The Pennsylvania State University
College of Earth and Mineral Sciences
Department of Energy and Geo-Environmental Engineering
Thermo-Hydro-Mechano-Chemical Couplings that
Define the Evolution of Permeability in Rock Fractures
A thesis summary
by
Hideaki Yasuhara
1
Attribution
This thesis comprises a series of six papers either published or in-submittal in journals. In
chronological order these papers are:
1.
Yasuhara, H., Elsworth, D., and Polak, A. (2003), A mechanistic model for compaction of
granular aggregates moderated by pressure solution. J. Geophys. Res. 108(B11), 2530,
doi:10.1029/2003JB002536.
2.
Yasuhara, H., Elsworth, D., and Polak, A. (2004), The evolution of permeability in a natural
fracture: significant role of pressure solution. J. Geophys. Res., Vol. 109, B03204,
doi:10.1029/2003JB002663
3.
Yasuhara, H., Marone, C., and Elsworth, D. (2005), Fault zone restrengthening and
frictional healing: the role of pressure solution. J. Geophys. Res. 110(6), B06310,
doi:10.1029/2004JB003327.
4.
Yasuhara, H., Polak, A., Mitani, Y., Grader, A., Halleck, P., and Elsworth, D. (2006),
Evolution of fracture permeability through fluid-rock reaction under hydrothermal
conditions. Earth Planet. Sci. Lett. 244. 186–200.
5.
Yasuhara, H., Elsworth, D., Polak, A., Liu, J., Grader, A., and Halleck, P. (2006),
Spontaneous permeability switching in fractures in carbonate: lumped parameter
representation of mechanically- and chemically-mediated dissolution. In press. Trans.
Porous Media. 30 pp.
6.
Yasuhara, H and Elsworth, D. (2006), A numerical model simulating reactive transport and
evolution of fracture permeability. In press. Int. J. Numer. Anal. Meth. Geomech. 49 pp.
In all cases, first-authorship establishes my principal role in developing and executing the
analyses.
Abstract
Coupled thermal-hydraulic-mechanical-chemical (THMC) processes exert significant
influences on the evolution of the mechanical and transport properties of rocks. The competition
between agents that reduce porosity (grain compaction, pressure solution and precipitation) and
those that generate porosity (dissolution and dilation) controls rates, magnitudes, and senses of
permeability modification, strength gain, and stiffness change. In turn, these processes are
important in defining the evolution of porosity and permeability in subsiding basins, in
2
geothermal and petroleum reservoirs, around repositories for the entombment of radioactive
wastes, and in defining rates or strength gain on faults that in turn impact recurrence times and
magnitudes of earthquakes.
Significantly, coupled THMC processes strongly control changes in fracture permeability.
The sense of permeability change is controlled by a complex interaction of the evolution of
effective stress and temperature in the sample, and species concentrations within the transiting
fluid. Currently, there is no conclusive view of whether fractures subject to applied stresses and
chemical net dissolution will seal or alternately gape. Also, recent flow-through experiments
examining permeability evolution in limestone [Polak et al., 2004] and novaculite [Yasuhara et al.,
2006a] samples show spontaneous switching between permeability reduction and permeability
increase, without any change in stress or fluid influent chemistry (Figure 1) – and only benign
changes in temperature. These observations are attributed to result from the spontaneous
switching between a regime where contacting asperity dissolution by pressure solution is
dominant, to one where free-face etching prevails. Correspondingly, anticipated senses, rates, and
magnitudes of evolution in permeability remain ill-constrained. Furthermore, the critical
conditions of stresses, temperatures, and chemistry in triggering the spontaneous switching are
poorly defined. Resolving this knowledge-deficiency is the main focus of the current work.
In this work, quantitative models are developed to explain significant and anomalous
observed changes in the permeability of fractures circulated by hydrothermal fluids.
Permeabilities are shown to reduce under net dissolution, and to anomalously switch between
permeability reduction and enhancement with no change in ambient conditions. This anomalous
behavior is attributed to the significant role of pressure solution in driving the redistribution of
mineral mass within fractures. A lumped-parameter model is developed to represent this behavior.
This model linked processes of dissolution at the stressed interfaces of grain-to-grain contacts,
3
diffusive transport of dissolved matter from the interface to the pore space, and finally
precipitation at the less-stressed surface of the grains. This behavior has been applied to both
hydraulically closed and open systems, and: (i) to describe controls on the compaction of quartz
aggregates, as analogs for compacting sedimentary basins, and to follow the evolution of porosity
and related permeability (Chapter 1), (ii) to explain the significant observation that under certain
circumstances net dissolution from contacting rock fractures results in a net decrease in porosity
and permeability (Chapter 2), and (iii) to examine the frictional restrengthening attributed to the
growth and welding of grain contact areas in the simulated fault gouge (Chapter 3). The model
defines and constrains the significant roles of temperature, applied stress, and fluid chemistry on
the compaction of quartz aggregates, the observed reduction in permeability of fractures in
novaculite, and the frictional restrengthening in quartz aggregates, mediated by pressure solution.
