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Analytical and Numerical Modelling of Thermal
Conductive Heating in Fractured Rock
by
Daniel Peter Baston
A thesis submitted to the
Department of Civil Engineering
in conformity with the requirements for
the degree of Master of Science (Engineering)
Queen’s University
Kingston, Ontario, Canada
April 2008
Copyright c© Daniel Peter Baston, 2008
Abstract
Analytical and numerical modelling studies were conducted to assess the performance of
thermal conductive heating (TCH) systems for the purpose of contaminated site remedia-
tion. Modelling was conducted in a fractured bedrock environment containing a system of
parallel, equally-spaced horizontal fractures.
A semi-analytical solution to the two-dimensional heat conduction equation was devel-
oped and used to study temperature distributions between two thermal wells. A sensitivity
analysis was conducted to assess the relative importance of hydrogeological parameters (hy-
draulic gradient, fracture aperture, fracture spacing) and rock material properties (density,
thermal conductivity, heat capacity). Hydrogeological parameters were far more important
than rock material properties in determining treatment zone temperature distributions.
Knowledge of the bulk groundwater influx may be sufficient to predict the temperature
within the treatment zone for low to moderate values of influx.
To further the analysis, numerical modelling was employed. A three-dimensional domain
was constructed, representing a symmetrical portion of a heater well cluster. Simulations
were run for different combinations of bulk permeability, fracture spacing, matrix perme-
ability, and matrix porosity. Flow concentration in fractures had a significant effect on
treatment zone temperature distributions when bulk permeability was high. For low values
of bulk permeability (kb ≤ 10−14 m2), the minimum treatment zone temperature changed
by less then 7% when modelling the fractured medium as an equivalent homogeneous porous
i
medium.
Fracture spacing significantly influenced the time needed reach complete steam satura-
tion, even in cases where it did not affect temperature distributions. A pressure rise may
occur in the matrix as water expands thermally, elevating the boiling point of water. The
magnitude of the pressure rise is affected by the distance to the nearest fracture, as well as
the matrix permeability and porosity. For a given bulk permeability, the time needed to
reach complete steam saturation will be lengthened by an increase in fracture spacing, an
increase in matrix porosity, or a decrease in matrix permeability. Of these parameters, the
matrix permeability is the most significant.
The time needed to reach complete steam saturation in the matrix cannot be predicted
if the fracture spacing, matrix permeability, and porosity are not known. Further, a clear
temperature plateau is not observed during boiling in the matrix, posing a difficulty in mon-
itoring thermal treatment, where temperature measurements may be the only information
available.
ii
Acknowledgements
The work presented in this thesis was supported through a Discovery Grant from the Natural
Sciences and Engineering Research Council of Canada (NSERC), contract ER-0715 from
the U.S. Department of Defense Environmental Security Technology Certification Program
(ESTCP), and Queen’s University through scholarships to the author.
I would like to thank my advisor, Bernie Kueper, for providing me with his guidance
and support, while giving me the independence to chart my own path in this research. Kent
Novakowski was of great help to me in the early phase of my work – I am very thankful
that there was someone out there willing to talk shop about integral transforms. Ron Falta
and Karsten Pruess both provided me with a great deal of assistance in my numerical
modeling work, the results of which would have been difficult to analyze if not for the
data management ideas given to me by Rob Harrap and Gerry Barber. And I could have
accomplished nothing on the mathematical front without the foundations given to me by
Joe Siddiqui, Pat Farrell, Duncan Innes (deceased), and Guy Kember.
I am greatly indebted to the other students in my research group. In particular, Mike
West was especially generous with his knowledge about numerical modeling, scientific pub-
lishing, and the ins and outs of grad school at Queen’s. Tom Gleeson is thanked for his
valuable insights on my research, numerous edits to my papers, and letting me raise poul-
try in his backyard. Morgan Schauerte and Stephanie Villeneuve were both a great help in
my modeling work. Everything would have been more difficult had I not had the fun and
iii
supportive environment provided by my office-mates: Luis Bayona, Stephanie Grell, John
Kozuskanich, Eric Martin, Justin Matthew, Titia Praamsma, David Rodriguez, and Shawn
Trimper.
Many thanks to my friends, my family, and my partner, Valerie, for your patience,
support, and convincing me that it would all be worth it someday.
iv
Forward
Chapters 3 and 4 in this thesis have been written as self-contained manuscripts intended
for publication in Advances in Water Resources and Ground Water, respectively. Daniel
Baston is the senior author of both publications. Bernard Kueper is a co-author of both
publications. Supporting information for these chapters is provided in the appendices.
v
Table of Contents
Abstract i
Acknowledgements iii
Forward v
Table of Contents vi
List of Tables viii
List of Figures ix
Nomenclature xi
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Chapter 2: Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Thermal Properties of Rock . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Subsurface Remediation by Thermal Conductive Heating & Soil Vapour Ex-
traction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Laboratory Studies of Thermal Remediation . . . . . . . . . . . . . . . . . . 282.5 Heat Transfer in Fractured Media: Analytical Solutions . . . . . . . . . . . 312.6 Heat Transfer in Fractured Media: Numerical Models . . . . . . . . . . . . 33
Chapter 3: Screening Calculations . . . . . . . . . . . . . . . . . . . . . . . 413.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 Outline of Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
vi
3.8 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Chapter 4: Numerical Modelling . . . . . . . . . . . . . . . . . . . . . . . . 634.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Chapter 5: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.1 Semi-Analytical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 Numerical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Appendix A: Semi-analytical Solutions . . . . . . . . . . . . . . . . . . . . . 97A.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97A.2 Single Fracture Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101A.3 Fracture Set Solution (Bessel Function Solution) . . . . . . . . . . . . . . . 106A.4 Fracture Set Solution (Improved Solution) . . . . . . . . . . . . . . . . . . 110
Appendix B: Verification of Simplified Heat Balance . . . . . . . . . . . . 119
Appendix C: Numerical Discretization . . . . . . . . . . . . . . . . . . . . . 122C.1 Discretization of Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122C.2 Radial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Appendix D: Calculated Numerical Model Input Parameters . . . . . . . 128D.1 Fracture Zone Physical Properties . . . . . . . . . . . . . . . . . . . . . . . 128D.2 Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132D.3 Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
vii
List of Tables
3.1 Base case parameters for sensitivity analysis . . . . . . . . . . . . . . . . . 523.2 Summary of sensitivity testing trials . . . . . . . . . . . . . . . . . . . . . . 523.3 Rock material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4 Parameters for runs used to assess correlation between bulk influx and treat-
ment zone temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1 Base case properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2 Parameters varied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
A.1 Parameters used for verification against the Gringarten et al. (1975) solution 115A.2 Numerical model material properties . . . . . . . . . . . . . . . . . . . . . . 118
B.1 Parameters used for verification of heat storage term omission . . . . . . . . 121B.2 Comparison of solution times when heat storage is included and neglected . 121
D.1 Capillary pressure parameters used in TOUGH2 simulation . . . . . . . . . 134D.2 Relative permeability parameters used in TOUGH2 simulations . . . . . . . 134
viii
List of Figures
2.1 Photograph of thermal wells at a field site. . . . . . . . . . . . . . . . . . . . 21
3.1 Conceptual model of fractured rock environment. . . . . . . . . . . . . . . . 453.2 Schematic of model domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Temperature distribution resulting from heating by a single thermal well. . 543.4 Summary of computed one-year interwell fracture temperatures. . . . . . . 553.5 Transient fracture temperature profiles at the centre of the treatment zone
for high and mid-level values of groundwater influx. . . . . . . . . . . . . . 563.6 Early and late-time fracture temperature profiles for various rock types. . . 583.7 Transient fracture temperature profiles at midpoint between heater wells,
showing the influence of hydraulic gradient. . . . . . . . . . . . . . . . . . . 593.8 Effect of increased heat production rate on the time needed to reach a target
temperature of 100 C and total energy consumption. . . . . . . . . . . . . 60
4.1 Plan view of model domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2 Section of model domain in r - z plane . . . . . . . . . . . . . . . . . . . . . 674.3 Isometric view of model domain . . . . . . . . . . . . . . . . . . . . . . . . . 674.4 Pressure as a function of distance from the fracture for a location just outside
the treatment zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.5 Temperature vs. time for a reference block in the rock matrix and fracture,
at the boundary of the treatment zone. . . . . . . . . . . . . . . . . . . . . . 704.6 Impact of matrix permeability (km) on magnitude of pressure spike at centre
of rock matrix and steam saturation within treatment zone. . . . . . . . . . 724.7 Impact of porosity (φ) on magnitude of pressure spike at centre of rock matrix
and steam saturation within the treatment zone. . . . . . . . . . . . . . . . 734.8 Minimum treatment zone temperature profiles for various values of fracture
spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.9 Relationship between steam saturation and distance from the fracture in base
case simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.10 Sensitivity of treatment zone boiling time to bulk medium properties. . . . 76
A.1 Heat balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98A.2 Determining an appropriate number of fracture sets . . . . . . . . . . . . . 110A.3 Comparison of new solution with that of Carslaw and Jaeger (1959, p. 263)
for the case of zero fracture aperture. . . . . . . . . . . . . . . . . . . . . . . 114
ix
A.4 Comparison of the present solution with that of Gringarten et al. (1975) . . 115A.5 Comparison of temperatures in fracture after one year of heating, as calcu-
lated using the De Hoog et al. (1982) and Weeks (1966) algorithms. . . . . 117A.6 Fracture temperature after 4 months of heating, computed using semi-analytical
solution and TOUGH2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
C.1 Comparison of semi-analytical solution, discrete fracture numerical solution,and “fracture zone” numerical solution before and after boiling. . . . . . . . 123
C.2 Comparison of CPU time and error in temperature for different values offracture zone thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
C.3 Fracture primary variable profiles for three domain sizes . . . . . . . . . . . 126C.4 Effect of radial discretization on pressure, steam saturation, and temperature
at two reference points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
D.1 Outline of calculation of derived input parameters . . . . . . . . . . . . . . 129
x
Nomenclature
Latin Letters
af anisotropy factor - page 18Anm interfacial area of elements n and m m2 page 36B Stefan-Boltzmann constant W/m2·K4 page 9c gravimetric heat capacity J/kg·K page 8D internodal distance m page 36e fracture aperture m page 44F mass flux vector kg/m2·s page 34F radiative configuration factor - page 10G heat flux vector W/m2 page 35g gravitational acceleration vector m/s2 page 11g internal heat generation W page 8H fracture half-spacing m page 44Hcc Henry’s Law constant - page 23h specific enthalpy J/kg page 35∇h hydraulic gradient - page 51J Jacobian matrix - page 39K thermal conductivity W/m·K page 7k intrinsic permeability m2 page 11kr relative permeability - page 35L characteristic length m page 12M mass kg page 34m pore-size distribution parameter - page 131N number of volume elements - page 37n direction vector m page 7P fluid pressure Pa page 11Pcap capillary pressure Pa page 131Pd displacement pressure Pa page 132Pe fracture entry pressure Pa page 131Pmax maximum capillary pressure for van
Genuchten (1980) functionPa page 131
Pe Peclet number - page 12
xi
q bulk groundwater influx L/m2·day page 54qh heat flux W/m2 page 7qm mechanical flux m/s page 8qmass mass source kg/s page 34q(x′, s) fracture-matrix heat exchange function W page 49R residual in equation varies page 37r radial coordinate in numerical model m page 66s Laplace variable s page 49Se effective liquid saturation - page 131Sgr residual gas saturation - page 133Sl liquid saturation - page 131Slr residual (irreducible) liquid saturation - page 131Sls maximum liquid saturation - page 131T temperature C page 7T Laplace-space temperature C page 49
Tmax maximum interwell fracture temperature C page 53Tmin minimum interwell fracture temperature C page 53U internal energy J page 35V volume m3 page 34v fluid velocity m/s page 12wj quadrature weights at point j - page 50x coordinate in direction of groundwater flow in
fracturem page 44
y coordinate normal to fracture plane in two-dimensional solutions
m page 45
∆zfz fracture zone thickness m page 127z coordinate normal to fracture plane in three-
dimensional modelm page 66
Greek Letters
α thermal diffusivity m2/s page 8αvg parameter in van Genuchten (1980) equation 1/L page 131Λ radiative reflectance - page 9λ pore size distribution index - page 132Θ radiative absorptance - page 9θ angular coordinate in numerical model page 66µ dynamic fluid viscosity Pa·s page 11ξ radiative energy flux W/m2 page 9ρ density kg/m3 page 8σ interfacial tension N/m page 132φ porosity - page 14Ω radiative transmittance - page 9
xii
Subscripts
b property of bulk medium page 67dry “dry” value of parameter (Sl = 0) page 130f fluid phase page 14
frac property of fracture page 129fz property of fracture zone page 127i node number page 50k time step number page 37m node number page 36m property of rock matrix page 67max maximum value of parameter page 15min minimum value of parameter page 14n node number page 36p Newtonian iteration number page 38s solid phase page 14v vapour phase page 35w water phase page 35wet “wet” value of parameter (Sl = 1) page 130
xiii
Chapter 1
Introduction
The vast majority of the world’s unfrozen fresh water is in the form of groundwater. Surface
freshwater sources such as lakes and rivers contain less than 4% of the volume present in
groundwater (Heath, 2004). Despite the importance of groundwater as a natural resource,
it is only recently that concerted efforts to protect groundwater quality have emerged.
Pollutants can enter groundwater by a multitude of pathways, including septic tanks,
landfills, waste lagoons, leaking storage tanks, urban runoff, and agricultural applications
(e.g. Fetter, 1993). Particularly problematic are non-aqueous phase liquids (NAPLs) such
as chlorinated solvents. These compounds, which are largely immiscible with and insoluble
in water, constitute widespread and persistent sources of contamination. Often trapped in
place by capillary barriers, NAPLs dissolve extremely slowly and may be a source of ground-
water contamination for hundreds of years (e.g. Kueper et al., 2003). Non-aqueous phase
liquids that are more dense than water (DNAPLs) present an especially difficult challenge
for removal. DNAPLs can migrate to great depths, often entering bedrock. Fractured rock
impacted by DNAPLs is considered to be among the most difficult environments to clean
up (National Research Council, 1994).
In recognition of this difficulty, recent efforts to mitigate risks from difficult sites have
been focused on either containment or in-situ destruction of the contaminant. In the past
1
CHAPTER 1. INTRODUCTION 2
decade, a strong shift was observed towards in-situ destruction methods such as chemi-
cal oxidation, thermal remediation, and bioremediation, and away from extraction-based
methods such as pump-and-treat, surfactant flushing, and air sparging (Simon, 2006).
Recently, several techniques have been developed to use heat to assist remediation ef-
forts. Heat can be useful through its effect on a variety of physical and chemical processes.
An increase in temperature typically causes a lowering of both NAPL viscosity and interfa-
cial tension with water, allowing pooled NAPL to be more easily removed. Most commonly,
however, in-situ thermal techniques seek to increase mass transfer out of the source zone
without moving the NAPL. For most organic contaminants, the processes of diffusion, disso-
lution, vapourization, volatilization, and desorption are all enhanced at higher temperatures
(Davis, 1997). At very high temperatures, contaminants may be destroyed in-situ through
the processes of oxidation and pyrolysis (Baker and Kuhlman, 2002).
Several approaches exist to deliver heat to the subsurface. One approach is to pass
an electrical current through the subsurface, generating heat through electrical resistance.
Electrical resistive heating has been used commercially at approximately fifty sites (Beyke
and Fleming, 2005). Because the pore water provides the bulk of the electrical conductance,
practical application of electrical resistive heating is limited to temperatures at the boiling
point of water. This temperature may be sufficient to vapourize many common volatile or-
ganic contaminants such as chlorinated solvents and gasoline compounds, but is inadequate
to initiate full boiling in less volatile compounds such as polychlorinated biphenyls (PCBs)
and polycyclic aromatic hydrocarbons (PAHs).
Heating has also been performed by the injection of steam into the subsurface. Steam
injection has been used at several research sites (e.g. Newmark et al., 1998). One limitation
of steam injection is its inability to penetrate low-permeability regions, although these
regions may be heated by conduction from adjacent high-permeability regions (Gudbjerg
et al., 2004). However, recent research has shown that it may be difficult to significantly
raise the temperature of a rock mass using steam injection alone (Davis et al., 2005).
CHAPTER 1. INTRODUCTION 3
Thermal conductive heating (TCH) systems are capable of reaching high temperatures
through the direct heating of the subsurface material. Arrays of “thermal wells,” each
consisting of an electrical resistive heating element inside of a steel casing, are installed
throughout a treatment area. The heater wells may reach temperatures as high as 800 C,
creating strong thermal gradients and allowing for a high rate of heat delivery. Heating
is continued until the boiling point of the target compound is reached. The produced
vapours are collected in vacuum extraction wells and treated ex-situ. In some cases, target
compounds may be destroyed in-situ through breakdown reactions such as pyrolysis and
oxidation (Stegemeier and Vinegar, 2001).
