1. J.M. Abril Department of Applied Physics (I); University of Seville (Spain) IAEA Regional Training Course on Sediment Core Dating Techniques. RAF7/008 Project J.M. Abril, University of Seville 1 Lecture 6: Introduction to the compaction theory and tracer conservation equation Compaction and diffusive processes in sediments Analytical solutions

2. Bulk density versus depth profiles in sediment cores 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 05101520  (g/cm 3 ) Depth [cm] z J.M. Abril, University of Seville 2

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4. v (L T -1 ) is the sedimentation rate, t (T) is the time, D (L 2 T -1 ) is the diffusion coefficient, and (T -1 ) is the radioactive decay constant. (1) (2) Classical formulation of the advection-diffusion equations in sediments s is the specific activity of a particle-associate radiotracer, (in Bq kg -1 or similar units). The classical differential equations for the conservation of solids, pore water and the particle-associated tracers, are given by Berner (1980). Based on Berner’s equations: Christensen and Bhunia (1986), Robbins (1986) J.M. Abril, University of Seville 4

5. (1) (4) (3) Eq.1 cannot account for typical bulk density profiles as those of Eq. 3 under constant sedimentation rate and constant (and positive) diffusion. Eq.4 is the steady-state solution of Eq. 1 with profiles given by Eq.3. The increase of bulk density with depth results in an upwards-directed diffusional flux of solid matter. As result, at the sediment-water interface, the sediment should be ejecting material to the water column. This seems to be a physically inconsistent situation. J.M. Abril, University of Seville 5

6. What is diffusion ? J.M. Abril, University of Seville 6

7. The compaction energy potential and the continuity equation for solids As result of the accumulation of new material, the sediment- water interface displaces up. From a framework anchored to this boundary, the sediment moves down as a whole with the sedimentation velocity v. Independently of this displacement, there are four different elemental processes involving mass exchanges between two adjacent layers ( a conceptual division). J.M. Abril, University of Seville 7

8. The exchange of solid particles between two adjacent sediment layers does not result in changes in  (if the exchanged particles have similar volumes), but they can change s (if they are carrying different specific activities). The only exchanges affecting  are those involving solid particles by pore water (and reciprocally). Nevertheless, these exchanges do not take place at the same rate in the two directions (up and down), since they may be subject to a forcing term (solids tend to move down unless other forces compensate the gravity). Consequently, they cannot be treated as diffusion. J.M. Abril, University of Seville 8

9. Thus we can, at least conceptually, introduce a specific potential energy for solid particles, , which decreases when water pores are occupied by solids. Let us call  the compaction potential. It is defined as energy per unit weight [L]. When sediments are perturbed (e.g., by mechanical waves of by the action of organisms), the system tries to restore the equilibrium (decreasing its energy) and the large water pores tend to be again occupied by solids. Conceptually, the spatial gradients of  can only be upwards directed and they represent a forcing term resulting in a downwards-directed flux of matter. J.M. Abril, University of Seville 9

10. (5) Eq. 5 is similar to the linear transport equations of classical physics, being q the velocity (L T -1 ) associated to the flow of solids. Thus, K(z) can be interpreted as a conductivity function. It has dimensions of M L -2 T -1. (6) Thus, taking into account the mass flow associated to the sedimentation rate v, one can introduce the continuity equation J.M. Abril, University of Seville 10

11. Eq. 6 can be formally written as an advection-diffusion equation expanding the  gradient by the chain rule and introducing a diffusivity function D(z) as This way, Eq. 6 can be rewritten formally as Eq. 1. A similar treatment for the diffusivity function can be found in Hillel (1971: 110-111) for water movement in soils. Nevertheless, we have to note that this is only a formal writing, and we must remember that the process of movement of solids in sediments is not one of diffusion but of mass flow. Thus, diffusivity takes negative values as seen further. J.M. Abril, University of Seville 11

12. (7) Under steady state compaction General boundary conditions: w ( ,t) =   v(t) ; w(0,t) =  o (v(t)+q 0 ) J.M. Abril, University of Seville 12

13. Steady state compaction J.M. Abril, University of Seville 13

14. These steady-state mass flows associated to compaction can be generated, either under constant conductivity and depth dependent spatial gradients of Ψunder constant conductivity and depth dependent spatial gradients of Ψ Ψ= A-BρΨ= A-Bρ or under constant spatial gradients of Ψ and depth-dependent conductivities k(z)or under constant spatial gradients of Ψ and depth-dependent conductivities k(z) K(z) = A e -αzK(z) = A e -αz These steady-state mass flows associated to compaction can be generated, either under constant conductivity and depth dependent spatial gradients of Ψunder constant conductivity and depth dependent spatial gradients of Ψ Ψ= A-BρΨ= A-Bρ or under constant spatial gradients of Ψ and depth-dependent conductivities k(z)or under constant spatial gradients of Ψ and depth-dependent conductivities k(z) K(z) = A e -αzK(z) = A e -αz This mass flows may involves: “Cuasi-homogeneous” reduction of pore spaces “Intra-advection” of small size particles This mass flows may involves: “Cuasi-homogeneous” reduction of pore spaces “Intra-advection” of small size particles J.M. Abril, University of Seville 14

