The application of mathematical models of the transport of chemical substances in the remediation of consequences of the uranium mining

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1 The application of mathematical models of the transport of chemical substances in the remediation of consequences of the uranium mining Michal Beneš, Martin Stýblo Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Prague Trojanova 13, Prague Czech Republic Phone: Fax: JiříMaryška, JiříMužák Department of Control Engineering, Faculty of Mechatronics, Technical University of Liberec Hálkova 6, Liberec Czech Republic Phone: /268 Fax: Abstract A large contamination of the Cenomanian aquifer in the area of the Stráž deposit in Northern Bohemia due to chemical mining of uranium is one of the most important environmental problems in the Czech Republic. The paper deals with mathematical modelling of transport of dissolved chemical substances in this area. The model consists of computation of porous media flow, convection-diffusion transport of chemicals, and simulation of selected chemical reactions. Several finite element formulations (i.e. primal, mixed-hybrid and mixed) of flow and transport problems are considered. For spatial discretization, trilateral prismatic elements are used to describe the structure of stratified layers. Results of numerical experiments and practical computations including simulation of remediation scenarios in the area of leaching fields and modelling of pollution leakage from waste ponds are discussed. 1

2 2 1 Situation in the Stráž deposit Since 1968, more than 4 million tons of sulphuric acid (H 2 SO 4 ), 300 thousand tons of nitric acid (HNO 3 ), 120 thousand tons of ammonia (NH 3 ) and smaller quantities of other chemicals have been injected into the Stráž deposit. A part of the contaminants (about 0.5 %) is situated in the upper Turonian aquifer, which is a source of drinking water for the surrounding region. This contamination is caused by the escape of solutions either from surface technologies or from damaged casings. The wells of older construction had no doublecasing in the Turonian aquifer. In the Cenomanian aquifer, two different areas can be defined: area of leaching fields and area of solution excursion towards the deep mine. The lower part of the geological profile in the leaching fields area with uranium ores (washout horizon, friable sandstones) is filled with solutions of total salinity g/l. High concentrations and large vertical extents of contamination (in some places the full thickness of the fucoid sandstones, i.e. 40 meters) can be found in the area of older leaching fields where extremely high dosages of sulphuric acid were used. Similar solutions also are in more permeable parts of the profile, in the neighbourhood of the leaching fields ( meters). In the area of the solution escape, the thickness of contaminated horizon is decreasing from the hydraulic barrier to the drainage system of the deep mine. The purpose of this drainage system is to hold back acid water contaminated from the leaching process. The salinity of solutions varies from 1 to 10 g/l; ph is usually from 1.8 to 3.5. In the whole Cenomanian aquifer, there are more then 180 millions m 3 of contaminated water. For details, see [1]. 2 Mathematical models of transport The most general mathematical model under consideration consists of porous-media flow problem, transport of chemical substances with diffusion and dispersion, and thermodynamical modelling of chemical processes. Porous-media flow problem is described by the Darcy s law and the continuity equation u = A p ;.u = q in Ω, (1) where p denotes piezometric head, u is filtration velocity, A is symmetric positive definite second rank tensor of hydraulic permeability of the porous medium, q represents density of potential liquid sources in the medium. The boundary conditions are given by p = p D on Ω 1, n.(a p) =0on Ω 2 (impermeable part of the boundary) and n.(a p)+σp = σp D on Ω 3, where n is unit vector of outer normal (see [2]). The transport of chemical species dissolved in the technological solution is described by the following law l i = D(u) l c ; [D] ij = D m δ ij + α T. u.δ ij +(α L α T ) u iu j, (2) u where l i is the flux of l-th substance, l c denotes its concentration, and D is the diffusivity-dispersivity tensor. D m denotes the coefficient of molecular diffusion, α T and α L the transversal and the longitudinal dispersivity, respectively. The equation of mass balance for l-th substance is l c t + u c+ li+ l l cq l r( l c, i c,...) = l c q + ; l=1,..., L. (3) where the first term describes storage of the l-th substance, second term describes convection, and third term reflects the influence of diffusion and dispersion. The term l cq describes the influence of sink of the l-th species and the term l c q + determines source of l-th species. Last term on the left hand side equation, l r( l c, i c,...), describes chemical processes. The boundary conditions are l c = 0 on the part of boundary Ω +, where u.n 0and l i= 0 on the remaining part Ω (see [3]). The reaction term l r( l c, i c,...) describes the influence of all reactions where the l-th species is present. In general, it can be rewritten as l r( l c, i c,...) =,l r( l c, i c,...)+ +,l r( l c, i c,...) (4)

