J.M. Abril Department of Applied Physics (I); University of Seville (Spain)

J.M. AbrilDepartment 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 Seville1

Lecture 6: Introduction to the compaction theory and tracer conservation equation

•Compaction and diffusive processes in sediments

• Analytical solutions

Bulk density versus depth profiles in sediment cores

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20

(g

/cm

3)

Depth [cm]

ze 1

ze 1z



tzD

zv

z

)()( tz

Dz

vz

)()(

t

ss

z

sD

zsv

z

)()(

)( t

ss

z

sD

zsv

z

)()(

)(

v (L T-1) is the sedimentation rate, t (T) is the time, D (L2 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)


zezDrv

1)(1

tzD

zv

z

)()(

ze 1

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.


What is diffusion ?What is diffusion ?


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).


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.


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.


zzKq

)(z

zKq

)( (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))( qv

zt

)( qv

zt

Thus, taking into account the mass flow associated to the sedimentation rate v, one can introduce the continuity equation


Eq. 6 can be formally written as an advection-diffusion equation expanding the gradient by the chain rule

zz

and introducing a diffusivity function D(z) as

)()( zKzD

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.


)( qvw )( qvw (7)

Under steady state compaction

)(),( ztz ),0(),( twtzw

General boundary conditions: w (,t) = v(t) ; w(0,t) = o (v(t)+q0)


Steady state compaction


These steady-state mass flows associated to These steady-state mass flows associated to compaction can be generated, either compaction can be generated, either

•under constant conductivity and depth dependent under constant conductivity and depth dependent spatial gradients of Ψ spatial gradients of Ψ

•Ψ= A-BΨ= A-Bρρ

•or under constant spatial gradients of Ψ and or under constant spatial gradients of Ψ and depth-dependent conductivities k(z)depth-dependent conductivities k(z)

•K(z) = A eK(z) = A e--ααzz

These steady-state mass flows associated to These steady-state mass flows associated to compaction can be generated, either compaction can be generated, either

•under constant conductivity and depth dependent under constant conductivity and depth dependent spatial gradients of Ψ spatial gradients of Ψ

•Ψ= A-BΨ= A-Bρρ

•or under constant spatial gradients of Ψ and or under constant spatial gradients of Ψ and depth-dependent conductivities k(z)depth-dependent conductivities k(z)

•K(z) = A eK(z) = A e--ααzz

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


Chronology

If an age T = 0 is assigned to the sediment-water interface at a given time ts (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= ts-T(z,ts) till time t=ts , where T(z,ts) is the age of formation of the layer at a depth z if intra-advection was neglected.

z

s

t

tzTt

zmSdztzSdttwSs

ss 0),(

)('),'(),0(

z

s

t

tzTt

zmSdztzSdttwSs

ss 0),(

)('),'(),0(

where m(z, ts) is the cumulative mass thickness or the mass depth.Differential instead of integral relationships also applies.

For the particular case of w being constant:

z

sss w

tzmdztz

wtzT

0

),('),'(

1),(

zs

ss w

tzmdztz

wtzT

0

),('),'(

1),(


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.

Z

Solid Pore water

Diffusion Mass flow

1 2 3 4

== SEDIMENT

v

Advection and diffusion processes for a particle-associated tracer in sediments.


Z

Solid Pore water

Diffusion Mass flow

1 2 3 4

== SEDIMENT

v

z

szv

zt

sD )(

zvD Dcharacteristic mixing (or diffusion) length, LD

)()(

swzz

sD

zs

t

s

)()(

swzz

sD

zs

t

s

vD

21 )

2()

2()( z

zD z

sz

zzAt

zzvzAs

22 )

2()

2()( z

zD z

sz

zzAt

zzvzAs

Az s(z,t)


The third point could be regarded as a minor question. Nevertheless …


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,

),0()(0

tswz

sDt

z

0),(lim

tzsz

)()(

swzz

sD

zs

t

s

)()(

swzz

sD

zs

t

s

(21)

)(t 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).


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.


a) Spatial gradients in q and lateral reallocations

•1D approach is using cross-section averaged values for q

q’

q’


b) Two (or more) solid species with distinct concentrations and relative compaction velocities

+ …

With dimensions of a diffusion term


1 2


“Virtual” particles as mathematical equivalents for exchanges of radionuclides through the liquid phase



New notation:

•Bulk density ρm ( 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 kb [ D ]


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 CONDITIONSBOUNDARY CONDITIONS

Fundamental equationsFundamental equations



Particular solutions: Constant rate of supply CRS model

Steady state inventories


Steady-state activity density versus mass thickness profiles

ma(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 ma) with a constant diffusion coefficient


with boundary conditions


The general solution is



Constant Flux with Constant Sedimentation Rate (CF-CSR) Model



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:



Laplace’s transformation for general equation and boundary condition:


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:


Complete Mixing Zone (CMZ) Model (with CSR)

Corresponds to the CMZ model for 210Pb. Mixing mass depth ma

Solution in the Laplace’s space:



Incomplete Mixing Zone (IMZ) Model (with CSR)


0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25 30

Unsu

pport

ed 2

10

-Pb (

pC

i/g)

Depth (cm)

IMZ Model

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25 30

Unsu

pport

ed 2

10

-Pb (

pC

i/g)

Depth (cm)

IMZ Model

0

10

20

30

40

50

60

0 5 10 15 201

37-C

s (m

Bq

/g)

Mass thickness (g/cm^2)

IMZ Model

0

10

20

30

40

50

60

0 5 10 15 201

37-C

s (m

Bq

/g)


IMZ Model

0

10

20

30

40

50

60

0 5 10 15 201

37-C

s (m

Bq

/g)


IMZ Model

Incomplete mixing zone model

g= 0.65 ± 0.04, w= 0.374 ± 0.01 ma= 6.0 ±0.3 g cm-2 g= 0.65 ± 0.04, w= 0.374 ± 0.01 ma= 6.0 ±0.3 g cm-2


Constant Diffusion Model (IMZ) Model (with CSR)


Find here more details for numerical solutions


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

We need to learn how to read the history in these profiles


Documents

J.M. Abril Department of Applied Physics (I); University of Seville (Spain)