Download Mathematical modelling of gravitational compaction and clay

Geophys. J . Inf. (1995) 122,283-298
Mathematical modelling of gravitational compaction and clay
dehydration in thick sediment layers
D. Marc Audet
University of Oxford, Deparfment of Earth Sciences, Parks Road, Oxford OX1 3PR, U K
Accepted 1995 February 8. Received 1995 January 24; in original form 1994 October 5
SUMMARY
Gravitational compaction is an important process in sedimentary basins which
controls the reduction of porosity with burial depth and the development of high
pore fluid pressures. Often, sediments contain hydrated, expandable clay (smectite)
that can undergo a transition to a dehydrated, non-expandable clay (illite). During
dehydration, structural water bound within the sheet layers of smectite is released
into the pore space, which can increase the pore pressure and influence geological
processes such as solute transport, hydrocarbon migration and hydrothermal
fracturing.
A multicomponent continuum mechanics model that accounts for Darcy’s Law,
Terzaghi’s principle of effective stress and a thermally activated dehydration
reaction is derived and solved numerically. A closed-form solution is available in the
limiting case of hydrostatic pore pressure and no dehydration. The results show that
excess pore-pressure development is controlled by the sedimentation parameter, the
dimensionless ratio of the hydraulic conductivity to the sedimentation rate. For
relatively impermeable sediments, chemically released water can increase the excess
pore pressure by as much as 30 per cent, and the excess pressure persists over
geological time-scales. The pressure contribution due to the excess pore water is
important provided that the dehydration goes to completion at sufficient burial
depth, which depends in part on the activation energy of the reaction. If sediments
overlie a permeable basement, fluid can flow out of the sediments and relieve pore
pressure throughout the sedimentary column.
Key words: compaction, layered media, sedimentation.
1
INTRODUCTION
This paper presents a continuum mechanics model for the
l - D gravitational compaction of a sediment layer whose
thickness increases with time. The sediment grains undergo
a thermally activated dehydration reaction that releases
water, which increases the pore fluid pressure above normal
hydrostatic values. The original contribution of this work is
the derivation and numerical solution of a mathematical
model that describes the coupling of compaction with
concurrent sediment dehydration, along with the associated
effects on pore pressure and porosity. High pore pressures in
sedimentary basins influence many geophysical processes,
such as the tectonic deformation of the Earth’s crust, the
transport of solutes in groundwater flow, and the migration
of hydrocarbons in petroleum reservoirs.
The mathematical model is based on the continuum
theory of mixtures (Atkin & Craine 1976) and consists of a
series of mass balances for the solid components and the
pore fluid. The overall momentum balance is quasi-static,
and the fluid flow obeys Darcy’s Law. The sediment
mechanical behaviour is assumed to follow Terzaghi’s
principle of effective stress (Terzaghi 1943). The constitutive
laws for the sediment rheology and permeability are
adopted from soil mechanics theory. The equations lead to a
non-linear parabolic equation for the porosity as a function
of time and depth, which can be solved using a
finite-difference method and a Newton-Raphson scheme.
Since the thickness of the sediment layer is determined as
part of the solution, the model represents a moving
boundary value problem (Crank 1984).
Since the study of compaction is multidisciplinary, it is
useful to review the geological jargon used to describe
mechanical and chemical alteration of sediments. Diugenesis
refers to the sum of all physical, chemical and biological
processes that alter sediments after they are deposited.
Some of these processes, compaction and cementation,
convert loose. soil-like material into hard rock-like material.
283
284
D. M . Audet
This hardening process is called lithification or consolidation. Consolidation is often used in geotechnical engineering
and implies the reduction of pore space by mechanical
loading. Lirhification, a geological term, refers to the
combined effects of mechanical loading and cementation.
Znduration refers to the hardening of rock material by heat,
pressure and cementation, that is, a more advanced state of
lithification. In the extreme case, indurated sediments
undergo metamorphism, that is, substantial changes to the
mineral structure of the original rock material caused by
high temperatures and pressures. In geology, the distinction
between diagenesis and metamorphism is clear and the two
subdisciplines rarely overlap.
This paper concerns two specific diagenetic processes:
compaction and clay dehydration. Compaction. the reduction of pore space by mechanical effects, increases sediment
strength. In this paper, the role of clay dehydration is simply
to generate additional pore fluid. It is assumed that
dehydration does not affect the sediment mechanical
properties. An example of clay dehydration is the
conversion of smectite to illite. Smectite is an expansive clay
that can swell and retain excess water. Under suitable
conditions, smectite transforms to illite, and the excess water
is liberated into the pore fluid. The dehydration mechanism
used in this paper is only an analogue for the real process.
Geochemically realistic reaction mechanisms are much more
complicated.
This paper is organized as follows. The background
literature concerning gravitational compaction and the
modelling of smectite-illite transformation is reviewed in
Section 2. The geological system is described in Section 3,
and the coupled compaction-dehydration model is derived
in Section 4. Details of the numerical solution scheme are
shown in Section 5. Section 6 summarizes the choice of
parameters in the model, Results are presented in Section 7
and discussed in Section 8. Conclusions are given in Section
9.
2
BACKGROUND
The roles of fluid flow and pore pressure in soils and rocks
are important for understanding tectonic deformation of
sediments, especially near faults and thrust zones. The
interaction of pore pressure and fracturing of rocks relates
to earthquake mechanisms. These geophysical applications
have been reviewed by Mase & Smith (1987). On a regional
scale pertaining to hydrology, fluid flow in sediments is
important for understanding groundwater flow and solute
transport, as recently reviewed by Mangold & Tasang
(1991). The work in this paper pertains specifically to
sedimentary basins.
The gravitational compaction of sediments has received
much attention in geotechnical engineering. One of the first
1-D compaction models was by Gibson (1958), who
proposed a linear model for the compaction of thin layers of
clays (fine-grained sediments consisting of clay minerals).
Gibson’s (1958) model is in terms of an evolution equation
for the void ratio. The linear model assumes that the
sediment mechanical properties do not depend on the
compaction, that is, the void ratio or the porosity.
Subsequent models were fully non-linear and were solved
numerically (Gibson, England & Hussey 1967; Gibson,
Schiffman & Cargill 1981). The original linear model was
extended by Gibson & Sills (1990) to account for active fluid
removal from the base of the sediment layer. One of the
important ideas from geotechnical engineering is Terzaghi’s
principle of effective stress, which states that the strength of
the sediments depends only on the difference between the
total stress and the pore pressure, which is known as the
effective stress. For a given material, the effective stress
depends only on the porosity (or the void ratio). Terzaghi’s
principle forms the basis of soil mechanics theory.