The lumped parameter model, incorporating the role of pressure (dis)solution, diffusion, and
precipitation, has replicated the observed monotonic closure of a natural fracture in novaculite
under constant effective stresses and at moderately elevated temperatures. However, companion
experiments of permeability evolution in limestone (Chapter 4) and another novaculite (Chapter
5) sample do not show similar monotonic decreases in permeability; the spontaneous switching in
permeability change from aperture reducing to aperture increasing without significant changes in
environmental conditions, and are not explicable by the same simple model. To accommodate
this switching the model has been extended to accommodate free-face dissolution or precipitation
that may be significant in some circumstances. The extended model enables the principal
characteristics of observed behavior (i.e., switching from net closure to net gaping) to be
followed with time although is incapable of replicating the rapid observed growth in permeability,
that likely results from the development of the through-going flow conduit (Chapter 4).
The inability of lumped models to consistently follow the evolution in fracture permeability
4
where dissolution dominates suggests the significant control exerted by fracture topography and
void structure. A distributed parameter model is developed and applied to represent the evolution
in fracture aperture mediated by the significant processes of pressure solution and free-face
dissolution (Chapter 6). The model employs an initial realization of fracture topography to
generate a distribution of fracture apertures. From this the evolving flow field is determined using
Reynolds equation, and this in turn is used to calculate the combined influence of chemical
processes of pressure solution and free-face dissolution/precipitation in sequentially modifying
the initial aperture distribution. Significantly, where advective flows dominate, as is often the
norm, the transport equation is solved in mixed Lagrangian-Eulerian form, enabling high Peclet
number flows to be accommodated. The distributed model shows good agreement in tracking
changes in fracture aperture and solute concentration observed within a well-constrained
flow-through test completed on a natural fracture in novaculite (Chapter 5) - although a sharp
reduction of the fracture aperture during the early experiment period is unable to be followed.
5
Figure 1. Spontaneous switching of fracture aperture (or related permeability).
(a) Limestone [Polak et al., 2004]; (b) Novaculite [Yasuhara et al., 2006a]
6
Chapter 1: A Mechanistic Model for Compaction of Granular Aggregates Moderated by
Pressure Solution
A mechanistic model is proposed to describe the compaction of porous aggregates (see
Figure 2) that incorporates the important effects of grain interpenetration that accompany the
redistribution of mass due to the three serial processes of dissolution, diffusion, and precipitation,
which are respectively defined as,
2
2
dM diss 3π V m (σ a − σ c ) k + ρ g d c
,
=
dt
4R T
dM diff
dt
dM prec
dt
=
2π ω Db
C int − C pore ,
d
ln⎛⎜ c ⎞⎟
⎝ 2a ⎠
= Vp
(
)
A
k − (C pore − C eq ) ,
M
(1)
(2)
(3)
where dMdiss/dt, dMdiff/dt, and dMprec/dt represent the dissolution, diffusion, and precipitation
mass fluxes, respectively. In Eq. (1) Vm is the molar volume of the solid, σa is the disjoining
pressure equal to the amount by which the pressure acting at contact area exceeds the hydrostatic
pore pressure, σc is the critical stress, which defines the stress state where the compaction of
grain aggregates will effectively halt, k+ is the dissolution rate constant of the solid, ρg is the grain
density (2650 kg m-3 for quartz), dc is the diameter of the asperity contact, R is the gas constant,
and T is the temperature of the system. In Eq. (2) ω is the thickness of the water film trapped at
the interface, Db is the diffusion coefficient, a is an infinitesimal length substituted to avoid a
singularity in integrating Fick’s first law to the center of a circular contact area, and (C int ) x = a
and (C pore ) x=dc / 2 are mineral concentrations in the interface fluid and pore space, respectively. In
Eq. (3) Vp is the pore volume, k- is the precipitation rate constant of the dissolved mineral, and
7
Ceq is the equilibrium solubility of the dissolved mineral. A is the relative grain surface area, and
M is the relative mass of the fluid, which are dimensionless quantities defined by Rimstidt and
Barnes [1980]. The incorporation of these processes, at a micromechanical level allows the
compaction of porous aggregates to develop in a natural manner, and the dominant process to
develop naturally, at any particular effective stress or temperature regime. The dominant
compaction rate-limiting process may switch during the progress of compaction, as the local
grain-to-grain contact geometry is modified with interpenetration. As interpenetration occurs,
diffusive transport pathways are lengthened, and the rate of mass transfer to the water-filled pore
is reduced.
Although relatively simple, the model is capable of representing the compaction process,
including defining the important contribution of mass to the evolving pore fluid mass
concentration, as shown in Figures 3 and 4. This specific attribute enables differences in the
compaction behavior of both hydraulically closed and open systems to evolve naturally. For
closed systems, porosity reduction increases, in both rate and ultimate magnitude as temperature
of the system increases; the process of compaction resulting from grain interpenetration that may
operate over many centuries at 75 °C, are shown to take but a fraction of a year at 300 °C
(Figures 5 and 6). For this range of temperatures, the effect of hydraulic circulation is to reduce
the ultimate diminution of porosity, as aqueous precipitates are advected from the system, but
with negligible effect on evolving compaction strains. This is apparent, even for unreasonably
high replenishment at rates, defined as tc, of the order of 0.1 to 1 pore volumes per day. The
diminution of concentrations of pore fluids by flushing is greatest where diffusive transport to the
pore fluid is smallest, at low temperatures. The asymptotic porosity is controlled by the
8
magnitudes of the heat and temperature of fusion of quartz, the ambient temperature, and the
form of packing of the aggregate. The rate that this asymptotic porosity is attained is controlled
by temperature and stress magnitudes. For temperatures in the range 75°C to 300 °C and stresses
of the order of 35 MPa considered here, the process is dissolution controlled. Temperature exerts
a dominant influence on compaction rate, due to the exponential form of the Arrhenius-type
dependence of dissolution rate on temperature. This thermal control is much stronger than that of
stress level.
