1. 1 Transport of suspensions in porous media Alexander A. Shapiro* Pavel G. Bedrikovetsky** * IVC-SEP, KT, Technical Univ. of Denmark (DTU) ** Australian School of Petroleum, Univ. of Adelaide

2. 2 Applications - petroleum Injectivity decline (e.g. under sea water injection) Formation damage by drilling mud (creation of the filter cakes) Migration of reservoir fines in unconsolidated rocks Thermal reservoirs?

3. Applications within EOR Erosion of the rock –e.g. under injection of carbon dioxide Filtration of large molecules –e.g. under polymer flooding Propagation of bacteria in porous media –e.g. under microbial recovery (to some extent) Behavior of drops/emulsions in porous media Similar models describe flow of tracers in porous media 3

4. 4 Micro-Physics Straining Gravity Bridging Electric forces Van Der Waals forces Sorption Gravity - + + + + + + - - - - - Transfer of particles in the flow Complex interaction with the flow Multiple mechanisms of capturing Variation of particle sizes

5. 5 Micro-Physics (2) -Motion of particles in porous medium is to some extent similar to ”motion in a labirynth”; -It is characterized by the different residence times and steps in the different capillaries/pores -Dispersion of the times and steps requires stochastic modeling

6. 6 Traditional model Iwasaki, 1937; Herzig, 1970, Payatakes, Tien, 1970-1990; OMelia, Tufenkij, Elimelech, 1992-2004 porosity suspended concentration filtration coefficient flow velocity dispersion coefficient -Advection-dispersion particle transfer -”First-order chemical reaction” type particle capture mechanism -Empirical equations for porosity/permeability variation -No particle size or pore size distributions -No residence time dispersion

7. Experimental observations (After Berkowitz and Sher, 2001) The observed profiles do not correspond to predictions of the traditional model

8. Breakthrough times The traditional model predicts breakthrough after 1 porous volume injected (p.v.i.)

9. 9 Problems with the traditional model Contrary to predictions of the model: -Particles may move (usually) slower and (sometimes) faster than the flow; -There may be massive ”tails” of the particles ahead of the flow; -The distributions of retained particles are ”hyperexponential” 00.20.40.60.81 1 2 3 4 XXlab  Tufenkji and Elimelech, 2005 Bradford et al., 2002

10. 10 Goals Creation of a complete stochastic model of deep bed filtration, accounting for: –Particle size distribution –Dispersion of residence steps and times Averaging of the model, reduction to ”mechanistic” equations Clarification of the roles of the different stochastic factors in the unusual experimental behavior

11. 11 Previous work (all > 2000) ApproachAuthors Particle size distrib. Step dispers. Empirical distributed model for the capture coefficient Tufenkji and Elimelech √ Boltzmann-like population balance models Bedrikovetsky, Shapiro, Santos, Medvedev √ Continuous-time random walk (CTRW) models Cortis, Berkowitz, Scher et al. √ Elliptic transport CTRW-based theory Shapiro, Bedrikovetsky √

12. Essence of the approach Random walk in a lattice Random walk in one dimension -The particles jump between the different points of the ”network” (ordered or disordered) -They spend a random time at each point -The step may also be random (at least, its direction) Einstein, Wiener, Polia, Kolmogorov, Feller, Montrol…

13. Direct numerical experiment: 13 Random walks with distributed time of jump τ 1D walk 2-point time distribution Expectation equal to 1 Original Smoothed

14. Direct numerical experiment: 14 Random walks with distributed time of jump τ Classical behavior with low temporal dispersion Anomalous behavior with high temporal dispersion: More particles run far away, but also more stay close to the origin

15. ”Einstein-like” derivation Probability to do not be captured Joint distribution of jumps and probabilities Expansion results in the elliptic equation: New terms compared to the standard model A stricter derivation of the equation may be obtained on the basis of the theory of stochastic Markovian semigroups (Feller, 1974)

16. 16 Monodisperse dilute suspensions Maximum moves slower than the flow The ”tail” is much larger - Qualitative agreement with the experimental observations Pulse injection problem

17. 17 Generalization onto multiple particle sizes Monodisperse: For the particles of the different sizes For concentrated suspensions coefficients depend on the pore size distributions. The different particles ”compete” for the space in the different pores. Let be numbers of pores where particles of the size will be deposited. Then it may be shown that

18. 18 General theory Monodisperse: Polydisperse: The transport equation becomes: Distribution of the particles by sizes All the coefficients are functions of They depend also on the varying pore size distribution Length of one step and radius of a pore

19. 19 Initial and boundary conditions Consider injection of a finite portion of suspension pushed by pure liquid. Presence of the second time derivative in the equation requires ”final condition”. After injection of a large amount of pure liquid, all the free particles are washed out and their concentration becomes (efficiently) zero. This gives the final condition at liquid suspension liquid

20. 20 Numerical solution (SciLab) Solve the system of the transport equations under known coefficients: Solve equations for pore size evolution under known concentrations: Determine the coefficients: Normally, convergence is achieved after 3-4 (in complex cases, 5-6) iterations.

21. 21 Results of calculations x t x t Concentration Porosity (The values are related to the initial value)

22. 22 Retention profiles Temporal dispersion coefficient Low temporal dispersion High temporal dispersion 1=2 total 1=2

23. 23 Different capturing mechanisms vs temporal dispersion 1=2 total 12 High temporal dispersion Different capturing mechanisms

24. 24 Conclusions A stochastic theory of deep bed filtration of suspensions has been developed, accounting for: –Particle and pore size distributions –Temporal dispersion of the particle steps Temporal dispersion leads to an elleptic transport equation Pore size distribution results in a system of coupled elliptic equations for the particles of the different sizes Coupling appears in the coefficients: particles ”compete” for the different pores The temporal dispersion seems to play a dominating role in formation of the non-exponential retention profiles Difference in the capture mechanisms may also result in the hyperexponential retential profiles, but the effect is weaker No ”hypoexponential” retention profiles has been observed

25. Future work A new Ph.D. student starts from October (Supported for Danish Council for Technology and Production) Collaboration with P. Bedrikovetsky (Univ. of Adelaide) More experimental verification (not only qualitative) More numerics –Scheme adjustement and refinement –Softwareing Incomplete capturing Errosion 25