The above 1-D models have been used to study the
development of excessively high pore pressures in
sedimentary basins. Bredehoeft & Hanshaw (1968) and
Hanshaw & Bredehoeft (1968) used a linear compaction
model to study the pore pressure in thick shale layers. Smith
(1971) derived a non-linear model assuming a two-phase
continuum model along with the effective stress concept. A
further investigation using linear models was made by
Bishop (1979). Bethke (1986) presented a compaction model
that included a variety of effects, including water release due
to clay hydration. However, since Bethke’s model does not
incorporate Terzaghi’s principle of effective stress, the
mechanics of compaction are not treated rigorously.
Recently, a number of authors have shown renewed
interest in I-D compaction models, namely Audet & Fowler
(1992); Audet & McConnell (1992); Wangen (1992); Luo &
Vasseur (1992). The paper by Audet & Fowler gives a brief
review of the literature, poroelasticity (Biot 1941), and
critical state theory (Chen & Mizuno 1990). The compaction
model gives a porosity evolution equation similar to that of
the model by Smith (1971). The main extension is that the
equations are put into dimensionless form, which allows the
compaction process to be characterized by the sedimentation parameter, denoted by A , which is the ratio of the
hydraulic conductivity of the sediments to the sedimentation
rate. Asymptotic solutions for large and small As are given
along with numerical results. A detailed parameter study is
presented in Audet & McConnell (1992). Audet & Fowler
(1992) formulated a simplified model for clay dehydration
but did not actually solve it. The purpose of this paper is to
extend and solve the coupled compaction-dehydration
model from Audet & Fowler (1992) and to present results
pertinent to geophysical applications.
Using a void ratio formulation like that of Gibson (1958),
Wangen (1992) included a heat equation in the compaction
model and studied the relative importance of non-linear
conduction versus advection. Wangen scaled his equations
and derived the sedimentation parameter, which he denotes
by A(). The coupled effects of kerogen maturation and
hydrocarbon migration on compaction are included
(Wangen 1993). England et al. (1987) studied a similar
problem in the context of petroleum entrapment in oil
reservoirs.
Shi & Wang (1986) formulated a 1-D compaction model
in which the governing equation is in terms of the pore
pressure instead of either the porosity or the void ratio. 4
similar model was used by Palciauskas & Domenico (1989)
and by Luo & Vasseur (1992). Shi & Wang (1986) and Luo
& Vasseur (1992) studied the relative importance of
aquathermal pressuring, that is, the development of excess
pore pressure due to the thermal expansion of the pore
fluid. Both papers seem to agree that compaction
Gravitational compaction and clay dehydration
disequilibrium is the main cause of overpressuring, and the
thermal effects on fluid density are less important. However,
it is not likely that everyone will ever agree on this issue; see
the discussions by Miller & Luk (1993) and Luo & Vasseur
(1 993).
The focus of this paper is the transition of smectite to illite
and the concurrent release of water at depth. The release of
extra pore water may lead to excess pore pressure (Powers
1967; Burst 1969). For this reason, there has been interest in
understanding the kinetics of the smectite-illite transition.
There are two hypotheses concerning illitization; first,
smectite converts to illite by a series, either parallel or
sequential, of dissolution-precipitation reactions; second,
illitization takes place as a solid reaction due to ionic
substitution of aluminium cations for silicon atoms.
Assuming a first-order dissolution-precipitation model,
Eberl & Hower (1976) estimated an effective activation
energy of 82.0 kJ mol-' for the overall reaction. This value
is within the range of activation energies measured for a
variety of mineral dissolution reactions (Lasaga 1984).
Bethke & Altaner (1986) used a Monte Carlo method to
study the solid-state transition of smectite to illite.
Preliminary results suggest that the Monte Carlo simulations
may be able to predict more realistic illitization rates, but. as
pointed out by Bethke & Altaner. this does not preclude
dissolution-precipitation mechanisms. Field studies in the
Gulf Coast region of the United States (Freed & Peacor
1989) and in the North Sea (Pearson & Small 1988) suggest
that illitization occurs at burial depths between 1.5 and
3.5 km, corresponding to temperatures between 69 "C and
W)
I
1 5 1 1
Depositional
Surface
z
= h(t)
I"
Compacting
Sediment Layer
285
116 "C. However, the field data show appreciable variablity
and further theoretical and observational studies are
required.
3
DESCRIPTION OF T H E MODEL
The idealized geological system being modelled is depicted
in Fig. 1. A basin consists of a basement composed of
relatively incompressible rocks. Overlying the basement,
there is a layer of compressible sediments. The top of the
sediment layer is called the depositional surface. A space
coordinate, z , is defined such that z = 0 is the basement and
z = h ( t ) is the depositional surface. The thickness of the
sediment layer is h ( t ) , where r is time. For convenience, the
burial depth d is defined by d = h ( r ) - z . Sediments are
added to the layer at the rate V ( t ) , which means that h ( t )
increases with t. Gravity acts in the negative z direction. A
heat flux, Q, enters the sediment layer at z = 0 and sets up a
temperature gradient across the layer. A fluid flux, &), can
enter or leave the sedimentary column through the
basement. If the basement is impermeable, &,=O. As the
sediment layer thickness increases with time, the sediments
compact under their own weight, which leads to a reduction
in the pore space of the sediments, which causes the pore
fluid to migrate upward towards the depositional surface.
This process is called gravitational compaction. If the
sediments have low enough permeability, then the fluid flow
through the sediment layer is hindered and the pore
pressure increases above hydrostatic. Sediments with excess
pore pressure resulting from hindered fluid flow are said to
be in a state of compaction disequilibrium, since the excess
pore pressure can dissipate, leading to further compaction.
The solid portions of the sediments are assumed to
contain a clay such as smectite which can change chemically
to a non-expandable clay such as illite. As part of this
clay-to-clay transition, water that is bound in the structure of
the smectite is released and increases the amount of fluid
within the pore space of the sediments. Under lowpermeability conditions, this excess pore water can lead to
higher than normal pore fluid pressures. One of the aims of
the model in this paper is to quantify how much excess pore
pressure results from fluid release at depth.
4
MATHEMATICAL FORMULATION
4.1 Geochemistry
19
Basement
Figure 1. Diagram of a compacting sediment laycr overlying an
incompressible basement. The basement is at z = 0 and the
depositional surface is at z = h(r). where h ( t ) is the sediment
thickness. The sediment deposition rate is V ( f ) . The depth
coordinate d is measured from the depositional surface. A heat flux,
Qo, enters the sediment laycr at z = O and establishes thc
geothermal gradient. T h e basement fluid flux, 6,.
may either seep
into or out of the layer. Gravity, denoted by g , acts in the - z
direction.
