An alternative approach to simulate transport based on the Master equation

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1 Tracers and Modelling in Hydrogeology (Proceedings of (lie TraM'2000 Conference held at Liège, Belgium, May 2000). IAHS Publ. no. 262, An alternative approach to simulate transport based on the Master equation MARTIN SPILLER Institute of Hydraulic Engineering and Water Resources Management, Technology, Mies-van-der-Rohe-Str. 1, D Aachen, Germany Aachen University of RACHID ABABOU Institut de Mécanique des Fluides de Toulouse, Allée du Professeur Camille F Toulouse, France JÙRGEN KÔNGETER Institute of Hydraulic Engineering and Water Resources Management, Technology, Mies-van-der-Rohe-Str. 1, D Aachen, Germany Soula, Aachen University of Abstract A stochastic particle-in-cell algorithm for advective-diffusivereactive (A-D-R) transport processes based on a discrete-state Master equation is presented. The Master equation is equivalent to the macroscopic concentration-based partial differential equation. This approach is now being implemented and compared with a previous 3-D Lagrangian particle model of A-D-R processes using the example of Fisher's equation, a second order reaction-diffusion equation. INTRODUCTION The transport of fluids and chemical species in geological porous media is commonly described by advection-diffusion-reaction (A-D-R) equations. In the following, a stochastic method for simulating A-D-R equations based on a microscopic description of the transport processes via a corresponding Master equation is outlined. Physical transport processes are directly identified with particles, which move or change their state by advection, diffusion, and various reaction processes such as sorption. In this paper, the time-evolution of A-D-R systems is simulated by a stochastic sequence of single Markov steps using the minimal process method, which we also call the Gillespie algorithm (Gillespie, 1976). This algorithm is a general approach to the simulation of reaction and/or transport processes which can be described by a Master equation. In a nutshell, it consists in solving the Master equation stochastically by implementing a random walk process in the space of all configurations of the system. This approach has the potential for being efficient enough to handle millions of particles. A large class of problems can be addressed, including nonlinear kinetics, and sorption/desorption with first-order kinetics (Spiller et al, 1998). LAGRANGIAN PARTICLE TRACKING SIMULATION OF A-D-R PROCESSES The Lagrangian method is gridless but requires the processing of many particles, each being governed by a discrete time random walk approximating the Wiener process

2 122 Martin Spiller et al. with an additional drift for advection. The method is limited by the storage capacity of the computer, as it requires storing the positions of all particles. Thus, for a threedimensional (3-D) transport problem with four species (4x4 bytes plus 12 bytes for a 3-D position) the maximum number is about 2 x 10 6 particles for a 64 MB computer memory. Furthermore, the Lagrangian approach is difficult to implement when species are exchanged among different phases, i.e. sorption/desorption. A common approach to the simulation of reactive transport is to apply operatorsplitting techniques where the A-D-R equation is split up into an A-D transport operator and a reaction operator. The advantage of this method is that the best-suited method for each of the operators can be applied. This idea can be used in a Lagrangian particle tracking framework (e.g. Spiller et al, 1998). However, an inevitable time lag error is made since either transport processes or reaction processes are simulated first, and a fine time step is required in relation to the time scale of the fastest reaction (Kaluarachchi & Morshed, 1995). A stochastic Lagrangian model was previously developed for simulating A-D-R transport, where advection-diffusion processes were simulated by a Lagrangian particle tracking algorithm (LPT-3D code of Fadili et al., 1999) and reaction kinetics were simulated by the Markovian "minimal process" approach or Gillespie algorithm which is described below. The concentration distribution is represented by the number of particles (and their mass) within one discretization interval. MASTER EQUATION FORMULATION OF A-D-R PROCESSES In general, two fundamental aspects of the Master equation approach to nonequilibrium reaction-transport systems have to be considered. Firstly, the Master equation must be consistent with the macroscopic description which is usually given in terms of balance equations for the conserved quantities. Consistency is achieved if it can be shown that in the macroscopic limit the expectation values of the corresponding stochastic process are governed by these balance equations. This equivalence holds for an advection-diffusion system with linear sorption reactions, if the mesh Peclet number AxV/D is less then two (Spiller et al., 1999). Furthermore, the fluctuations of the simulated variables have to be consistent with theoretical predictions. Secondly, it has to be taken into account that the Master equation is not solved numerically as a differential equation. Instead, a sample of realizations of the stochastic process is obtained, where each realization corresponds to a random walk in the space of the number of particles in cells. In the last decade the Master equation approach has been used by several research groups. The crucial step with respect to the performance of the algorithm is the choice of the next reaction and the bookkeeping connected with this step. To overcome this problem, two different approaches can be distinguished (Breuer et al., 1996): (a) One single transition changes a large number of transition rates. This is typical for complex chemical reactions with a large number of species and many interactions; e.g. aggregation or polymerization which implies expensive bookkeeping. In this case, the algorithm developed by Breuer et al. (1996) is efficient. (b) At each transition, only a small number of reaction rates are changed. This is the case in reaction-diffusion systems with many cells and a small number of different