3 3 where,l r expresses the decrease of the l-th species by its reaction with other species, and,l r expresses the increase of the l-th species by reactions of other species (see [4]). The chemical situation in the mining field is very complex. Therefore, the diffusion-convection problem is solved on some simple testing examples whereas the real situation is simulated as follows: the stationary porous-media flow problem is solved; the convective transport over finite volumes is computed; the diffusion and dispersion is computed; the chemical situation is computed on each volume separately. The following types of reactions are considered: fast reactions in the technological solution described by the Guldberg-Waag law schematically written as: k G( i c)=0 (5) Such equilibrium relations represent a coupling (constraint) which should be considered when solving dynamical relations as shown below. kinetic reactions with the rock (dissolving); kinetic reactions of precipitation; sorption on the surface of the rock. Kinetic relations describing previous three cases can be written in the following form ([5]): d l c dt = ν l,i k i,l ( l c) ν l,i ( i c) ν i,l i I l + ν(i,j) l k ij( i c) ν ij ( j c) ν ji. (6) i,j l Here, I l denotes the set of indeces of species reacting with the l-th species, ν l,i is stechiometric factor of l-th species if reacting with the i-th species, and k i,l is the reaction rate. The coefficient νl,i is the modified stechiometric factor. 3 Mathematical models and numerical methods Considering several different requirements of the application, the following mathematical problems are solved: Steady-state porous media flow with free boundary is modelled by primal FEM. By an iteration process, the grid is deformed according to the phreatic surface (see [2]). For transport modelling, the mixed-hybrid FEM model of porous media is computed on a grid obtained in the previous step in order to get a conservative approximation of the flow field (see [6]). The system of convection-reaction equations is solved by an explicit finite-volume upwind method using the flow field computed in the previous step (see [7]). The chemical kinetics is computed on each element separately. The discretization of the problem is given by physical and chemical characteristics of the geological region. There are used trilateral prismatic elements with vertical faces and general nonparallel basis (cnf. Figure 1). The five-face elements and the simplices are used to model closures of the sedimented layers. These elements describe character of stratified sedimented layers and complicated horizontal profile with many wells. In case of real problems, it is necessary to consider large area of several km 2. The thickness of cenomanian sandstones is several tens of meters. Consequently, real elements have considerable size disproportions in horizontal and vertical nodal distances. The influence of these disproportions on the spectrum of the resulting system of linear equations and on the rate of convergence of numerical methods is studied.

4 4 x 6 x 4 n e 2 x 5 n e n e n e x 2 z x 1 n e 1 x 3 y x Figure 1: The RT 0 1,h and H1 h elements. x 4 x 5 t c 2 x 12 6 c23 c 13 x 1 t 1 x 3 x 2 The following approximation spaces are used: The Raviart-Thomas space RT 1,h 0 of linear vector valued functions for mixed-hybrid formulation. The finite space Hh 1 for primal and mixed formulation (6 nodal degrees of freedom). The Raviart-Thomas space of vector valued functions was generated on the following type of elements for mixed-hybrid formulation vj e := kj e [ x1 αj1 e, x 2 αj2, e βjx e 3 αj3 e ]. Parameters of these functions are determined by the conditions n e j vi e ds := δ ij, i,j =1,..., 5 f e j where fj e denotes j-th face of the element e n e j is its unit normal (with respect to the element e ). The Raviart-Thomas approximation of the porous-media flow problem yields the system of linear algebraic equations with symmetric indefinite matrix. The finite space Hh,ψ 1 (Ω) was generated for primal and mixed formulation. Any element is decomposed by intersections of diagonals of vertical faces and centres of gravity of bases into 11 subelements (simplices). On these simplices, there are defined linear functions in such a way, that base function on whole element is continous and piece-wise linear. The approximation of primal formulation leads to a system of linear algebraic equations with sparse positively definite matrix (for details, see [8], [9]). 4 Modelling of the remediation process In 1996, the uranium mining has been stopped, and the major interest of the Czech government is in the removal of consequences of former mining activity - in the optimal remediation scenario. For this purpose, the model is used to produce short- and long-term predictions of behaviour of whole area under different outer conditions. The transport equation is discretized in time with timestep that corresponds to the size of time interval in which the whole problem is solved. For the space discretization of the transport problem the same mesh and the same spaces of scalar and vector valued functions are used. The solution of chemical species transport problem in the convective field of filtration velocities leads to the large nonsymmetric sparse system of equations. The transport equation is solved on each timelevel by superimposing the influence of convection solved in the first stage and the influence of diffusion and dispersion. Obtained results of concentration distribution create an input for thermodynamical model of chemical processes. This procedure exhibits certain time stability, which is very important for calculations in long-time intervals (several decades). One of such predictions is shown in the Figure 2. In the first phase, the problem of pumping-out concentrated solutions is solved. Evolution of pressure in the Cenomanian aquifer is solved by a local regional hydraulic model (see [1]), which is a two-layer model with grid covering an area about 300 km 2. This area is mostly limited by tectonic lines. The horizont is splitted vertically into 9-13 layers. The grid consists of 1400 plane elements and spatial elements. Filtration parameters are determined from hydrogeological by experiment, and their

5 Figure 2: The situation of the water level in the Cenomanian aquifer. The situation after starting the evaporation station (in 1996) with less injection into the barrier, and finally the situation after flooding the mine. 5