The sediments are modelled as a two-phase system of solids
and pore fluid. The solid phase consists of three
components: a large-grained, coarse material: a fine-grained
hydrated clay; and a fine-grained dehydrated clay. The pore
fluid is assumed to be pure water, even though, in reality,
pore fluid contains dissolved mineral species. At this point,
the model does not consider the liquid side of the
geochemistry, and it ignores the effects of any dissolved
cations. The hydrate undergoes a simple dehydration
reaction given by
H ~ " D + ~ w .
(1)
where H, D and W represent hydrated clay, dehydrated clay
and water, respectively. It is assumed that n moles of W are
D. M. Audet
286
liberated for every mole of D formed. The coarse material is
assumed to be inert and it does not participate in the
reaction. The overall reaction rate kr, for the process is
assumed to follow an Arrhenius temperature dependence:
k,,
=
Ea
v exp - RT'
where E , is the activation energy for the dehydration
process, v is the frequency (pre-exponential) factor, R is the
gas constant and T is the absolute temperature.
The following kinetic model is postulated. Let [HI, [D]
and [W] denote the molar concentrations of hydrate,
dehydrate and water respectively. Assuming a first-order
reaction mechanism, the rates of formation of each species
are
d
- - [HI =
dt
+
d
I d
- [D] = + - - [W] = + k,,[H].
dt
n dt
(3)
For a unit volume of sediment material, the volume fraction
of each of the four species, water, hydrate, dehydrate and
coarse, are denoted by &, c $ ~ ,+d and dC,respectively. By
definition, the volume fractions satisfy the condition
4,'
4 h + 4 d S 4 c = 1,
4.2
Mass conservation
The following dimensionless variables, shown as quantities
with overbars, are defined:
The time- and length-scale factors are taken so that
y l = Y / F where V, is the nominal value of the sediment
deposition rate. The temperature-scale TI is the temperature
at the depositional interface. The dimensionless reaction
rate lrx
is defined by
where kf, is the reaction rate at TI and E,, = E,/(RT,) is the
scaled activation energy. Hatted symbols denote dimensionless functions such as the reaction rates, the effective stress,
the permeability and the pore fluid viscosity.
Assuming that pw, ph, pd and pc are constant, scaling eqs
(9)-( 12) gives
(4)
and the bulk density of the mixture, p, is given by
P = 4wPw + 4hPh + 4 d P d + ~ c P c ,
(14)
(5)
where pw, phr pd and pc are the densities of the four
components.
The molar concentrations can be written in terms of the
volume fractions as follows:
=Ph4hlMh,
LD1
=Pd4dlMd,
LW1
=Pw4wlMw,
(6)
where Mh, Md and M, are the molecular weights of H, D
and W respectively. The rate at which mass of H is
transformed is r, (kgm-3s-') and this is balanced by the
production of D at a rate r, and of W at rate r,. The
reaction rates are thus
rh = krxPhdhr
rd = (Md/Mh)rh.
rw= (nMw/Mh)rh.
(7)
For a closed system, conservation of mass implies that
rh = r,
+ r,,
(8)
which is equivalent to Mh = Md + nM,.
The model assumes that the motion is only in one
dimension, the z direction, and that the three solid phases
move with velocity u, and that the pore fluid moves with
velocity u,. Thus, the 1-D mass conservation equations for
the four components are
(9)
where a , and a2 are defined by
The water denisty pw depends both on temperature and
pressure. However, Luo & Vasseur (1993) showed that this
dependence is not a major control on overpressure
development, so it is not considered in this paper. The
specific molar volumes of components W, D and H are
qr
W
=-M W
M d
qr,,=--,
Pw
Pd
Mh
7"
h ---.
Ph
Thus a , = nVwlYh and a2 = Td/Vh represent ratios of
specific molar volumes. The constant R,, = k:x9, the
reaction parameter, is the ratio of the sedimentation
time-scale to the reaction time-scale.
The following result will be useful later. Combining eqs
(14), (15) and (16) gives
a
y[l -
at
a
4 w I +azy [ (-l~ W > ~ =, I-(I
- a2)Rrxkrx4h,
(19)
which can be combined with eq. (13) and then integrated ro
give
where F,
= [4wVw,z=o,
a function of i only, is the flux of
Gravitational compaction and day dehydration
water prescribed at the basement. Eq. (20) uses the
boundary condition 6, = 0 at Z = 0 (see Section 4.7).
4.3 Evolution equation for sediment thickness
Define m ( t ) to be the solid mass per unit area of the
sedimentary column at time t:
287
used to calculate h(?)at the first iteration of a time-step.
For each subsequent Newton-Raphson iteration, eq. (29) is
used to update h(7).The use of (29) helps to maintain the
accuracy of the overall mass balance of the solid phase
throughout the time-stepping history of the numerical
simulation.
4.4 Momentum conservation
The momentum balance in the
where q5s=
defined by
~ - c # I ~
P545 = 4hPh + 4dPd + 4cPc.
(22)
Differentiating rn with respect to t gives
where ( 4 , ~ ~means
) ’ &p5 evaluated at z = h ( t ) . Using eqs
(10)-(12) and the definition of u2 (eq. 17) gives
a
-[
at
a
P , ~ S+] i)z [ P S ~ ? =
~ ,- (]r h - ‘d) = - ( P h
Substituting eq. (24) into eq. (23), integrating, using u, = 0
at z = 0, and scaling leads to
where V(T) = ri?/[(pS4J1V,] represents the dimensionless
sedimentation rate based on the flux of solids deposited on
the column, and 4; means 41T
evaluated at z = h ( t ) , and so
on for
4; and 6;. The constants Ap, yh and yd are
defined by
+A,
Instead of integrating eq. (25) directly, it is more accurate to
calculate h(t)_from an integral formulation. Using the
definition for V ( i )from eq. (25),
JI:
Lr
ri? dt = ( + s p s ) ’ v ) . T
where nvis the vertical stress, g is gravitational acceleration
and p is the bulk density given by eq. (5). Darcy’s law is
where k is the permeability, p is the fluid viscosity and pf is
the pore fluid pressure. The effective stress, denoted by uc,
is
- PduZ)krx4h.
(24)
m(t) =
z direction is
and p s is the average solid density
From soil mechanics theory, a: is a function of the porosity.
For many types of fine-grained sediments, the effective
stress correlates with the porosity according to
where u , ( ~
is a reference value of the effective stress
l &) is the void ratio, el,,,, is
(usually 100 kPa), e = + w / ( the void ratio of the sediments when u: = glo0and Cc is the
compression index (Burland 1990). Correlations such as eq.
(33) are sometimes called compressibility laws. Since
lithology and mineral composition control sediment
compressibility, the mechanical parameters C, and
el,,,, should depend explicitly on
4d and 4=. However,
the relationship between the mechanical parameters and the
solid-phase composition is not well established, and for this
reason C, and el(xlare assumed to be constant.