All methods of thermal remediation are potentially limited by the flow of heat away
from the treatment area. Conductive losses occur at the boundaries of the treatment area,
as heat diffuses outward towards lower-temperature areas. Groundwater flow through the
treatment area may cause convective losses of heat, as heated water is replaced by incoming
cool groundwater, or incoming cool groundwater is boiled.
Very little research exists about the significance of these convective heat losses from the
treatment area (National Research Council, 2004). In one study, numerical modelling was
used to study the influence of groundwater influx on the rate of contaminant removal in a
saturated porous medium (Elliott et al., 2003). It was found that the success of treatment
was highly dependent on the amount of groundwater flow into the treatment area. Another
modelling study examined the effectiveness of impermeable barriers in managing ground-
water influx (Elliott et al., 2004). No similar studies have been conducted for fractured rock
environments. Groundwater flow in fractured rock has the potential to be rapid and highly
localized, relative to flow in unconsolidated deposits. For this reason it is suspected that
the influence of convective cooling in fractured rock environments may be different than
in unconsolidated deposits. Despite this, almost no research has been conducted into heat
transfer in fractured rocks at small scales where equivalent continuum and dual-continuum
models may not hold (National Research Council, 1996).
CHAPTER 1. INTRODUCTION 4
1.1 Research Objectives
The objective of this study was to use analytical and numerical modelling to determine if
there are cases in which incoming groundwater may inhibit heating of a fractured rock mass
using TCH and, if so, if this effect may be overcome through careful consideration in the
system design.
Throughout this study, a conceptual model was used wherein a body of fractured rock
was considered to have a system of parallel, equally-spaced discrete fractures. Using this
conceptual model, new semi-analytical solutions were developed to the governing equations
of heat transfer. Using one of these solutions, a sensitivity analysis was performed to
determine the relative importance of rock material properties and hydrogeological properties
on the rate of heating in the treatment zone at temperatures up to the boiling point of water.
Three material properties were studied: rock density, thermal conductivity, and specific heat
capacity, along with three hydrogeological properties: fracture aperture, fracture spacing,
and hydraulic gradient.
Using the results of this sensitivity analysis, numerical modelling was performed to
assess the performance of thermal conductive heating in fractured rock at temperatures
above the boiling point of water, where latent heat effects and two-phase flow preclude
the development of semi-analytical solutions. In this analysis, the rock matrix is no longer
considered to be completely impermeable, permitting an investigation of the importance of
matrix permeability and porosity.
1.2 Organization
In the following chapter, a literature review is presented that summarizes much of the
research to date on heat transfer in fractured rock, specifically in the context of thermal
conductive heating systems. Chapter 3 is a stand-alone manuscript intended for publication
CHAPTER 1. INTRODUCTION 5
in Advances in Water Resources and describes a sensitivity analysis of material and hydro-
geological properties, performed using a new semi-analytical solution to the heat equation.
Derivations and verifications of the new solution are provided in Appendices A and B.
Discussion of the numerical modelling is given in Chapter 4, which is a manuscript
prepared for publication in Ground Water. Supporting information for this work is included
in Appendices C and D.
Chapter 2
Background
The process of thermal conductive heating in fractured rock is a complex process, drawing
on research in a variety of fields. This chapter provides an overview of this research, which is
lumped into several categories. First, a discussion is given of the fundamental concepts and
equations used to describe heat transfer, with a particular focus to hydrogeological systems.
Next, a summary is provided of the thermal properties of rock and their relationship with
physical properties, mineralogical properties, and thermodynamic conditions. An overview
of thermal conductive heating is then provided, with attention given to the many physical
and chemical processes proposed to contribute to the removal of contaminants from the
subsurface. Finally, a summary is given of previous analytical and numerical modelling
efforts focused on heat transfer in fractured media. Although very few studies have given
explicit consideration to thermal conductive heating, relevant research is presented from
the fields of nuclear waste disposal and “hot dry rock” geothermal systems.
6
CHAPTER 2. BACKGROUND 7
2.1 Heat Transfer
2.1.1 Conduction
The transfer of heat by conduction is perhaps the simplest and most direct form of heat
transfer. Conductive heat transfer occurs at the molecular scale, by means of collisions
and interactions between molecules at different energy states. In nonmetallic solids these
interactions typically take the form of lattice vibrational waves. Heat may also be conducted
by the movement of free electrons, this behaviour being characteristic of metals (Incropera
and DeWitt, 2002).
Rigorous quantitative study of heat conduction began with Joseph Fourier’s 1822 pub-
lication of The Analytical Theory of Heat (Fourier, 1955, translation). In this work Fourier
presented the basic law concerning the rate of heat movement at a steady state, now known
as Fourier’s Law:
qh ∝dT
dn(2.1)
where T is the temperature, and qh is the heat flux in a direction n. Fourier also noted
that the rate of heat conduction is dependent on the material through which it travels.
Thermal conductivity, written as K in this text, can be inserted in to the above relation as
a proportionality constant, giving:
qh = −KdT
dn(2.2)
Like specific heat, thermal conductivity is often assumed to be a constant, though in
many cases it exhibits a dependence on temperature. At very high temperatures, solids
may also conduct heat radiatively – transferring energy by radiation between the particles
of the lattice or matrix. This will cause the apparent thermal conductivity to increase.
An equation to describe transient heat conduction can be constructed by performing a
heat balance on a small control volume, using Fourier’s Law to account for heat flow through
the surfaces of the volume. The result is a second-order partial differential equation, often
CHAPTER 2. BACKGROUND 8
called the “heat diffusion equation” or simply “the heat equation”:
∇ · (K∇T ) + g = ρc∂T
∂t(2.3)
where g represents heat generated internally, ρ is the density , and c is the gravimetric
specific heat capacity. If the heat is conducted through an isotropic medium (in which the
thermal conductivity is equal in all directions) then the equation can be expanded as follows
in three-dimensional Cartesian coordinates:
∂
∂x
(K∂T
∂x
)+
∂
∂y
(K∂T
∂y
)+
∂
∂z
(K∂T
∂z
)+ g = ρc
∂T
∂t(2.4)
If the medium is also homogeneous, having a spatially constant value of thermal con-
ductivity, then the equation may be further simplified:
∂2T
∂x2+∂2T
∂y2+∂2T
∂z2+
g
K=ρc
K
∂T
∂t=
1α
∂T
∂t(2.5)
where α represents the thermal diffusivity, the ratio of thermal conductivity to volumetric
heat capacity:
α =K
ρc
2.1.2 Convection
Heat may also be transferred through the movement of heated matter. For example, heat
is transferred by convection when hot water flows through a pipe. Convective heat transfer
is further classified into free and forced convection. When water is heated, its density
decreases, causing it to expand and rise. As the water rises, heat is transferred upwards by
free convection. If, in contrast, the warm water is injected through a pumping well, heat is
transferred away from the well by forced convection.
The convective heat flux is directly proportional to the mechanical flux of the fluid (qm),
CHAPTER 2. BACKGROUND 9
and is given by qmcρT . When a fluid moves with a constant velocity, the effects of forced
convection can be modelled by the addition of this convective flux into the heat equation:
∇ · (K∇T )−∇ · (ρcqmT ) + g = ρc∂T
∂t(2.6)
2.1.3 Radiation
All objects constantly emit energy as electromagnetic radiation, of an intensity determined
by the object’s temperature and surface characteristics. This energy can be absorbed by
other objects, also as a function of their temperature and surface characteristics. The
transfer of heat by electromagnetic radiation is termed radiative heat transfer.
Not all electromagnetic radiation that reaches a body is absorbed. The energy may be
either absorbed, reflected, or transmitted through the body, at fractions described by the
absorptance Θ , reflectance Λ, and transmittance Ω. These coefficients, whose sum must
equal unity, will vary depending on the wavelength of the incoming radiation.
The Stefan-Boltzmann law describes the maximum amount of energy that may be emit-
ted from a body at a given temperature. The law, which is valid for a theoretical “black
body” having Θ = 1, predicts a maximum energy flux (ξ) of ξ = BT 4 per unit emitting
area. The proportionality constant B is referred to as the Stefan-Boltzmann constant and
is equal to 5.67× 10−8 W/m2·K4.
When radiation is exchanged between two objects, the rate of heat transfer is a function
of the difference in the fourth power of their temperatures. The heat flux from object 1 to
object 2 is given by ξ(T1), while the heat flux from object 2 to object 1 is given by ξ(T2).
The net heat flux from object 1 to object 2 is thus described by ξ(T1)−ξ(T2) = B(T 4
1 − T 42
).
The actual rate of heat transfer from object 1 to object 2 is a function of the heat
fluxes of the two objects, the area through which heat is emitted and absorbed, and their
configuration in space. The rate can be calculated according to:
CHAPTER 2. BACKGROUND 10
QR = A1F1−2B(T 4
1 − T 42
)(2.7)
where A1 is the area from which radiation is emitted and absorbed, and F1−2 is the
configuration factor, a function of the emittances and absorptances of the two objects, as
well as their arrangement in space.
2.1.4 Heat Transfer in the Subsurface
A variety of mechanisms contribute to the transfer of heat in a porous medium, making it
a difficult system to model mathematically. Bear (1972) identifies six mechanisms of heat
transfer in a porous medium:
1. Heat transfer through the solid phase (considered as a continuum) by
conduction;
2. Heat transfer through the fluid phase (considered as a continuum) by con-
duction;
3. Heat transfer through the fluid phase (considered as a continuum) by con-
vection;
4. Heat transfer through the fluid phase by dispersion;
5. Heat transfer from the solid phase to the fluid phase;
6. Heat transfer between solid grains by radiation, when the fluid is a gas.
Local thermal equilibrium
At the pore-scale, temperature differences may exist between soil grains and pore fluids.
However, these temperature differences are typically small and require consideration only
when rapid thermal transients or highly concentrated heat sources are present in one phase
CHAPTER 2. BACKGROUND 11
but not the other (Kaviany, 1995). In porous media, where fluids move quite slowly, it is
appropriate to assume local thermal equilibrium between the fluid and solid phases. This
very useful assumption permits the definition of an effective medium thermal conductivity,
removing the need to separately examine heat conduction in the solid and liquid phases.
In fractured media, fluid is typically assumed to be in thermal equilibrium with the
fracture walls, although temperature gradients may be present within the rock matrix itself.
In some geologic systems, such as geysers and phreatic eruptions, the assumption of local
thermal equilibrium may be inappropriate (Ingebritsen et al., 2006).
Conduction and convection
In the heat equation presented above (2.6) the effect of convection was included in the
∇ · (ρcqmT ) term. In environments where large temperature differences are not present,
flow equations can be solved to determine the fluid mechanical flux qm. This approach has
the advantage of preserving the linearity of (2.6); however, it may be a source of error if
large temperature differences are present.
If the fluid flow is assumed to follow Darcy’s Law, then the flux term qm can be expanded
as:
qm = −kµ
(∇P − ρg) (2.8)
where k is the permeability of the medium, µ is the dynamic fluid viscosity, P is the fluid
pressure, and g is the gravitational acceleration vector. Because the fluid dynamic viscosity
and density are dependent on temperature, the rate of fluid flow will be affected by changes
in temperature. Further, the increase in fluid density that accompanies an increase in
temperature will cause an increase in fluid pressure. In high temperature environments,
these fluid property variations can not be ignored (Ingebritsen et al., 2006).
Conduction and convection are typically the dominant forms of heat transfer in porous
media, to the extent that other forms are typically neglected (Domenico and Schwartz,
CHAPTER 2. BACKGROUND 12
1998). For example, radiation is insignificant at temperatures below 600 C, and dispersion
is almost always regarded as negligible due to the rapid rates of heat diffusion in geologic
materials (Ingebritsen et al., 2006). The Peclet number, a dimensionless parameter, can be
used to describe the relative importance of conduction and convection (Bear, 1972):
Pe =Lv
α(2.9)
where L is a characteristic length, often taken as the mean grain diameter, v is the fluid
velocity, and α is the thermal diffusivity. As Pe increases, so does the importance of heat
convection.
Heat dispersion
Differences in pore size, shape, and friction can cause microscale variations in fluid velocity.
These variations tend to cause fluid to spread out, or disperse, as it moves along a path
defined by the hydraulic gradient. Hydrodynamic dispersion plays a large role in the transfer
of solute throughout the subsurface, and the same mechanism affects heat transport through
convection. However, because heat diffusion (conduction) is so rapid relative to molecular
diffusion, the effect on heat transport is far less significant than for solute transport, and is
typically neglected (Ingebritsen et al., 2006).
Radiative heat transfer
At temperatures about 600 C, radiation can begin to play a significant role in inter-grain
heat transfer. This effect is typically modelled by adding a radiative contribution to the
thermal conductivity of the medium (Clauser and Huenges, 1995; Ingebritsen et al., 2006).
CHAPTER 2. BACKGROUND 13
2.2 Thermal Properties of Rock
As seen in equation (2.3), the rate of transient heat conduction through rock is a function of
both thermal gradient and thermal diffusivity. Though thermal diffusivity may be measured
directly in a laboratory, historical research has tended to focus on the thermal conductivity
of rocks, and to a lesser extent, their heat capacity. The following section provides a
summary of this research and its implications for heat transfer in rocks.
2.2.1 Thermal Conductivity
A large range of thermal conductivities can be observed in geologic materials. An impure
diamond may have a thermal conductivity on the order of 1000 W/m·K, while many soils
have thermal conductivities below 1 W/m·K(e.g. Abu-Hamdeh and Reeder, 2000). Com-
mon thermal conductivities for rocks range from 1 to 6 W/m·K. Most rocks have anisotropic
thermal conductivity, exhibiting a preference for heat flow in a particular direction. This is
typically due to the preferential alignment of crystals within the rock. However, rocks that
have no preferred orientation of mineral grains may behave as an isotropic body (Bunte-
barth, 1984).
Much of the early data on the thermal conductivity of rocks comes from a two-part 1940
paper by Harvard University researchers Birch and Clark (1940a). In addition to providing
a wealth of new thermal conductivity measurements, the paper provides a comprehensive
description of the mechanisms of heat transfer in crystals, as predicted by Debye theory
(Birch and Clark, 1940b).
The thermal conductivity of rocks is neither constant nor readily predicted from simple
measurements such as density. Nonetheless, a large body of research has elucidated the
major factors influencing the thermal conductivity of rocks, and a number of relationships
have been proposed to permit a degree of extrapolation from existing measurements. The
most reliable estimate of a rock’s thermal conductivity, apart from direct measurement, is
CHAPTER 2. BACKGROUND 14
given by its mode, or mineral composition (Buntebarth, 1984; Birch and Clark, 1940b).
The thermal conductivity of sedimentary rocks is largely dependent on porosity and
fluid saturation. Crystalline rocks, on the other hand, are more sensitive to mineral grain
orientation, temperature, and pressure effects. These relationships are discussed in greater
detail later in this section. More comprehensive presentations are given by by Robertson
(1988) and Clauser and Huenges (1995).
Dependence on Porosity and Fluid Saturation
Increases in fluid saturation typically result in increased thermal conductivity, as water
begins to occupy pore spaces formerly filled with air (e.g. Cermak and Rybach, 1982). As
a result of increasing interest in thermally enhanced oil recovery, petroleum researchers in
the 1950s began conducting extensive research into the relationships between the thermal
conductivity of porous rocks samples and their porosity and fluid saturation. These parame-
ters were found to correlate well with thermal conductivity, and a number of equations were
developed to describe this dependence. The result was a series of “mixing law” formulas
that predict the thermal conductivity of a porous rock, given the thermal conductivities of
its solid and fluid components.
In a paper outlining many of the mixing law models, Woodside and Messmer (1961)
used an analogy to electric circuits to describe the two limiting cases of the mixing law
models. The minimum value of thermal conductivity is described by a series model, where
heat passes from the solid phase, then to the liquid phase. This is mathematically described
by the weighted harmonic mean of the two conductivities, or:
Kmin = KsKf/[φKs + (1− φ)Kf ] (2.10)
where φ is the porosity and the subscripts f and s denote the fluid and solid phases, re-
spectively. The maximum value of thermal conductivity corresponds to the parallel circuit
CHAPTER 2. BACKGROUND 15
model, mathematically expressed as the weighted arithmetic mean of the conductivities:
Kmax = φKf + (1− φ)Ks (2.11)
Woodside and Messmer (1961) also proposed the use of the geometric mean to describe the
conductivities, based on the mathematical fact that Kharmonic < Kgeometric < Karithmetic:
Kgeo = KφfK
1−φs (2.12)
While Woodside and Messmer (1961) do not propose a physical justification for the use
of the geometric mean thermal conductivity, it is appears to be one of the more widely
used mixing law models (Beck, 1976). Other researchers have extended the analogy of
electrical conductivity, developing equations based on Maxwell’s theoretical equation for
the electrical conductivity of a system consisting of solid spheres randomly distributed
throughout a continuous medium at sufficient spacing so that they do not interact (Woodside
and Messmer, 1961; Beck, 1976). Translated into thermal quantities, this is expressed as:
K = Kf
[2φKf + (3− 2φ)Ks
(3− φ)Kf + φKs
](2.13)
Kunii and Smith (1960) developed a mixing law model for porous sandstones based on a
theoretical packed bed reactor, with an empirically determined “consolidation parameter”
to account for the additional conduction between the cemented grains. In studies where
predicted and measured thermal conductivities have been compared, most of the mixing
law models have been found to be able to predict thermal conductivity to within 10-15%
accuracy, depending on the data set (Clauser and Huenges, 1995).