9
σ
Δv
σ
grain
dissolution
Time
precipitation
diffusion
grain
σ
σ
Δb
σ
σ
Time
fracture
σ
σ
Figure 2. Schematic of pressure solution for twin contacting grains comprising an
aggregate and fractured rock comprising a discontinuity. At the contacts the mineral
dissolves due to high localized stresses, and dissolved mass diffuses from the interface
into the pore space. Finally, precipitation occurs on the free-faces of the pore walls.
10
Figure 3. Comparison of porosity reduction with time between the experimental data of
Elias and Hajash [1992] and the predictions of the model. Open triangles represent the
prediction using parameters shown in Table1 and open squares represent the prediction
using modified values, (a) with Db = 1.0 × (1.12 ×10-9) m2 s-1, k+ = 1.8 × (2.51 ×10-9)
mol m-2 s-1, and k– = 13.0 × (1.42 ×10-7) s-1 for σeff = 69.9 MPa and T = 150 °C, (b)
with Db = 1.0 × (1.12 ×10-9) m2 s-1, k+ = 3.0 × (2.51 ×10-9) mol m-2 s-1, and k– = 60.0 ×
(1.42 ×10-7) s-1 for σeff = 34.96 MPa and T = 150 °C, and (c) with Db = 1.5 × (1.12
×10-9) m2 s-1, k+ = 1.0 × (2.51 ×10-9) mol m-2 s-1, and k– = 1.0 × (1.42 ×10-7) s-1 for σeff
= 17.2 MPa and T = 150 °C.
11
Figure 4. Comparison of silica concentration in the pore fluid between the
experimental data of Elias and Hajash [1992] and the predictions of the model. (a)
For σeff = 69.9 MPa and T = 150 °C, open squares represent the prediction using
modified values of Db = 1.0 × (1.12 ×10-9) m2 s-1, k+ = 1.8 × (2.51 ×10-9) mol m-2 s-1,
and k– = 13.0 × (1.42 ×10-7) s-1. (b) For σeff = 34.96 MPa and T = 150 °C, open
squares represent the prediction using modified values of Db = 1.0 × (1.12 ×10-9) m2
s-1, k+ = 3.0 × (2.51 ×10-9) mol m-2 s-1, and k– = 60.0 × (1.42 ×10-7) s-1.
12
Figure 5. Comparison between closed and open system on long-term compaction
behavior for σeff = 34.96 MPa and T = 75 °C. Predictions are conducted utilizing
modified values of Db = 1.0 × (4.90 ×10-10) m2 s-1, k+ = 3.0 × (3.19 ×10-11) mol m-2 s-1,
and k– = 60.0 × (6.61 ×10-9) s-1. Calculations of open system for low and high
velocities of inlet flow are made with tc = 10 and 1 day, respectively. (a) Porosity
reduction with time. (b) Strain increase with time. (c) Silica concentration change with
time.
13
Figure 6. Comparison between closed and open system on long-term compaction
behavior for σeff = 34.96 MPa and T = 300 °C. Predictions are conducted utilizing
modified values of Db = 1.0 × (3.06 ×10-9) m2 s-1, k+ = 3.0 × (5.05 ×10-7) mol m-2 s-1,
and k– = 60.0 × (5.67 ×10-6) s-1. Calculations of open system for low and high velocities
of inlet flow are made with tc = 10 and 1 day, respectively. (a) Porosity reduction with
time. (b) Strain increase with time. (c) Silica concentration change with time.
14
Chapter 2: The Evolution of Permeability in a Natural Fracture: The Significant Role of
Pressure Solution
A mechanistic model previously developed in Chapter 1 [Yasuhara et al., 2003] is extended
to represent the processes of aperture change and related changes in permeability in a natural
fracture that incorporates the significant role of pressure solution (see Figure 2). This behavior is
represented by the three serial processes of dissolution, diffusion, precipitation, that redistribute
mineral mass from contacting asperities either to the fracture walls, or to the system exterior. The
interaction of these processes at a representative contacting asperity (Figure 7) allows the closure
of a contacting fracture to be followed with time under arbitrary conditions of effective stress and
temperature. A simple relation between fracture aperture and fracture contact-area ratio (Figure
8) is defined to represent the geometric modification of the fracture surfaces as compaction
proceeds, and the fracture walls attempt to interpenetrate.
The extended model is capable of representing the compaction process, and changes in
fracture aperture and mineral mass concentration in the effluent fluid may be followed with time.