Let u1be the stress scale. Then, the dimensionless vertical
stress, pore pressure and effective stress are
u,= u,a,,
V ( i ) dT.
(27)
p f=
^I
ff=
;( T I U ” .
In keeping with Audet & Fowler (1992), the stress-scale is
taken to be
Eq. (21) and Z = z / h ( t ) (see Section 5) gives
m ( t ) = K(7)Y
JT:
u1= ApgY.
+,p, d Z .
(34)
(28)
The vertical stress, u,, and the pore pressure, pf, can be
eliminated from Darcy’s law by using eqs (30) and (32):
In the numerical implementation (see Section S ) , eq. (25) is
where eq. (4) was used to eliminate 4= from the
Combining eqs (27) and (28) gives
D.M . Audet
288
formulation. The quantity J(4,) is the volume-averaged flux
of water relative to the moving matrix, and it depends
primarily on 4,. The dimensionless group A , the
sedimentation parameter, is given by
where k , is the permeability at z = h ( t ) , p I is the fluid
is the hydraulic
viscosity at TI, and K~ = k,p,g/pl
conductivity, which has units of velocity. Therefore, A
represents the ratio of the permeability-controlled fluid
velocity versus the sediment deposition rate. For large
values of A , the sediments are highly permeable or
deposited at slow rates, and pore pressures tend to be
hydrostatic. Conversely, relatively impermeable sediments
deposited at fast rates (small As) lead to high pore pressures.
The dimensionless permeability is taken to be
4.5 Exact solution A +
+
00
The limiting case of large A represents a normally pressured
sediment layer for which u, and u , are zero. For the case of
F;, = R,, = yh = yd = 0, an impermeable basement, no
dehydration, homogeneous solid composition, one can
obtain a closed-form expression for 2 as a function of e:
(43)
Eq. (43) is useful because it connects geotechnical,
laboratory-scale parameters (C, and e,,,,) to observable
field-scale data (4w versus d ) . It was derived earlier in
Audet (1995) and used t o study the compaction trends of
normally pressured, pelagic sediments in the Pacific Ocean.
4.6 Non-linear heat conduction
with
= 4;,
C#I~
=
The geothermal gradient is assumed to be due primarily to
non-linear, quasi-steady heat conduction resulting from a
specified basement heat flux. Since the Peclet number for
this system tends to be small,
and the dimensionless viscosity is
[
= exp q(+111
I)].
and 17 are constants. If u ,is the effective stress at
4w= 4;, then
where
,.
CY
a:
gv=-=
WI
exp
I
&I)
el - e
~
P
T=I+Q,,
df
7 .
k,,,
(44)
The dimensionless, bulk thermal conductivity of the
sediment mixture is taken to be
(39)
where P =C,/ln10, using the notation of Audet &
McConnell (1992), and e , = +1/(1 Substituting eq. (20) into eq. ( 3 9 , multiplying by
(1 - &,), differentiating with respect to Z and using eq. (19)
gives
Eq. (40) is a non-linear diffusion equation for 4, with a
combination of point and distributed source terms.
Using eqs (20) and (35). V 5 and V , can be written in terms
of the volumetric flux J:
based on the work of Woodside & Messner (1961). and k;h
and k:h are the thermal conductivities of the solid and fluid
phases respectively. The dimensionless basal heat flux, Q,),
is defined by
where q,, is the unscaled geothermal heat flux (in mW m-*)
at Z= 0. In this study, q,, = 37.9 mW m -', which corresponds
to Q,, = 0.02. In practice, this yields a temperature gradient
of 29-31 "C km-', in keeping with observations (Deming &
Chapman 1989).
Heat flow in sedimentary basins is an important topic and
there is an extensive body of literature devoted to it, of
which Deming & Chapman (1989) and Gallagher (1987) are
only two representative works. An overview of the topic is
given by Furlong & Chapman (1987) from a global,
geophysical point of view.
4.7 Boundary conditions
At z = h ( t ) , values of 4,, 4 h , 4dare specified. At z = 0, the
prescribed velocity boundary conditions are u, = 0 and
4,uW = &), both of which are satisfied identically if
6,= J ( 4,) at z = 0 is satisfied, which represents a non-linear
boundary condition for I$,.,. For initial conditions, at t = 0,
all the volume-fraction profiles are uniform. Because of eq.
Crauitutional compaction and clay dehydration
(33), specifying c $ ~ at the depositional surface (4;) is
equivalent to specifying the effective stress (cT,).
5
289
the volume fractions are rescaled by
NUMERICAL SCHEME
The following coordinate transformation is used:
(47)
This has the advantage of fixing the moving boundary, the
depositional surface, to Z = 1, which simplifies the
numerical treatment. From the chain rule,
a
dz
-
l a
hdZ’
._-
a’
_1 _a’
a
i)z2
h’dZ’’
dl
-
ij
at’
~h a
h aZ’
(48)
where h = dhldi. Since there is no difference between t and
t ’ except when taking partial derivatives of functions that
depend on both t’ and Z , there is no need to distinguish
between t and t‘ in the rest of the paper. Even though the
moving boundary is now fixed at Z = 1, the sediment
thickness appears as a time-dependent coefficient in the
relevant equations, and the evolution equation for h ( f) still
has to be solved.
The coupled compaction-dehydration model is a system
of partial differential cquations consisting of a non-linear
parabolic equation for 4w (eq. 40) and three non-linear
first-order hyperbolic equations for 4,,, +d, +c (eqs 14, 15,
16). In addition, there are two quadratures, one for t% (eq.
29) (see comments in Section 4.3) and one for 7 (eq. 44).
The solution procedure begins by solving eq. (40) for 4w
assuming suitable guesses for the other unknowns (&, 4d,
4c,T, h,.A t a given time-step, the unknowns are evaluated
at the previous time-step. The non-linear parabolic equation
is put into discrete form using second-order, centred
finite-difference approximations for the Z derivatives, and a
second-order, backward difference for the T derivative
(Smith 1985). The time-steps are constant and the space
mesh is uniform. The non-linear algebraic equations are
solved using a Newton-Raphson scheme.
After obtaining 4wat the current time-step. t% and 7 are
updated using eqs (29) and (44), respectively, and the
solid-phase velocity comes from eq. (41). Using secondorder finite-difference approximations for the space and
time derivatives (forward in Z , backward in T), the
hyperbolic equations are integrated from Z = 1 to Z = 0 in
order to obtain the current values of (bh, (bdr +c.
Owing to numerical dispersion and dissipation in the
solution of the hyperbolic equations, the sum of the volume
fractions d o not sum t o unity, that is,
The notation (.), means some quantity evaluated at the kth
step of the iteration procedure. Note that S, varies with 2.
