Unit 06 : Advanced Hydrogeology
Poroelasticity
Poroelasticity
• Poroelasticity is a continuum theory for the analysis
of a porous media consisting of an elastic matrix
containing interconnected fluid-saturated pores.
• In physical terms, the theory postulates that when a
porous material is subjected to stress, the resulting
matrix deformation leads to volumetric changes in the
pores.
• Since the pores are fluid-filled, the presence of the
fluid not only acts as a stiffener of the material, but
also results in the flow of the pore fluid (diffusion)
between regions of higher and lower pore pressure.
• If the fluid is viscous the behavior of the material
system becomes time dependent.
Biot
• Biot in a series of classic papers spread over
a 20 year period (Biot, 1941a, 1955, 1956a,
1962; Biot and Willis, 1957) proposed the
phenomenological model for such a material
generally adopted today.
• The application of the theory has generally
concerned soil consolidation (quasi-static)
and wave propagation (dynamic) problems in
geomechanics.
Biot Diffusion-Deformation Model
• The classical linear model of transient flow and deformation of a
homogeneous fully saturated elastic porous medium depends
on an appropriate coupling of the fluid pressure and solid stress.
• The total stress consists of both the effective stress, given by
the strain of the structure, and the pore-pressure, arising from
the fluid.
• The local storage of fluid mass results from increments in the
density of the fluid and the dilation of the structure.
• The combinations of the fluid mass conservation with Darcy’s
law for laminar flow, and of the momentum balance equations
with Hooke’s law for elastic deformation, result in the Biot
diffusion-deformation model of poroelasticity.
Constitutive Equations
• The poroelastic constitutive equations are simple
generalizations of linear elasticity whereby the fluid
pressure field is incorporated in a fashion entirely
analogous to the manner in which the temperature
field is incorporated in thermo-elasticity.
• Two basic phenomena underlie poroelastic behavior:
– solid-to-fluid coupling occurs when a change in applied
stress produces a change in fluid pressure or fluid mass;
– fluid-to-solid coupling occurs when a change in fluid
pressure or fluid mass is responsible for a change in the
volume of the porous material.
Uncoupled Problem
• The magnitude of the solid-to-fluid coupling depends
on the compressibility of the framework of the porous
material, the compressibility of the pores, the
compressibility of the solid grains, the compressibility
of the pore fluid, and the porosity.
• If only fluid-to-solid coupling were important, the flow
field can be solved independently of the stress field.
• The stress field (and hence strain and displacement
fields) can be calculated as functions of position and
time once the flow field has been determined as a
function of position and time.
• This one-way coupling known as the uncoupled
problem and allows some groundwater flow models
to successfully predict subsidence.
Coupled Problem
• When the time-dependent changes in stress
feed back significantly to the pore pressure,
two-way coupling is important, and is called
the coupled problem.
• Applied stress changes in fluid-saturated
porous materials typically produce significant
changes in pore pressure, and this direction
of coupling is significant.
• For this reason, it may be necessary to
consider the loading effects of large piles of
waste materials in groundwater flow models
employed in the mining industry.
Effective Stress
• Before proceeding to general poroelasticity, we will
review the simple case of 1-D consolidation.
• The effective stress principle gives:
s = s’ + p
where s is the total stress, s’ is the effective stress
and p is the pore pressure
• Under constant total stress conditions, a change in
pore pressure generates an equal and opposite
change in effective stress:
ds = ds’ + dp = 0
ds’ = -dp = -rwgdh
Water Compressibility
• For isothermal compressibility of water:
bw = 1/Kw = -(1/Vw)Vw/p
where bw is the compressibility of water, Kw is the
bulk modulus of compression of water, Vw is the
volume of water and p is the pore pressure.
• Mass conservation requires:
rwdVw + Vwdrw = 0
• Using the definition of compressibility:
dVw = -Vwdrw/rw = -Vw bwdp
• dVw is the volume change due to compression of
water as a result of a pore pressure increase dp.