Predictions of the evolution in fracture aperture replicate experimental measurements, but
aqueous concentrations of silica are underestimated, even with the inclusion of roughness factors
to account for the mismatch between macroscopic and microscopic contact areas (Figures 9 and
10). This mismatch may result from a potentially dominant contribution of free-face dissolution
to aqueous concentrations in the silica-undersaturated permeant, relative to a mechanical
response dominated by minor mass removal from limited areas of contact - fracture closure is
amplified by a small asperity contact area. Rates of pressure solution are indexed by magnitudes
of the roughness factors, modified in this chapter to accommodate unknown and unmeasured
15
microscopic reactive surface areas that exist at fracture asperity contacts. The multipliers are
significant, but are premised on the contrasting and undefined magnitudes of the fracture surface
areas obtained at “macroscopic” and “microscopic” levels. These magnitudes are consistent
between comparisons in terms of the rates of aperture reduction, db/dt, between the experimental
measurements and the predictions based on the dissolution mass flux that show nominal
agreement with the thousandfold (80°C) and hundredfold (120 and 150°C) roughness factors.
For a closed system, predictions indicate the insignificant contributions of precipitation to the
compaction process through the potential to occlude the fracture void. In practice, precipitation
may localize close to its source at the contacting asperities, and result in a more significant
impact on transport behavior, than if uniformly distributed over the fracture wall.
The rate of aperture reduction increases as temperature and applied stress increase. For the
dissolution-dominated system examined here, the dependency of aperture reduction rate on stress
is roughly linear – doubling the effective stress roughly doubles the closure rate. However, the
dependency on temperature is more pronounced due to the linkage of dissolution rate with its
Arrhenius-type (exponential) dependence. Where evolving changes in contact area are ignored,
the mass removal rate, scales roughly as
dM diss
k0
∝ + exp(− Ek + / R T )
dt
RT
(from Eq. (1)) and the
overriding control on temperature is apparent. Durations for the completion of aperture reduction
are reduced from a few centuries to approximately a year at temperatures in the range 80 to
200°C, and from a few years to a fraction of a year at effective stresses in the range 3 to 10 MPa
(Figure 11).
Aperture reduction as compaction proceeds may result in healing and related strength gain in
fractures. This consideration is important in considering strength gain in reservoir rocks, and the
16
development of sealing and healing on active faults. In this chapter simple relations between
fracture contact area and fracture aperture are defined, but quantitative evaluations for strength
recovery are not considered. For this, further geometric constraints of two rough surfaces in
contact are required.
17
dc
Asperity contacts
Local contact area, Acl
Figure 7. Idealized representation of the asperity contact condition. One representative
contact area, Alc (on right side), represents the area of each asperity in contact (on left
side), and is considered circular in shape, of diameter, dc.
18
Figure 8. Relation between aperture and contact-area ratio. Two different relations are
set. The curve with open circles shows a sharper reduction of aperture with an increase
of contact area the relation represented by the open squares.
19
Figure 9. Comparison of aperture reduction with time between experimental data of
Polak et al. [2003] and predictions of the current model. (a) For a = 1.0 ×10-2, open
triangles represent the prediction using parameters shown in Table 1 and open circles
represent the prediction using modified values with fr = 300.0, 40.0, and 40.0 for T =
80, 120, and 150 °C, respectively. (b) For a = 7.0 ×10-2, open triangles represent the
prediction using parameters shown in Table 1 and open squares represent the prediction
using modified values with fr = 3000.0, 300.0, and 300.0 for T = 80, 120, and 150 °C,
respectively.
20
Figure 10. Comparison of change of silica concentration with time between
experimental data of Polak et al. [2003] and predictions of our model. For a = 1.0
×10-2, open circles represent the prediction using modified values with fr = 300.0, 40.0,
and 40.0 for T = 80, 120, and 150 °C, respectively. For a = 7.0 ×10-2, open squares
represent the prediction using modified values with fr = 3000.0, 300.0, and 300.0 for T
= 80, 120, and 150 °C, respectively.
21
Figure 11. Predictions of aperture reduction in a fracture with time under various
system conditions of (a) temperature, and (b) effective stress.
22
Chapter 3: Fault Zone Restrengthening and Frictional Healing: The Role of Pressure
Solution
Slide-hold-slide (SHS) experiments have been conducted to examine the effects of solution
transfer processes on frictional healing in simulated quartz gouge layers. Data at 20 °C (Figure
12) show a log-linear relation between healing and hold periods and exhibit a frictional healing
rate of 0.0082 per decade. At mildly elevated temperature (65 °C), frictional healing is faster than
suggested by the log-linear dependency for hold times above a threshold of ~1000 s. The
temperature and hold-time threshold for this strength gain suggests pressure solution as a
plausible mechanism – strength gain being related to the growth in grain-to-grain contacts within
the particulate assemblage. The feasibility of this pressure solution mechanism is reinforced by
observations of an increase in the magnitude of stable sliding friction (Figure 12c) and in the
generation in post-hold dilation that dominates compaction during healing (Figure 13).