Using the rescaled volume fractions for the solid
components, and the current temperature profile, bWis
recalculated by resolving eq. (40), followed by resolving the
relevant equations for T, h, 4,,. +d, 4c. The changes in
these five unknowns are monitored until the differences
between any two iterations are sufficiently small. Let r, be
the residual of one of the five unknowns at the ith grid
point. For each unknown, the iterations were deemed to
have converged when llr, 5 2 X lo-’. As the iterations
proceed, the renormalization factor S, -+ 1 over all mesh
points. The renormalization step does not affect the
accuracy of the numerical scheme because it is used only to
prevent numerical noise from the solution of the hyperbolic
equations from feeding back into the non-linear parabolic
equation.
Test cases using various mesh sizes were made in order to
check the convergence of the numerical scheme. The
numerical model was verified by comparison with
asymptotic cases given in an earlier paper (Audet & Fowler
1992). Typically, the space mesh was AZ=0.01 (100 grid
points) and the time-step was AT= 0.025. In a few cases
(Section 7.5), up to 400 grid points (AT = 0.0025) were used
in order to obtain good accuracy on the overall solid mass
balance on the sediment layer (eq. 29). The numerical
scheme was implemented in FORTRAN using doubleprecision arithmetic on a Sun S P A R C station ipx.
6
PARAMETER VALUES
The kinetics of the dehydration reaction are controlled by
two parameters, E , and v. Values for E , range from 40 to
80 kJ mol for mineral dissolution reactions (Lasaga 1984).
In this study, E,, = 60 kJ mo1-l was used as a representative
value. The pre-exponential is more difficult to quantify
because it depends on many factors, such as local fluid-phase
chemistry, the details of which are beyond the scope of this
paper. In order to assess the scope of the model, it is
postulated that the reaction goes to completion when
R,,k^,, = Sy l ) , and that this happens when T = T,, some
specified temperature. Under these conditions, v is given by
1
E,,
Try
Y = -exp e ,
The volume fractions must be corrected to prevent the
solution from deteriorating. The correction scheme used
consists of renormalizing the volume fractions of the solid
components. It is assumed that 4,, is correct since the
non-linear parabolic equation is not prone to the numerical
problems associated with the hyperbolic equations. Defining
the renormalization factor Sk for the kth iteration:
3
(54)
and R,, is
&
R,, = exp E()(
-
1
).
The advantage of this choice of parameter is that R,, is
independent of the time-scale. The specific values o f the
parameters are given in Table 1.
After examining the effects of temperature-dependent
290
D. M . Audet
Table 1. Summary of model parameters
Parameter
Nominal Value
Reference
1,050 kg/m3
Hamilton (1976)
2,740 kg/m3
Lambe & Whitman (1979)
2,840 kg/m3
,I
2,700 kg/m3
I,
770 g/mol
5-10
Deer et al. (1962, p. 213)
(using KyAl~(Si~-yA1,)02~(OH)4
with 1 < y < 1.5)
Freed & Peacor (1989)
18 g/mol
2.02 W/m K
Sharp & Domenico (1976, Table 1)
0.631 W/m K
I,
60 kJ/mol
Lasaga (1984)
1.62 MPa
Burland (1990)
0.5
I,
500 m/Ma
100 m
0.2 Ma
14
lo-" m/s
37.9 mW/mz
Olsen (1960)
I1
Deming & Chapman (1989)
300 K
viscosity on the solutions, it was concluded that viscosity
effects were comparable to the effects of varying the
permeability for
md 5 k , 5 10F2md (note: 1 md =
lO-'m s-' = lo-'' m2). The effects of a, which controls the
sensitivity of permeability to porosity, were studied in Audet
& McConnell (1992). From these considerations, it was
decided to use a constant viscosity with 7 = 0 (see eq. 38)
since the main objective of the paper is to study the effects
of clay dehydration on pore pressure. Recent in-depth
studies (Luo & Vasseur 1992; Wangen 1992) have examined
the role of temperature-dependent fluid density and
viscosity on overpressuring.
T h e effect of yh and yd (eq. 26) were tested. For
y h = +0.25, the overall effects on the volume fractions and
the pore pressure were comparable to the effects of varying
a, (eq. 17; results in Section 7.4). Using the density values
given in Table 1, typical values of the density ratios are
yh = 0.0242 and yd = 0.0848, both of which are smaller than
the value of 0.25 used in the test cases. Since the density
ratios seemed to be of secondary importance, the values
yh = yd = 0 are used for all the cases presented in Section 7.
7
RESULTS
7.1 The effects of the sedimentation parameter: A
This section considers the simple case in which there is no
dehydration ( R r x= 0) and the basement is impermeable
(4)= 0). The sedimentation rate is constant throughout the
deposition history [V(T)= 11, which means that, for equal
time spans, the same mass of solid material has accumulated
in the sediment layer. Fig. 2 shows the porosity (water
volume fraction, 4w) and the pore pressure (pf)as a
function of burial depth (d) for 0.5 5 A 5 10. The porosity
profiles (Fig. 2a) begin with 4w=0.5 at d=O and the
porosity decreases to 0.2 < 4,,, < 0.3 at the basement depth
of 37 < d< 43. Each of the profiles corresponds t o T = 50,
and in each case the same amount of solid mass has
accumulated between the basement and the depositional
surface. For reference, the porosity profile for normally
pressured sediments (eq. 43) is shown. A s A increases, the
porosity approaches the normally pressured profile. The
Gravitational compaction and clay dehydration
291
and the lithostatic pressure is
-50
--
A = 0.5
1
1
1
1
,
1
1
1
1
~
1
1
1
1
where 4; is c $ ~ at d=O, the depositional surface. The
lithostatic pressure represents the pressure that the pore
fluid would obtain if the sediments were not compacted. In
this study, the variation of A is interpreted as a change in
sediment permeability. As A decreases, the sediments are
less permeable, which hinders the expulsion of pore fluid by
compaction, and the sediments retain a higher porosity
compared to the normally pressured situation. Since each
profile represents the same amount of solid mass, the total
thickness (basement depth) of the sediment decreases with
increasing A since the average porosity decreases with
increasing A. A more detailed study of porosity evolution is
given in Audet & McConnell (1992), in which the effects of
time, rheology and permeability are examined.