Pore Compressibility
• The bulk compressibility of a poroelastic material
under one dimensional compression is given by:
bb = 1/Kb = -(1/Vb)Vb/s’ = -(1/Vp)Vp/s’ = bp = 1/Hp
where Vb is the bulk volume and Vp is the pore
volume. Kb is the bulk modulus of compression and
Hp is a vertical bulk modulus of pore compression.
• For incompressible grains: Vb = Vp
• For or a total volume change dVb:
dVb = dVp = -bpVbds’ = bpVpdp
• dVp is the pore volume change as a result of an
effective stress change -ds’ = dp
Total Volume Change
• The total volume change is:
dVp - dVw = bpVbdp + bwVwdp
• The water volume Vw = nVb so the volume change is:
dVp - dVw = bpVbdp + nbwVbdp = Vb(bp + nbw)dp
• From the expression for total head where z is a
constant:
h = z + p/rwg or p = rwgh - rwgz
dp = rwgdh
• Note the implicit assumption in this conversion is that
rw is not a function of pressure.
• Hence:
dVp - dVw = Vb(bp + nbw)rwgdh
Total Stress Change
• The effective stress principle for hydrostatic
conditions gives:
s = s’ + p
• Consider an excess pore pressure Dp as a result of
an applied total stress increment Ds:
s + Ds = s’ + (p + Dp)
• Flow occurs in order to dissipate the excess pore
pressure increment Dp and over the drainage period
the effective stress is increased from s’ to s’ + Ds’
such that:
s + Ds = s’ + Ds’ + p
Conservation Statement
• In a deforming poroelastic medium there are two conservation
statements. One for fluid mass conservation and one for solid
mass conservation. Restricting the discussion to the 1-D case:
• For the fluid mass: -[nrwvw]/z = [nrw]/t
• For the solid mass: -[(1-n)rsvs]/z = [(1-n)rs]/t
where vw and vs are the average velocities with respect to a
static frame of reference.
• The Darcy flux is defined relative to the solid matrix so:
qz = n(vw-vs)
• Now we introduce a material derivative (not a partial derivative)
that follows the motion of the solid phase (ie any subsidence):
dn/dt = n/t + vs n/z
• The total change in porosity with time includes components due
to pore compression within the reference volume and pore
displacement with respect to the reference volume.
Grain Incompressibility
• Now we assume the grains are incompressible, so rs
is constant and all derivatives of rs are zero and:
• For the solid mass: -rs[(1-n)vs]/z = rs(1-n)/t
-[(1-n)vs]/z = (1-n)/t
-(1-n)vs/z + vsn/t = -n/t
(1-n)vs/z = n/t + vsn/t
(1-n)vs/z = dn/dt
• For the mass fluid flux, derived from Darcy flux:
rwqz = nrw(vw-vs)
-(rwqz)/z = -(nrwvw)/z + (nrwvs)/z
-(rwqz)/z = (nrw)/t + vs(nrw)/z + nrwvs/z
-(rwqz)/z - nrwvs/z = (nrw)/t + vs(nrw)/z = d(nrw)/dt
-(rwqz)/z - nrwvs/z = d(nrw)/dt
Approximate Conservation Equation
• The two derived equations are:
(1-n)vs/z = dn/dt
-(rwqz)/z - nrwvs/z = d(nrw)/dt
• From the first equation: vs/z = [1/(1-n)]dn/dt
• Substituting in the second equation:
-(rwqz)/z - rw [n/(1-n)]dn/dt = d(nrw)/dt
-(rwqz)/z = rw[n/(1-n)]dn/dt + rwdn/dt + ndrw/dt
-(1/rw)(rwqz)/z = [n/(1-n)]dn/dt + dn/dt + (n/rw)drw/dt
-(1/rw)(rwqz)/z = [1/(1-n)]dn/dt + (n/rw)drw/dt
• If the volumetric strain of the solid matrix is small, then the total
derivatives can be replaced by the partial derivatives:
-(1/rw)(rwqz)/z = [1/(1-n)]n/t + (n/rw)rw/t
• Assuming rw is independent of z, this reduces to
-qz/z = [1/(1-n)]n/t + (n/rw)rw/t
Consolidation Equation
• Conservation Equation:
-qz/z = [1/(1-n)]n/t + (n/rw)rw/t
where the first RHS term represents pore volume change and
the second term represents fluid volume change
• Rewriting the pore compression/expansion term:
-qz/z = bp(p/t - s/t) + (n/rw)rw/t
where p is the excess pore pressure increment as a result of the
(vertical) total stress increment s.