CHAPTER 2. BACKGROUND 16
Dependence on Temperature
Since at least the early 20th century, researchers have been aware of the significant cor-
relation between temperature and thermal conductivity measurements in rocks. However,
much of the early research into the thermal properties of rocks is of limited quantitative
value, due to poor characterizations of both the rock samples and the ambient conditions
(temperature, pressure) during testing. An extensive study published by Birch and Clark
(1940a) set a new standard for research into the thermal conductivity of rocks, tabulating
thermal conductivity values over the 0 C to 400 C range for a series of well-characterized
rock samples.
A more recent compilation of the now-extensive experimental data is provided by Clauser
and Huenges (1995). As seen in this work, which includes the data of Birch and Clark
(1940a) as well as many others, the thermal conductivity of most rocks decreases by a
factor of 1.5 to 4 as temperature is increased from 0 C to 800 C. The temperature depen-
dence of thermal conductivity is generally largest for sedimentary and metamorphic rocks,
and weakest for plutonic and volcanic rocks. For some rocks, the temperature-dependent
variation in thermal conductivity may be large enough to be significant in the modelling of
terrestrial heat flow, geothermal reservoirs, and subsurface heating for soil and groundwater
remediation. This dependence on temperature presents difficulties in the modelling of heat
conduction, as it introduces non-linearity to the heat equation (e.g. Carslaw and Jaeger,
1959).
According to Cermak and Rybach (1982), two separate effects explain the change in
thermal conductivity with increasing temperature. Lattice conductivity, mentioned in Sec-
tion 2.1.1 as the primary mechanism of conduction in non-metals, is inversely proportional
to T . Theory predicts that the radiative component of thermal conductivity should increase
with T 3, although empirical evidence suggests that the increase is proportional to T rather
than T 3 and does not become significant until temperatures are high (Buntebarth, 1984).
CHAPTER 2. BACKGROUND 17
Exceptions are present to the general rule of decreasing thermal conductivity with tem-
perature. The thermal conductivities of rocks with high feldspar content are often unaf-
fected or may even increase with rising temperature. In addition, rocks with an amorphous
phase, such as fused (noncrystalline) quartz may have thermal conductivities that rise with
temperature (Ratcliffe, 1959; Abdulgatov et al., 2006; Cermak and Rybach, 1982). Several
researchers have proposed equations to relate the conductivity a rock at an elevated tem-
perature to its conductivity at a lower temperature. For crystalline rocks, Seipold (1998)
proposed an expression of the form:
K(T ) =T
F × T + E(2.14)
where E and F are constants. Vosteen and Schellschmidt (2003) proposed an a slightly
more complex relationship:
K(T ) =K(0)
0.99 + T a−bK(0)
(2.15)
where a and b are constants.
Dependence on Pressure
Several references provide data on the thermal conductivity of rocks as a function of pres-
sure. In the data compiled by Clauser and Huenges (1995), increasing pressure generally
causes an increase in thermal conductivity up to about 20%, as fractures are closed and
fluid is forced out of the rock. Above 15 MPa, little change is observed in the thermal
conductivity. Cermak and Rybach (1982) suggest that the pressure effect is somewhat
less important, observing a maximum thermal conductivity increase of 10%. A more pro-
nounced effect was observed by Woodside and Messmer (1961) in tests on a sandstone with
φ = 22%. Increases in thermal conductivity were observed until approximately 20 MPa, by
which time the thermal conductivity had risen by about 35%. Combining their original data
CHAPTER 2. BACKGROUND 18
on pressure dependence with that from a compendium of previous works, Abdulgatov et al.
(2006) propose a much higher range of 50 MPa to 100 MPa for the “cross-over pressure” at
which further pressure increases cease to affect thermal conductivity significantly. In sum-
mary, the extent of the pressure effect is strongly dependent on the minerology, porosity,
and density of the rock sample. Although the value of the cross-over pressure varies widely
between data sets, there seems to be agreement that there is an upper limit to the pressure
effect on thermal conductivity.
Dependence on Orientation
Many rocks are thermally anisotropic, showing a preferred direction of thermal conductivity.
Mineral-anisotropy, or microanisotropy, refers to anisotropy in rock thermal conductivity
that arises from a dominant orientation of an anisotropic mineral. Shape-anisotropy, or
macroanisotropy, arises from bedding and foliation. For example, a metamorphic rock may
have alternating layers of quartz and feldspar, causing a dependence on the direction of
foliation (Vosteen and Schellschmidt, 2003).
Cermak and Rybach (1982) define the anisotropy factor as the ratio of parallel thermal
conductivity to normal thermal conductivity, or:
af =Kp
Kn
2.2.2 Heat Capacity
The specific heat of rocks tends to rise with temperature (Robertson, 1988). Because
the thermal diffusivity of a rock is calculated as the thermal conductivity divided by the
volumetric heat capacity, an increase in c has the same effect as a decrease in thermal
conductivity. The temperature effects in both c andK generally drive the thermal diffusivity
CHAPTER 2. BACKGROUND 19
in the same direction - toward a slower conductive transfer of heat at higher temperatures.
For multi-phase systems, such as porous rocks saturated with water, aggregate specific
heat can be calculated simply from a weighted arithmetic mean of the specific heats of the
different components. Rocks with large water contents may have very high values of specific
heat, due to the very high specific heat of water.
The same reasoning can be used to calculate the specific heat of a rock from the specific
heats of its mineral components. Somerton (1958) performed a chemical analysis on several
sedimentary rocks to determine their mineral composition, then computed a value for the
specific heat of the rock based on the specific heats of the minerals. Heat capacities of the
rocks were also measured directly using a calorimeter, and the calculated and measured
values agreed within 2%. According to Robertson (1988), for most purposes the specific
heat capacity can be calculated with reasonable accuracy from an estimate of the mineral
composition determined only by hand-lens inspection of the rock.
2.3 Subsurface Remediation by Thermal Conductive Heat-
ing & Soil Vapour Extraction
In-situ Thermal Desorption, or ISTD, is a method of groundwater and soil remediation
in which a combination of conductive heating (TCH) and soil-vacuum extraction (SVE)
is used to treat subsurface contamination. Outside of the United States and Canada, the
combination of thermal conductive heating and vacuum extraction is occasionally referred
to as TEVE (thermally enhanced vacuum extraction) or THERIS (thermischen In-Situ
Sanierung). More recently, the term “thermal conductive heating” has been used to refer
to the combination of conductive heating and vacuum extraction.
In a TCH system, heat is delivered to the formation by heater elements installed in steel
casings. Heat is transfered by radiation from the heater element to the casing, from where
it can conduct into the formation. Conduction is the primary mechanism of heat transfer
CHAPTER 2. BACKGROUND 20
away from the thermal wells (Stegemeier and Vinegar, 2003). However, in formations with
significant movement of groundwater, convection may also play a role in heat transfer. Very
close to the heater wells, where the temperature may exceed 600 C, radiation may make
an additional contribution to the heat transfer away from the well.
Thermal conductive heating is relatively insensitive to formation heterogeneities, as the
thermal conductivities in the subsurface vary by a factor of less than five. By contrast, the
transfer of heat into the subsurface by steam injection is limited by the intrinsic permeability
of the soil and rock, which may vary by over ten orders of magnitude (Ingebritsen et al.,
2006).
Heater wells are typically installed in a hexagonal configuration. In some cases, the well
at the centre of the hexagon may be a heater-vacuum well, while the wells along the edges
of the hexagon are heater-only wells. In other cases, all of the wells may be heater-vacuum
wells. In another approach, the combination heater-vacuum wells may be omitted from the
design, instead allowing contaminants to migrate upwards through a permeable sandpack
installed around the heater-only wells (LaChance et al., 2006). The sandpack can be used to
direct contaminant vapours to horizontal vacuum extraction wells, where they are extracted
and directed to the treatment facility. An example TCH configuration is shown in Figure
2.1.
Spacing of the wells is determined by the amount of time available for treatment, as
well as the target temperature for treatment of the compounds of interest. At the upper
end of the treatment temperature range, semivolatile organic compounds (SVOCs) such as
polychlorinated biphenyls (PCBs) require treatment temperatures in the range of 300 C
to 400 C (Uzgiris et al., 1995). In this case a well spacing of 2-3 m is typically used.
More volatile compounds such as chlorinated solvents (CVOCs) can be treated at 100 C,
allowing for a well spacing of 5 m to 8 m (LaChance et al., 2006). Treatability studies on
polycyclic aromatic hydrocarbons have found the level of removal to be affected by both
the treatment temperature and the treatment time (Hansen et al., 1998).
CHAPTER 2. BACKGROUND 21North Adams-MGP Site.jpg (JPEG Image, 1640x642 pixels) - Scaled (75%) http://www.terratherm.com/site%20Photos/Well%20Field%20Overvie...
1 of 1 24/04/2008 12:25 PM
Figure 2.1: Photograph of thermal wells at a field site. (TerraTherm photo)
Although remediation using thermal conductive heating is energy intensive, it applica-
tion may represent a long-term savings in energy relative to treatment methods that operate
over a longer period of time. Using a life-cycle analysis methodology, Hiester and Schenk
(2005) compared the energy use of a “cold” soil vapour extraction system with a combined
TCH-vapour extraction system. They found that, although the thermally enhanced system
required much more power to operate, it used 58% less energy than the “cold” due to its
shorter operation time.
2.3.1 Mechanisms of Remediation
Heat can aid in the removal of contaminants from the subsurface through its effect on a
number of physical and chemical processes (e.g. Davis, 1997). Physical changes resulting
from elevated temperatures can include enhanced contaminant desorption from soil, in-
creased dissolution, increased vapourization and volatilization, increased rates of diffusion,
and an increase in soil permeability. On a chemical level, heat contributes to contaminant
breakdown through oxidation and pyrolysis.
CHAPTER 2. BACKGROUND 22
Not all effects of added heat may be beneficial to remediation. NAPL viscosity and
interfacial tension with water both tend to decline with increasing temperature, increasing
the mobility of NAPL present. In a worst-case scenario, this could enable NAPL to migrate
downwards, spreading the source zone.
A number of these effects are reported to have significant effects on contaminant removal
during subsurface heating by TCH (Baker and Kuhlman, 2002). However, little research has
been conducted to demonstrate the relative importance of these mechanisms in a subsurface
environment.
Desorption of Contaminants from Soil
Thermal desorption has long been recognized as an effective means of extracting contami-
nants from soil, allowing for subsequent treatment of the vapours by combustion or other
means. In earlier applications of thermal desorption, soil was excavated prior to thermal
treatment in a rotary kiln or other ex-situ heating device (Lighty et al., 1988). The applica-
tion of thermal desorption in-situ is more recent. Iben et al. (1996) used heater blankets to
conductively deliver heat to a shallow area of PCB contamination, reporting a thousand-fold
reduction in PCB concentration after 24 hours of heating. The kinetics of thermal desorp-
tion are complex, with a portion of the contaminant mass being released rapidly from the
soil, and a “recalcitrant fraction” requiring a much longer treatment time (Uzgiris et al.,
1995).
Increased Dissolution
When thermal treatment is applied to a NAPL source zone, the rate of overall mass transfer
may be affected by increased partitioning from the NAPL phase into the aqueous phase.
Knauss et al. (2000) presented new data on the solubility of TCE and PCE at temperatures
ranging from 21 C to 117 C and 22 C to 161 C, respectively. For both TCE and PCE,
solubility is increased by a factor of approximately 2 when the system is heated to just
CHAPTER 2. BACKGROUND 23
below the boiling point of water.
Increase in NAPL mobility
She and Sleep (1998) measured imbibition and drainage curves for a PCE-water system at
varying temperatures and found that water-PCE capillary pressure decreased by a factor of
about two when temperature was increased from 20 C to 80 C. A variety of mechanisms
may contribute to capillary pressure lowering, including changes in NAPL-water interfacial
tension, NAPL-water contact angle, residual wetting phase saturation, and the presence of
small amounts of vapour in soil pores (She and Sleep, 1998).
Vapourization and Steam Stripping
At equilibrium, the concentration of contaminant in the vapour phase is related to the
concentration of dissolved contaminant by the Henry’s Law constant: Hcc = Cg/Cw. Heron
et al. (1998a) found that the Henry’s Law constant for TCE increased by a factor of 20
when temperature was increased from 10 C to 95 C.
Consequently, as steam is produced, the concentration of contaminant in the gas phase
will decrease, while the concentration in the aqueous phase is relatively unchanged. In
order to maintain equilibrium, the rate of volatilization will increase (e.g. Fair, 1987). This
phenomenon is known as “steam stripping.”
In-Situ Contaminant Destruction: Pyrolysis
When subjected to very high temperatures, many gaseous phase organic contaminants will
undergo pyrolysis, a unimolecular decomposition reaction. Interpretations of field studies
of TCH have suggested that pyrolysis reactions may occur in the superheated zones directly
adjacent to thermal wells (Baker and Kuhlman, 2002). The pyrolysis of chlorinated solvents
can be expressed by the following reaction (e.g. Frenklach, 1990):
CHAPTER 2. BACKGROUND 24
CxHyClz → CxHy−1Clz−1 + HCl
Baker and Kuhlman (2002) write the overall reaction for the complete pyrolysis of TCE
in the presence of water as follows:
C2HCl3 + 4H2O→ 2CO2 + 3HCl + 3H2
The first equation provides a more complete description of the pyrolysis mechanism.
Dehalogenation does not occur in one step; while the original contaminant may be almost
completely removed, it will likely be transformed into a number of intermediary products
in addition to CO2, HCl, and H2.
Yasuhara (1993) and Yasuhara and Morita (1990) studied the pyrolysis of PCE at tem-
peratures between 310 C and 940 C and of TCE at temperatures between 300 C and
800 C. Measurable amounts of 15 compounds were detected from the pyrolysis of PCE,
while 23 were detected from the pyrolysis of TCE.
A similar study was performed by Thomson et al. (1994) on the high-temperature ox-
idation of 1,1,1-trichloroethane. Vapour phase 1,1,1-TCA gas was burned at a controlled
temperature. It was found that, at temperatures near to 530 C, significant amounts of
1,1,1-TCA were converted into phosgene (COCl2), a toxic gas. Phosgene concentration
increased as temperature was increased, reaching a peak at a temperature of 890 C, when
fully 29% of the 1,1,1-TCE was converted into phosgene in 0.1 s. Only at temperatures
exceeding 1200 C did phosgene cease to be a major oxidation product. These results
suggest that phosgene may be a product of in-situ combustion under thermal remediation
conditions (Costanza et al., 2003).
Conditions favourable to pyrolysis may occur in the superheated zone that occurs close
to heater wells. Costanza (2005) calculated that gaseous TCE will be 99% destroyed in
the superheated zone with a residence time of about 7 days at 500 C or with a residence
CHAPTER 2. BACKGROUND 25
time of 7 seconds at 700 C. Assuming a 1-foot superheated zone with no preferential flow
pathways, the residence time of vapours in the superheated zone was calculated to be on the
order of 30-70 seconds for typical gas extraction rates. It should be noted that treatment in
the superheated zone is not a necessity; vapours extracted from the subsurface are treated
above-ground.
While destruction of contaminant by pyrolysis reduces the need for above-ground treat-
ment, the formation of gaseous HCl can cause excessive corrosion in the vapour extraction
system. The problem may be even more severe if the HCl vapours are allowed to condense,
as HCl in the liquid form may be 20 times more corrosive than in the gaseous form (U.S.
EPA, 2002).
In its analysis of the failure of an ISTD system installed at the Rocky Mountain Arsenal
“ hex pit”, the Rocky Mountain Arsenal Remediation Venture Office describes the complete
degradation of heater elements, casings, and collection equipment as a result of corrosion
from condensed HCl vapours (U.S. EPA, 2002).
In-Situ Contaminant Destruction: Oxidation
In addition to the destructive reactions presented in the previous section, the elevated
temperatures present in thermal remediation conditions can enhance in-situ oxidation re-
actions. The following reaction for the aqueous oxidation of TCE is presented by Knauss
et al. (1999):
2C2HCl3(aq) + 3O2(aq) + 2H2O→ 4CO2(aq) + 6H+(aq) + 6Cl−(aq)
While this reaction is thermodynamically favoured (∆Gr < 0), the reaction rate con-
stant is so low at standard groundwater temperatures that the process becomes negligible.
However, Knauss et al. (1999) report that the rate constant increases by a factor of about
2500 from 25 C to 90 C.