7.2 The effcts of stoichiometry: n
0.0
10.0
20.0
30.0
40.0
50.0
60.0
Pf
Figure 2. The porosity, & (a), and the pore pressue, jif (b), as
functions of burial depth, 2, at ?=SO for four values of the
sedimentation parameter, A =0.5. 1, 2 and 10. The dashed line in
(a) is the porosity versus depth profile for normally pressured
sediments (eq. 43). In (b), the hydrostatic pressure is denoted by
phydro
and the lithostatic pressure is F,,,,,", as defined by eqs (56)
and (57) in the text.
corresponding pore-pressure profiles (Fig. 2b) reflect the
dependence on A. The case of A = 10 is near hydrostatic, and
the case of A = 0.5 is closer to the lithostatic profile. The
hydrostatic pressure is
This section looks at the effects of dehydration for the case
of R,, =0.01 with A = 1 and &,=0. For E,,= 24 (corresponding to E;,= 60 kJ mol-'), R,, = 0.01 corresponds to
7;.x= 100°C. Assuming a , + a, = 1, the sediments undergo
no overall specific volume change, which means that the
integral term in eq. (40) drops out. The number of moles of
water released per mole of hydrate formed, n , is varied such
that n = 2.5 ( a , = 0.14198), n = 5 ( a , = 0.26910) and n = 10
( a , = 0.48722). The values of u , are based on Md = 770,
M, = 18, c $ ~ = 4c= 27oO kg m-3 and 4w= 1050 kg rK3,
with a , = 1 - a , by assumption.
The initial composition of the sediments is +h = 0.2,
4d= 0.0, $c = 0.3 and c$w = 0.5. These values are based on
the composition of North Sea shales given in Table 1 of
Pearson & Small (1988). Referring to Table 1 of Pearson &
Small (1988), sample P1 has about 26 wt per cent of fraction
less than 2 p m , of which at least 83 per cent is smectite.
Since 83 per cent of 26 per cent is about 22 per cent, 4hwas
set to 0.2. Since it is not known what size fraction of the
sediments undergoes dehydration, using 2 p m to identify the
hydrated clay fraction is not very precise. Even though the
initial value of 4, does affect the amount of excess pore
pressure predicted by the model, this detail does not alter
the overall conclusions derived from the model.
Figure 3(a) shows the volume fractions of the hydrate
phase (H), the dehydrate (D) and the water (W) phase as a
function of burial depth for 5 = 50. The profile of 4wversus
d for no dehydration is also shown (dashed curve). As n
increases, more water is released during the reaction, and
4,,, increases for a given burial depth. This is due to the
reaction term u , R , , ~ ^ , ,in~ ~eq. (40). The 4h versus 2
profiles do not vary much with n. This is because the
evolution equation for
eq. (14), does not depend on a ,
or a,. In contrast, the c $ ~profiles vary with n , because eq.
(15) depends on u 2 . Since u , = 1 - a , . as n increases, a ,
increases and a , decreases, which is why, at a given depth,
4ddecreases as n increases.
The corresponding excess pore-pressure 6,
- jihydrC,)
profiles are shown in Fig. 3(b). Results are shown for A = 1
and 10. The results show that, for A = 1, a water-releasing
+,,
D. M . A d e t
292
parameters are used: a , = 0.25: a , = 0.75, corresponding
approximately to n = 5 ; h = 1: F;, = O ; E,,= 24. Results are
shown for T = SO. A temperature, T,,, is picked and R,, is
chosen so that the reaction source term is O(1) at that
,, #, and #,, as a function of
temperature. Fig. 4(a) shows #
2 for R,, = 0.05, 0.01 and 0.002 corresponding t o T,, = 70,
100 and 130 "C, respectively. The volume-fraction profiles
for the hydrate and dehydrate phases (Fig. 4a) show that the
dehydration reaction goes to completion at greater depth
with decreasing Rrx. By decreasing R,,, the pre-exponential
0.0
0.2
0.1
4w
0.3
or
4d
0.4
or
0.6
0.5
4 h
0.0
03
0.2
0.1
d
0.4
4 w or 4 d or
-
50
X=l
I
1
1
1
1
1
1
1
I
l
l
0.6
0.5
d'h
-
1
Pf - P h y d r o
Figure 3. (a) The volume fractions of hydrate (&), dehydrate (4d)
and water (4,) as a function of burial depth d a t f = 50 and A = 1.
The dashed curve shows r # ~for~ t h e case of no dehydration. The
profiles are shown for three valucs of the stoichiometric coefficient,
n = 2.5, 5, 10. (b) shows the excess pore pressure, P; - flhydrC,.
as a
function of 2 for the three values of n , for A = 1 and 10.
reaction can increase excess pore pressure by about 30 per
cent at the basement compared to the case of no water
release (dashed curves). The extra amount of overpressuring
due to dehydration increases as n increases.
7.3 The effect of the reaction parameter: R ,
In this section, R,, is varied while keeping the activation
energy constant. This is equivalent to varying the
pre-exponential factor v independently of E,. The following
50
( c ) 0.002
1
1
1
1
1
1
1
1
I
I
I
I
I
Gravitational compaction and clay dehydration
factor decreases, which means that higher temperatures are
required for the reaction t o go to completion, which, in this
system, are found at greater burial depths. As the reaction
takes place at greater depth, the porosity (&) is perturbed
further away from the reference profile corresponding to no
dehydration. The change in porosity is reflected in the
excess pore pressure (Fig. 4b). A s pore fluid is released at
deeper levels (Rrx = 0.002), the excess pore fluid increases
by about 10 per cent with respect to the case of R,, = 0.01.
d
7.4
293
The effects of sediment expansion and contraction: a2
The purpose of this section is to examine how the integral
source term affects the porosity and the excess pore
pressure. The integral source term is important if there is a
net change in the specific volume of the sediments during
dehydration. The stoichiometric coefficients are chosen such
that a , = 0.25 and a2 = 0.6, 0.7.5 and 0.90, corresponding to
relative specific volume changes (i.e. AY"/Yh= u 1 + a2 - 1,
with AY"=nVw+ Vd- Yh) of -0.15, 0.0 and +0.15
respectively. Fig. 5(a) shows the porosity profiles for A = 1
and 4, = 0 at T = 50 for the three values of a2. The kinetic
parameters are R,, = 0.01 and E,, = 24. The results show
that when the reaction products shrink during dehydration
(a2 =0.6), that is, the combined specific volume of the
dehydrate component and the extra, released water is less
than the specific volume of the hydrate component, the
porosity increases relative to the isovolume case ( a 2 = 0.75).
and the excess pore pressure (Fig. Sb) increases by about 5
per cent near the basement. Because n'Vw+ 'Vd is less than
'Vh when a , + a , < 1, extra void space is created by the
dehydration reaction. Water originally present in the
sediments fills the extra void space. The extra void space
increases 4,.,, which decreases g:. causing p , to increase in
order to balance the overburden, w,,. In contrast, when the
reaction products expand (a2 = 0.90), the porosity decreases
relative to the isovolume case and the excess pore pressure
decreases by about 5 per cent.