• Rewriting the fluid compression/expanision term:
-qz/z = bp(p/t - s/t) + nbwp/t
• Rearranging:
-qz/z = (bp+ nbw)p/t - bps/t
• This is a form of the 1-D consolidation equation in terms of the
pressure dependent variable.
Companion Strain Equation
• 1-D consolidation equation:
-qz/z = (bp+ nbw)p/t - bps/t
• Rearranging again:
-qz/z = nbwp/t – bp(s/t - p/t)
-qz/z = nbwp/t – bp(s – p)/t
-qz/z = nbwp/t – bps’/t
• Recognizing the stress-strain relationship for pore
deformation, s’ = e/bp where e is the 1-D (vertical)
volumetric strain:
-qz/z = nbwp/t – e/t
• The is the companion equation to the 1-D
consolidation equation written in terms of strain.
More Familiar Forms
• 1-D consolidation equation:
-qz/z = (bp+ nbw)p/t - bps/t
• 1-D companion strain equation:
-qz/z = nbwp/t – e/t
• The equations become more familiar if we recognize:
qz = -Kzh/z = -(Kz/rwg)p/z
• 1-D consolidation equation:
Kz2p/z2 = rwg(bp+ nbw)p/t - rwgbps/t
Kz2p/z2 = Ssp/t - rwgbps/t
Kz2h/z2 = Ssh/t - bps/t
• 1-D companion strain equation:
Kz2p/z2 = n rwgbwp/t – rwge/t
Kz2h/z2 = nbwp/t – e/t
General Poroelasticity
• A more complete model assumes that all components
of the porous medium are compressible, the bulk
volume (bb), the solid grains (bs), the fluids (bw), and
the pores (bp = bb - bs).
• The key concepts of Biot’s 1941 poroelastic theory,
for an isotropic fluid-filled porous medium, are
contained in just two linear constitutive equations, for
the case of an isotropic applied stress field s.
• In addition to s, the other field quantities are the
volumetric strain e = dV/V, where V is the bulk
volume, the increment of fluid content , and the fluid
pressure p.
Rice and Cleary
• Rice and Cleary’s 1976 reformulation of Biot’s
linear poroelastic constitutive equations has
been adopted widely for geophysical
problems.
• Rice and Cleary chose constitutive
parameters that emphasized the drained
(constant pore pressure) and undrained (no
flow) limits of long- and short-time behavior,
respectively.
Alternate Formulations
• Rice and Cleary defined fluid mass content (mf ) to be the fluid
mass per unit reference volume.
• The change in fluid mass content relative to the reference state,
Dmf = mf - mfo
• is related to increment of fluid content  by:
 = Dmf / rfo
where rfo is the fluid density in the reference state.
• Fluid mass content is a state property, whereas the increment of
fluid content  must be viewed in the hydrogeologic sense as
the volume of fluid transported into or out of storage.
• Jacob (1940) also defined storage in terms of fluid mass.
• The great advantage of the original Biot formulation using fluid
increment  as a primary variable is that it is dimensionless, like
strain, and the constitutive equations do not have to include a
density factor.
Biot Formulation
• The volumetric strain dV/V is taken to be positive in expansion
and negative in contraction.
• Stress s is positive if tensile and negative if compressive.
• Increment of fluid content  is positive for fluid added to the
control volume and negative for fluid withdrawn from the control
volume.
• Fluid pressure (pore pressure) p greater than atmospheric is
positive.