A lumped parameter model (Chapter 1; Yasuhara et al., 2003) is applied to examine the
growth of contact area with time, linking this observation as a proxy for strength gain observed
during periods of healing. This model incorporates the process of pressure solution, representing
the three serial processes of grain-contact dissolution, grain-boundary diffusion, and free-wall
precipitation. This enables the evolution of contact area to be followed in time under arbitrary
prescribed stress and temperature conditions. As shown in Figure 14, the model underestimates
experimental measurements of fault healing for short hold durations (less than 1000 s), due to the
dominant influence of other frictional mechanisms of strength gain. However, for hold durations
longer than ~1000 s, rapid growth in strength with hold time is well represented by a model for
pressure solution if the unknown magnitude of microscopic asperity contact area is increased
23
over the nominal macroscopic contact area.
For pressure solution, the prediction for strength gain does not show the same log-linear
dependence between healing and time, apparent in other frictionally based mechanisms. Thus, if
healing is rate-controlled by pressure solution, extrapolation by the log-linear relation may
significantly underestimate actual healing rates in natural faults. The model is applied to the
long-term prediction of the behavior of fault healing. Predictions show augmentation of healing
in both rates and magnitudes with depth (Figure 15), and indicate that pressure solution may
form impermeable fault zones and result in pore-fluid over-pressurization that likely plays a key
role in the rupture of fault zones.
24
Figure 12. Friction data for T = 20 (solid circles) and 65 °C (open circles). (a) Δμc, (b)
Δμ, (c) changes in steady state friction (Δμss) are plotted against log hold time. The
log-linear regression lines are evaluated for Δμc and Δμ. At 65 °C a distinct transition
of Δμ is observed between 1000 and 3000 s hold periods.
25
Figure 13. Changes in normal strain between pre- and post-healing with log hold time.
Data at 65 °C show that net-dilation arise for hold times greater than 3000 s, which
likely results from increase of resistance for shearing due to welded grains.
26
Figure 14. Contrast of the growth of contact area with the evolution of strength
observed in the experiments. Strength gain is normalized as the ratio of peak shear
stress to the shear strength for initial sliding friction (μ0 = 0.68). The growth in contact
area is normalized relative to the initial Hertzian contact area, Ac/Ac0. The predictions
underestimate the observed strength gain for hold times less than 1000 s, but
subsequently follow observations reasonably well especially for sixty-fold increase.
27
Figure 15. Evolution in contact area, normalized by initial contact area, with healing
time at depths from 1 to 5 km. Solid and dotted lines represent predictions without any
modifications and with hundred-fold increases in reactions rates constants (k+ and k-),
respectively. Healing rates and magnitudes increase with depth (i.e., applied
temperatures and stresses).
28
Chapter 4: Spontaneous Permeability Switching in Fractures in Carbonate: Lumped
Parameter Representation of Mechanically- and Chemically-Mediated Dissolution
The previous mechanistic model (Chapter 1; Yasuhara et al., 2003) is extended to explain the
spontaneous switching in permeability modification from net reduction to net gaping, absent any
changes in environmental conditions of ambient stress or the chemistry of the influent fluid
[Polak et al., 2004]. This change in observed behavior is accommodated through the significant
process of free-face dissolution. , which allows the fracture to gape as dissolution on the fracture
void walls dominates the response. The observed switch from net fracture closure, to net gaping,
is shown compatible with competing mechanisms of pressure solution and free-face dissolution.
As the fracture closes, flow is localized within a progressively smaller cross-section of the
fracture, reducing the contribution from pressure solution, and isolating the source of the mass
removed. This source region for dissolved mass ultimately becomes a ‘wormhole.’
The evolution of this process may be followed by representing the chemical interactions of
the individual contributing processes, and transport within the system. These important processes
include dissolution at asperity contacts, interfacial diffusion of dissolved mass along the
asperity-to-asperity contact boundary, followed by either advection out of the system, or
precipitation at the fracture-void surface. Where the water circulating in the fracture-void is
undersaturated, an additional mass will be removed from the fracture-void wall. These processes
are linked to the evolution of fracture porosity, and related fluid transport parameters, by a
geometric model of the fracture that relates fracture aperture and contact-area (Figures 7 and 16).
Together, the transport system and geometric model enable the progress of fracture aperture and
mass concentration in the pore fluid to be followed under arbitrary conditions of applied stress,
29
temperature, and pH.
As shown in Figure 17, this model is capable of replicating experimental observations
recovered from circulation of water in a stressed fracture in limestone [Polak et al., 2004].
Predictions of changes in sample permeability with time, and related aperture change, are
congruent with observations, although calcium concentrations in the effluent fluid are
underestimated during the initial circulation of groundwater (stage I), but are better constrained in
the late test during the circulation of distilled water (stages II and III); the predicted and measured
mass volume effused from the system are similar (168 and 160 mm3, respectively).
In the first stage of groundwater circulation (0-932 hr) the fracture aperture monotonically
reduces. The process is controlled only by the dominant process of pressure solution, with
free-face dissolution contributing no change in the impedance. The calcium concentrations in the
effluent fluid are predicted to be slightly larger than the equilibrium solubility, and result in the
activation of precipitation. However, the precipitated mass is trivial and causes no observable
change in aperture. In the second stage of distilled water circulation (932-1082 hr) the lower pH
in the pore fluid (pH ~6) results in accelerated rates of aperture reduction, and the low calcium
concentrations (~0.5 ppm) activate free-face dissolution. The maximum flow impedance is
predicted around 1028 hr, close to the experimentally measured time of 1082 hr, as, for the first
time, the effect of free-face dissolution exceeds that of pressure solution. This results as local
stresses reduce, in response to the widening of asperity contacts, even though applied mean stress
remains constant.