50
0.2
0.4
0.3
0.5
7.5 Dissipation of excess pore pressure
4w
The previous sections examined how chemically released
pore fluid affected the excess pore pressure in a thick layer
of low-permeability sediments. This section cxamines how
long the excess pore pressure will be maintained if
sedimentation ceases and the pore pressure dissipates to its
hydrostatic value. Dehydration is omitted since the primary
control on overpressuring is through A, and the basement is
impermeable (4,= 0).
Consider a sedimentary layer with A = 1 that has been
uniformly deposited for O S T S S O .
For ?>SO, sedi= O . Fig. 6(a) shows the
mentation ceases, that is,
evolution of the porosity as a function of depth for T = 5 0 ,
75, 100 and 12.5. As T gets larger, the porosity profile
approaches the normally pressured equilibrium profile
(dashed curve) denoted by T = +x. The excess porepressure profiles (Fig. 6b) show a similar trend. After the
first time interval (7 = 75), about 50 per cent of the excess
pore pressure has dissipated. After each subsequent time
interval, the excess pore pressure decays by about SO per
cent, which is reminiscent of the exponential decay
associated with parabolic equations such as eq. (40).
As the pore pressure approaches hydrostatic (zero excess
pore pressure), both the sediment and the fluid velocities go
to zero (Fig. 6c). T h e pore fluid velocity is monotonic at
T = SO. After sedimentation stops, u, near the depositional
surface (0 < d< 20) decreases faster than near the basement
region. A t T = 75, V , reaches a maximum value near d = 15.
v(T)
50
0.0
5.0
10.0
15.0
pf - p h y d r o
Figure 5. The effects of specific volume change on porosity (a) and
excess pore pressure (b). The dashed curves arc the case of no
dehydration, and the profiles are for T = 50 and h = 1. The profiles
are for uz = 0.60, 0.75 and 0.90.
D . M . Aiidet
294
d
-
I
-
50
0.2
0.3
4W
0.4
0.5
50
0.0
5.0-
10.0
15.0
50
I
-
I
1
1
1
1
I
I
1
normally pressured sediment layer having the same amount
of solid mass as the porosity profiles shown in Fig. 2(a). As T
increases, h(T)approaches a uniform value. This is clear for
A = 1, 2 and 10. For A = 0.5, the asymptotic limit is reached
for some time greater than 350. The final asymptotic values
calculated from solving eq. (40) agree to within 0.2 per cent
of the theoretical value, determined by integrating the
hydrostatic porosity-depth profile (eq. 43). The agreement
improves as the resolution of the Z mesh is increased, so the
agreement between h(T)and h, for large 7values is limited
by the accuracy of the numerical solution.
The time needed to approach h, depends on A. For
normally pressured sediments, A = 10, the column relaxes
quickly, and for smaller As, relaxation times increase, which
reflects the effects of relative decreases in the sediment
permeability.
pf - phydro
Figure 6. The relaxation of porosity, excess pore pressure and
phase velocities as a function of time following the end of active
sedimentation. (a) Shows & w versus 2 for T=50 (end of
Sedimentation), 75, 100 and 12.5, and part (b) shows the
corresponding profiles of - P;lydr,,versus ;% The solid velocity (17,)
and fluid velocity (6,) profiles are shown in part (c). The dashed 4,,,
profile in (a) is the normally pressured profile (eq. 43). The case
corresponds to A = 1. R,, = 0, fil = 0.
In contrast, the velocity of the solids remains monotonic
with depth.
Figure 7 shows the sediment thickness, h(T)as a function
of time (?>SO) for different values of the Sedimentation
parameter, A = 0.5, 1, 2 and 10. In each case, the sediment
layer has been uniformly deposited for 0 5 Ts 50, followed
by no sedimentation for T>50. The dashed horizontal line
(labelled h,) represents the theoretical thickness of a
7.6 The effects of basement fluid flux: F,
All the previous cases have assumed an impermeable
basement (6)
= 0). This section examines the role of pore
< 0)
fluid either seeping into (Cl > 0 ) or draining out of (6)
the sediments at the basement for the case of no
dehydration (Rrx = 0).
Figure 8(a) shows the porosity profiles (A = 1 and T = 50)
for different values of
For &,>O, water seeps into the
sediments and the porosity increases relative to the case
with an impermeable basement ( f i ) = O ) . For 6,<0, the
sediments are draining at the basement and the sediments
are more compacted. For Fo< -0.05, it is possible for the
sediments to be overconsolidated, that is, at a given burial
depth, the porosity is less than that given by the normally
pressured porosity profile (eq. 43).
The pore-pressure profiles (Fig. 8b) show an increase in
pore pressure due to fluid seeping into the sediments
Grauitationul compaction und clay dehydration
205
&(t)
50.0
100.0
150.0
200.0
-
250.0
300.0
350.0
t
Figure 7. The sediment thickness, &(T) as a function of time following the end of active sedimentation at T = SO. for four values of the
Sedimentation parameter, A = 0.5, 1. 2 and 10. The horizontal. dashed line, denoted by i xis, the thickness of the normally pressured sediment
column which the curves approach as T - t +=.
(4)>0)and a decrease in pt for drainage (F;,<O). The
overconsolidated region in Fig. 8(a) corresponds to driving
the pore pressure below the hydrostatic value. which occurs
for 6)< -0.05.
Figure 8(c) shows the pore fluid velocity as a function of
depth. The solid-phase velocity is shown for 6)
= id.10.
These results show that the basement flux condition affects
the fluid velocity but has little effect on the solid-phase
velocity.
8
DISCUSSION
The sedimentation parameter, A, is the most important
parameter in the I-D compaction model because it controls
whether or not the pore pressure is greater than hydrostatic.
This is clear from Fig. 2, and consistent with results from
other similar models (Wangen 1992: Luo & Vasseur 1992;
Bethke 1986; Smith 1971). The sedimentation rate is an
important control, but only relative to the hydraulic
conductivity of the sediments. If the sediment permeability
is sufficiently high, then the pore pressure will essentially be
hydrostatic regardless of the sedimentation rate.
A larger problem concerns the constitutive laws used for
the effective stress and the permeability. Permeability
measurements are often done on sediment samples several
centimetres in length, and yet the model in this paper is
concerned with compaction trends on the length-scale of
kilometres. The same can be said for the effective stress.
More work is required concerning how to scale up
laboratory data in order to use it in macro-scale models for
basin analysis.
Attention has been given to the role of temperature
effects on overpressure development. One effect is
aquathermal pressuring, that is, excess pore-pressure
development due to the thermal expansion of pore fluid.
Related to this is temperature-dependent viscosity. A
second effect is the role of chemically released pore fluid at
depth, an example of which is water released due to the
transformation of smectite to i l k . The results in this paper
address these problems. Chemically released water is
probably more significant then thermal expansion of pore
fluid. The results of Fig. 3 are important because they
quantify when fluid release is important. To have an
appreciable increase in excess pore pressure, the dchydration mechanism has to release at least 5-10 moles of water
per mole of reaction product when d,, = 0.2 initially.