CHAPTER 2. BACKGROUND 26
Increase in Soil Permeability
At very high temperatures, soils may undergo structural changes that affect the intrin-
sic permeability. Vinegar et al. (1997) heated a PCB-contaminated soil for 42 days and
compared soil samples taken before and after heating. They reported an increase in soil
porosity from about 30% to about 40%. In addition, reported air permeabilities increased
from 3×10−3 md to 50 md (horizontal) and from 1×10−3 md to 30 md (vertical). The au-
thors propose that permeability was increased due to the removal of moisture from the pore
spaces, clay dessication, fracturing, and the removal of organic material. This increase in
permeability allowed vapour-phase contaminants to travel more rapidly to the SVE system.
Some data on thermally-induced structural changes of rocks are available from petroleum
research. For example, Somerton et al. (1965) heated several sandstone samples to 800 C
in an oven and allowed them to cool to room temperature, observing permeability increases
on the order of 200% to 400% and fracture index increases (the number of fractures observed
in a cross-section) of about 800%.
2.3.2 Heat Losses
Heat losses from the treatment area are an important consideration in the design of a TCH
system. To compensate for “end effects” – conductive losses of heat out the top and bottom
of the treatment zone – the heating elements are extended at least two feet above and below
the treatment area and may be designed to produce about 25% additional heat output in
these sections. This can be accomplished by reducing the cross-sectional area of the heater
element or adding an additional element in parallel (Stegemeier and Vinegar, 2001, 2003).
During thermal applications below the water table, heat may be lost due to the flow
of groundwater through the treatment area. When the temperature in the treatment zone
is below the boiling point of water, groundwater flow can cause heated water to flow out
of the treatment zone. The incoming cool water, which must be continually heated, can
CHAPTER 2. BACKGROUND 27
in some cases overwhelm the heater wells, preventing the attainment of boiling tempera-
tures (National Research Council, 2004). Above the boiling point of water, incoming cool
groundwater will be boiled, causing a loss of energy into the latent heat of vapourization.
Baker and Heron (2004) state that an injection of steam may also be used to prevent
groundwater influx from outside of the treatment zone. The injection of steam at high
pressure serves to provide a pressure barrier to groundwater flow and to reduce the relative
permeability of water in the formation. In addition, control of groundwater influx may be
possible through the use of extraction wells or impermeable barriers.
Modelling Studies
In one of the few published model studies on ISTD, Elliott et al. (2003) used the commer-
cially available STARS reservoir model to study the cooling influence of groundwater influx
to the treatment zone. The model comprised a two-dimensional grid with a NAPL source
zone spread between two heater wells spaced at 3.048 m, with a vacuum-heater well in the
centre of the space between the two wells. Using TCE, a heavy hydrocarbon, and a PCB,
the authors measured the heating time necessary for treatment (defined as 99% removal).
When the permeability of the simulated porous medium was increased, the treatment
time decreased, as vapourized contaminants were allowed to move more quickly through
the soil. However, this trend was reversed as permeability was increased beyond a certain
threshold (k = 9.86 × 10−11 m2 for the parameters of the study), as the increased water
influx from outside of the treatment zone prevented the soil inside the treatment zone from
reaching the treatment temperature.
A similar effect was observed as the hydraulic gradient across the treatment zone was
increased. Above a certain gradient, the heaters were unable to deliver sufficient energy to
counter the cooling influence of groundwater, and contamination was removed only from
the downgradient side of the treatment zone, after the incoming groundwater had been
preheated by the heater well on the upgradient side of the treatment zone. In order to
CHAPTER 2. BACKGROUND 28
ensure an adequate temperature rise throughout the treatment zone, the authors proposed
using a combination of tighter well spacings, higher heater temperatures, and impermeable
barriers.
The effectiveness of impermeable barriers was studied by Elliott et al. (2004), who used
a three-dimensional numerical model to study several impermeable barrier designs. A 20 m
× 20 m square was used as the model domain, constant head boundaries at the edges of
the square. At the centre of the domain, a hexagon of heater wells was placed, with a
single heater-extraction well at the centre of the hexagon. It was found that with a medium
permeability of k = 9.86× 10−12 m2, 99.8% of the TCE could be removed after 19 days of
heating. However, when the intrinsic permeability was increased to k = 9.86 × 10−11 m2,
TCE removal did not exceed 50%, even after 150 days of heating.
Using this value of intrinsic permeability, three barrier configurations were modelled to
test their effectiveness in preventing influx cooling. When a barrier was placed along one
edge of the treatment zone, the TCE removal rate was increased to slightly above 60%.
With barriers on two sides of the treatment zone, the removal rate reached 80%. When
the treatment zone was entirely enclosed by impermeable barriers, the removal rate reached
97%.
2.4 Laboratory Studies of Thermal Remediation
A number of laboratory studies have been conducted to show the ability of heat to remove
contaminants from soil, using an oven to heat a contaminated soil sample (e.g. Merino
and Bucala, 2007; Burghardt, 2007). However, there have been very few laboratory-scale
implementations of the technologies most commonly used for in-situ thermal treatment:
steam injection, electrical resistive heating, and thermal conductive heating. Research on
these studies has primarily been conducted at the field-scale, where only limited monitoring
of the treatment area is possible.
CHAPTER 2. BACKGROUND 29
Nonetheless, a few laboratory-scale studies have been conducted. Heron et al. (1998b)
applied resistive heating to a two-dimensional soil tank filled with a TCE-contaminated
silty soil. After 37 days of heating, 99.8% of the TCE mass was removed through vapour
extraction. Gudbjerg et al. (2004) simulated steam injection into a two-dimensional soil
tank with a fine and a course layer. The injected steam was not able to enter the fine layer;
however, the fine layer was heated by heat conduction from the coarse layer, albeit at a
much longer time scale than the heating of the coarse layer.
A few laboratory studies have been conducted of thermal conductive heating and are
discussed at length in the following sections. Kunkel et al. (2006) studied the applicability
of TCH to removal of elemental mercury from soil. Hiester et al. (2004, 2006) conducted a
laboratory-scale study of TCH for the removal of trimethylbenzene from a partially satu-
rated soil.
2.4.1 University of Texas Column Studies
While field studies have quantified the removal potential of thermal conductive heating
for organic DNAPLs such as halogenated solvents and PCBs, little data is available for
other DNAPLs, such as elemental mercury. In an effort to fill this gap, researchers at the
University of Texas performed a series of thermal well experiments in a 5 cm diameter
column of dry Ottawa sand (Kunkel et al., 2006). In one set of experiments, soil in the
column was contaminated with a known mass of perfluorodecalin, a surrogate compound for
Hg(0). Air was passed through the column and the mass of contaminant in the effluent was
measured. The experiments were conducted at varying temperature and air flow rate, and
it was concluded that temperature had a far greater impact on clean-up time than did flow
rate. In addition, three trials were performed using elemental mercury as a contaminant,
and similar results were achieved. Both sets of experiments were modelled numerically using
the STARS simulator, and good agreement was seen between the predicted and measured
results. In the case of both contaminants, rapid removal was achieved at temperatures well
CHAPTER 2. BACKGROUND 30
below the boiling point.
2.4.2 VEGAS Laboratory Studies
Some larger scale laboratory studies of ISTD are ongoing at the VEGAS Research Facility
for Subsurface Remediation at the University of Stuttgart. In a recent study, an ISTD
system was used to remove 30 kg of trimethylbenzene, a semivolatile organic compound,
which had been released into a 6 m×6 m×4.5 m container filled with a partially saturated
soil (Hiester et al., 2004).
Two vacuum extraction wells were installed in the soil, on one side of the TMB source
zone. Two air inlets were installed on the opposite side of the source zone. For two months,
vapours were extracted at a rate of 35 m3 per hour. Concentrations of TMB in the vapour
remained nearly constant for the two months over which the system was run.
After two months, four heater wells installed in the periphery of the source zone were
turned on. The temperature of the heater wells remained constant, at about 500 C. After
20 days of operation, the centre area between the wells had reached a temperature of slightly
under 100 C, and all of the remaining TMB had been removed. The authors calculated that
had the contaminant been removed by SVE alone, remediation time would have exceeded 8
months. The use of heater wells thus improved remediation time significantly, and resulted
in large energy savings.
Since the initial study at VEGAS, additional studies in a modified container have been
performed (Baker et al., 2006; Hiester et al., 2006). Recent experiments have been conducted
using a constant head boundary along the perimeter of the container, permitting a study
of ISTD behaviour in the saturated zone. Currently, no contaminant has been added to
the modified container, but researchers have activated the thermal wells and monitored
the temperature distribution throughout the container. The results from the experiment,
yet unpublished, are being used to calibrate the STARS model for use in designing future
experiments.
CHAPTER 2. BACKGROUND 31
2.5 Heat Transfer in Fractured Media: Analytical Solutions
Heat transfer in fractured media is difficult to model analytically. Separate governing equa-
tions are used for the matrix, in which there is no convection, and the fracture, where heat
is both conducted and convected. The two domains are coupled by a Cauchy boundary
condition (simultaneous Type I and II) that assumes thermal equilibrium and conservation
of energy between the two domains. In this section, an overview is given of previous analyt-
ical solutions to the heat equation in this environment. Consideration is given both to fully
analytical (closed-form solutions) as well as semi-analytical solutions (in which an inverse
Laplace transform is performed numerically, etc.)
Although solution of these equations is tractable only for very simple geometries, analyti-
cal solutions allow rapid and exact calculations of temperature in these cases. Consequently,
analytical solutions have an important application in sensitivity analysis and are indispens-
able for determining the governing parameters in a particular system. Further, they provide
a reliable solution against which numerical solutions can be verified.
Several analytical solutions have been developed for the problem of heat exchange be-
tween water in a single fracture and an impermeable matrix. These solutions do not consider
heat generation within the rock matrix; the system under consideration is typically cold wa-
ter flowing through hot rock.
For the case of a single discrete fracture, solutions are provided in 2D Cartesian coor-
dinates by Lauwerier (1955), Carslaw and Jaeger (1959, p. 396), Bodvarsson (1969), Yang
et al. (1998), and Kocabas (2004). Some 3D Cartesian solutions exist, though few are
published in English. Some examples are the approximate solution of Alishaev (1979) and
the semi-analytical solution of Heuer et al. (1991), who developed solutions for an arbi-
trary flow field in a planar fracture. In addition, solutions can be readily adapted from the
equivalent problem of matrix solute diffusion; such solutions are provided by Grisak and
Pickens (1981) and Tang et al. (1981). Solutions to the heat transport problem were found
CHAPTER 2. BACKGROUND 32
in radial coordinates by Bodvarsson (1972) and to the equivalent solute transport problem
by Feenstra et al. (1984).
Extension of the problem to a series of equally-spaced parallel fractures was presented
first in the geothermal literature by Gringarten et al. (1975) and Lowell (1976), who devel-
oped Laplace-space solutions to the problem using Cartesian coordinates in two dimensions.
Bodvarsson and Tsang (1982) solved the problem in Laplace space using radial coordinates.
Sudicky and Frind (1982) presented a solution to the equivalent solute problem, using
Cartesian coordinates.
All of the aforementioned solutions assume that diffusion of heat or solute in the rock
matrix occurs only in the direction normal to the fracture plane. Even in the geothermal
applications for which many of these solutions were intended, numerical modelling has
shown that multidimensional conduction in the rock matrix can play a significant role in
determining the final temperature distribution (Kolditz, 1995).
There are no published closed-form analytical solutions that incorporate two-dimensional
diffusion in the rock matrix. However, some relevant approximate and semi-analytical solu-
tions exist in the literature. Cotta et al. (2003) used a radial lumping procedure to develop
an approximate analytical solution to the equivalent solute transport problem. Cheng et al.
(2001) developed a Laplace-space semi-analytical solution for multidimensional conduction
from a single fracture by formulating the problem as an integral equation which can then
be solved numerically. In a later paper (Ghassemi et al., 2003) this was extended to include
three-dimensional conduction from a single fracture, although a finite difference procedure
must first be used to calculate the temperature on the fracture plane. Although the inte-
gral equation formulations allow for rapid calculation of the temperature at any point after
the temperature in the fracture is known, accuracy is generally poor at points close to the
fracture-matrix boundary.
Analytical solutions are very useful for developing an understanding of the important
CHAPTER 2. BACKGROUND 33
parameters governing heat transfer in a particular environment. For problems where ge-
ometry is simple or little information is known about heterogeneity, they can be an ideal
method of solution. Yet, as the geometry of a problem becomes more and more complex, the
applicability of analytical solutions becomes limited. For a true representation of the mul-
tidimensional heat flow effects present when groundwater flows in to a thermal treatment
area, it is necessary to solve the three-dimensional heat conduction-convection equation. At
the present time, numerical models present the only means of solving these equations.
2.6 Heat Transfer in Fractured Media: Numerical Models
Several numerical models are available to model the concurrent flow of heat and groundwater
in fractured media. However, few are designed around a discrete fracture conceptual model.
Many codes have been developed to model the single-phase coupled transfer of heat and
fluid in the subsurface (e.g. Molson and Frind, 2002; Kipp, 1997; Therrien et al., 2007).
Other codes allow for multiphase flow, including unsaturated flow and the boiling of pore
water (e.g. Pruess et al., 1999; White and Oostrom, 2006).
Of the aforementioned models, only HydroGeoSphere (Therrien et al., 2007) is designed
to work with a discrete fracture conceptual model. The code uses a finite element dis-
cretization to efficiently model planar fractures. However, the model is presently not able
to handle the boiling of water.
TOUGH2 (Pruess et al., 1999) uses a finite-difference discretization. Using finite dif-
ferences, a discrete fracture approach is possible, but the very small gridblocks that result
impose a limitation on the range of possible timesteps, lengthening the simulation time con-
siderably (Pruess et al., 1990a). Consequently, modelling of fractured media in TOUGH2 is
typically conducted using a dual-porosity or multiple interacting continua approach (Pruess
and Narasimhan, 1982; Pruess, 1992). Support does not exist in released versions of the
code for hydrodynamic dispersion or temperature-dependent thermal conductivity.
CHAPTER 2. BACKGROUND 34
2.6.1 The TOUGH Family of Codes
The name “TOUGH” refers to “transport of unsaturated groundwater and heat,” a code
developed at Lawrence Berkeley National Laboratory in the 1980s. The code was developed
primarily to model transport processes in the unsaturated formations at Yucca Mountain,
then (and still) under consideration for development as a high-level nuclear waste disposal
site. Several more recent models have been developed using TOUGH as a base. The most
widely used of these is TOUGH2, which is applied to problems such as geothermal reservoir
modelling, nuclear waste storage, and geologic CO2 sequestration (Pruess, 2004).
Governing Equations
TOUGH2 numerically solves the governing equations for fluid and heat flow using a so-
called “integral finite difference” (IFD) approach (Narasimhan and Witherspoon, 1976),
commonly referred to in other implementations as a “finite volume” method (Ferziger and
Peric, 1996; Kolditz, 2002). In an IFD or finite volume approach, the integral forms of the
conservation laws are used as the governing equations, rather than the more-commonly used
differential forms. One advantage of IFD over traditional finite differences is its flexibility in
problem geometry; any grid satisfying the Voronoi conditions can be used with IFD (Palagi
and Aziz, 1994).
In integral form, the governing equation for fluid flow is given for a one-component
system by (Pruess and Narasimhan, 1982):
d
dt
∫Vn
MdVn =∫
Γn
F · ndΓn +∫Vn
qmassdVn (2.16)
where M is the mass contained in volume element Vn, F is the mass flux vector, and
qmass is a mass source term. For heat transport, the governing equation is (Pruess and
CHAPTER 2. BACKGROUND 35
Narasimhan, 1982):
d
dt
∫Vn
UdVn =∫
Γn
G · ndΓn +∫Vn
QdVn (2.17)
where U is the internal energy in volume element Vn, G is the heat flux vector, and Q is
a heat source term. The mass flux F is given by Darcy’s Law (Pruess and Narasimhan,
1982):
F = −kkrwµw
ρw(∇P − ρwg)− kkrvµv
ρv(∇p− ρvg) (2.18)
where the subscripts w and v denote the liquid water and vapour phases, respectively. Each
phase has an absolute permeability k, relative permeability kr, dynamic viscosity µ, density
ρ, and pressure P . Although it is assumed in this formulation that only one liquid phase
(water) is present, extended versions of TOUGH2 have support for both single-component
and multi-component non-aqueous phase liquids (Falta et al., 1995; Pruess and Battistelli,
2002).
The heat flux G is given by the conduction-convection equation:
G = −K∇T + hwFw + hvF v (2.19)
where K is the thermal conductivity of the rock-fluid mixture, T is the temperature, hw is
the specific enthalpy of the water phase, and hv is the specific enthalpy if the vapour phase.
