Igor Faoro
Pressure solution around two cylindrical rods in contact
1 -6
Pressure solution around two cylindrical rods in
contact
By using the Femlab program, we introduce and study a finite element model capable to
describe the rate of mass diffusion due to a progressive compaction of two infinitely long
cylindrical quartz grains, that are subjected to a constant normal stress uniformly
distributed along their diametric section area. We investigate the dissolution process up to
the reaching of the plastic deformation state of the mineral and we display its behavior as
a function of time.
1. Introduction
In recent works it has been shown that
the evolution of the mechanical and
transport proprieties of rocks containing
fractures at a variety of scales is strongly
influenced both by the mechanical
effects of crack formation, dilation, and
closure, and their interaction with
chemical effects of stress-mediated
dissolution and precipitation. Specifically, these effects seem to be rather
common either in microcracks [e.g,
Tenthorey et all., 2003] or faults
[Tadokoro and Ando, 2002; Yasuhara et
al., 2005]. Many data are current
available at high temperatures in granite
[Moore et al., 1994], at lower
temperatures in tuff [Lin et al.,1997] and
at high and low temperatures in
composite aggregates of quartz [Elias
and Hajash, 1992], halite [Gratier,
1993], calcite [Zhang et al., 1994] and
albite [Hajash et al.,1998] [e.g, Zoback
and Byerlee, 1975; Siddqi et al., 2001] .
On the contrary, there is only a few
number of studies on fractures [Moore et
al., 1994; Lin et al., 1997; Durham et
al.,2001; Polak et al.,2003; Yasuhara et
al., 2004], that can account for the
changes in the transport parameters
resulting when a fracture is subjected to
pressure solution by circulating hydrothermal fluids. Main reason for this is
due to the fact that the phenomenon of
pressure solution in fractures is not well
defined as for the case of granular
aggregates [Weyl, 1959; Coble,1963;
Rutter, 1976; Raj, 1982; Tada et al.,
1987], where it represents a dominant
mechanism for the deformation and the
compaction within the upper earth crust
[Palciauskas
and
Domenico,1989;
Stephenson et al., 1992].
In this paper we aim to study the mass
dissolution process due to the pressure
solution that occurs between the
fracture’s walls. To this aim, we
introduce a punctual and simplified two
dimensional model, that is independent
from the temperature of the circulating
fluid, where we consider only twin
cylindrical quartz rods in contact that are
subjected to a constant normal compressive stress acting along their diametric
cross section (see figure 1).
Figure 1. Schematic of the geometry
model configuration.
2. Governing equations
The rate of compaction and the final
steady configuration for the two quartz
Igor Faoro
Pressure solution around two cylindrical rods in contact
rods in the model are obtained by using
the following formula:
G u
 G 2 u  0
(1   ) y
(1)
here is thePoisson coefficient, G
denotes the modulus of rigidity and u
represents the displacement . As it will
be clarified in the following, in our
treatment only two dimensions (x,y) are
relevant.
The rate of mass diffusion equation is
given by:
M  K * 
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example a spherical shape for the grains
is adopted.
In order to analyze the problem in two
dimensions, we assume to neglect the
strengths, and the relative deformations
in the z direction produced by the load.
As a result, the cylindrical grains must
be considered as infinitely long.
Furthermore due to symmetry argument,
only the quarter part of the all system
has been evaluated. (See figure 2.)
(2)

where M denotes the dissolution mass
rate, K is the dissolution rate constant
and  is mean value of the stress.
Notice that Eq. (2) provides a link
between
the
structural-mechanical
problem and the diffusion one. Indeed, it
allows to obtain the dissolution mass rate
as a function either of the flow rate or of
the amount of strengths between the
contact boundaries of the rods.
The rate of diffusion along the x
direction is given by:
c
 2c
D 2
t
x
(3)
where D is the diffusion coefficient.
3. Formulation
Geometry assumption
The geometry of the model consists of
two cylindrical grains and the pore
volume between them. The choice of the
cylindrical shape is not casual. Indeed,
by using this geometry, it is possible to
neglect the void space around the
diametric section area of the grains that
would be otherwise present if for
Figure 2. Schematic of the final geometry model configuration, and boundary
numeration
 The plain strain problem has been
solved using the “Structural Mechanics
Module /plane strain / non linear static
analysis” application provided by the
Femlab program. In particular, the
expressions necessary in order to define
the contact boundary forces and compute
the analytical values are (applied for the
boundaries 3and 4):
Fc  t n  (en  gap) gap  0
(4)
e

Fc  t n  exp  n  gap  gap  0
 tn

(5)
(1   2 )
E 
(6)
2F
 b2  x2
  b2
(7)
b  2 F R
P
here gap denotes the distance between
the grain and the base, PB are the
contact forces, tn is the input estimate of
Igor Faoro
Pressure solution around two cylindrical rods in contact
the contact force, en represents the
penalty stiffness, b is half of the total
length, F is the applied load, R denotes
the radius of grain, is the Poisson’s
coefficient, E is the Young modulus, x
is
the
distance
along
x-axis.