Reactions with n < 5 will not influence pore-pressure
development to any significant degree. Even though
sediment dehydration may increase pore pressure, the
effects are significant only if A = O(1). Chemically released
water in highly permeable sediments will not cause
overpressuring, especially when A > 10, as shown in Fig.
3(b).
The geochemical model used in this paper is very simple
and ignores the pore fluid chemistry. As shown in Fig. 4, for
a given value of the activation energy, the pre-exponential
factor (i.e. R J determines at what depth the dehydration
reaction goes to completion. which in turn influences the
enhancement of the pore pressure. If this model is to be
quantitatively useful for studying real sedimentation
systems, then the geochemical details, that is the kinetics and
the thermodynamics of the relevant reactions, have to be
included in the mechanical compaction model. With a more
robust geochemical formulation, a coupled compactiondehydration model may be used to discern the effective
activation energy of the rate controlling reaction from the
~ c#J,, versus depth data. In addition. the depth
slope of $ J or
range over which the reaction goes to completion can
provide some information about the pre-exponential factor
D.M . Audet
296
Figure 8. (Continued.)
50
f
0.0
1
1
~
1
10.0
1
1
1
11
1
~
20.0
1
1
.\.
1
30.0
~
1
1
1
40.0
1
1
1
50.0
Pf
Figure 8. The porosity (a) and the pore-pressure (b) profiles at
T = 50 and A = 1 for different basement seepage conditions. Values
of 4,> 0 correspond to water entering the sediments at the
basement and 4)<0 corresponds to water draining out of the
sediments at the basement. The normally pressured q5w and jic
profiles are denoted by dashed lines. (c) shows the fluid-phase
velocity (V,) versus 2 for -0.1 5 4 ) s +0.1 (solid lines). For
comparison, the solid-phase velocity profiles (V,) are shown for
6,= k0.1 (dotted lines).
in terms of the concentrations of the cations that control the
mineral dissolution rates, which in turn control the overall
reaction rate.
The results of Fig. 5 illustrate that the exact details of the
specific volume changes between the reactants and the
products do not have a controlling effect on the pore
pressure, at least for overall volume changes less than k15
per cent. This further supports the idea that the thermal
expansion of the pore fluid exerts only a minor control on
overpressuring, in keeping with the more detailed study by
Luo & Vasseur (1992).
In addition to understanding the mechanisms leading to
excess pore pressure, it is important to understand the
time-scale over which the excess pressure can be maintained
in a geological system. The results from Fig. 6 provide some
insight. For a length-scale d;p= loom, the profiles in Fig.
6(a) correspond to about 4 k m of sediments. For
V,, = 500 m Ma-', the relevant time-scale is .T = 0.2 Ma. Fig.
6(b) shows that excess pore pressure will be maintained
from T = 50 to T = 125, which can be interpreted as a time
interval of 15 Ma for A = 1. Thus, the excess pore pressure is
maintained longer than it took to accumulate the sediment
column, which took i= 50, that is, about 10 Ma. The results
from Fig. 7 demonstrate that the time for excess pressure
dissipation scales with A. For the case of A = 0.5, dissipation
will take about twice as long, say of the order of 30Ma.
Likewise, for A = 10, the excess pore pressure, which is not
much to begin with, will dissipate in less than 1.5 Ma. These
estimates are important because they give some insight into
how long high pore fluid pressures are capable of causing oil
migration in sedimentary basins. This has implications for
the timing of petroleum migration into reservoir traps. In a
more general context, in order to understand how fluids
move in any sedimentary environments, it is important to
model the pore pressure rigorously, which is why the
fundamental equations given in this paper are important.
To date, many of the 1-D compaction models have
assumed an impermeable basement. The results shown in
Fig. 8 are the first attempt at studying the effects of a pore
fluid flux at the base of a thick sedimentary sequence. One
relevant application is the common situation in which a thick
layer of low-permeability sediments (e.g. shales or
mudstones) overlies a highly permeable material such as a
sandstone layer. If the sandstone layer is not hydrodynami-
Gravitational compaction and clay dehydration
cally isolated, then it is possible for fluid to seep out of the
shale layer and into the sandstone, which would relieve
excess pore pressure as shown in Fig. 8(b). Thus, in addition
to A, the overpressuring is influenced by the basement water
flux boundary condition,
which up to now has not been
considered in most basin fluid-flow models published to
date.
e,,
9
CONCLUSIONS
This paper develops a 1-D model for the gravitational
compaction of thick sediment layers in which the sediments
undergo a dehydration reaction that releases pore fluid at
depth. Using a finite-difference method to solve the
governing partial differential equations, a series of case
studies illustrates the effects of the model parameters on the
porosity and pore-pressure profiles. This work presents
three original accomplishments: a self-consistent set of
equations that describes the dehydration reaction; a study of
the time-scale over which excess pore pressure dissipates
after sedimentation stops; a study of the effects of pore fluid
flux at the basement.
A dehydration reaction, such as smectite transforming to
illite, can increase overpressuring by about 30 per cent
provided that the sediments are relatively impermeable, that
is, A = O( l), and that sufficient water is released, n 2 5. The
exact details of the density changes of the components are
not crucial to the conclusions. However, the geochemical
details that govern the pre-exponential factor of the kinetic
rate law are important because they control the depth at
which the dehydration goes to completion, which controls
the enhancement of the overpressuring by the released fluid.
For similar reasons, the activation energy of the ratecontrolling step is also important.
The results show that, for a 4 km thick sediment layer,
with K~ = lop" m s p ' , excess pore pressure can exist over
the course of 1.5 Ma, provided that A = ql).The excess pore
pressure dissipates more quickly for large A , as expected
since this is interpreted as sediments of higher permeability.
The basement fluid-flux boundary condition can also
control the pressure profile in the sediments. If the pore
fluid drains out of the sediment layer at the basement, the
pore pressure in the bottom half of the layer is reduced,
sometimes to subhydrostatic values, which means that it is
possible for the,sediments to be overconsolidated.
The model can be improved by improving the scale-up of
constitutive laws for the permeability and the effective
stress. The coupled dehydration-compaction model can be
made more realistic by adding more kinetic and
thermodynamic details. Further studies in these areas will
help to improve the understanding of pore pressure and
fluid flow in sedimentary systems.
ACKNOWLEDGMENTS
This work was supported by the Natural Environment
Research Council through grant No. Dl/G1/189/01, along
with continuing support from Amoco (UK) Exploration
Company. The author thanks Prof. J. D. C. McConnell and
Dr A. C. Fowler for useful discussions throughout the
course of this work. T h e author is especially grateful to M.
Wangen and B. KrooR for reviewing the manuscript.
297
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