Discretization
In order to solve for the primary variables of temperature and pressure throughout the
solution domain, it is assumed that the volume integrands within equations (2.16) and
(2.17) are constant throughout each volume element. Further, the surface Γn is broken
down into discretized surfaces between adjacent volume elements Vn and Vn, having an
CHAPTER 2. BACKGROUND 36
area of Anm. With this simplification, the mass conservation equation can be rewritten as
follows:d
dtVnMn =
∑AnmFnm + qVn (2.20)
A similar treatment is given to the heat conservation equation. Individual mass flux
terms are approximated using a first-order finite difference. For example, the flow of water
is approximated as (Pruess et al., 1999):
Fw = −knm(krwρwµw
)nm
(Pwn − Pwm
Dnm− ρwgnm
)(2.21)
In this equation, the subscripts n and m refer to nodes n and m; Dnm is the distance
between these two nodes. The value of quantities such as permeability, density, and relative
permeability (mobility) must be approximated at the element interface using an average
between their defined values at nodes n and m. Several weighting schemes may be used, in-
cluding the arithmetic mean, harmonic mean, and upstream weighting. The appropriateness
of the various schemes has been a subject of considerable study. Earlier references (Peace-
man, 1977; Aziz and Settari, 1979) generally recommend upstream weighting of relative
permeabilities to minimize error. Still, these authors consider that absolute permeabilities
may be weighted with a simple arithmetic or harmonic mean. More recent references (e.g.
Pruess et al., 1999) recommend the use of upstream weighting for both absolute and relative
permeabilities in two-phase flow simulations.
Solution of Discretized Equations
The time derivatives in equations (2.16) and (2.17) are also approximated by a first-order
finite difference. TOUGH2 uses a backward difference expression taken at time step k + 1,
which gives the following series of equations:
CHAPTER 2. BACKGROUND 37
Mk+1n = Mk
n −∆tVn
∑m
AnmFk+1nm + Vnq
k+1n
for mass flow (2.22)
Uk+1n = Ukn −
∆tVn
∑m
AnmGk+1nm + VnQ
k+1n
for heat flow (2.23)
Because the backward difference is taken from time step k + 1, these time-discretized
equations are written entirely in terms of the flux, source, and sink terms at the “new” time
step k+ 1. This “fully implicit” treatment causes the numerical solution to be uncondition-
ally stable, meaning that it will approach the exact solution as the discretization is refined
(Jaluria and Torrance, 2003).
Because equations (2.22) and (2.23) can be written for each of the N volume elements,
a 2N × 2N system of nonlinear algebraic equations can be developed. The 2N unknowns
in the system are the “primary variables,” a set of variables which can completely define
the state of the flow system. In a water-only system in TOUGH2, the primary variables X1
and X2 are pressure and temperature for single-phase conditions, and gas-phase pressure
and gas saturation for two-phase conditions (Pruess et al., 1999). The system is nonlinear
because the flux terms Fnm and Gnm are themselves functions of the primary variables.
An alternative way to write equations (2.22) and (2.23) is in terms of residuals Rk+1n :
Rk+1n = Mk+1
n −Mkn −
∆tVn
∑m
AnmFk+1nm + Vnq
k+1n
, n = 1...N for mass flow (2.24)
Rk+1n = Uk+1
n − Ukn −∆tVn
∑m
AnmGk+1nm + VnQ
k+1n
, n = N + 1...2N for heat flow
(2.25)
When the equations are written in this way, the solution can be seen as a root-finding
problem; for a “perfect” solution, Rk+1n would be equal to zero for all n. The strong
CHAPTER 2. BACKGROUND 38
nonlinearity in equations (2.24) and (2.25) makes their solution difficult. The approach of
TOUGH2 is to use Newton-Raphson iteration to linearize these equations. Newton-Raphson
iteration is based on the idea that, given x0 an estimated value of the root of f(x), a better
estimate of the root can be obtained by considering the first two terms of a Taylor series
expansion of f(x) about x0:
f(x) ≈ f(x0) + f ′(x0)(x− x0) (2.26)
where x0 is the initial estimate of the root of f(x). When f(x) is set equal to zero, equation
(2.26) can be written as:
xp = xp−1 − f(xp−1)f ′(xp−1)
(2.27)
where xp−1 is the previous estimate of the root, and xp is the improved estimate. The very
same process can be used to find an approximate solution to equations (2.24) and (2.25);
only now, a multi-variable Taylor series must be used (e.g. Ferziger and Peric, 1996). The
multi-variable equivalent to equation (2.27) is given by:
Rp,k+1n (xp1, x
p2, ..., x
p2N ) = Rp−1,k+1
n (xp−11 , xp−1
2 , ...xp−12N )
+2n∑j=1
(xpj − xp−1j )
∂Rp−1,k+1n (xp−1
1 , xp−12 , ...xp−1
2n )
∂xp−1j
(2.28)
where p refers to the pth Newtonian iteration. Equation (2.28) provides a 2N × 2N system
of linear algebraic equations. The system can be written in a concise matrix form as:
Jp−1d = Rp−1 (2.29)
where d = xp − xp−1 and:
Jij =∂Ri∂xj
, i = 1...2N, j = 1...2N (2.30)
CHAPTER 2. BACKGROUND 39
Solution of the linear system of equations (2.29) provides a new estimate of the primary
variables at the new time step, xp,k+1i . Using the new values of the primary variables, the
residuals Rp,k+1n are again calculated. If the residuals are considered to be small enough,
no further Newton-Raphson iterations are conducted, and work advances to the next time
step. Specific convergence criteria are discussed in the TOUGH2 manual.
The number of Newtonian iterations needed to achieve convergence is of importance.
If p is very small, then the values of the primary variables are changing slowly. This is an
indication that the time step can probably be safely increased. In TOUGH2, the time step
is doubled when Newton-Raphson convergence is reached with p ≤ 3. On the other hand,
if the residuals are still too large at higher values of p (p > 8 in TOUGH2), the Newton-
Raphson iteration is considered to have failed to converge. In this case, the timestep is
reduced, and a new set of Newtonian iterations is initiated (Pruess et al., 1999).
In TOUGH2, equation (2.29) may be solved using either sparse direct solvers or iterative
matrix solvers. Iterative solvers, while less reliable than direct solvers, are far more efficient
in terms of both memory and computational effort; for this reason, they are preferred
(Moridis and Pruess, 1997). A maximum number of iterations is specified as a fraction of
the total number of equations 2N . Each solver has its own convergence criterion, which can
be controlled through the input parameter CLOSUR (Moridis and Pruess, 1995). Most of
the computational work of TOUGH2 is spent in the computation of the Jacobian matrix
and the solution of the resulting set of linear equations. The Jacobian matrix at each
Newtonian iteration can be included in the TOUGH2 output stream by specifying MOP(6)
= 7 in the input file. Software such as vismatrix, or the spy() function in MATLAB, may
then be used to view a graphical representation of the matrix.
Applications
TOUGH2 has been primarily applied to problems in geothermal reservoir modelling and
nuclear waste disposal (Pruess, 2004). Two derivatives of the code, T2VOC and TMVOC
CHAPTER 2. BACKGROUND 40
(Falta et al., 1995; Pruess and Battistelli, 2002), are able to model the behaviour of non-
aqueous phase liquids and have been used to model environmental remediation technologies.
Published applications of these codes have focused on air sparging at the bench scale (Hein
et al., 1997) and field scale (McCray and Falta, 1996), and steam injection at the bench
scale (Gudbjerg et al., 2004) and field scale (Gudbjerg et al., 2005; Kling et al., 2004).
Although no published studies show the use of TOUGH2 to model thermal conductive
heating systems, heater wells are easily handled through the definition of heat source terms
in the gridblocks.
TOUGH2 is often applied to fractured rock environments and provides a numerical
multi-continuum framework to handle this (Pruess and Narasimhan, 1982). Although
TOUGH2 may be used in conjunction with a discrete-fracture conceptualization, such ap-
plications of the code are rare. Pruess et al. (1984, 1990a,b) used discrete fractures to
model unsaturated flow in vertical fractures near heat-generating nuclear waste canisters.
In this particular context, it was concluded that the fluid and heat flow could be adequately
modelled by considering an equivalent continuum with specially constructed relative per-
meability and capillary pressure functions. In a study of tracer transport, Seol et al. (2005)
used a discrete fracture network model as a “base case,” the results of which the authors
then attempted to match with a modified dual-continuum model.
An alternative to discrete fracture models and equivalent continuum models is the “chan-
nelized flow” or “fracture zone” concept. Pruess and Tsang (1994) reasoned that, when
detailed information about fractures is not available, it is appropriate to approximate their
effect by introducing a porous zone of high permeability. This approach also permits the
use of much larger gridblocks, and is expected to be in between a discrete fracture and
equivalent continuum model in terms of computational demands.
Chapter 3
Thermal Conductive Heating in
Fractured Bedrock: Screening
Calculations to Assess the Effect of
Groundwater Influx
3.1 Abstract
A two-dimensional semi-analytical heat transfer solution is developed and a parameter sen-
sitivity analysis performed to determine the relative importance of rock material properties
(density, thermal conductivity and heat capacity) and hydrogeological properties (hydraulic
gradient, fracture aperture, fracture spacing) on the ability to heat fractured rock using ther-
mal conductive heating (TCH). The solution is developed using a Green’s function approach
in which an integral equation is constructed for the temperature in the fracture. Subsurface
temperature distributions are far more sensitive to hydrogeological properties than material
properties. The bulk groundwater influx (q) can provide a good estimate of the extent of
41
CHAPTER 3. SCREENING CALCULATIONS 42
influx cooling when influx is low to moderate, allowing the prediction of treatment zone
temperatures without specific knowledge of the aperture and spacing of fractures. Target
temperatures may not be reached or may be significantly delayed when the groundwater
influx is large.
3.2 Introduction
Many of the chemical and physical properties of organic chemicals frequently encountered
at hazardous waste sites exhibit a functional dependence on temperature. Elevated temper-
atures often bring about a decrease in non-aqueous phase liquid (NAPL) viscosity resulting
in an increase in NAPL mobility, a decrease in the organic carbon partition coefficient re-
sulting in decreased sorption, an increase in vapour pressure resulting in increased NAPL-air
mass transfer (vapourization), and an increase in the Henry’s constant leading to increased
water-air mass transfer (volatilization) (e.g. U.S. EPA, 2001; Sleep and McClure, 2001; Na-
tional Research Council, 2004). At high temperatures (> 100 C), heat may also stimulate
processes such as aqueous oxidation and pyrolysis that destroy contaminants in-situ and
reduce the need for above-ground treatment (Baker and Kuhlman, 2002). Although these
in-situ destruction mechanisms are typically significant only at the high temperatures used
to treat non-volatile compounds such as polychlorinated biphenyls (PCBs), they may alone
provide 95-99% of the removal in these cases (Baker and Kuhlman, 2002).
Heat can be delivered to the subsurface using several different approaches. Steam-
enhanced extraction (SEE), originally developed by the petroleum industry, has been ap-
plied at both pilot and full scales in unconsolidated deposits (e.g. Newmark et al., 1998).
However, field pilot testing of steam injection in fractured rock has demonstrated the dif-
ficulty of achieving large temperature increases throughout a treatment area (Davis et al.,
2005). Electrical resistive heating (ERH) achieves heating by passing an electrical current
CHAPTER 3. SCREENING CALCULATIONS 43
between electrodes inserted in-situ throughout the treatment area (e.g. Beyke and Flem-
ing, 2005). The amount of resistive heat produced is relatively uniform throughout the
treatment area, providing heat to low-permeability areas that may be by-passed by injected
steam. Because ERH relies on pore water to conduct electrical current, it only generates
temperatures below and at the boiling point of water. Thermal conductive heating (TCH)
systems employ arrays of wells containing resistive heating elements to provide heat to the
treatment area (e.g. Stegemeier and Vinegar, 2001). The resistive heating elements radiate
heat to the well casing, from where it is transferred away by conduction. One principal
difference between TCH and both SEE and ERH is the ability to heat to temperatures
of up to approximately 800 C, which allows the technology to target higher boiling point
compounds such as PCBs.
Several mechanisms may cause a loss of heat from the treatment area during thermal
applications. Strong vertical temperature gradients may cause heat to be lost through
conduction; in the case of TCH, wells are typically extended a minimum of two feet (0.6 m)
beyond the limit of the treatment zone to mitigate this effect (Stegemeier and Vinegar,
2001). In addition, insulating blankets may be placed on the ground surface above the
treatment area. The influx of cool groundwater may present another source of heat loss.
When the temperature of the treatment area is below 100 C, groundwater flow may cause
heated water to be carried out of the treatment area, representing a loss of energy. At
higher temperatures, cool incoming water must be boiled, causing a delay in the attainment
of target temperatures. In the presence of large groundwater influxes, the cooling influence
may be lessened by steam injection or the installation of an impermeable barrier at the
periphery of the treatment zone (Baker and Heron, 2004). Alternatively, an extra row of
heater wells could be used to preheat incoming groundwater before it enters the treatment
area.
Although the cooling effect of incoming groundwater may be a critical parameter in the
design of TCH systems, few published studies have quantitatively examined its importance
CHAPTER 3. SCREENING CALCULATIONS 44
in porous media, and none has done so in fractured rock. Elliott et al. (2003) used a commer-
cial reservoir simulator to study the cooling influence of groundwater in saturated porous
media. They found that the remediation time was largely governed by soil permeability and
hydraulic gradient; when these parameters were increased above certain threshold values,
treatment temperatures were not reached. A second modelling study (Elliott et al., 2004)
examined the effectiveness of several impermeable barrier designs in managing groundwater
influx.
The objective of this study is to present a screening-level model that can be used to
assess the effect of inflowing groundwater on the ability to heat a treatment zone in frac-
tured rock using TCH. A new semi-analytical solution is developed and used to model
different scenarios in a sensitivity analysis. A base case was established from which six
properties were varied to assess their relative importance to treatment time: hydraulic gra-
dient, fracture aperture, fracture spacing, rock density, rock thermal conductivity, and rock
heat capacity. Hydrogeological parameters were varied independently; rock properties were
varied as a group using measured values from literature. The potential for the mitigation
of groundwater influx cooling by both installation of an upgradient preheating well and
increases in thermal well power was also assessed.
3.3 Model Development
The fractured rock environment is conceptually modelled using a discrete fracture approach
whereby the location and aperture of fractures are specified directly. Fractures, which
have an aperture of e, are assumed to be parallel and evenly spaced by a distance of
2H. A schematic of the conceptual model is shown in Figure 3.1. The screening model
simulates heat transfer within a two-dimensional vertical cross section (x - y plane) oriented
perpendicular to the fractures and in line with the direction of groundwater flow (x) which
occurs at a specified velocity (v) in the fractures only. Thermal wells (line sources of heat)
CHAPTER 3. SCREENING CALCULATIONS 45
are placed within the cross section at a specified spacing. The screening model does not
simulate heat losses from the top or bottom of the overall target zone, implying that lines
of symmetry exist at a distance H above and below each fracture.
Baston 6
2.0 Model Development 119
The fractured rock environment is conceptually modeled using a discrete fracture approach whereby the 120
location and aperture of fractures are specified directly. Fractures, which have an aperture of e, are 121
assumed to be parallel and evenly spaced by a distance of 2H. A schematic of the conceptual model is 122
shown in Figure 1. The screening model simulates heat transfer within a two‐dimensional vertical cross 123
section (x – y plane) oriented perpendicular to the fractures and in line with the direction of 124
groundwater flow (x) which occurs at a specified velocity (v) in the fractures only. Thermal wells (line 125
sources of heat) are placed within the cross section at a specified spacing. The screening model does 126
not simulate heat losses from the top or bottom of the overall target zone, implying that lines of 127
symmetry exist at a distance H above and below each fracture. 128
129
Figure 1: Conceptual model of fractured rock environment. Groundwater flows at velocity v within equally 130
spaced (2H) fractures of aperture e. The model plane (x – y) is oriented perpendicular to the fractures and in line 131
with v. 132
Two‐dimensional heat transfer within the model plane may be described by two coupled differential 133
equations. In the rock matrix, the heat conduction equation is written as (e.g., Özişik 1980): 134
tT
Kyxg
yT
xT
rr ∂∂
=+∂∂
+∂∂
α1),(
2
2
2
2
(1) 135
2H
Figure 3.1: Conceptual model of fractured rock environment. Groundwater flows at velocityv within equally spaced (2H) fractures of aperture e. The model plane (x - y) is orientedperpendicular to the fractures and in line with v.
Two-dimensional heat transfer within the model plane may be described by two coupled
differential equations. In the rock matrix, the heat conduction equation is written as (e.g.
Ozisik, 1980):∂2Tm∂x2
+∂2Tm∂y2
+g(x, y)Kr
=1αr
∂Tm∂t
(3.1)
where Tm is the temperature in the rock matrix, Kr is the thermal conductivity of the rock
[W/m·K], αr is the thermal diffusivity of the rock [m2/s] and g is the strength of energy
generation [W/m3] at the point (x, y). Performing a heat balance on a control volume of
water in a fracture in the x-direction gives the following equation for the temperature in
CHAPTER 3. SCREENING CALCULATIONS 46
the fracture (Appendix A):
ρwcw∂Tf∂t
= −vρwcw∂Tf∂x
+2Kr
e
∂Tf∂y
∣∣∣∣y=e/2
(3.2)
where ρw is the density of water [kg/m3], cw is the specific heat of water [J/kg·K], v is the
average linear velocity of groundwater in the fracture [m/s], and e is the aperture of the
fracture [m]. Heat exchange between the fracture and the matrix is handled through the
g(x, y) term as follows (Cheng et al., 2001):
g(x, y) =Kr
e/2∂Tm∂y
∣∣∣∣y=e/2
(3.3)
For the dimensions and velocities typical of flow in fractured rock, it has been shown
that the heat storage term may be omitted from (3.2) without significant error (Appendix
B). This allows the governing equation for the fracture to be simplified to:
∂Tf∂x
=2Kr
ρwcwve
∂Tm∂y
∣∣∣∣y=e/2
(3.4)
Because the fracture and the edge of the rock matrix are assumed to be in thermal
equilibrium, Tm(x, 0, t) is equal to Tf (x, t) for all x, t. Consequently, no further distinction
is made between Tm and Tf in this derivation.
