its applicability in accordance with experimental laboratory results. The variability of the permeability due to the sorption of the surfactant is incorporated in the model according to Renshaw et al. (1997): k
1? k ! ke (c) = kcoarse clay
= clay + b (c)
ke (c)kr (p) =: K (p; c)
s
(5)
represents the e ective permeability, kcoarse and kclay the saturated permeabilities of the coarse respectively the clay fraction of the soil, clay is the volume fraction of the clay component and b and s are the densities of the bulk soil and the surfactant, respectively. Surfactants sorb preferentially to the clay and therefore increase the volume fraction of the clay-surfactant conglomerate, thus the e ective permeability decreases due to the sorption of the surfactant. These mechanisms couple the water ow to the surfactant transport. On the other hand, the current water content and Darcy ux in the transport equation (3) are determined by the Richards equation. As the surfactants have an in uence on the ow regime, they do also a ect the fate of a contaminant which is transported in the subsurface. ke
3 Advanced modeling techniques 3.1 Combination of modules
The modular design of the software on the one hand guarantees an easy incorporation of new features in the existing framework. On the other hand the existing components can be easily combined to perform a more complex simulation and take the interactions of di erent processes into consideration. Combining is understood in the following sense: we establish a rst problem that is independent of the subsequent problem(s). The information of the rst problem is now available and may enter the model equations of any succeeding problem. We name two examples of combining the aforementioned problems, of course the reader may think of any other possible connection. In the formulation of the reactive solute transport (3), the ux ~q and the 6
water content have to be determined. Instead of imposing a steady state scenario by de ning constant values for the ux and the water content, a transient simulation can be run by simultaneously solving the Richards equation (1) ('(un-)saturated ow') and transfering ~q and into (3) of the 'solute transport'. A second important issue is the combination of two transport problems as it is necessary for the modeling of carrier facilitated transport: The migration of a substance can be substantially enhanced or reduced by its sorption to a mobile carrier (e.g. dissolved organic carbon, DOC). Under the assumption that the transport behaviour of the carrier itself is not in uenced by the migration of the contaminant we establish rst a transport problem (3) for the carrier. The transport equation of the contaminant is modi ed by introducing the so-called e ective isotherm that takes into account the sorption of the free contaminant as well as the sorption of the carrier-bound contaminant. As this isotherm depends on the current carrier concentration, it becomes a time and space dependent isotherm. Its ingredients can be assessed in laboratory column experiments without the necessity to model every single chemical reaction involved in the complex process. Fig. 2 illustrates the relevant aspects of carrier facilitation. See Knabner et al. (1996) and Totsche et al. (1996) for the derivation of the model and Knabner and Schneid (1996) for a description of its properties. If dynamic water ow should be incorporated in this case, one may again combine these 'solute transport' problems with the (same) Richards equation ('(un-)saturated ow').
dissolved contaminant solid matrix mobile carrier sorbed carrier sorbed contaminant
Fig. 2: Schematic representation of a porous medium with colloidal transport. 7
3.2 Parameter identi cation
The aforementioned equations contain a number of coecient functions (like the sorption isotherms or the water retention curve) that have to be speci ed by the hydrogeologist, normally with the aid of experimental results. As direct measurements are not available for all parameters so-called inverse problems related to speci c column experiments have to be solved and we provide algorithms for the identi cation of these model parameters. In particular, the sorption isotherms can be determined from breakthrough experiments (based on the work of Igler, 1998 ) as well as the characteristics of the hydraulic functions, namely the water retention curve and the unsaturated hydraulic conductivity from out ow experiments with arbritary pressure decrease (see Bitterlich and Knabner, 2000 ).
3.3 Database
The presented model equations require the knowledge of various physicochemical properties of the substances in question (e.g. density, di usion coecient) as well as the knowledge of di erent soil properties (e.g. porosity, clay content). In order to ease this task we supply a database with the physico-chemical properties of 113 common contaminants and 40 di erent soil textures. The user may of course alter the provided data or amend his own information.