In the final stage (1082-1497 hr), impedance abruptly decreases until the experiment is
terminated, with free-face dissolution as the dominant mechanism of mass removal. A single
through-going conduit is observed to form in the experiment, although the lumped geometry
30
assumed in the analysis is incapable of appropriately representing this process. Despite this,
representation of the fracture by the behavior of a single representative asperity within a tributary
area is capable of recording the spontaneous switch from net fracture closure to net widening. To
accurately replicate observed response, the evolution of the fracture surface area must be
artificially augmented to increase the contribution of free-face dissolution on the evolving
wormhole (the boundary conditions and modification of parameters used in the analysis are
summarized in Table 1). The evolution of a wormhole may only be represented if the distributed
response of dissolution over the fracture contact area is adequately represented – a feature not
apparent in this lumped-parameter representation.
31
Figure 16. Relation between aperture and contact-area ratio.
32
Figure 17. Comparisons between experimental data of Polak et al. [2004] and
predictions of the current model, (a) aperture change with time and (b) change in
calcium concentrations (outlet minus inlet) with time. Open squares represent the
predictions using original parameters and modified values; with k+ = 0.01×(1.15×10-6)
mol m-2 s-1 in stage I; with k+ = 2.0×(1.15×10-6) mol m-2 s-1 in stages II and III, and the
area of a fracture void, Apore, is decreased by a factor of 1/200 in all stages. Open circles
show the predictions using modified values; with k+ = 0.01×(1.15×10-6) mol m-2 s-1 in
stage I; with k+ = 2.0×(1.15×10-6) mol m-2 s-1 in stages II and III, and Apore is decreased
by a factor of 1/200 in stages I and II and the factors of Apore is linearly increased from
1/200 to 60/200 in stage III. These modifications are summarized in Table 1.
33
Table 1. Boundary conditions and modification of parameters used in the analysis.
Parameter
Influent concentration [Ca], ppm
Influent pH
Precipitation rate constant, k-
I
40
~8
k-(pH~8) ×10-2
Dissolution rate constant, k+
k+(pH~8) ×10-2
Contact area, Ac
Fracture void area, Apore
Ac
(At-Ac) ×1/200
Test Stages
II
0
~6
k-(pH~6) ×10-2
k-(pH~6)= k-(pH~8) ×200
k+(pH~6) ×10-2
k+(pH~6)= k+(pH~8)×200
Ac
(At-Ac) ×1/200
III
0
~6
k-(pH~6) ×10-2
k-(pH~6)= k-(pH~8) ×200
k+(pH~6) ×10-2
k+(pH~6)=k+(pH~8)×200
Ac
(At -Ac) ×1/200
(At -Ac) ×1/200→60/200
34
Chapter 5: Evolution of Fracture Permeability through Fluid-Rock Reaction under
Hydrothermal Conditions
The response of a natural fracture in novaculite to the circulation of a hydrothermal fluid is
measured under moderate effective stresses of 1.4 MPa and temperatures of 20, 40, 80, and
120°C. Experimental measurements of fluid and mineral mass effluxes, constrained by
non-destructive imaging by X-ray CT and digital radiography, and by destructive imaging of a
fracture cast, allow the evolution of fracture aperture to be followed. For isothermal flow at low
temperature (20°C), the fracture closed monotonically as the bridging asperities were
progressively removed (Figure 18). This amounts to a 60 % reduction in fracture aperture over
the initial 1500 hr duration of the experiment. Subsequently, free-face dissolution dominates over
the effects of dissolution at contacting asperities, and results in an augmentation in fracture
aperture. The rate of gaping increases with an increase in temperature, corresponding to the
exponential (or Arrhenius-type) dependence on the mineral effluxes measured. The activation
energy constrained in this work is 22.2 kJ mol-1, congruent with that obtained from Iler [1979]
(i.e., 21.7 kJ mol-1).
Post-experiment observations by X-ray CT, digital radiograph, and by Wood’s metal casting
of the fracture void (Figures 19-21) revealed the detailed distribution of local apertures and
regions in contact within the fracture. The three independent measurements imply no presence of
a preferential flow path generated during the experiments. The average aperture evaluated from
the CT images is sevenfold larger than the hydraulic aperture at the end of the experiment. This
mismatch is likely due to limitations in the resolution of the CT imaging, and in the choice of
threshold CT number.
35
A process-based model is applied to examine the experimental observations – that aperture
closure spontaneously switches to aperture growth, but that net dissolution prevails throughout.
Employing a simple relation between fracture aperture and contact-area ratio (Figure 22), and
assigning the sources of mineral dissolution evaluated from measured Si concentrations to
contact area and free void walls with certain ratios, the model is capable of following the rates
and magnitudes of aperture changes measured in the experiment (Figure 23). However, the
ultimate contact area that is predicted significantly overestimates that constrained in the
fracture-plane profile recovered in a Wood’s metal cast of the fracture (i.e., 40 % v.s. 14.5 %).
