ࡱ> SURY \bjbjWW ݓ==`C]|||||DL|(pNPPP2V*$A5 :NMN(N||NNN [R||N About adsorbing boundaries in random walk simulations of groundwater contaminant transport G. Gjetvaj Faculty of Civil Engineering, University of Zagreb, Ka ieva 26, 10 000 Zagreb, Croatia E-Mail: Goran@master.grad.hr Abstract The common approach to the modelling of solute transport by random walk particle method presumes that the particle that crosses domain boundary, cannot reenter the model. A series of authors e.g. Kinzelbach,5 Thomson & Gelhar,11 have noticed that the adsorbing boundary conditions cause an error in the solute transport models based upon a random walk particle method. The error can be significant in case the domain is not big enough, or in case the significant amount of solute moves close to the adsorbing boundary. In this paper the necessary volume of space enclosed by the model has been analysed, in order to obtain results within the acceptable, i.e. given accuracy, as well as supplement of the existing program based on random walk particle method, in order to reduce the effect of adsorbing boundary conditions. 1 Introduction In order to predict the spreading of pollutants, numerical models are often used. In the engineers practice, in establishing numerical models, only final space segment in which pollutant transport is carried out is considered. In those cases by numerical model not the e EMBED Package ntire water layer is enclosed, but only the space with the most of the observed solute. In the engineers practice while modelling by random walk particle method the adsorbing boundary conditions are frequently used e.g. Pricket,8 Kinzelbach,5 Thomson and Gelhar.11 For instance Thomson and Gelhar11 observe solute transport with enforcement of adsorbing boundaries on the six faces of the domain c. The adsorbing boundary is an approximation in which mass breakthroughs can be observed. It is approximate because particles that move across a boundary to a region outside c will be removed from consideration and not be allowed to reenter the domain (from, for example, an upstream dispersive displacement).11 In order to reduce influence of the adsorbing boundary conditions it is necessary to select enough space enclosed by the model and/or redefine the way particles move. In this paper both problems have been observed. The approach to the research is based on the analysis of the solute quantity which is situated within the model and defining the solute quantity that is passing through the boundaries of the domain. 2 Distribution of pollutant in two dimensions We consider first a confined homogenous aquifer of constant thickness with parallel stream-lines and solute transport of ideal tracer with dispersion coefficient constant in time. Under the assumed conditions there are analytical solutions of transport equation. The cases with instantaneous and constant injection rate of the ideal tracer at one node will be observed, and solutions obtained by analytical and numerical method will be compared. 2.1 Constant injection rate If spreading of solute being constantly released at a location (x,y) is observed, one could notice that for the distance long enough from the injection position and after the stationary process, concentration distribution perpendicular to the direction of flow is Gaussian. Figure 1. Relative concentration of tracer in the case of continuous injection at one node (vertical scale exaggerated by factor 20, after Verruijt, 1971) Standard deviation is then proportional to the square root of convective movement, which means that the width of the zone of dispersion is proportional to the square root of the distance along which dispersion across the interface occurs.12 Lines of constant concentration are, for large distances from the place where the tracer is injected, parabolas which is shown in Fig. 1. For the figure elaboration relative concentration mark has been used. 2.2 Instantaneous injection of pollutant The spreading of tracer caused by convection, dispersion and diffusion for the instantaneously injected solute at a location can be, for the observed solute transport case, described by use of Gaussian function N(1,2,12,22,). The physical significance of that distribution in describing the solute transport process can be interpreted by assuming that 1,2 describe the plume center position (movement caused by the convection) and that 1 and 2 mark standard deviation, that is diffusion of solute from mean value (movement caused by the dispersion). Distribution function graph is a bell-shaped plane, whose intersections with planes z=c (0<c<cmax) are ellipses which in this case present lines of equal concentration. The ellipses have a common center at the node (1,2) in which there is maximum concentration of the observed solute c1 (1 , 2)=cmax , and with increasing distance from the node concentration of the observed solute is reduced. Figure 2: Two-dimensional concentration distribution due to a instantenous injection at location (x=0, y=0) and time t=0 (after Kinzelbach 4). 