When heat conduction in the rock matrix occurs primarily in the direction normal to the
fracture plane, the one-dimensional form of the heat conduction equation may be substituted
in place of (3.1), and the system is reduced to a more tractable system of ordinary differential
equations. Using this assumption, Lauwerier (1955) developed a solution applicable to heat
transfer between a body of rock and a single fracture; Yang et al. (1998) published a similar
solution that includes the effect of longitudinal conduction in the fracture. Gringarten et al.
(1975) developed a Laplace-space solution for heat transfer between a body of rock and a
set of parallel fractures. Lowell (1976) simplified that solution by showing that, for the
CHAPTER 3. SCREENING CALCULATIONS 47
modelling of hot dry rock geothermal systems where fracture spacing is typically very large,
little error is introduced by considering only a single fracture.
When the rock is heated directly, as is the case in TCH, a solution must consider multi-
dimensional conduction and heat generation within the matrix. To the authors’ knowledge,
the semi-analytical solution of Cheng et al. (2001) is the only solution to consider multidi-
mensional heat conduction in the matrix. However, there does not appear to be a published
solution to the case where heat is generated within the rock matrix.
Several features distinguish the present solution from previous works. First is the explicit
modelling of multiple parallel fractures with two-dimensional heat conduction in the rock
matrix. Although previous solutions have modelled parallel fractures or multidimensional
heat conduction, no solution has included both. Second, the present solution provides for
heat generation within the matrix, allowing the inclusion of an unlimited number of heater
wells, located at arbitrary coordinates. Third, the solution is given in terms of elementary
functions rather than special functions such as Bessel functions. This reduces computation
time and provides improved accuracy when evaluating the temperature at points inside
the rock matrix. The solution is not capable of modelling boiling within the fracture, or
thermally-induced changes in rock properties. Although the two-dimensional formulation
is adequate for an exploration of the important parameters in this system, use of a three-
dimensional numerical model may be preferable for design purposes.
A schematic of the two-dimensional model domain is presented in Figure 3.2. The
domain comprises a two-dimensional strip of rock of infinite dimension in x and of finite
width (H) in y. At y = 0, water flows through a fracture of aperture e at an average
linear velocity determined from a specified hydraulic gradient using the cubic law (e.g.
Witherspoon et al., 1980). The rock matrix is assumed to be impervious to the flow of
groundwater.
Type I zero-temperature boundary conditions are assigned at x = ±∞, and homoge-
neous Type II (no-flux) boundaries are assigned at y = 0 and y = H to represent lines of
CHAPTER 3. SCREENING CALCULATIONS 48
x
y
Symmetry boundary
Symmetry boundary
• • • • • • • • • • • • • • • • • • • • • • discretized
fracture,
x=0 to x=L
heater well
locations, x = Wi
y = H
y = 0
V
Figure 3.2: Schematic of model domain. Fractures are spaced at distance 2H, groundwaterflows through the fracture at velocity v, and heater wells are located at x = Wi.
symmetry. Prior to heating, the temperature is equal to zero throughout the domain. It
is important to note that the temperature rise will be computed; therefore, the computed
temperature rise can be added to any desired uniform initial temperature. At t = 0, heater
wells located at x = Wi begin generating heat at a constant rate of g [W/m]. Although the
rate of heat generation is assumed here to be constant, it will be seen that any function
g(t) may be used, provided that its Laplace transform is known.
The solution to (3.1) is given in terms of Green’s functions as (Beck et al., 1992):
T (x, y, t) =αrKr
∫ t
τ=0
∫ H
y′=0
∫ ∞x′=−∞
G(x− x′, y − y′, t− τ)g(x′, y′, τ)dx′dy′dτ (3.5)
where G(x−x′, y−y′, t−τ) is a Green’s function corresponding to the domain and boundary
conditions described above and shown in Figure 3.2, and g(x′, y′, τ) is the strength of an
instantaneous point source. A more complete discussion of the Green’s function used in
this solution is found in the appendix. The Laplace transform of (3.5) can be taken using
the convolution property, giving:
T (x, y, s) =αrKr
∫ H
y′=0
∫ ∞x′=−∞
G(x− x′, y − y′, s)g(x′, y′, s)dx′dy′ (3.6)
CHAPTER 3. SCREENING CALCULATIONS 49
where s is the Laplace variable, and the overbar denotes a transformed quantity. Two
vehicles of heat transfer to/from the rock matrix are present in this problem: heat generated
by the thermal wells and heat exchange with groundwater in the fracture. The source
function g(x′, y′, s) can therefore be expanded as:
g(x′, y′, s) = q(x′, s)δ(y′) +2∑i=1
gwsδ(Wi − x) (3.7)
where q(x′, s) is an as yet unknown function of x′ representing heat exchange between the
rock matrix and groundwater in the fracture, and gw is the constant strength of the thermal
wells, expressed in W/m. Because the two heat source terms are geometrically linear and
orthogonal, the two integrals of (3.7) may be separated, giving:
T (x, y, s) =αrKr
∫ ∞x′=−∞
q(x′, s)G(x− x′, y− 0, s)dx′ +gwαrKrs
2∑i=1
∫ H
y′=0G(W − x, y− y′, s)dy′
(3.8)
By substituting the fracture heat balance (3.3) into the first integral of (3.8), integrating
by parts, and evaluating at y = 0, the problem is converted into a Fredholm integral equation
of the second kind:
T (x, 0, s) =veρwcw
2αrKr
∫ ∞x′=−∞
T (x′, 0, s)∂G
∂x′(x− x′, 0, s)dx′ + T (−∞, 0, s)G(x, 0, s)
+ T (∞, 0, s)G(x, 0, s)
+gwαrKrs
2∑i=1
∫ H
y′=0G(W − x, y′, s)dy′ (3.9)
The derivative of the Green’s function, ∂G/∂x′, is singular as (x′ − x) → 0. In order
to provide accurate results, (3.9) must be regularized. A subtraction method is used (e.g.
CHAPTER 3. SCREENING CALCULATIONS 50
Delves and Mohamed, 1985). The regularized form of (3.9) is given by:
T (x, 0, s) =veρwcw
2αrKr
∫ ∞x′=−∞
[T (x′, 0, s)− T (x, 0, s)
] ∂G∂x′
(x− x′, 0, s)dx′
+ T (x, 0, s)G(∞, 0, s)− T (x, 0, s)G(−∞, 0, s)− T (−∞, 0, s)G(−∞, 0, s)
+ T (∞, 0, s)G(∞, 0, s)
+gwαrKrs
2∑i=1
∫ H
y′=0G(W − x, y′, s)dy′ (3.10)
A wide array of techniques may be used to solve (3.10) numerically (e.g. Delves and
Mohamed, 1985). Like Cheng et al. (2001), our approach is to use a quadrature method
to approximate the first integral. The result is an n × n system of linear inhomogeneous
equations. The ith equation is given by:
Ti =veρwcw
2αrKr
n∑j=1
wj [Tj − Ti]∂G
∂x′(xi − xj , 0, s) + T1G(xi, 0, s)− TnG(xn − xi, 0, s)
+ TiG(xn − xi, 0, s) + Ti(xi, 0, s)
+gwαrKrs
2∑i=1
∫ H
y′=0G(W − x, y′, s)dy′, i = 1 . . . n
(3.11)
where Ti is the Laplace-space temperature at xi, and wj are the values of the nodal weights.
The values of x and w are dependent on the choice of quadrature; the formula is sufficiently
general to allow the choice of any x,w pair. A quadrature method designed for infinite
integration limits, such as Gauss-Hermite quadrature, may be used, but this requires that
the temperature be scaled by a weighting function. An alternative is to evaluate only a
finite portion of the integral, from x′ = 0 to L. This approach is valid provided that that T
is equal to zero everywhere outside the evaluated portion. This condition will be satisfied if
L is chosen to be sufficiently large, and the condition may be easily checked by examining
the computed values of the temperature at the endpoints. Once the temperature on the
CHAPTER 3. SCREENING CALCULATIONS 51
boundary is known, the temperature at any point in the domain may be determined by:
T (x, y, s) =veρwcw
2αrKr
n∑j=1
wj[Tj − T (x, 0, s)
] ∂G∂x′
(x− xj , 0, s)
+ T1(x, y, s) + TnG(L, y, s) + T (x, 0, s)G(L− x, 0, s) + T (x, 0, s)G(x, 0, s)
+gwαrKrs
2∑i=1
∫ H
y′=0G(W − x, y − y′, s)dy′, i = 1 . . . n (3.12)
A number of numerical Laplace inversion routines may be used to determine time-domain
temperatures from the values calculated in (3.11) and (3.12). The authors have had success
using the algorithms proposed by Weeks (1966) and De Hoog et al. (1982). The special case
of zero fracture aperture was verified against an analytical solution by Carslaw and Jaeger
(1959, p. 263). In addition, the numerical simulator TOUGH2 (Pruess et al., 1999) was
used to model the base case scenario (Table 3.1). The maximum difference in computed
fracture temperatures between the semi-analytical solution and the numerical solution was
6% (Appendix A).
3.4 Outline of Simulations
The base case scenario (Table 3.1) consists of shale with 500 µm horizontal fractures spaced
at 1 m. Groundwater flows through the fractures subject to a hydraulic gradient (∇h)
of -0.005 , resulting in an average linear velocity of 67 m/day (computed using µ = 1.31×
10−3 Pa·s). Heater wells, located at x = 30 m and x = 33 m, each provide a constant
heat output (gw) of 100 W/m in the y direction. This output is equivalent to the spatially
averaged heat flux generated by a row of heater wells perpendicular to the model plane,
spaced at 3 m, each providing 300 W/m - well within the range attainable by the heater
elements in current use (Stegemeier and Vinegar, 2001). Use of an analytical solution
(Carslaw and Jaeger, 1959, p. 263) shows that, in the absence of cooling from fractures,
CHAPTER 3. SCREENING CALCULATIONS 52
a target temperature of 100 C would be reached throughout the interwell zone after 17
weeks of heating.
Table 3.1: Base case parameters for sensitivity analysisParameter Unit Value
Rock Type - ShaleHeater Well Locations m x = 30, 33Heater Well Power (gw) W/m 100Initial Temperature C 10Fracture Aperture (e) µm 500Fracture Spacing (2H) m 1Hydraulic Gradient (∇h) - -0.005Influent Temperature C 10
Parameter sensitivity was assessed through four sets of trials (Table 3.2). Three sets
involved the variation of a hydrogeological parameter: hydraulic gradient, fracture aperture,
or fracture spacing. In the fourth set, the host rock type was changed. The hydraulic
gradient values selected range from those representative of a natural gradient (0.0001) to
those representative of pumping conditions (0.05). Values for the thermal conductivity,
density, and specific heat capacity of each rock type used (Table 3.3) were taken from
literature (Cermak and Rybach, 1982). Although the thermal conductivity of sedimentary
rocks tends to decline with temperature (e.g. Clauser and Huenges, 1995), all rock properties
were assumed to remain constant with temperature in order to preserve the linearity of the
governing equations (3.1) and (3.4).
Table 3.2: Summary of sensitivity testing trialsParameter Range of Values Bulk Influx Range (L/m2·day)
Hydraulic Gradient (∇h) 0.0001 to 0.05 0.674 to 337Fracture Aperture (e) 10 µm to 2000 µm 2.69× 10−4 to 2160Fracture Spacing (2H) 0.25 m to 4 m 8.42 to 135Rock Type shale, limestone,
dolomite, sandstone
CHAPTER 3. SCREENING CALCULATIONS 53
Table 3.3: Rock material properties (Cermak and Rybach, 1982)
Rock Type Thermal Conductivity Density Specific Heat Capacity(W/m·K) (kg/m3) (J/kg·K)
Shale 2.98 2757 1180Sandstone 3.03 2391 960Limestone 2.40 2520 890Dolomite 2.87 2536 920
3.5 Results and Discussion
In order to facilitate direct comparison of the various trials, the temperature distribution
in the system is summarized by two values: the minimum (Tmin) and maximum (Tmax)
temperatures in the fracture between the two thermal wells after one year of heating. This
time was chosen as a point for comparison because transient results (discussed later) show
that the effect of changes in some parameters may not be apparent during the earlier part
of the heating period.
Although groundwater influx generally skews the distribution of temperature contours
near thermal wells, the contours remain primarily normal to the fracture plane. This is
a consequence of the rapid thermal diffusion that occurs in the direction normal to the
fracture plane. Figure 3.3(a), for example, illustrates the two-dimensional distribution of
temperatures after one year of heating in the vicinity of a single heater well for the case
of e = 1 mm, 2H = 1 m, and v = 20 m/day. Figure 3.3(b) illustrates the temperature
distribution in the absence of groundwater flow. Because the temperature difference between
the fracture and the centre of the matrix is typically less than one degree, Tmin and Tmax
provide good indicators of the overall temperature distribution in the treatment zone.
3.5.1 Sensitivity to Hydrogeological Parameters
Minimum and maximum interwell fracture temperatures (Tmin) and Tmax) after one year
of heating are plotted in Figure 3.4(a) for the variation of hydraulic gradient between
CHAPTER 3. SCREENING CALCULATIONS 54
Distance Along Fracture Plane (m)
Dista
nce f
rom
Frac
ture P
lane (
m)
0 10 20 30 40 50 60 700
0.1
0.2
0.3
0.4
0.5
Distance Along Fracture Plane(m)
Dista
nce f
rom
Frac
ture P
lane (
m)
0 10 20 30 40 50 60 700
0.1
0.2
0.3
0.4
0.5
20
40
60
80
100
20
40
60
80
100
(b)
(a)T (°C)
T (°C)
Figure 3.3: Temperature distribution resulting from heating by a single thermal well, show-ing the general effect of groundwater influx. (a) v = 20 m/day, 2H = 1 m, e = 1 mm,gw = 100 W/m (b) no flow, gw = 100 W/m.
−10−4 and −5 × 10−2. The semi-analytical solution is not capable modelling the boiling
of water, so temperatures above 120 C are not shown. Computed temperatures for the
hydraulic aperture trials and fracture spacing trials are found in Figures 3.4(b) and 3.4(c),
respectively. It can be observed that a high hydraulic gradient, or the presence of large-
aperture or closely-spaced fractures can significantly inhibit heating in the treatment zone.
A variation in any of the hydrogeological parameters will have an impact on the bulk
groundwater influx (q). Since the matrix is considered to be impermeable, q can be calcu-
lated as:
q =ρge3(∇h)
24Hµ(3.13)
where µ is the dynamic viscosity of water. For each of the hydrogeological parameter
variation trials plotted in Figures 3.4(a-c), the bulk groundwater influx was calculated.
Figure 3.4(d) presents Tmin and Tmax plotted against the calculated bulk groundwater
influx values.
CHAPTER 3. SCREENING CALCULATIONS 55
60
80
100
120
ature (°C) Tmin
Tmax
0
20
40
0 0.01 0.02 0.03 0.04 0.05 0.06
Tempe
r
Hydraulic Gradient
(a) Hydraulic Gradient
60
80
100
120
ture (°C)
Tmin
Tmax
0
20
40
0 500 1000 1500 2000 2500
Tempe
ra
Fracture Aperture (μm)
(b) Aperture
60
80
100
120
ature (°C)
0
20
40
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Tempe
r
Fracture Spacing (m)
TminTmax
(c) Fracture Spacing
60
80
100
120
ature (°C)
TminTmax
0
20
40
0 50 100 150 200 250 300 350 400
Tempe
r
Ground Water Influx (L/m2∙day)
(d) Bulk Influx
Figure 3.4: Summary of computed one-year interwell fracture temperatures.
It is apparent from the temperatures plotted in Figure 3.4(d) that there is some degree of
correlation between bulk influx and treatment zone temperatures. To test the robustness of
this correlation, a series of tests was performed in which several combinations and/or values
of hydrogeological parameters were employed to arrive at a single value of bulk groundwater
influx. The semi-analytical solution was applied by assigning each combination of influx
and spacing presented in Table 3.4 and adjusting the aperture accordingly, with a gradient
fixed at -0.005. Results for the cases of q = 33.7 L/m2·day and q = 3.23 L/m2·day are
summarized in Figure 3.5, where transient temperature profiles are shown for the point in
the fracture at the centre (x = 31.5 m) of the treatment zone (not necessarily the location
of Tmin or Tmax).
