Transport of suspensions in porous media

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

2. Applications - petroleum Injectivity decline (e.g. under sea water injection) Migration of reservoir fines in unconsolidatedrocks Formation damage by drilling mud (creation of the filter cakes) 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 • Similarmodels describeflow of tracers in porous media

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

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. Traditional model dispersion coefficient porosity suspended concentration flow velocity filtration coefficient Iwasaki, 1937; Herzig, 1970, Payatakes, Tien, 1970-1990; OMelia, Tufenkij, Elimelech, 1992-2004 • 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. Experimentalobservations (After Berkowitz and Sher, 2001) The observedprofiles do not correspond to predictions of the traditional model

8. Breakthroughtimes The traditional model predictsbreakthroughafter 1 porousvolumeinjected (p.v.i.)

9. Problems with the traditional model Tufenkji and Elimelech, 2005 Bradford et al., 2002 4 3 2 1 0 0.2 0.4 0.6 0.8 1 X , Xlab • 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”

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. Previous work (all > 2000)

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. Directnumericalexperiment: Randomwalkswithdistributed time of jump τ • 1D walk • 2-point time distribution • Expectation equal to 1 Original Smoothed

14. Directnumericalexperiment: Randomwalkswithdistributed time of jump τ Classicalbehaviorwithlow temporal dispersion Anomalousbehaviorwithhigh temporal dispersion: More particles run far away, but also more stayclose to the origin

15. ”Einstein-like” derivation Joint distribution of jumps and probabilities Probability to do not be captured Expansion results in the elliptic equation: New terms comparedto the standard model A stricter derivation of the equationmaybeobtainedon the basis of the theory of stochasticMarkoviansemigroups (Feller, 1974)

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

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

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

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

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

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

24. Conclusions • A stochastictheory of deep bed filtration of suspensions has beendeveloped, 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 coupledellipticequations for the particles of the differentsizes • Couplingappears in the coefficients: particles ”compete” for the different pores • The temporal dispersion seems to play a dominatingrole in formation of the non-exponential retention profiles • Difference in the capturemechanismsmayalsoresult in the hyperexponentialretentialprofiles, but the effect is weaker • No ”hypoexponential” retention profiles has beenobserved

25. Future work • A new Ph.D. student starts from October (Supported for Danish Council for Technology and Production) • Collaborationwith P. Bedrikovetsky (Univ. of Adelaide) • More experimentalverification (not onlyqualitative) • More numerics • Schemeadjustement and refinement • Softwareing • Incompletecapturing • Errosion