• The constitutive equations simply express e and  as a linear
combination of s and p:
e = a11s + a12p
(1)
 = a21s + a22p
(2)
• Generic coefficients aij are used in equations (1) and (2) to
emphasize the simple form of the constitutive equations.
Linear Equations
• The first constitutive equation is a statement of the
observation that changes in applied stress and pore
pressure produce a fractional volume change.
• The second constitutive equation is a statement of
the observation that changes in applied stress and
pore pressure require fluid be added to or removed
from storage.
• This second statement implies that applied stress
and/or pore pressure might be treated as fluid source
terms in groundwater flow equations.
Poroelastic Constants
• Poroelastic constants are defined as ratios of field
variables while maintaining various constraints on the
elementary control volume.
• The physical meaning of each coefficient in equations (1)
and (2) is found by taking the ratio of the change in a
dependent variable (e,) relative to the change in an
independent variable (s, p), while holding the remaining
independent variable constant:
•
•
•
•
a11 = e/s |p=0 = 1/K = bb
a12 = e/p |s=0 = 1/H = bp
a21 = /s |p=0 = 1/H = bp
a22 = /p |s=0 = 1/R = Ss = bp + nbw
constant pore pressure
constant total stress
constant pore pressure
constant total stress (3)
Drained Compressibilty bb
• The coefficient bb =1/K is obtained by
measuring the volumetric strain due to
changes in applied stress while holding pore
pressure constant.
• This state is called a drained condition.
• Therefore, bb is the compressibility of the
material measured under drained conditions,
and K is the drained bulk modulus.
Poroelastic Expansion Coefficient bp
• The coefficient 1/H is a property not
encountered in ordinary elasticity.
• It describes how much the bulk volume
changes due to a pore pressure change while
holding the applied stress constant.
• It is the pore compressibility, that is, the
drained bulk compressibility less the grain
compression, 1/H = bp.
• By analogy with thermal expansion, it is
called the poroelastic expansion coefficient.
Unconstrained Specific Storage Coefficient Ss
• The coefficient 1/R is a specific storage coefficient
measured under conditions of constant applied
stress; it is the ratio of the change in the volume of
water added to storage per unit aquifer volume
divided by the change in pore pressure.
• Here the specific storage coefficient at constant
stress is also called the unconstrained specific
storage coefficient and is designated Ss .
• The specific storage (Ss) used in the familiar
groundwater flow and consolidation equations is
given by Ss = rwgSs.
• In soil mechanics Ss may be better known as mw.
Biot Coefficients
• The introduction of three coefficients :
– drained compressibility
• bb = 1/K
– poroelastic expansion coefficient
• bp = 1/H
– unconstrained specific storage coefficient
• Ss = 1/R
• completely characterizes the poroelastic
response for isotropic applied stress. The Biot
formulation is both simple and elegant.
Coefficient Matrix
• These three coefficients are the three independent
components of the symmetric 2 x 2 coefficient matrix
relating strain and fluid volume increment to stress and
pore pressure.
• Using equation (3) in equations (1) and (2) yields:
e = b b s + bp p
(4)
 = bp s + Ss p
(5)
• The drained compressibility (bb) and the unconstrained
specific storage coefficient (Ss) are the diagonal
components.
• The poroelastic expansion coefficient (bp) is the offdiagonal matrix component.
Constrained Specific Storage Coefficient Se
• Biot also introduced the coefficient 1/M, which
is the specific storage coefficient at constant
strain.
• It is called the constrained specific storage
coefficient and designated Se.
• It is convenient here define the constrained
specific storage Se as rwgSe in an exactly
similar manner to Ss.
Se = p/s |e=0 = 1/M
(6)
Skempton’s Coefficient B
• Skempton’s coefficient (B) is defined to be the ratio of
the induced pore pressure to the change in applied
stress for undrained conditions - that is, no fluid is
allowed to move into or out of the control volume:
B = - p/s |=0 = R/H = bp/Ss
(7)
• The negative sign is included in the definition
because the sign convention for stress means that an
increase in compressive stress inducing a pore
pressure increase implies a decrease in s for the
undrained condition, when no fluid is exchanged with
the control volume.