2.3 Pollutant distribution In analysing dispersion of the observed solute plume in direction perpendicular to the flow direction it is necessary to know probability distribution of the solute particles move. In two-dimensional constant probability distribution one-dimensional constant distribution with the probability density function perpendicular to the flow direction is called conditional probability distribution of the Y component with fixed value of the x component X (Paue 7). It can be showed that the conditional distribution of the Y component for the fixed value x of the component X in two-dimensional normal distribution N(1,2,12,22,) is one-dimensional normal distribution N(,2). One-dimensional normal distribution has got the density function of the form: EMBED Equation.3 where i is the random variable. Figure 3: Gaussian curve of the probability density distribution. Fig.3 shows the curve of normal distribution for  = 5 and  = 1. While modelling solute transport in ground water flow, parameters determining normal distribution can be defined by the equations: EMBED Equation.3EMBED Equation.3 (2) where t stands for the time period from the moment the tracer starts moving,  for dispersivity coefficient and R is retardation factor. From the diagram in the Fig. 3 it is obvious that in normal distribution N(,2) in fact the whole probability load is in the interval <-3,+3> i.e. in case of extending the tracer by only 0.3% of the observed solute it would be beyond that domain. In the interval <-2,+2> there is 95% of the solute mass (4.6% of the solute mass is beyond the model), while in the interval <-,+> there is somewhat more than 68% of the solute. 3 Minimum dimensions of the domain comprised by the model Based on the Gaussian distribution minimum width of the domain c can be determined on condition that the given percentage of initially injected solute remains within the domain. Based on the described characteristics of the Gaussian distribution (Fig 3.), and presumption that for the engineers needs suffice 95% of solute within the domain, it is proposed that equation of the necessary width of the domain comprised by the model should be BM=4 i.e. in the form: EMBED Equation.3 Analogously it can be assumed that the length of the domain comprised by the model should be EMBED Equation.3 If coefficient of the transversal dispersivity T is considered as a constant value in time and space (and if retardation factor R=1 is assumed), the domain in which the tracer is transported has the form shown in the Fig.4. For construction of diagram, the relation (3) and transversal dispersion coefficient T =1, 5, 10, 20 m and flow direction parallel to the x-axis. Tracer injection at a location is mathematically defined concept of tracer injection into aquifer. In the engineers practice it is impossible to inject solute in material point. During the injection in the immediate vicinity of the injection point initial parallel stream lines are deformed. From a practical point of view it is more convenient to define the zone of the tracer injection, which must be considered separately when defining model dimensions. Figure 4: Necessary semi-width of the model for various values of the dispersivity coefficient T. 4 Adsorbing boundaries Most of the researchers modelling by random walk particle method use adsorbing boundaries eg. Pricket,8 Kinzelbach,5 Thomson & Gelhar.11 For example Thomson and Gelhar11 have in their research formed the model in which six faces of the computational region c were treated as adsorbing boundaries so that particles could freely cross through them and exit the domain. When this occurred, particles were removed from consideration and the total mass within c was reduced accordingly. This is an approximate procedure which does not allow the mass to reenter c. Generally, particles exited the domain for one of the two reasons: (1) upstream movement through upstream face and (2) outflow and downstream boundaries. In most cases, the initial loss of mass (if any) will occur for a small period of time 0 t tu, because of dispersion upstream out of the domain. The mass in c will than remain constant for a relatively long period of time tu t tD after which it will flow out of the domain through the downstream face, or perhaps disperse out of one of the side faces.11 Particle movement caused by dispersion could be considered as Brown motion, for which it is possible to calculate particle oscillation around most probable distribution.10 As the particles are constantly in motion, there is always a probability that part of the particles exiting the domain will reenter it. That was the reason to consider domain significantly larger than the plume of the observed solute. Thus the error caused by the adsorbing boundary most frequently becomes lesser than the errors resulting from assuming parameters which described flow and transport. 5. Particle motion The aim of this research is to supplement the algorithm by which particle motion is described. Therefore probability that particle is out of domain and reenters it has been tested. If X (t) marks the position of particle in an instant t(x(0)=0), trajectory of the observed particle {X(t):t>0} can be considered as random variable. Rao9 have concluded that motion of every single particle can be described as homogeneous Brown motion. Gupta (after Rao9 ) considered particle motion as non-homogeneous Markov process. Several researchers have so far used Markov processes and their properties in modelling solute transport through groundwater flow for describing transport of chemical solutes e.g. Knighton and Wagenet6 and transport of adsorbing solutes eg. Andri evi and Georgiou.1 Dispersion of particles carried by the groundwater flow can be thus considered as Brown motion. Brown motion is a stochastic process with independent stationary increases which is named Wiener process and which satisfies Markov property therefore belongs to the group of Markov processes. Markov property describes the fact that probability distribution of the random variable Xt, in the moment t=tn depends only on value xn-1 of the process in the moment xn-1, and does not depend on values xn-2, ..., x1 of the process in previous moments tn-2>tn-3>...