Furthermore we assume that there are no
displacements along the x directions, and
no load along the y-direction (applied for
boundary 1 and 2).
 The dissolution mass rate is
computed manually using equation 2,
after calculating the average value 
 The diffusion process is solved by
using
the ”Comsol Multiphysics /
convection and diffusion / diffusion
/transient analysis” application; the flux
equation reads:
DC  No  Kb(Cb  C )
3 -6
his plasticization limit; after that, no
further deformations can occur.
Figure 3. Von Mises stress [N/m2]
(8)
here No is the inward flux, Kb is the
mass transfer coefficient and Cb denotes
the bulk concentration.
4. Solution
The results of the plain strain problem
are reported in Figure 3 and Figure 4.
The load applied is equal to 10 KN. In
particular, in Figure 4 it is also possible
to see the final deformation line of the
quartz grain at the steady state. The total
displacement distribution inside the
surface and the development of the
surface tractions along the contact
boundary between the surface grain and
the base are reported respectively in
Figure 5 and Figure 6. Specifically,
Figure 6 shows that the tensile tensions
vary from an initial value equal to 7*108
Pa down to 0 Pa in the first portion of the
boundary (see also figure7). This
behavior can be explained only by
considering that the grain has reached
Figure 4. Particular Von Mises stress
[N/m2]
Figure 5. Total displacement distribution [m]
Igor Faoro
Pressure solution around two cylindrical rods in contact
4 -6
Figure 8. Concentration [mol/m3]
Figure 6. Surface Traction in y-direction
[Pa]
In Figure 9 we report the complete
variation of concentration as a function
of time, by assuming an initial
concentration equal to 1 [mol/m3] applied on the boundary 3 and 4. Notice
that, even from such a small concentration it is possible to understand how
small is the contribute due to the mass
dissolution compare to the complete
diffusion process.
Figure 7. Surface Traction in y-direction
[Pa], plasticization limit.
As it is evident from Figure 6, the mean
stress value recorded is 3.5*108 Pa; by
considering a unitary quartz grain
volume is possible to estimate that the
value of the dissolution mass rate is
equal to 5.8*10-5[mol/s]. Furthermore by
assuming a constant value for the
dissolution rate equal to 10-12 [gr/(sPa)],
the inward flux and the bulk concentration can be estimated and we find that
they are respectively equal to 0.058
[mol/(m2s)] and 3.6*106 [mol/m3] .
In Figure 8 we report the diffusion
distribution.
Figure 9. Concentration [mol/m3]
5. Validation-parametric study
The model that we study in the first part
simulates the stress and the relative
displacements distribution that occurs in
all those structural problems where
more or less complex elements are
subject to particular load configurations.
Because of its assumptions it is almost
impossible to compare the results we
obtained with the ones given in literature. Nevertheless, we can view this
model as a first step in order to analyze
Igor Faoro
Pressure solution around two cylindrical rods in contact
5 -6
all those structural problems in which
the plasticization phenomenon need to
be investigated. In this respect, some
possible applications may be planning
studies of tunnels, tubes, circular walls,
just to mention a few (see Figure 10)
b
Figure 10. Schematic plasticization zone
for planning .
Different results can be obtained by
changing the main variables of the model, like for example material characteristics (Young modulus, Poisson modulus,
density, stiffness..), loads, and values of
relative concentrations.
In Figure 11 we study the model in the
case progressive increasing loads (500
N, 1 KN, 10 KN) are applied.
This example is particularly interesting
because it follows the reaching of the
grain plasticization limit during the time.
Indeed, starting from an initial total
elastic behavior, one can see how the
steady plasticization configuration is
progressively reached by increasing the
load.
a
c
Figure 11. Surface Traction in y-direction [Pa] at load values respective equal
to: (a) 500 N,(b) 1 KN, (c) 10 KN.
