IAEA Regional Training Course on Sediment Core Dating
Techniques. RAF7/008 Project
J.M. Abril
Department of Applied Physics (I); University of Seville (Spain)
Lecture 6: Introduction to the compaction theory
and tracer conservation equation
•Compaction and diffusive processes in sediments
• Analytical solutions
1
J.M. Abril, University of Seville
Bulk density versus depth profiles in sediment cores
0.7
z
 (g/cm3)
0.6
0.5
0.4
0.3
     1 e
0.2
0.1
0
0
5
10
Depth [cm]
2
 z
J.M. Abril, University of Seville
15
20
3
J.M. Abril, University of Seville
Classical formulation of the advection-diffusion equations in sediments
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)



z

z
(v s ) 
(v ) 

z

D

z
(D
(  s)
z

z
)

(1)
t
   s

(  s)
t
(2)
s is the specific activity of a particle-associate radiotracer, (in Bq kg-1 or
similar units).
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.
4
J.M. Abril, University of Seville


z
(v ) 

z
(D

z
)
     1 e
v
1

r  D ( z ) 

t
 z
e
1
 z
(1)
(3)

(4)
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.
5
J.M. Abril, University of Seville
What is diffusion ?
6
J.M. Abril, University of Seville
The compaction energy potential and the continuity equation for solids
As result of the accumulation of
new material, the sedimentwater 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).
7
J.M. Abril, University of Seville
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.
8
J.M. Abril, University of Seville
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.
9
J.M. Abril, University of Seville
q   K ( z )

z
(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.
Thus, taking into account the mass flow associated to the
sedimentation rate v, one can introduce the continuity equation

t
10
J.M. Abril, University of Seville
 

z
( v   q)
(6)
Eq. 6 can be formally written as an advection-diffusion equation
expanding the  gradient by the chain rule

z

 
 z
and introducing a diffusivity function D(z) as
D(z)  K (z)


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.
11
J.M. Abril, University of Seville
w   (v  q )
(7)
General boundary conditions:
w (,t) =  v(t)
;
w(0,t) = o (v(t)+q0)
Under steady state compaction
 ( z, t)   ( z)
12
J.M. Abril, University of Seville
w ( z , t )  w (0, t )
Steady state compaction
13
J.M. Abril, University of Seville
These steady-state mass flows associated to
compaction can be generated, either
•under constant conductivity and depth dependent
spatial gradients of Ψ
•Ψ= A-Bρ
•or under constant spatial gradients of Ψ and
depth-dependent conductivities k(z)
•K(z) = A e-αz
14
This mass flows may involves:
•“Cuasi-homogeneous” reduction of pore spaces
• “Intra-advection” of small size particles
J.M. Abril, University of Seville
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.
ts
S
z
 w ( 0 , t ) dt
S
t s  T ( z ,t s )
  ( z ', t
s
) dz '  S m ( z )
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:
T ( z, ts ) 
1
z
 (z', t

w
0
15
J.M. Abril, University of Seville
s
) dz ' 
m ( z, ts )
w
Advection and diffusion processes for a particle-associated
tracer in sediments.
Let be s(z,t) the concentration
(the mass of tracer per unit dry
mass of solids) of a particleassociated tracer
==
SE D IM E N T
M ass flow
D iffusion
v
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.
16
J.M. Abril, University of Seville
1
Z
Solid
2
Pore w ater
3
4
 1 (  s Az)  vD ( z 
z
) t A  ( z 
2
z
) z
2
s
z
==
z
z
SE D IM E N T
M ass flow
D iffusion
2
v
vD
A
z
s(z,t)
 2 (  s Az)  vD (z 
( s)
t

z
) t A  ( z 
2
s 
 
v

z

 D

z 
z 
D  vD z
characteristic mixing (or
diffusion) length, LD
17
1
Z
J.M. Abril, University of Seville
z
2
) z
s
z
z
Solid
z
2
3
4
Pore w ater
2
( s)
t
   s 
s  
 
D

(w s)


z 
z  z
The third point could be regarded as a minor question. Nevertheless …
18
J.M. Abril, University of Seville
( s)
t
s 
 

   s 
(w s)
D 

z 
z  z
(21)
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,
 (t )   D 
s
z
lim s ( z , t )  0
z0
 w s (0, t )
z 
 (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).
19
J.M. Abril, University of Seville
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.
20
J.M. Abril, University of Seville
a) Spatial gradients in q and lateral reallocations
•1D approach is using cross-section averaged values for q
q’
q’
21
J.M. Abril, University of Seville
b) Two (or more) solid species with distinct concentrations and
relative compaction velocities
1
2
+…
With dimensions of a
diffusion term
22
J.M. Abril, University of Seville
23
J.M. Abril, University of Seville
“Virtual” particles as mathematical equivalents for exchanges of
radionuclides through the liquid phase
24
J.M. Abril, University of Seville
25
J.M. Abril, University of Seville
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 ]
26
J.M. Abril, University of Seville
Fundamental equations
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
27
J.M. Abril, University of Seville
28
J.M. Abril, University of Seville
Particular solutions: Constant rate of supply CRS model
Steady state inventories
29
J.M. Abril, University of Seville
Steady-state activity density versus mass thickness profiles
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
(I)
ma
(II)
30
J.M. Abril, University of Seville
with boundary conditions
31
J.M. Abril, University of Seville
The general solution is
32
J.M. Abril, University of Seville
33
J.M. Abril, University of Seville
Constant Flux with Constant Sedimentation Rate (CF-CSR) Model
34
J.M. Abril, University of Seville
35
J.M. Abril, University of Seville
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:
36
J.M. Abril, University of Seville
37
J.M. Abril, University of Seville
Laplace’s transformation for general equation and boundary condition:
38
J.M. Abril, University of Seville
Constant Sedimentation Rate without diffusion
Corresponds to the CF-CSR model. Solution in the Laplace’s space:
With the inverse Laplace’s transformation:
An example of application of this model can be found in Abril and García-Leon (1996)
39
J.M. Abril, University of Seville
Complete Mixing Zone (CMZ) Model (with CSR)
Corresponds to the CMZ model for 210Pb. Mixing mass depth ma
Solution in the Laplace’s space:
40
J.M. Abril, University of Seville
41
J.M. Abril, University of Seville
Incomplete Mixing Zone (IMZ) Model (with CSR)
42
J.M. Abril, University of Seville
Incomplete mixing zone model
60
3
IMZ Model
IMZ Model
50
2.5
40
137 -Cs ( mBq /g)
Unsupported 210-Pb (pCi/g)
2
1.5
1
30
20
10
0.5
0
0
0
5
10
15
Depth (cm)
20
25
30
0
g= 0.65 ± 0.04, w= 0.374 ± 0.01 ma= 6.0 ±0.3 g cm-2
43
J.M. Abril, University of Seville
5
10
Mass thickness (g/cm^2)
15
20
Constant Diffusion Model (IMZ) Model (with CSR)
44
J.M. Abril, University of Seville
Find here more details for numerical solutions
45
J.M. Abril, University of Seville
Bulk density profiles : The never seen history
K(z) ??
Ψ (z) ??
46
We need to learn
how to read the
history in these
profiles
J.M. Abril, University of Seville