The measured Si concentrations are significantly below equilibrium solubilities at all
temperatures, due to the high prescribed flow rates, and the resulting short residence time in the
fracture. Examining this far-from-equilibrium behavior is important in considering the rates and
magnitudes of permeability-enhancement within rock fractures stimulated by chemical permeants
in geothermal and petroleum reservoirs. The current experimental measurements showed first
permeability reduction, followed by enhancement as temperature of the circulating fluid was
increased. However, the interaction of important factors (i.e., temperature, stress, and chemistry
in the pore fluid) triggering this switching remain insufficiently constrained.
36
Figure 18. Experimental measurements; (a) Changes in hydraulic aperture and
normalized differential pressure with time and (b) change in Si concentration with time.
Measured differential pressure is normalized by the flow rates and fluid viscosities.
37
c
a
b
Figure 19. Images of the CT scan obtained at the end of the flow experiment. The left
images show two CT images of the rock sample taken at the top (a) and bottom (b)
parts of the sample. The fracture is clearly visible at the center of the image where the
large black pores are areas where pieces of the fracture surface was missing. The right
image (c) shows a 3D reconstruction of 1477 CT images representing a complete view
of the fracture surface. The white areas are fracture voids demarked with a threshold
CT number of 2650.
38
a
b
Figure 20. A digital radiograph (DR) of the fracture cast of Wood’s metal; (a) plane view
and (b) side view. The white areas represent the filled pores.
39
(a)
(b)
Figure 21. Wood’s metal fracture extracted through chemical dissolution using
hydrofluoric acid; (a) whole fracture – white region is residual rock sample, and (b)
black-and-white representation of the boxed area in (a). Black area represents regions
in contact and calculated contact area is 14.5 %.
40
Figure 22. Relation between mean aperture and contact-area ratio. Circles are evaluated
from point-by-point subtraction using the profiling data, and the dotted line is the
regression curve of <b> = 2.5 + 16.0 exp(-(Rc-Rc0)/20) with the correlation coefficient, R2 =
0.92.
41
Figure 23. Comparisons of changes in (a) aperture and (b) normalized differential
pressure with time between experimental results and predictions. Solid line represents
the experiments, and open circles are predictions.
42
Chapter 6: A Numerical Model Simulating Reactive Transport and Evolution of Fracture
Permeability
A distributed parameter model is developed to represent the evolution in fracture aperture
mediated by the significant processes of pressure solution, and incorporating the serial processes
of dissolution, diffusion, precipitation, and free-face dissolution. From the prescribed initial
aperture distribution within the fracture, and assuming steady state conditions, the fluid-field is
calculated from the Reynolds approximation – this flow-field is used to calculate the influence of
chemical processes of pressure solution and free-face dissolution/precipitation in sequentially
modifying the initial aperture distribution. Significantly, the model accommodates the solution
for advection-dominant transport systems by employing the Lagrangian-Eulerian method. The
model is capable of predicting the evolution in aperture and solute concentration for a single
fracture under arbitrary prescribed stress, temperature, and flow rate conditions. Predictions of
both lumped parameter [Yasuhara et al., 2004 (Chapter 2) & 2006a (Chapter 4)] and distributed
parameter representations of permeability evolution in a single facture [Yasuhara et al., 2006b
(Chapter 5)] are examined to test the adequacy of each model in representing experimentally
observed behavior (Figures 24 and 25 and Tables 2 and 3). Notably, if the controlling parameters
of reaction rate constants are increased by a factor of only 30, the numerical model shows
excellent agreement with the experimental observations although a sharp reduction of the fracture
aperture during the early experiment period is unable to be followed. This mismatch is likely
attributed to processes of mechanical creep, that are not represented in the model. Both
observations by X-ray CT (Figure 19), and model predictions (Figure 26), conclude that fluid
flow at the conclusion of the experiment (3150 hr) is broadly distributed throughout the fracture -
43
no preferential flow paths are generated.
The model is applied to examine the evolution of fracture aperture (or permeability) where
injection is concentrated at a point, as an approximation of injection into a reservoir. As shown in
Figure 27, the predictions show the propensity to develop channeled dissolution features that
concentrate flow. This exercise portends the potential to determine the form, plausible rates,
magnitudes, and senses of permeability enhancement at field scale.
44
Figure 24. Comparisons of changes in (a) aperture and (b) Si concentration between the
experimental results [Yasuhara et al., 2006b] and the predictions by the lumped
parameter model. Open circles represent the predictions using modified values of
reaction rate constants k+/─ shown in Table 2.
45
Table 2. Experimental conditions and modification of parameters used in the analysis by the
lumped parameter model.
Test stages
Parameters
Temperature [°C]
Flow direction
σc (Eqs. (2), (3))
k+ (Eq. (2))
k+/─ (Eqs. (4),(5))
I
II
III
IV
V
VI
20
Original
σc×0.1
k+×106
k+/─×104
20
Reversed
σc×0.1
─
─
20
Original
σc×0.1
k+×104
k+/─×104
40
Original
σc×0.1
k+×104
k+/─×104
80
Original
σc×0.1
k+×500
k+/─×500
120
Original
σc×0.1
k+×200
k+/─×200
46
Figure 25. Comparisons of changes in (a) aperture and (b) Si concentration between the
experimental results [Yasuhara et al., 2006b] and the predictions by the distributed
parameter model developed in this work. Open circles represent the predictions using
modified values of reaction rate constants k+/─ shown in Table 3.