6 6 vertical distribution is given by data of the Geotechnological Information System (GTIS). The initial conditions (the material concentrations in solution) in the leaching field area are given by monitoring wells. A vertical distribution of the contaminants is controlled by induction logging and the results are stored in the GTIS. Chemical processes are simulated by two different methods. A decay of caolinit by the sulphuric acid, which causes decrease of salinity and increase of aluminium contents in solution, is solved by the thermodynamical kinetic model. This model was developed from the balanced thermodynamical model for solutions and rock of the Stráž deposit. The thermodynamical system of the balanced model consists of 18 main components of solution, 32 minerals, 110 chemical reactions between the solution components and 44 reactions between minerals and solution. The main components of the solution may occur in 184 different forms. In the first stage, the kinetic model was reduced to 5 components and 11 equations. The model solves a leaching of Al from the rock and its precipitation in the form of alunit or aluminium hydroxid. The concentration of H 2 SO 4 and the total amount of sulphur ion varies at the same time. The kinetics of uranium leaching is solved by kinetic functions given by the experiment. The kinetics is simplified in comparison with the detailed models for uranium mining optimization, because it is required to simulate the situation in the whole volume for a period of several decades with reasonable time steps (see [10]). 5 Modelling of the influences of the Stráž wa-ste pond on the groundwater The Stráž wastepondisabout180ha (1.8 km 2 ). It is divided into two parts: the 1st and the 2nd stage. The first stage was built in It is filled-up to height of about 25 m above the former surface. In its upper part, there is a water surface, in which waste muds are floated. The second stage was built in 1988, when the future development of uranium production was planned. After a short term use of this second stage, production was significantly limited. At present, this part of the waste pond forms a lake with an area of about 90 ha (0.9 km 2 ). The lake contains water contaminated by waste from the Stráž uranium mill. The water surface in this lake is approximately on the same level as groundwater. The main contamination is 3 to 5 g/l of SO 4 ions. Underlying rocks of the pit are created by permeable coniacian sandstones with a filtration coefficient k f = m/s. The boundary conditions are determined by the level of water surface in the pit and in the Ploučnice river in the South (Dirichlet boundery conditions). The part of the boundary formed by impermeable rocks is described by homogenous Neumann boundary condition. In addition, the rain dotation during the season, and the dotation from surrounding hills in the North are considered (Newton boundary condition). Details can be found in [1]. 6 Conclusion The models of transport of contaminants in the area of the Stráž deposit are very important for monitoring and prediction of the environmental situation and help in decision of an appropriate remediation scenario for the mine. The real situation is simulated by a multi-stage model including porous-media flow, convective and diffusive transport, and chemical reactions. Such models will be improved by solving some of these stages simultaneously. As the description of situation near waste ponds and in subsurface layer is required, the models will be adapted for this purpose, too. Acknowledgement. 205/96/0921. The reserch is supported by the Czech Grant Agency under the project No. References [1] P. Mareček, J. Novák, and V. Wasserbauer. Mathematical modelling of the influences of the Stráž mill waste pond on the groundwater. In IAEA Technical Committee Meeting on Computer Application in Uranium Exploration and Production Case History and Current Status, Vienna, November 1994.

7 Figure 3: The time development of the concentrations of TDS (total dissolved solids) near the bottom of the friable sandstones before beginning evaporation in 1995, and a cross-sections through the deposit. 7

8 8 [2] J. Maryška. Mathematical models of the underground water flow and chemical transport problem. PhD thesis, FNSPE CTU Prague, [3] J. Maryška and J. Mužák. Mathematical modelling of the transport of chemical species in the contaminated underground water. Short communication at Congres milieux poreux, St. Etienne, [4] M. Stýblo, J. Maryška, and J. Mužák. Mathematical modelling of groundwater flow and transport of chemicals. In Proceedings of the 9th Summer School Software and Algorithms of Numerical Mathematics, pages , Železná Ruda, [5] R. Brdička, M. Kalousek, A. Schütz. Introduction into Physical Chemistry. SNTL, Prague [6] J. Maryška and J. Mužák. Hybrid mixed model of the transport of the chemical substances. In Proceedings of Numerical Modelling in Continuum Mechanics, volume Part II., pages , [7] J. Maryška, J. Mužák, and M. Stýblo. Mixed-hybrid formulation of the transport of chemicals in the contaminated underground water. In S. Míka, M. Míková, and Brandner M., editors, Lecture Notes of IMAMM 95, pages , Srní, June Univ. of West Bohemia in Plzeň. [8] J. Maryška, M. Rozložník,andM.Tůma. Mixed-hybrid finite element approximation of the potential fluid problem. J. Comput. Appl. Math., 63: , [9] J. Maryška, M. Rozložník, andm. Tůma. The potential fluid flow problem and the convergence rate of the minimal residual method. Submitted to Num. Lin. Alg. Appl., [10] M. Stýblo, J. Maryška, J. Mužák, J. Skokan, M. Tůma, J. Drkošková, M. Rozložník, Z. Strakoš, J. Novák, P. Mareček, and V. Wasserbauer. Mathematical modelling of porous media flow and transport of chemicals in underground water. In WORKSHOP 96, pages Czech Technical University in Prague and Technical University in Brno, 1996.

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