4 Discretization and adaptive techniques
The model equations in Richy are discretized by the backward Euler method in time and by nite elements in space. The equations describing the ow of uid phases are discretized by mixed nite element methods to ensure the local conservation of mass and the continuity of the ux (also for heterogeneous media), a crucial quality for subsequent transport processes depending on that uid ow. The standard nite element method is used for the discretization of 'heat conduction' and 'solute transport'. Richy supports the usage of adaptive strategies to control the sizes of discretization parameters. The usage of such techniques ensures the ecient utilization of the available performance of a computer, that otherwise would be restrictive for complex scenarios. Using error indicators for the nite element discretization of the model equations, the grid representing the underlying domain of the simulation is automatically re ned and coarsened, corresponding to the form of the solution. Additional indicators for the error of the time discretization allow for an adaptive time stepsize control. This 8
automative adaption of discretization parameters is currently implemented for (un-)saturated uid ow and will be applied also to the remaining model components. Fig. 3 shows an example of adaptive grid re nement in a two layer medium with a higher conductivity in the upper layer (taken from Schneid, 2000b ). Pressure (left) and ux (right) pro les for subsequent time points during in ltration into the soil are depicted in the two plots at the top. The plot below shows the spatial distribution of the grid points over simulation time. At the beginning of the simulation the areas of important pressure variations 1
0.8
0.8
t
1
x
t
0.6
= 200 000
Coordinate
0.4
0.4
0.2 0
=0
0.6
0.2
t
=0
0 -0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
t
-0.2
= 200 000
-1
-1 -0.4
-0.3
Pressure
-0.2
-0.1
0
-2.5e-06
-2e-06
p
-1.5e-06
Flux
-1e-06
-5e-07
0
~ q
x
1
Coordinate
0.5
0
-0.5
-1 10
100
1000
Time
10000
100000
t
Fig. 3: Pressure (top left) and ux (top right) pro les and distribution of grid points (bottom) for the in ltration in a two layer soil column. 9
above the interface between the layers and above the outlet are discretized particularly ne. Along with the in ltration process the re ned areas are changing due to the evolution of the pressure distribution within the soil. At the end of the simulation the pressure pro le in the deeper layer is very smooth and thus is discretized with only a few grid points.
5 Implementation The implementation of the algorithms is realized in the programming language C, the graphical interface is based on Tcl/Tk and OpenGL. This guarantuees the portability of Richy to di erent platforms including Microsoft Windows and Linux on personal computers and various unix workstations. The adjustable menu structure enables an interactive data input, but the usage by script les is possible, too. The simulation results may be visualized in time or space dependent plots of the model variables and stored in ASCII data les.
6 Applications Subsequently two examples of the application of the presented code are given to demonstrate the importance of the nonstandard e ects that have been incorporated in the models. It is not the scope of the authors to line out all the parameter settings of the simulations in detail but to give an idea of the presented model capacities.
6.1 Carrier facilitated contaminant transport
We study the reactive transport of phenanthrene in the presence of DOC in a multilayer soil column based on experimental data. Phenanthrene is a well investigated, neutral hydrophobic organic compound with a strong sorption tendency to natural materials and therefore a candidate for colloid enhanced transport in natural, porous media (Magee et al., 1991 ). The colloid enhanced simulation is compared to predictions without carrier facilitation. The porous medium is composed of six distinct, homogeneous layers of soil with a total depth of 9 m, where we suppose the groundwater level. We simulate a spill during the rst 100 days, after that, no more contaminant is added in the previously uncontaminated soil. We assume a constant water
ux corresponding to an annual groundwater recharge of 250 mm/a, a value reasonable for a temperate European climate. Phenanthrene may sorb to the solid matrix as well as to the mobile carrier 10
0 -100
depth [cm]
-200 -300
t = 12 a t = 200 a t = 400 a t = 600 a
-400 -500 -600 -700 -800 -900 0.0E+00
3.0E-03
6.0E-03
9.0E-03
1.2E-02
1.5E-02
1.8E-02
c [mg/kg]
Fig. 4: Concentration c of sorbed phenanthrene at di erent times t when no carrier is present. DOC which itself should not sorb to the soil in our scenario. As phenanthrene is not predicted to travel signi cant distances in the subsurface when there is no carrier present, the migration of the contaminant plume is very slow. See the concentration pro les of sorbed phenanthrene at t = 12; 200; 400; 600 a in g. 4 in the absence of DOC. It takes about 500 years until the peak of the breakthrough curve passes the groundwater level (see g. 5). But if we respect a linear sorption of phenanthrene to DOC with partition coecients based on experimental ndings by Roy and Dzombak (1998) we observe the peak of the breakthrough curve after only 8 years at the groundwater level with concentrations augmented by two orders of magnitude (see g. 6). This realistic numerical example demonstrates the importance of carrier facilitation and suggests strongly to take this aspect into account. The observed di erences concerning residence times would lead to substantially di erent decisions concerning a site remediation strategy like natural attenuation.
11
7.0E-04 6.0E-04
c [mg/l]
5.0E-04 4.0E-04 3.0E-04 2.0E-04 1.0E-04 0.0E+00 0
100
200
300
400
500
600
t [a]
Fig. 5: Breakthrough curve of the concentration c of dissolved phenanthrene at x = ?900 cm when no carrier is present. 3.5E-02 3.0E-02
c [mg/l]
2.5E-02 2.0E-02 1.5E-02 1.0E-02 5.0E-03 0.0E+00 0
2
4
6
8
10
12
t [a]
Fig. 6: Breakthrough curve of the concentration c of dissolved phenanthrene at x = ?900 cm whith carrier facilitation.