3 An alternative approach to simulate transport based on the Master equation 123 interacting species. Only the reaction rates in one or two adjacent cells will change. Therefore, the update is computationally inexpensive and the time needed for bookkeeping is negligible, compared with the time-consuming choice of the next reaction. Using the logarithmic class routine by Fricke & Wendt (1995), about Markov steps per second (on a SUN Ultra 60) can be performed. The effort for the choice of the next reaction is inextensive with an average rejection rate of only about 75%. Since the time and space evolution of the system is of interest, it is necessary to divide the multi-dimensional total volume Cl into m = Q/co cells of size co. To ensure a homogeneous concentration in each cell it is required that the diffusion time scale be "faster" than the reaction time scale. Each cell is labelled by an index a and contains a discrete number of particles. This is similar to particle in cell methods used in physics and fluid mechanics. The current state of the system is represented by the number of different species N, where N = (N t,...,n m ). The next step depends only on the current state of the system and is independent of the course of the reaction and transport processes in the past; therefore the process is Markovian. It has to be noted that the system and the corresponding Markov process are discrete in state and continuous in time. The simulated system is the continuum limit of a discrete-state Markov process governed by a Master equation of the form: (1) where P(N,t) is the probability of finding the system in the state N at time t, and -^Af<_w' is the probability rate of a transition from state N' to N. The minimal process time algorithm, which provides a mathematically exact simulation of one realization of the Master equation, is described below in pseudo code form (Gillespie, 1976). It has to be emphasized that the result obtained is purely based on the transition probabilities, without solving the Master equation numerically. (a) t = 0: Initialization: initial configuration and computation of transition rates such as RD, RA, PC corresponding to Rfi^jj. for diffusion, annihilation and creation. (b) t = t + x: Increment time by a random "time step" (or waiting time) x with an exponential distribution p(x) = R-exp(-Rx) where R = XIÂW'^W/V' s t a n d s f r t n e total transition rate or reaction rate, for example R = zzr D, for pure diffusion. (c) For each waiting time event, choose randomly a cell (a) and a "reaction" (e.g. diffusion of a mobile particle in cell a), according to its relative probability of occurrence R^^^, / R (algorithm of Fricke & Wendt, 1995). (d) Perform the transition for the selected reaction (e.g. for diffusion, one mobile particle is transferred to one of the nearest neighbour cells). (e) Update the configuration N and the transition rates R^p, (RD, RA, Re)- (f) End the simulation, if time "t" exceeds a maximum time, or else continue with step (b).

4 124 Martin Spiller et al. TEST CASE: SIMULATION OF FISHER'S EQUATION As a test case we consider Fisher's nonlinear reaction-diffusion equation: c = -^- T c + e-(l-c) (2) dt dx' and obtain results which are in a good agreement with known analytical solutions (Spiller et al., 1998). Fisher's equation describes the diffusion of a single species with a nonlinear second order homogeneous reaction. Asymptotically, a quasi-stationary evolution of concentration profiles with constant velocity is achieved. We compare Lagrangian and Master equation methods. Due to the reaction, the gridless Lagrangian approach was modified as follows: a fixed grid (Ax) was introduced in order to compute the reaction rate R(C) as a function of concentration. Apart from this, all processes were simulated in a Lagrangian framework. The results shown in Figs 1 and 2 were obtained for a domain length L = 200 with Ax = 0.5 and about 5000 particles per interval Ax. The observable wave velocity discrepancy is due to a diffusive error caused by the explicit nature of the discrete time random walk method. Nevertheless, the result shows that the algorithm is capable of simulating reactive transport processes with second order reactions fairly accurately D 0) a. "> tt> > < t h analytic At =0.05 At =0.1 A t = time Fig. 1 Velocity of the concentration wave for different time discretizations. 1.1 t=o t=10 t=20 t=30 t=30 analytic Fig. 2 Spread of the concentration wave at four different times. X 100