Meaning of the B Coefficient
• If compressive stress is applied suddenly to a small
volume of saturated porous material surrounded by
an impermeable boundary, the induced pore pressure
is B times the applied stress.
• Skempton’s coefficient must lie between zero and
one and is a measure of how the applied stress is
distributed between the skeletal framework and the
fluid.
• It tends toward one for saturated soils because the
fluid supports the load. It tends toward zero for gasfilled pores in soils and for saturated consolidated
rocks because the framework supports the load.
Biot-Willis Coefficient a
• The ratio of the pore compressibility to the bulk
compressibility bp / bb is known as the Biot-Willis
coefficient a or Skempton’s A parameter and
represents the ratio of the volume of water squeezed
out of a rock to total volume change for deformation
at constant fluid pressure:
 a = bp / bb = K / H = 1 - bs / bb
(8)
• Values of a tend toward one for porous materials
where grain compressibilities are insignificant, such
as unconsolidated sediments, but tend towards much
lower values for consolidated rocks with more rigid
skeletons where pore compression is inhibited.
Specific Storage Coefficients
• The (constrained) specific storage
coefficient at constant strain (Se) is
smaller than the (unconstrained)
specific storage coefficient at constant
stress (Ss) due to the constraint that the
bulk volume remains constant.
• Algebraic manipulation shows that:
Ss = Se + abp
(9)
Relationships
• The following equations help to clarify the
relationship between Ss and Se :
Ss = bp + nbw
(10)
Se = (1 - a)bp
+ nbw
(11)
• Notice that Ss - Se = abp and that when a
approaches one, as it does for many
unconsolidated materials, the constrained
specific storage coefficient is simply nbw.
• When a approaches zero, Se approaches Ss,
as is the case for bedrock aquifers, rigid
consolidated rocks with low porosities.
Typical Values
bb
-1
Pa
Clay
1.6E-08
Mudstone 4.6E-10
Sandstone 1.1E-10
Limestone 3.0E-11
Basalt
2.2E-11
Lithology
bp
-1
Pa
1.6E-08
4.4E-10
7.8E-11
2.4E-11
5.1E-12
Ss
-1
Pa
1.6E-08
5.3E-10
1.2E-10
9.4E-11
4.3E-11
Ss
-1
m
1.6E-04
5.2E-06
1.2E-06
9.2E-07
4.3E-07
a
Ratio
1.00
0.96
0.72
0.80
0.24
B
Ratio
0.99
0.83
0.66
0.25
0.12
• Data are derived from published information on the
compressibilities of various materials from a large number of
sources. Palciauskas and Domenico (1989) and Kümpel (1991)
are the key references but Domenico and Schwartz (1997) is
the most accessible.
Flow Equations
• Palciauskas and Domenico (1989) presented the flow
equations for completely deformable porous media.
• Equation (12) incorporates stress changes over time
(the second RHS term) as a mechanism to generate
excess pore pressures:
 p =   [ K p] + B s
(12)
t
Ss
t
• The companion equation (13) incorporates pore
volume strain (the second RHS term) as a
mechanism to generate excess pore pressures:
 p =   [ K p] + a e
(13)
t
Se
Se  t
Loading and Barometric Efficiencies
• The equations can be used to predict both
drained and undrained response of water
levels to loading.
• Equation (12) predicts that loading efficiency
(tidal efficiency or pore pressure coefficient)
p/s = B and barometric efficiency (1 – B)
for undrained conditions.
• Equation (13) predicts the fluid pressure
response to earth tides  p/e = a / Se at
constant fluid mass.
Tidal Fluctuations
Earth Tides
Barometric Response
1-D Consolidation Theory
• Biot showed at an early stage that Terzaghi’s one-dimensional
consolidation problem is a special case of his theory. Onedimensional consolidation theory (Terzhagi consolidation) assumes:
• (1) the stress increment generating the excess fluid pressure is
vertical and
• (2) the solid components of the rock/soil matrix are incompressible.