6.2 Permeability changes by surfactant transport
Our second example demonstrates the permeabilty reduction due to surfactant migration in a homogeneous soil column. As surfactants preferentially sorb to the clay fraction of the soil, they change its conductivity. In a site remediation study, the simulation of surfactant migration helps us to estimate 12
the performance of SEAR. The permeability reduction of the porous medium may be an important factor hereby. We plot the breakthrough curves of the surfactant at the inlet, at the center and at the outlet of the soil column in g. 7. The permeability changes are illustrated by the graph of the unsaturated hydraulic conductivity at the center of the column in g. 8. 0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1 0
200000
400000
600000
800000
1e+06
0
200000
400000
600000
800000
1e+06
Fig. 7: Breakthrough curves of the dissolved surfactant concentration at the top of the soil column (red), at the center (green) and at the outlet (blue). 3e-06
2.5e-06
2e-06
1.5e-06
1e-06
5e-07
0
Fig. 8: Unsaturated hydraulic conductivity versus time at the center of the column. 13
7 Conclusions In the context of site remediation and natural attenuation we presented a numerical ow and transport model to attack the challenging task of evaluating and predicting the fate of contaminants. Not only the implemented model formulations but especially the exible modular and expandable software design guarantee the bene t of the proposed code now and concerning future developments. The advanced ecient mathematical kernel enables accurate solutions of the problems and possible extensions. Thus we introduced a practical tool to quantify and qualify natural groundwater hazards in site remediation problems.
References [1] Bitterlich, S. and Knabner, P. (2000) Formfree and cascadic identi cation of material laws in (un)saturated uid ow from column experiments. Submitted. [2] Helmig, R. (1997) Multiphase Flow and Transport Processes in the Subsurface. Springer, Berlin. [3] Igler, B. (1998) Identi cation of Nonlinear Coecient Functions in Reactive Transport through Porous Media. PhD Thesis, Institute of Applied Mathematics, University of Erlangen-Nuremberg, Erlangen. also: http://www.am.uni-erlangen.de/am1/publications/dipl phd thesis/ [4] Knabner, P. and Schneid, E. (1996) Qualitative properties of a model for carrier facilitated groundwater contaminant transport. In Scienti c Computing in Chemical Engineering (eds F. Keil et al.), pp. 129-135. Springer, Berlin. [5] Knabner, P., Totsche, K. U. and Kogel-Knabner, I. (1996) The modeling of reactive solute transport with sorption to mobile and immobile sorbents, 1. Experimental evidence and model development. Water Resour. Res., 32 (6), 1611-1622. [6] Magee, B. R., Lion, L. W. and Lemley, A. T. (1991) Transport of dissolved organic macromolecules and their e ect on the transport of phenanthrene in porous media. Environ. Sci. Technol., 25, 323-331. [7] Marsily, G. de (1986) Quantitative Hydrogeology. Academic Press, Orlando. 14
[8] Renshaw, C. E., Zynda, G. D. and Fountain, J. C. (1997) Permeability reductions induced by sorption of surfactant. Water Resour. Res., 33 (3), 371-378. [9] Roy, S. B. and Dzombak, D. A. (1998) Sorption nonequilibrium e ects on colloid-enhanced transport of hydrophobic organic compounds in porous media. J. Contam. Hydrol., 30, 179-200. [10] Schneid, E. (2000a) Richy Documentation. http://www.am.uni-erlangen.de/~schneid/RichyDocumentation/Main.html. [11] Schneid, E. (2000b) Hybrid-Gemischte Finite Elemente Diskretisierung der Richards-Gleichung. PhD Thesis, Institute of Applied Mathematics, University of Erlangen-Nuremberg, Erlangen. [12] Smith, J. E. and Gillham, R. W. (1994) The e ect of concentration dependent surface tension on the ow of water and transport of dissolved organic compounds: A pressure head-based formulation and numerical model. Water Resour. Res., 30 (2), 343-354. [13] Smith, J. E. and Gillham, R. W. (1999) E ects of solute concentrationdependent surface tension on unsaturated ow: Laboratory sand column experiments. Water Resour. Res., 35 (4), 973-982. [14] Spitz, K. and Moreno, J. (1996) A Practical Guide to Groundwater and Solute Transport Modeling. Wiley, New York. [15] Totsche, K. U., Knabner, P. and Kogel-Knabner, I. (1996) The modeling of reactive solute transport with sorption to mobile and immobile sorbents, 2. Model discussion and numerical simulation. Water Resour. Res., 32 (6), 1623-1634. [16] van Genuchten, M. T. (1980) A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J., 44, 892898.
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