5 An alternative approach to simulate transport based on the Master equation 125 When Fisher's equation is modelled with the Master equation approach, the following Markov processes have to be taken into account (Fig. 3) : (a) Diffusion of solute, simulated as a random walk with a hopping rate D r to neighbour cells. (b) Annihilation of a mobile particle with the annihilation rate Ica- (c) Creation of a mobile particle with the creation rate k c. The immobile particles (S) serve as a reservoir, but it is of course possible to incorporate the constant concentration of S in the reaction rate constants. The results obtained are shown in Fig. 4. The observed deviations are due to two sources of error: (a) Systematic error due to the finite "discretized" mass of reacting and transported particles. (b) Statistical error due to the finiteness of the sample generated in the simulation. Note that the time lag error inherent in any operator splitting technique is avoided. The comparison of the results for 5000 particles per unit mass obtained with the two different algorithms shows a good agreement. It can be concluded that the number of particles per unit mass has in both cases the most striking effect on the quality of the results. The Master equation method needs no external time discretization and avoids Diffusion Annihilation Creation C + C ^->C + {} s + C t ^ C + C o x>^ o a O Oj OOOOOO 0 OOQOO 0 OOOOOO oooeooo oooeooo oeooooo Q,# Q,#lOi#IO# 0,# Q»10,# Q,#1 0,»IO# Q,(!0,( ] a-2 a-] a a+1 a-2 a-î a a+1 a-2 a-1 a a+1 0 Rate for diffusion: R,, =2-D r-c a Rateforannihilation:R A= k A-C 0-(C a-l) Rate for creation: R c=k cs uc t l : Mobile particles C C : Number of mobile particles in cell a H : Immobile particles S S a : Number of immobile particles in cell a Fig. 3 Possible transitions of state in an advection-diffusion-sorption/desorption system p * f- 0 analytic N=50000 N=10000 N=1000 Fig. 4 Velocity of the concentration wave for different mass resolutions obtained with the Gillespie algorithm.

6 126 Martin Spiller et al. the difficulty of operator splitting. Although the improvement due to the simultaneous simulation of reactions and transport is neutralized by the influence of the mass discretization, the Master equation approach is suited for more complex chemically reacting flow, where the choice of the time step in "classical" algorithms is crucial. DISCUSSION AND CONCLUSION The algorithm outlined above provides a stochastic simulation of the Master equation that takes time correlations and fluctuations into account. No external time step At is needed, as the time step is automatically adapted to the characteristic time scales of the A-D-R system. The algorithm can be extended to multi-dimensional systems with spatially variable A-D-R coefficients and non-linear reactions. As demonstrated, no operator splitting is necessary. The proposed model is easy to implement and to apply to complex nonlinear or active transport problems. There is no limitation of the number of particles due to the storage capacity available. However, it should be noted that, due to the consistency criterion Pe < 2, the algorithm requires a fine grid for advection dominated A-D-R transport processes. In this case, the use of the Lagrangian particle tracking method may be recommended. Finally, if the reactions have a much shorter time scale than transport processes, the algorithm becomes ineffective. In such cases, it is appropriate to use local equilibrium rather than kinetics. The algorithm is currently being extended to 3-D heterogeneous porous media, and it can in fact be applied to transport in spaces with fractal dimensions, e.g. fractured media. REFERENCES Breuer, H.-P., Huber, W. & Petruccione, F. (1996) A fast Monte Carlo algorithm for non-equilibrium systems. Phys. Rev. Ser. E S3(4/B), 4232^1235. Fadili, A., Ababou, R. & Lenormand, R. (1999) Dispersive particle transport: identification of macroscale behavior in heterogeneous stratified subsurface flows. Math. Geol. 31, Fricke, T. & Wendt, D. (1995) The Markov automaton a new algorithm of simulating the time evolution of large stochastic dynamic systems. Int. J. Mod. Phys. C 6(2), Gillespie, D. T. (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, Kaluarachchi, J. i. & Morshed, J. (1995) Critical assessment of Ihe operator-splitting technique in solving the advectiondispersion-reaction equation: 1. First-order reaction. Adv. Wat. Resour. 18(2), Spiller, M., Ababou, R. & Kongeter, J. (1999) Particle modelling of active transport processes based on stochastic simulations of Master equations. In: 14e Congrès français de Mécanique (Toulouse, août-septembre 1999), 1-6. ENSAE (CD-Rom), Toulouse, France. Spiller, M., Jansen, D., Bodarwé, H. & Kôngeter, J. (1998) A stochastic algorithm for simulating transport of kinetically reacting solutes. In: Hydrolnformatics '98 (ed. by V. Babovic) (Proc. Third Int. Conf. Hyroinformatics, Copenhagen, August 1998), Balkema, Rotterdam.