>t1. That property is very important in modelling solute transport by groundwater because it enables that a new position of a particle describing the solute transport is dependent only on the data on its position and previous positions need not to be known. Finding parameters describing particle motion is a very complicated process and they still have not been found. Therefore a series of numerical experiments has been carried out, with three basic aims: a) establishing probability of particle exiting the domain covered by the model in function of their distance from the model boundary, b) establishing probability of their reentering the domain covered by the model and c) describing probability for particle reentering the domain covered by the model . Based on series of numerical experiments it has been assumed that:3 a) probability that the particle which has in the first time interval exited the domain covered by the model in some other interval reenters it and b) probability that the particle which has in the first time interval exited the domain covered by the model, after repeated movement from the initial position stays within the domain covered by the model, are approximately equal. This statement practically means that reentering of the particles (in series of subsequent time intervals) that have exited could be replaced by repeated movement of the particles from the initial position. In case that in the second attempt particle moves across a boundary of the domain it is removed from the list of active particles. If in the second attempt particle remains within the domain, numerical procedure is continued with a new particle position. 6 Verification of the supplemented algorithm In order to verify the modification of the model, the modified algorithm has been compared to the other models as well as to the known analytical solutions. 6.1 Flow near the model boundary Adsorbing boundaries take effect in the models based on the random walk particle method when the particles moves parallel with the boundary of the domain covered by the model at small distance. Figure 5: Solute concentration on the distance of x = 200 m from the place of injection. 6.1.1 Concentration comparison After the modification of the program has been done, simulation of instantaneous tracer injection in the flow with parallel streamlines has been done. The results of modelling are shown in Fig.5 where concentration distribution in the node x = 200 m away from the place of injection and displaced y = 5 m from the transport axis have been compared. Analytical solution is shown by continuos line (curve A). Curves B and C present calculated modifications obtained by supplemented program while curve D shown by dotted line presents result of modelling obtained by standard random walk particle method. 7 Conclusion The models based on the random walk particle method remove particles that move across a boundary to a region outside domain from further consideration. This kind of approach in fact assumes existence of adsorption on the model boundary and neglects the possibility of particles reentering the domain. In some cases this kind of boundary condition can cause error in the results of modelling. In this paper the equation has been proposed to define minimum space volume covered by the model. In the paper supplement of the programs used so far was proposed introducing possibility of reentering of particles that have exited the domain covered by the model. Comparison of the results obtained by supplemented program and corresponding analytical model show correctness and validity of the performed supplement. References 1. Andri evi,R. & Foufoula-Georgiou,E. Modeling kinetic non- equilibrium using the first two moments of the residence time distribution, Stochastic Hydrology and Hydraulics, 1991, 5, 155-171. 2. Garabedian,S.P. LeBlanc,D.R. Gelhar,L.W. & Celia,M.A. Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts, 2. Analysis of spatial moments for a nonreactive tracer, Water Resources Research, 1991, 27 , 911- 924. 3. Gjetvaj, G. Modelling a solute transport by groundwater flow, Dissertation, Zagreb, 1986 (in Croatian) 4. Kinzelbach, W. Groundwater modelling, Elsevier, Amsterdam, 1986. 5. Kinzelbach,W. The Random Walk Method in Pollutant Transport Simulation, Advances in analytical and numerical groundwater flow and quality modeling, Lisabon, 1987 6. Knighton,R.E & Wagenet,G.J. Simulation of solute transport using a continuous time Markov proces, 1., Theory and steady state aplication, Water Resources Research ,1987,23 , 1911 - 1916. 7. Paue, _. Introduction into the mathematical statistics, kolska knjiga, Zagreb, 1993 (in Croatian). 8. Pricket , T.A., Naymik, T.G.& Lonnquist C.G. A 'random walk' solute transport model for selected groundwater quality evaluations. Illinois State Water Survey, Bulletin 65, 1981. 9. Rao,P.V, K.M.Portier & P.S.C.Rao, A Stochastic Approach for Describing Convective - Dispersive Solute Transport in Saturated Porous Media, Water Resources Research,1981, 17, 963 - 968. 10. Supek, I. Theoretical physics and structure of solids, First part, kolska knjiga, Zagreb, 1974 (in Croatian) 11. Tompson A.F.B.&Gelhar,L.W. Numerical simulation of solute transport in three -dimensional, randomly heterogeneous porous media, Water resources research, 1990,26,2541-2562. 12. Verruijt,A. 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