47
Table 3. Experimental conditions and modification of parameters used in the analysis by the
distributed parameter (numerical) model.
Test stages
Parameters
Temperature [°C]
Flow direction
σc (Eqs. (2), (3))
k+ (Eq. (2))
k+/─ (Eqs. (4),(5))
I
II
III
IV
V
VI
20
Original
σc×0.1
k+×5.0×106
k+/─×30
20
Reversed
σc×0.1
─
─
20
Original
σc×0.1
K+×30
k+/─×30
40
Original
σc×0.1
k+×30
k+/─×30
80
Original
σc×0.1
k+×30
k+/─×30
120
Original
σc×0.1
k+×30
k+/─×15
48
Figure 26. Flow field resulting from the finite element solution of the Reynolds
equation at (a) 0 hr and (b) 3150 hr. Vectors represent the relative magnitude and
direction of local flow. Note that the flow is distributed randomly with flow-excluded
zones growing with increased time, and related contact area.
49
Figure 27. Results of numerical flow-through simulation at 100 hr after flow started. (a)
Overlay of flow field on aperture contour. Note that several flow paths are formed. (b)
Contour of aperture difference between 0 and 100 hr. Lighter shading represent
dissolution (erosion) regions.
50
References
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.
Iler, R. K. (1979), The Chemistry of Silica, John Wiley, New York.
Polak A., D. Elsworth, J. Liu, and A. S. Grader (2004), Spontaneous switching of permeability
changes in a limestone fracture with net dissolution, Water Resour. Res., Vo. 40, W03502,
doi:10.1029/2003WR002717.
Rimstidt, J. D. and H. L. Barnes (1980), The kinetics of silica-water reactions, Geochim.
Cosmochim. Acta, 44, 1683-1699.
Yasuhara, H., D. Elsworth, and A. Polak (2003), A mechanistic model for compaction of granular
aggregates moderated by pressure solution, J. Geophys. Res., 108(B11), 2530,
doi:10.1029/2003JB002536.
Yasuhara, H., D. Elsworth, and A. Polak (2004), Evolution of permeability in a natural fracture:
Significant
role
of
pressure
solution,
J.
Geophys.
Res.,
109(B3),
B03204,
doi:10.1029/2003JB002663.
Yasuhara, H., D. Elsworth, A. Polak, J. Liu, A. Grader, and P. Halleck (2006a), Spontaneous
permeability switching in fractures in carbonate: Lumped parameter representation of
mechanically- and chemically-mediated dissolution, Trans. Porous Media, (in press).
Yasuhara, H., A. Polak, Y. Mitani, A. Grader, P. Halleck, and D. Elsworth (2006b), Evolution of
fracture permeability through fluid-rock reaction under hydrothermal conditions, Earth
Planet. Sci. Lett., 244, 186–200.
Energy and Geo-Environmental Engineering
College of Earth and Mineral Sciences
The Pennsylvania State University
110 Hosler Building
University Park, PA 16802-5000, USA
+1 814 865-2225
+1 814 865-3248 (Fax)
elsworth@psu.edu
www.egee.psu.edu
June 7, 2006
Rocha Medal Committee
ISRM
Dear Award Committee:
It is a pleasure to nominate Dr. Hide Yasuhara, an August 2005 graduate from our program, for the Rocha
Medal. His resume, thesis, thesis-abstract, degree certification, and copyright release form are all attached.
Hide is an extraordinarily inquisitive, creative, and talented scientist. Perhaps the most productive I have
had the pleasure of working with. In his work he has been seeking a consistent understanding of coupled
mechanical and chemical controls on the transport and mechanical properties of natural fractures. In this
work he has completed unusually well-constrained laboratory experiments which have provided new and
puzzling observations of permeability changes in fractures – that fractures may switch between gaping or
sealing with no change in experimental conditions of stress, temperature, or influent fluid chemistry. His
coordinated laboratory and modeling studies of these effects has implicated the central control of pressure
solution in these observations. He has discovered that these effects are both of much greater magnitude that
previously anticipated, and significantly faster – significant enough for net dissolution to decrease
permeabilities by two orders of magnitude within a period of a month. Significantly at this rate, these
behaviors not only participate in the progress of geological processes, but also influence the performance
of engineered structures.
Hide has developed important characterizations of these effects using micro-mechanical models, and
extended them via continuum modeling of the crucial active, and intrinsically coupled, processes. These
characterizations have added importantly to our understanding of both the transport properties of fractures
mediated by stress and chemistry, and also to the restrengthening of faults, implicated in the earthquake
cycle.
Hide has been the force majeure in moving these analytical characterizations ahead. He has the unusual
distinction of being a resourceful experimentalist with the imagination and creativity to translate these
observations into rigorously-prescriptive and eminently-quantifiable models.
For these contributions to a body of distinguished work, Hide is an excellent and deserving candidate for
this award.
Yours sincerely,
Derek Elsworth
Professor
An Equal Opportunity University