For the case of q = 33.7 L/m2·day, the temperature profiles are in good agreement
CHAPTER 3. SCREENING CALCULATIONS 56
Table 3.4: Parameters for runs used to assess correlation between bulk influx and treatmentzone temperature
Parameter Unit Values
Bulk Influx L/m2·day 33.7, 3.23, 0.323, 0.0323Fracture spacing m 0.25, 0.5, 1.0, 2.5, 5.0, 10.0
60
80
100
120
erture (°C)
q = 33.7 L/m2∙day
0.25 m0 50m
0
20
40
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Tempe
Time (years)
0.50 m1.0 m2.5 m5.0 m10 m
(a) high influx
60
80
100
120
erture (°C)
q = 3.23 L/m2∙day
0.25 m0 50m
0
20
40
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Tempe
Time (years)
0.50 m1.0 m2.5 m5.0 m10 m
(b) mid-level influx
Figure 3.5: Transient fracture temperature profiles at the centre of the treatment zonefor high (a) and mid-level (b) values of groundwater influx. Curves correspond to variousfracture spacings.
for fracture spacings of 2.5 m or less, but temperatures drop off when fracture spacing is
increased beyond this threshold. Similarly, for the case of q = 323 L/ m2·day, deviation
was observed for fracture spacings of 5 m and greater. When the bulk influx is decreased
to 3.23 L/m2·day (Figure 3.5(b)), fracture spacing does not appear to play a large role
in determining the temperature distribution. Results are not shown for the cases of lower
influx, where there is no significant difference between the temperature profiles at any
time during heating. Therefore, while the correlation between influx and Tmin and Tmax
appears to be strong in Figure 3.4(d), the temperature profiles in Figure 3.5(a) show that
the correlation becomes weaker when more extreme combinations of parameters are used.
For a screening-level estimate of the extent of fracture cooling at sub-boiling tempera-
tures for conditions of low to moderate influx, it may be necessary only to know the influx
CHAPTER 3. SCREENING CALCULATIONS 57
through the treatment area and not have knowledge of the particular combination of spac-
ing and aperture. This forms a distinction between the problem of heat extraction from hot
dry rock and subsurface heating using thermal wells. For example, Gringarten et al. (1975)
found that the number and size of fractures had a strong effect on heat extraction from a
hot dry rock reservoir, even when the total flow rate was kept constant. This difference in
behaviour may be due to the different time scales of the two problems; it is after a reser-
voir has been in operation for several years that the predicted outlet temperature becomes
dependent on the nature of the fractures carrying the flow.
3.5.2 Sensitivity to Host Rock Type
Using the four sets of rock properties (Table 3.3) and the base case properties (Table 3.1),
the temperature in the fracture between the two thermal wells was computed. Figure 3.6
illustrates the fracture temperature profiles for each rock type after one month and one year
of heating. Compared to the hydrogeological parameters, host rock material properties play
a relatively minor role in determining temperature distributions throughout the treatment
zone. This behaviour is not surprising, as the range of material properties is far smaller
than the range of hydrogeological parameters. Heating in rocks with low thermal diffusivity
will progress more slowly than in rocks with high thermal diffusivity; yet, this variation
does little to affect the shape of the steady-state temperature profile.
3.5.3 Transient Behaviour
As an example of transient behaviour during heating, the temperature in the fracture at
the midpoint between the two heater wells is plotted in Figure 3.7 for eight different values
of hydraulic gradient, using the base-case parameters of 500 µm fractures spaced by 1 m.
The transient temperature plot shows a correlation between the groundwater influx (in this
case determined by a varying hydraulic gradient) and the time needed to reach a steady-
state temperature profile. Similar results are seen when the influx is varied by a change in
CHAPTER 3. SCREENING CALCULATIONS 58
60
80
100
120
erature (°C)
Shale
Limestone
Dolomite
Sandstone
0
20
40
30.0 30.5 31.0 31.5 32.0 32.5 33.0
Tempe
Distance (m)
(a) t = 1 month
60
80
100
120
erature (°C)
Shale
0
20
40
30.0 30.5 31.0 31.5 32.0 32.5 33.0
Tempe
Distance (m)
Shale
Limestone
Dolomite
Sandstone
(b) t = 1 year
Figure 3.6: Early (a) and late-time (b) fracture temperature profiles for various rock types.
aperture or fracture spacing.
For a given value of groundwater influx, a temperature threshold exists below which
the fractures have little effect. When groundwater influx is very small (due to low-aperture
or widely spaced fractures, shallow gradient, etc.) the cooling effect is negligible until
temperatures approach 100 C or higher. For a mid-level groundwater influx, such as the
base case in this study, the cooling effect is negligible until a temperature of about 50 C,
where heating begins to lag before reaching a steady state. When the groundwater influx
is very high, the threshold temperature is so low that a steady state is reached almost
immediately, before any significant heating occurs.
3.5.4 Mitigation of Cooling Effect
The semi-analytical solution was employed to model two methods of overcoming the ground-
water influx cooling effect: the installation of a pre-heating well, and an increase in the
thermal well heat production. Using the base case parameters (Table 3.1), the time to
reach a target temperature of 100 C was calculated for thermal well heat production rates
between 70 W/m and 1000 W/m. The amount of energy required to achieve this target was
calculated by the product of the heat generation rate [W/m2] and the time [days] to reach
CHAPTER 3. SCREENING CALCULATIONS 59
60
80
100
120
erature (°C)
‐0.025
‐0.005
‐0.0075
0
20
40
0 1 2 3
Tempe
Heating Time (years)
‐0.010
‐0.015‐0.025
Figure 3.7: Transient fracture temperature profiles at midpoint between heater wells, show-ing the influence of hydraulic gradient.
the target. Computed values for the time to reach the target and energy consumption are
shown in Figure 3.8.
With these parameters, the target is not reached within three years at heat production
rates below 140 W/m. However, the time required to reach the target decreases sharply
above 140 W/m, reaching a minimum at approximately two weeks for heat fluxes above
900 W/m. Because the total power consumption is correlated to the heating time, it appears
to be generally advantageous to generate heat at a high rate, although for the parameters
used in this study, efficiency peaks at approximately 300 W/m. At peak efficiency, electrical
costs total approximately 94 dollars per well, per metre depth (assuming an electrical cost
of 15 cents / kW·h.)
Mitigation using a preheating well was also modelled (results not shown). Using the base
case parameters, a third thermal well was placed 3 m upgradient of the first well (x = 27 m),
and the temperature in the original interwell zone was monitored through time. Although
the preheating well caused a rise in the interwell temperatures, it was not sufficient to reach
the target temperature throughout this entire zone.
CHAPTER 3. SCREENING CALCULATIONS 60
1500
2000
2500
200
250
300
350
mption (kW‐h/m
2 )
e (days)
Time to Reach 100 °C (days)
Power Consumption (kW‐h)
0
500
1000
0
50
100
150
0 200 400 600 800 1000 Total Ene
rgy Co
nsum
Time
Thermal Well Heat Production (W/m)
Figure 3.8: Effect of increased heat production rate on the time needed to reach a targettemperature of 100 C and total energy consumption.
3.6 Conclusions
Groundwater influx may prevent or delay the heating of fractured rock during application
of thermal conductive heating (TCH). When bulk groundwater influx is high, temperatures
in the fractures are influenced by the aperture and spacing of fractures. For medium and
low values of influx, fracture properties do not appear to be important in determining the
temperature in fractures. In these cases, it appears not to be important to characterize dis-
crete fracture features in the treatment zone; only a quantification of the total groundwater
influx through the treatment zone is necessary.
Variations in material properties (rock density, rock thermal conductivity, and rock heat
capacity) amongst rock types do have a small effect on the early-time temperature distri-
bution in the rock, but on the whole are less significant than variations in hydrogeological
parameters (hydraulic gradient, fracture aperture, and fracture spacing). It is noted that the
range of variation in material properties is much smaller than the range of hydrogeological
properties, which may vary by several orders of magnitude.
Transient analysis shows that influx cooling affects treatment zone temperatures only
CHAPTER 3. SCREENING CALCULATIONS 61
once a certain temperature threshold has been passed during heating. It is possible that, if
target treatment temperatures are low, influx cooling may not pose a problem.
One solution to the problem of groundwater influx cooling is to simply increase the power
delivered to the thermal wells. In the case where this may not be done due to equipment
limitations or other concerns, preheating wells installed outside of the treatment zone may
be used to partially mitigate the cooling effects.
3.7 Acknowledgements
The authors would like to thank the U.S. Department of Defence Environmental Security
Technology Certification Program (Contract ER-0715), the Natural Sciences and Engineer-
ing Research Council of Canada (Discovery Grant), and Queen’s University for financial
support of this work. Tom Gleeson is thanked for many helpful conversations and sugges-
tions.
3.8 Green’s Function
The Green’s function used in the presented solution is determined by the product of two
one-dimensional Green’s functions: GX00(x, t|x′, τ), for an infinite domain in x, where the
temperature vanishes as x→ ±∞; and GY 22(y, t, y′, τ), for finite domain in y of length H,
with Type-II boundary conditions at y = 0 and y = H. These functions are given by Beck
et al. (1992) as:
GX00(x, t|x′, τ) =
(1
2√πα(t− τ)
)exp
(− (x− x′)2
4α(t− τ)
)(3.14)
GY 22(y, t|y′, τ) =1H
[1 + 2
∞∑m=1
exp(−m2π2α(t− τ)
H2
)cos(mπyH
)cos(mπy′
H
)](3.15)
CHAPTER 3. SCREENING CALCULATIONS 62
The effect of a point source function g(x′, y′, τ) at (x, y, t) is given by:
T (x, y, t) =α
K
∫ t
τ=0
∫ a2
x′=a1
∫ b2
y′=b1
GXIJ(x, t|x′, τ)GYMN (y, t|y′, τ)g(x′, y′, τ)dy′dx′dτ
(3.16)
where G is the product GX00GY 22. From the convolution property of the Laplace transform,
the Laplace transform of (3.16) is given by:
T (x, y, s) =α
K
∫ a2
x′=a1
∫ b2
y′=b1
L[GXIJ(x, t|x′, τ)GYMN (y, t|y′, τ)]L[g(x′, y′)]dy′dx′ (3.17)
where G, the Laplace transform of GX00GY 22, is given by:
G(x, y|x′, y′) =1
2H√sα
exp(−|x− x′|
√s
α
)
+∞∑m=1
exp(−|x− x′|
√sα + m2π2
H
)cos(mπyH
)cos(mπy′
H
)√H2s+m2π2α
(3.18)
Chapter 4
Numerical Modelling of Thermal
Conductive Heating in Fractured
Bedrock
4.1 Abstract
Numerical modelling was employed to study the performance of thermal conductive heating
(TCH) systems in a fractured shale under a variety of hydrogeological conditions. Model
results show that the effect of concentrated flow in fractures does not significantly affect
treatment zone temperature distributions, except near the beginning of heating or when
groundwater influx is high. Excluding these scenarios, there is little difference in the ability
of discrete fracture and equivalent porous media (EPM) simulations to model tempera-
ture distributions. However, fracture and rock matrix properties can significantly influence
the time necessary to reach complete steam saturation in the treatment area. A low-
permeability matrix, large porosity, or high fracture spacing can contribute to boiling point
elevation in the rock matrix. Consequently, knowledge of these properties is important for
63
CHAPTER 4. NUMERICAL MODELLING 64
the estimation of treatment times in these environments. Because of the variability in boil-
ing point throughout a fractured rock treatment zone, it may be difficult to monitor the
progress of thermal treatment using temperature measurements alone.
4.2 Introduction
Thermal methods of in-situ remediation attempt to take advantage of enhanced mass trans-
fer processes occurring at elevated temperatures in order to remove organic contaminants
from the subsurface, or destroy them in-situ. For many organic chemicals, elevated tem-
peratures bring about enhanced dissolution, vapourization, volatilization, desorption and
non-aqueous phase liquid (NAPL) mobility (e.g. National Research Council, 2004). A va-
riety of methods have evolved for heating the subsurface; in this study, consideration is
given to thermal conductive heating (TCH). TCH systems employ electrical heater wells
to directly heat the subsurface to temperatures of up to 800 C. The technology has been
applied at several field sites in porous media, but has been used only recently in fractured
rock environments.
The design of thermal conductive heating systems may be significantly affected by the
flow of cool groundwater into the treatment zone. This cooling effect has been examined in
modelling studies conducted in porous media (Elliott et al., 2003, 2004) and fractured media
(Baston et al., 2007). For high values of intrinsic permeability or hydraulic gradient, target
temperatures may not be attained unless the influx cooling is accommodated in the heating
system design through the use of components such as preheating wells and impermeable
barriers. Although the effect of influx on boiling has been studied in porous media (Elliott
et al., 2003, 2004), no published study has examined how boiling during thermal treatment
may be affected by the high degree of heterogeneity present in fractured rock environments.
Using a two-dimensional semi-analytical solution, Baston et al. (2007) found that for the
prediction of treatment zone temperatures, the size and location of fractures are important
CHAPTER 4. NUMERICAL MODELLING 65
when groundwater influx is high. However, the semi-analytical solution used in that study
was not able to model the boiling of water.
The goal of the present study is to analyze the performance of a TCH system in a
three-dimensional fractured shale environment where boiling of the water is considered. A
numerical model is utilized to compute temperature distributions and steam saturations
in a circular treatment zone heated by seven thermal wells. The importance of matrix
permeability, matrix porosity, bulk permeability, and fracture spacing is examined.
4.3 Numerical Model
4.3.1 Model Domain and Boundary Conditions
Numerical modelling was conducted using the TOUGH2 simulator, which is capable of
modelling nonisothermal multiphase flows (Pruess et al., 1999). The numerical simulations
were designed to model seven heater wells arranged in a hexagon, with a heater-extraction
well at the centre (Figure 4.1). This geometry permits the use of partial radial symmetry,
whereby only a slice of the hexagon is modelled. The rock is a fractured shale, with equal-
aperture fractures occurring in the horizontal plane only and at regular spacing. The shape
of the model domain in the horizontal plane is shown in Figure 4.1. In the vertical direction,
the domain comprises one half of a fracture and adjacent matrix block, making use of vertical
symmetry (Figure 4.2).
The control volume formulation of TOUGH2 permits the use of irregularly shaped grids
such as those used in this study (Figure 4.3). In order to represent the flow that would
occur in fractures, a row of 1 mm thick “fracture zone” cells with high permeability was
created along the z = 0 plane (Figure 4.2). It was shown through a grid sensitivity study
and comparison with a semi-analytical discrete fracture solution that this approach results
in temperature distributions that are very similar to those predicted by a discrete fracture
model, while achieving much more rapid convergence due to the larger cell sizes (Appendix
CHAPTER 4. NUMERICAL MODELLING 66
(Figure 1). This geometry permits the use of partial radial symmetry, whereby only a slice of the
hexagon is modeled. The rock is a fractured shale, with equal-aperture fractures occurring in the
horizontal plane only and at regular spacing. The shape of the model domain in the horizontal
plane is shown in Figure 1. In the vertical direction, the domain comprises one half of a fracture
and adjacent matrix block, making use of vertical symmetry (Figure 2).
Figure 1: Plan view of model domain
The control volume formulation of TOUGH2 permits the use of irregularly shaped grids such as
those used in this study (Figure 3). In order to represent the flow that would occur in fractures, a
row of 1 mm thick “fracture zone” cells with high permeability was created along the z = 0 plane
(Figure 2). It was shown through a grid sensitivity study and comparison with a semianalytical
discrete fracture solution that this approach results in temperature distributions that are nearly
identical to those predicted by a discrete fracture model, while achieving much more rapid
convergence due to the larger cell sizes (Baston, 2008).
heater wells (6)
symmetry boundary
θ = 30°
flow
3 m well spacing model boundary
symmetry boundary
x
y
heater‐extraction well (1)
T = To P = P0
Figure 4.1: Plan view of model domain
C).
Initially, the entire domain is at 10 C. The initial pressure distribution throughout the
model domain is hydrostatic, with the pressure at z = 0 corresponding to a depth of 15 m
below the water table. Pressures in the extraction well, which were held constant through-
out the simulation, were based on 1 m of drawdown from the initial pressure condition. At
the outer edge of the domain (r = 25m), both temperature and pressure were held constant.
A grid dependence analysis showed that increasing the size of the model domain in the ra-
dial direction did not significantly affect temperature or pressure distributions within the
heated region (Appendix C). The two thermal wells located within the model domain, both
assumed to produce heat at a steady rate of 800 W/m, were represented by line sources.
The equations of van Genuchten (1980) were used for capillary pressure and relative per-
meability functions. All properties of water and steam were computed using the EOS1
module of TOUGH2, which provides an implementation of the 1967 International Formu-
lation Committee steam table equations (e.g. ASME, 1979). The model does not consider
processes such as fracture creation or dilation that may occur under elevated pressures.