6. Conclusions
In this paper we introduced and
discussed a finite element model that can
be considered as a preliminary study of
the dissolution process caused by the
pressure solution between two cylindrical quartz rods that are subjected to a
constant normal stress distribution. The
reaching of the plasticization behavior
by the grain and the relative dissolution
mass rate have been investigated.
In the future, it could be interesting to
study the evolution of this phenomenon
when the rock starts to fail, and to
investigate how the dissolution mass rate
changes as a function both of the
temperature and of the circulating fluid.
Igor Faoro
Pressure solution around two cylindrical rods in contact
References
Coble,R.L.(1963), A model for boundary
diffusion controlled creep in polycrystalline
materials, J.Appl.Phys.,34,1679-1682.
Durham,W.B.,W.L.Boucrcier, and E.ABurton
(2001), Direct observation of reactive flow in a
single fracture, Water resour.Res.,37,1-12.
Elias,B.P., and A.Hajash (1992), Change in
quartz solubility and porosity change due to
effective stress: An experimental investigation of
pressure solution, Geology, 20,451-454.
Gratier,J.P.(1993),
experimental
pressure
solution of halite by an intender tecnique,
J.Geophisics.Res.,Lett.,20(15),1647-1650.
Hajash,A.,T.D.Carpenter, and T.A. Dewers
(199810, Dissolution and time dependent
compaction of albite sand: Experiments at 100 C
and 260 C in pH-buffered organics acid and
distilled water, Tectonophysics,295,93-115.
Lin,W.,J.Roberts, W.Glassey, and D.Ruddle
(199710,Fracture and matrix permeability at
elevated temperatures, paper presented at
Workshop on Significant Issues and Available
Data, Near-Field/Altered-Zone Coupled Effects
Expert Elicitation Project, U.S.Dep. of Energy,
San Francisco, Calif.,Nov
Moore, D.E., D.A.Lockner and J.A Byerlee
(1994), Reduction of the permeability in granite
at elevate temperatures, Science, 265 (5178),
1558-1560, 1994
Palciauskas,V.V., and P.A.Domenico (1989),
Fluid pressures in deformation porous rocks,
Water Resour.Res.,25,203-213.
Polak A, Elsworth D, Yasuhara H, Grader AS,
Halleck PM Permeability reduction of a natural
fracture under net dissolution by hydrothermal
fluid Geophysical research letters 30 (20): Art.
No. 2020 OCT 16 2003
Raj,R. (1982), Creep in polycrystalline
aggregates by matter transport though a liquid
fase, J.Geophisics.Res.,87,4731-4739.
Rutter,E.H (1976), The kinetics of rock
deformation by pressure soltution, Philos.
Trans.R.Soc.london,Ser.A, 283 ,203-219.
Siddiqi.G. ,B.Evans, G.Dresen, and D.Freund
(2001), Effect of semibrittle deformation on
transport proprieties of calcite rocks, J. Geophisics.Res.,106,8665-8686.
Tada,R.,RMaliva,and R.Siever (1987), Rate
laws for water-assisted compaction and stressinduced water-rock interaction in sandstones,
Geochic.Cosmochin.Acta,51,2295-2031.
Tadokoro,K,and M.Ando (2002), Evidence for
rapid fault healing derived from temporal
changes in S wave splitting, J.Geophisics.Res.,
Lett.,29(4),1047,doi:10.1029/2001GL013664.
6 -6
Tenthorey. E.,S.F.Cox, and H.F.Todd (2003),
Evolution of strength recovery and permeability
during fluid-rock reaction in experimental fault
zones, Earth Planet.Sci.Lett.,206,161-172.
Weyl, P.K., Pressure solution and force of
crystallization – A phenomenological theory.
Journal of geophysical research-solid earth
.64,2001-2025,1959
Yasuhara H, Elsworth D, Polak A Evolution of
permeability in a natural fracture: Significant
role of pressure solution Journal of geophysical
research-solid earth 109 (B3): Art. No. B03204
MAR 13 2004
Yasuhara H, Marone C, Elsworth D Fault zone
restrengthening and frictional healing: The role
of pressure solution, Journal of geophysical
research-solid earth 110 (B6): Art. No. B06310
JUN 24 2005
Zhang, S., S. Cox and M.Paterson, (1994), The
influence of room temperature deformation on
porosity and permeability in calcite aggregates,
J.Geophisics.Res.,99,761-755
Zoback, M., and J.Byerlee (1975), The effect of
microcracks dilatancy on the permeability of
Westerly Granite, J.Geophisics.Res.,80,752-755.