• The first assumption is implicit in the three-dimensional forms given
in equations (12) and (13) since no shear stress-strain components
are included.
• The second assumption means that bulk volume changes are
exactly equal to pore volume changes, that is, a is one. These are
reasonable approximations for many practical purposes but neither
assumption is strictly true.
Modelling Subsidence
• Most of the calculations used to estimate subsidence
are based on Terzaghi’s simple one-dimensional
analytical model (Poland and Davis, 1969).
Terzaghi’s principle of effective stress coupled with
Hubbert’s force potential and Darcy’s Law provided
the basis for one-dimensional subsidence modelling
(Gambolati et al, 1974).
 p =   [ K p] + Q
(14)
t
Ss
• When the change in mean total stress is assumed to
be negligible within a ground-water basin, equation
(12) reduces to equation (14) and becomes identical
to the traditional groundwater flow equation used in
many models.
San Joaquin Valley
Settlement vs Water Level Decline
Terzhagi’s Equation
• For the one-dimensional case writing Cv = K/Ss ,
reveals Terzhagi’s familiar consolidation equation:
 p = K  2p =
K  2p = Cv  2p
(15)
 t Ss  z2 mwrwg  z2
 z2
• This is perhaps the most famous equation
incorporating poroelastic effects but it conceals a
very large number of assumptions and simplifications
which make it inappropriate as a starting point for a
more rigorous analysis.
Las Vegas Subsidence Profile
Modelling Total Stress Changes
• Most numerical models used to simulate groundwater
flow (equation 14) were derived with the assumption
that the total stress imposed on the aquifer remains
constant with time.
• If the aquifer is subjected to time-varying total
stresses (loading), these models are inappropriate.
• In such cases, a more general model that accounts
for aquifer deformation is required.
• The effect of changes in total stress on an aquifer
system has been considered by many authors for a
variety of applications.
Total Stress Change Examples
• Gibson (1958) discusses the impact of increasing clay thickness
on the consolidation of a clay unit and introduces a source term
to the equation describing the excess pore-water pressure
increment associated with the loading.
• Abnormal pressure, both over-pressure and under-pressure, in
both aquifers and oil reservoirs, has been studied by Bredehoeft
and Hanshaw (1968), Neuzil (1993), and Neuzil (1995).
• For these systems, changes in the total stress applied to the
system occur as a result of the slow geologic processes of
sedimentation or erosion.
• Provost et al. (1998) considered the preconsolidation pressures
induced by changes in total stress due to glaciation in a long-term
model for a proposed nuclear waste repository.
• Gardner et al. (1998) found that inclusion of pore pressures
derived from tidal loading of peat in salt marshes was necessary
in order to match observed piezometric responses.
Rate of Pore Pressure Dissipation
• Equation (15) is readily solved for constant head
conditions (Terzhagi, 1943) and an analytical
expression can be derived to predict the time needed
for dissipation of a specified percentage of any
increment of excess pore-pressure by vertical
drainage:
tp = d2SsTv = d2Tv
(16)
K
Cv
where tp is the time, d is the drainage distance, and
Tv is consolidation time factor.
Vertical and Radial Drainage
• Setting Tv to a constant value corresponding to 90%
pore-pressure dissipation (average pore pressure
over the interval of thickness d ) gives:
t90 = 0.848 d 2
(17)
Cv
• A similar analysis, for radial drainage, yields a similar
equation (after Carslaw and Jaeger, 1947) when the
consolidation time factor (Tr) for 90% dissipation is
somewhat lower:
t90 = 0.333 r 2
(18)
Cr
Constant Loading Rate
• Abashi (1970) derived an addition useful analytic
solution for the vertical consolidation equation with a
constant rate of monotonic loading (rather than a
constant instantaneous load). The time factor Tc for
90% dissipation is larger than for the instantaneous
load case:
t90 = 0.946 d 2
(19)
Cv
• The constant loading rate solution is not used as
commonly at the Terzhagi solution in practice, but is
valuable for the analysis of embankment construction
and pile building.