15. Chronology If an age T = 0 is assigned to the sediment-water interface at a given time t s (the time of sampling), then the total mass accumulated below a surface of area S till a given depth z must be equal to the time integral of w(0,t) from t= t s -T(z,t s ) till time t=t s, where T(z,ts) is the age of formation of the layer at a depth z if intra-advection was neglected. where m(z, t s ) is the cumulative mass thickness or the mass depth. Differential instead of integral relationships also applies. For the particular case of w being constant: J.M. Abril, University of Seville 15

16. Let be s(z,t) the concentration (the mass of tracer per unit dry mass of solids) of a particle- associated tracer Processes 3 and 4 account for advective transport. For particle-associated tracers, the process 2 does not contribute to changes in  nor in s. Process 1 will result in changes in s if they are carrying different specific activities. These exchanges may be produced by bioturbation or other physical processes. Advection and diffusion processes for a particle-associated tracer in sediments. J.M. Abril, University of Seville 16

17. characteristic mixing (or diffusion) length, L D vDvD A zz s(z,t)s(z,t) J.M. Abril, University of Seville 17

18. The third point could be regarded as a minor question. Nevertheless … J.M. Abril, University of Seville 18

19. Under steady-state for bulk density, one has to provide appropriate initial conditions ( s(z,0) ), suitable parameter values ( D(z,t) and w(0,t) ) and boundary conditions, (21) is the flux of radionuclides entering the sediment at time t through the sediment-water interface. If is not steady state, then Eq. 21 has to be solved simultaneously with Eq. 6 (and its related initial and boundary conditions). J.M. Abril, University of Seville 19

20. Some aspects of physical diffusion in growing sediments When the governing equations involves spatial and temporal averaged values of dynamic variables, diffusion arises related with sub-grid scale advection. J.M. Abril, University of Seville 20

21. a) Spatial gradients in q and lateral reallocations 1D approach is using cross-section averaged values for q q’ J.M. Abril, University of Seville 21

22. b) Two (or more) solid species with distinct concentrations and relative compaction velocities + … With dimensions of a diffusion term J.M. Abril, University of Seville 22 1 2

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24. “Virtual” particles as mathematical equivalents for exchanges of radionuclides through the liquid phase J.M. Abril, University of Seville 24

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26. New notation: Bulk density ρ m ( ρ) Sedimentation rate or sedimentation velocity r ( v) (Mass) sedimentation rate w Mass depth m Concentration of a particle-associated tracer A(z,t) [s(z,t)] Diffusion coefficient k b [ D ] J.M. Abril, University of Seville 26

27. Situations where the tracer is partially carried by pore water or in presence of selective and/or translocational bioturation Eq. has to be reviewed. BOUNDARY CONDITIONS Fundamental equations J.M. Abril, University of Seville 27

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29. Particular solutions: Constant rate of supply CRS model Steady state inventories J.M. Abril, University of Seville 29

30. Steady-state activity density versus mass thickness profiles mama (I) (II) Let us consider the following particular case: Steady state for bulk density and activity concentration profile Constant sedimentation rate Two regions in the sediment, the first one (of mass thickness m a ) with a constant diffusion coefficient J.M. Abril, University of Seville 30

31. with boundary conditions J.M. Abril, University of Seville 31

32. The general solution is J.M. Abril, University of Seville 32

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34. Constant Flux with Constant Sedimentation Rate (CF-CSR) Model J.M. Abril, University of Seville 34

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36. Time dependent fluxes. General method For artificial fallout radionuclides, fluxes are time-dependent and concentrations unsteady. Initial conditions: For steady-state bulk densities, an elegant way of solution is to use the Laplace’s transformations: J.M. Abril, University of Seville 36

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38. Laplace’s transformation for general equation and boundary condition: J.M. Abril, University of Seville 38

39. An example of application of this model can be found in Abril and García-Leon (1996) Constant Sedimentation Rate without diffusion Corresponds to the CF-CSR model. Solution in the Laplace’s space: With the inverse Laplace’s transformation: J.M. Abril, University of Seville 39

40. Complete Mixing Zone (CMZ) Model (with CSR) Corresponds to the CMZ model for 210 Pb. Mixing mass depth m a Solution in the Laplace’s space: J.M. Abril, University of Seville 40

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42. Incomplete Mixing Zone (IMZ) Model (with CSR) J.M. Abril, University of Seville 42

43. 0 0.5 1 1.5 2 2.5 3 051015202530 Unsupported 210-Pb (pCi/g) Depth (cm) IMZ Model 0 0.5 1 1.5 2 2.5 3 051015202530 Unsupported 210-Pb (pCi/g) Depth (cm) IMZ Model 0 10 20 30 40 50 60 05101520 137 - Cs ( mBq /g) Mass thickness (g/cm^2) IMZ Model 0 10 20 30 40 50 60 05101520 137 - Cs ( mBq /g) Mass thickness (g/cm^2) IMZ Model 0 10 20 30 40 50 60 05101520 137 - Cs ( mBq /g) Mass thickness (g/cm^2) IMZ Model Incomplete mixing zone model g= 0.65 ± 0.04, w= 0.374 ± 0.01 m a = 6.0 ±0.3 g cm -2 J.M. Abril, University of Seville 43

44. Constant Diffusion Model (IMZ) Model (with CSR) J.M. Abril, University of Seville 44

45. Find here more details for numerical solutions J.M. Abril, University of Seville 45

46. Bulk density profiles : The never seen history K(z) ?? Ψ (z) ?? K(z) ?? Ψ (z) ?? We need to learn how to read the history in these profiles J.M. Abril, University of Seville 46