CHAPTER 4. NUMERICAL MODELLING 67
Initially, the entire domain is at 10 °C. The initial pressure distribution throughout the model
domain is hydrostatic, with the pressure at z = 0 corresponding to a depth of 15 m below the
water table. Pressures in the extraction well, which were held constant throughout the
simulation, were based on 1 m of drawdown from the initial pressure condition. At the outer
edge of the domain (r = 25 m), both temperature and pressure were held constant. A grid
dependence analysis showed that increasing the size of the model domain in the radial direction
did not significantly affect temperature or pressure distributions within the heated region
(Baston, 2008). The two thermal wells located within the model domain, both assumed to
produce heat at a steady rate of 800 W/m, were represented by line sources. The equations of
van Genuchten (1980) were used for capillary pressure and relative permeability functions. All
properties of water and steam were computed using the EOS1 module of TOUGH2 was used,
Figure 2: Section of model domain in rz plane
Figure 3: Isometric view of model domain
symmetry boundary
symmetry boundary
r
z 1 mm fracture zone
T = To P = P0
extraction well P = P0 – 9806 Pa
Figure 4.2: Section of model domain in r -z plane
Initially, the entire domain is at 10 °C. The initial pressure distribution throughout the model
domain is hydrostatic, with the pressure at z = 0 corresponding to a depth of 15 m below the
water table. Pressures in the extraction well, which were held constant throughout the
simulation, were based on 1 m of drawdown from the initial pressure condition. At the outer
edge of the domain (r = 25 m), both temperature and pressure were held constant. A grid
dependence analysis showed that increasing the size of the model domain in the radial direction
did not significantly affect temperature or pressure distributions within the heated region
(Baston, 2008). The two thermal wells located within the model domain, both assumed to
produce heat at a steady rate of 800 W/m, were represented by line sources. The equations of
van Genuchten (1980) were used for capillary pressure and relative permeability functions. All
properties of water and steam were computed using the EOS1 module of TOUGH2 was used,
Figure 2: Section of model domain in rz plane
Figure 3: Isometric view of model domain
symmetry boundary
symmetry boundary
r
z 1 mm fracture zone
T = To P = P0
extraction well P = P0 – 9806 Pa
Figure 4.3: Isometric view of model do-main
4.3.2 Rock Properties and Outline of Simulations
A series of simulations was conducted to examine the effect of matrix permeability (km),
matrix porosity (φ), bulk permeability (kb), and fracture spacing (2H) on the temperature
and time necessary to reach complete steam saturation within the treatment zone, defined
as the area within a 3 m radius of the centre well.
A base case set of rock matrix and bulk medium properties is defined in Table 4.1. To
evaluate the sensitivity to rock matrix properties, km and φ were varied independently over
the ranges in Table 4.2. The range of rock matrix permeabilities, 10−18 m2 to 10−22 m2,
is representative of literature values for the permeability of shale (e.g. Keith and Rimstidt,
1985; de Marsily, 1986; Hart et al., 2006). Rock thermal properties, variations in which have
been shown to play a minor role in this context (Baston et al., 2007) were held constant
throughout the simulations.
To evaluate the sensitivity to bulk medium properties, a simulation was conducted for
each combination of bulk permeability and fracture spacing in Table 4.2. In addition to
the five values of fracture spacing used, a simulation was run for each value of kb using an
equivalent porous medium (EPM) approximation.
CHAPTER 4. NUMERICAL MODELLING 68
For a given value of kb, an increase in the fracture spacing is associated with increased
fracture permeability and therefore a higher concentration of flow in the fracture zone cells.
The fracture zone permeability was calculated by considering the bulk permeability to be
a weighted average of the matrix and fracture zone permeabilities.
Table 4.1: Base case properties. Density, specific heat capacity, and thermal conductivityfrom Cermak and Rybach (1982).
Property Unit Value
Rock MatrixPermeability m2 10−18
Porosity - 0.03Density kg/m3 2757Specific heat capacity J/kg·K 1100Thermal conductivity W/m·K 2.98
Bulk MediumFracture spacing m 2.5Bulk permeability m2 10−13
Table 4.2: Parameters varied.Parameter Unit Values Tested
Rock MatrixPermeability m2 10−17, 10−18, 10−19,10−20,10−22
Porosity - 0.005, 0.010, 0.050, 0.100
Bulk MediumBulk permeability m2 1.0× 10−11, 1.0× 10−12, 1.0× 10−13, 7.5× 10−14,
5.0× 10−14, 1.0× 10−14, 1.0× 10−15, 1.0× 10−16
Fracture spacing m 0 (EPM), 1.0, 2.5, 5.0, 7.5, 10.0
4.4 Results and Discussion
The fluid pressure response to the heating of a saturated rock mass depends on the relative
influence of the thermal and hydraulic diffusivities of the rock (Palciauskas and Domenico,
CHAPTER 4. NUMERICAL MODELLING 69
1982). When the hydraulic diffusivity is much larger than the thermal diffusivity, the
thermal expansion occurs at near-constant pressure and the system responds as a drained
medium, with water being easily expulsed from the heated region. When the hydraulic
diffusivity is less than the thermal diffusivity, the system behaves as an undrained medium.
In this case, the heated water is not able to escape and a significant pressure rise occurs
(Palciauskas and Domenico, 1982).
The thermal diffusivity of the rock matrix is less than or approximately equal to the
hydraulic diffusivity for the rock considered in this study. In the fracture, however, the
hydraulic diffusivity is far in excess of the thermal diffusivity. Consequently, the fractures
act as drains for the fluid expansion in the rock matrix pores.
As boiling progresses in the treatment zone, a steam front propagates radially outward.
When a gridblock on the steam front is heated, water within the pores expands. The
mobility of this water is limited; it cannot flow radially inwards as a result of expanding
steam drive. Due to the low permeability of the rock matrix, the water can flow outward
only slowly; consequently, the pressure in the gridblock will rise. Because the pressure rise
causes the boiling point of water to increase, the process is self-promoting: the boiling point
elevation causes the water to expand more prior to boiling, with a concomitant increase in
pressure. Finally, the water does boil and the pressure is relieved as the less viscous steam
flows toward the extraction well. Still, some time is necessary for pressures in the rock
matrix to return to their “background” levels.
This pressure response is shown in Figure 4.4 for a location just outside the treatment
zone (r = 3.88 m, θ = 28.5), using the base case parameters (Table 4.1). The pressure dis-
tribution is very similar to that seen in a low-permeability medium subjected to an external
loading. For example, a similar pore pressure distribution was calculated by Nogami and Li
(2003), using an analytical solution for consolidation in a system of alternating horizontal
sand and clay layers. The magnitude of this pressure spike, which effectively determines
the time necessary to reach complete steam saturation in the matrix, is determined in part
CHAPTER 4. NUMERICAL MODELLING 70
by the fracture spacing, matrix permeability, and matrix porosity.
Because boiling in the rock matrix does not necessarily take place at constant pressure,
the temperature may not remain constant throughout the boiling period. Consequently,
a temperature plateau may not be observed in the rock matrix, as it typically is in the
fracture. In Figure 4.5, temperature is plotted against time for a reference block in the rock
matrix (r = 2.96 m, θ = 28.5, z = 1.17 m) and in the fracture zone (r = 2.96 m, θ = 28.5,
z = 0.5 mm) for the base case parameters (Table 4.1). Although a clear temperature plateau
is observed during the boiling period in the fracture, the boiling period in the rock matrix
can not be discerned from temperature alone.
0.8
1.0
1.3
rom
Fra
ctur
e (m
)
343314
292
37139
168226
285
0.0
0.3
0.5
0 5 10 15 20
Dis
tanc
e Fr
Pressure (bar)
Figure 4.4: Pressure as a function of dis-tance from the fracture for a location justoutside the treatment zone, computed us-ing the base case parameters. Heating time(days) is indicated for each line. Pressureprofiles prior to the beginning of boilingare indicated with a solid line; profiles af-ter the end of boiling are indicated with adashed line.
150
200
250
300
350pe
rature (°C)
Matrix Fracture
end of boiling in matrix
start of boiling
start of boiling in fracture
0
50
100
150
0 50 100 150 200 250 300 350
Temp
Heating Time (days)
start of boiling in matrix
end of boiling in fracture
Figure 4.5: Temperature vs. time for areference block in the rock matrix (dot-ted line) and fracture (solid line), at theboundary of the treatment zone.
CHAPTER 4. NUMERICAL MODELLING 71
4.4.1 Influence of Matrix Permeability
The lower the matrix permeability, the more slowly the heated water will move outwards.
Consequently, the pressure rise becomes larger as the matrix permeability is decreased,
as shown in Figure 4.6(a) for a reference gridblock at the centre of the rock matrix and
edge of the treatment zone (r = 2.96 m, θ = 28.5, z = 1.17 m). The degree of boiling
point elevation can be quite significant in a very low-permeability rock matrix; even at
treatment zone temperatures in excess of 300 C, water in the matrix pores may still be
in the liquid phase. It is conceivable that stresses resulting from the high pore pressure in
these cases could cause failure or microfracturing of the rock, thus attenuating the pressure
spike and reducing the boiling point elevation (Palciauskas and Domenico, 1982; Horseman
and McEwen, 1996).
In cases where the boiling point elevation is large, the time necessary to bring about
complete steam saturation throughout the treatment zone will increase significantly. In
Figure 4.6(b), the volume fraction of the treatment zone that has reached complete steam
saturation is shown for several values of matrix permeability. The significance of the boiling
point elevation may depend on the context of the thermal treatment. A boiling point of
300 C may not affect the treatment of low boiling-point compounds such as PCBs, where
the treatment temperature would need to exceed 300 C even in the absence of boiling point
elevation. However, boiling point elevation may significantly delay the treatment of volatile
compounds, which would be treated near 100 C in a porous medium (LaChance et al.,
2006).
4.4.2 Influence of Matrix Porosity
Figure 4.7(a) presents the transient pressure response for a reference block near the centre
of the rock matrix, at the outer radius of the treatment zone. The time necessary to reach
complete steam saturation is influenced by the matrix porosity, which not only determines
CHAPTER 4. NUMERICAL MODELLING 72
10
100
1000
10000
Pressure (b
ar)
1
0 50 100 150 200 250 300 350
Heating Time (days)
km = 1e‐17 km = 1e‐18 km = 1e‐19
km = 1e‐20 km = 1e‐22
(a)
0 2
0.4
0.6
0.8
1.0
1.2
ge Steam
Saturation
Treatm
ent Z
one
0.0
0.2
0 50 100 150 200 250 300 350
Averag
in T
Heating Time (days)
km = 1e‐17 km = 1e‐18 km = 1e‐19
km = 1e‐20 km = 1e‐22
(b)
Figure 4.6: Impact of matrix permeability (km) on magnitude of pressure spike at centre ofrock matrix (a) and steam saturation within treatment zone (b).
the quantity of water to be boiled, but also has an effect on the pressure distribution. For
a given value of intrinsic permeability, an increase in the porosity causes a larger volume of
water to undergo thermal expansion, without increasing the ability of the expanded water
to flow outward. Consequently, the magnitude of the pressure spike will increase. For the
range of porosities tested, the significance of matrix porosity is much less than permeability
in determining the magnitude of the pressure spike.
The effect of porosity on steam saturation in the treatment zone is shown in Figure
4.7(b). In addition to boiling point elevation due to an increase in pore pressures, an
increase in matrix porosity causes a larger amount of energy to be expended to overcome
the latent heat of vapourization, thereby resulting in prolonged treatment times.
4.4.3 Influence of bulk medium and fracture properties
The importance of bulk medium and fracture properties was assessed through an exami-
nation of treatment zones temperatures and the time necessary to reach complete steam
saturation within the treatment zone. For each combination of bulk medium properties in
CHAPTER 4. NUMERICAL MODELLING 73
1
20
25
30
35su
re (b
ar)
φ = 0.005
φ = 0.01
φ = 0.03
φ = 0.05
0
5
10
15
0 100 200 300 400 500
Pres
s
Heating Time (days)
φ = 0.10
(a)
0.6
0.8
1.0
1.2
am S
atur
atio
n in
m
ent Z
one
φ = 0.005
0.0
0.2
0.4
0 50 100 150 200 250 300 350
Ave
rage
Ste
aTr
eatm
Heating Time (days)
φ = 0.01
φ = 0.03
φ = 0.05
φ = 0.10
(b)
Figure 4.7: Impact of porosity (φ) on magnitude of pressure spike at centre of rock matrix(a) and steam saturation within the treatment zone (b).
Table 4.2, the minimum temperature and the average steam saturation in the treatment
zone were recorded at fixed time intervals. It is important to note that the location of
the minimum treatment zone temperature may vary throughout time; consequently, the
temperatures plotted do not correspond to one single location in space.
Treatment Zone Temperatures
Figure 4.8 plots the minimum treatment zone temperature versus time for four values of bulk
permeability. For each value of bulk permeability, the temperature is plotted as calculated
using four different values of fracture spacing, as well an equivalent porous medium. It is
important to note that the location of the minimum treatment zone temperature may vary
throughout time; consequently, the temperatures plotted do not correspond to one single
location in space.
At high values of bulk permeability, the temperature profiles are very dependent on the
heterogeneity of the system. When kb is equal to 10−12 m2, the treatment zone does not
reach boiling temperatures and comes to a steady state with a minimum temperature of
approximately 50 C (Figure 4.8(a)). The time needed to reach this steady state varies by
CHAPTER 4. NUMERICAL MODELLING 74
a factor of approximately five, depending on the fracture spacing. For each case of fracture
spacing, the final temperature is the same, illustrating that flow heterogeneity has an effect
only at early and mid-time in this scenario.
When the bulk permeability is decreased to 10−13 m2, the effect of fracture spacing is
somewhat different. Although the temperature profiles are closely spaced at early time,
the profiles diverge as boiling is initiated in the fracture, and begin to converge some time
after boiling has finished (Figure 4.8(b)). Once again, the differences between the trials
are diminished at late-time. After 150 days of heating, only the temperature profile for
the 10 m fracture spacing case is significantly different. This gap is gradually closed as
the simulation time increases. For the cases of high fracture spacing in Figure 4.8(b), a
temperature plateau can be observed. In these cases, the coldest point in the treatment
zone is located in the fracture, where a plateau does develop during boiling.
At smaller values of bulk permeability (kb ≤ 10−14 m2), the differences between the
trials become less significant. For both kb = 10−14 m2 and kb = 10−15 m2 (Figures 4.8(c-
d)), the difference in predicted temperatures between the simulations rises gradually to a
maximum of 7% at 226 days of heating, before declining.
CHAPTER 4. NUMERICAL MODELLING 75
300
350EPM
250
e (°C)
1 m
2.5 m
150
200
perature
5 m
10 m
100
50
Temp 10 m
0
50
0
0 50 100 150 200 250 300 350
Heating Time (days)
(a)
300
350
250
e (°C)
150
200
perature EPM
1 m
100
50
Temp
2.5 m
5 m
0
5010 m
0
0 50 100 150 200 250 300 350
Heating Time (days)
(b)
300
350
250
e (°C)
150
200
perature EPM
1 m
100
50
Temp
2.5 m
5 m
0
50 10 m
0
0 50 100 150 200 250 300 350
Heating Time (days)
(c)
300
350
250
e (°C)
150
200
perature EPM
1 m
100
50
Temp
2.5 m
5 m
0
5010 m
0
0 50 100 150 200 250 300 350
Heating Time (days)
(d)
Figure 4.8: Minimum treatment zone temperature profiles for various values of fracturespacing, with (a) kb = 10−12 m2, (b) kb = 10−13 m2, (c) kb = 10−14 m2, (d) kb = 10−15 m2
CHAPTER 4. NUMERICAL MODELLING 76
Treatment Zone Boiling
Fracture properties can have a large impact on boiling times within the treatment zone, even
in cases where temperatures seem to be little affected by flow heterogeneity. In the fracture,
where there is no significant pressure spike, boiling will begin at a lower temperature than
in the rock matrix. Consequently, for all parameters used in this study, boiling occurred in
the fracture at a significantly earlier time than in the matrix. An example of this behaviour
is shown in Figure 4.9, which shows the relationship between steam saturation and distance
from the fracture plane in the treatment zone, for the base case simulation (Table 4.1).
Although the entire fracture is saturated with steam after 117 days, the matrix does not
reach complete steam saturation until 263 days of heating.
Although fracture spacing and bulk permeability may not have a significant effect on the
amount of time needed to reach a target temperature, these factors can have a significant
impact on the amount of time necessary to bring about complete steam saturation in the
treatment zone, as seen in Figure 4.10. For a given bulk permeability, the treatment period
is prolonged more by an increase in fracture spacing than an increase in bulk permeability.
1.5
2.0
2.5
Fracture Plane
(m)
44
80
117153
190 263
0.0
0.5
1.0
0.0 0.2 0.4 0.6 0.8 1.0
Distance From
Average Steam Saturation in Treatment Zone
Figure 4.9: Relationship between steamsaturation and distance from the fracturein base case simulation. The heating time(days) is indicated for each curve.
250
300
n )
200
250
Boiling
ie (days)
150
End of B
ent Z
one
k = 1.0e‐15
k = 1.0e‐14
100
me to E
Treatm
e
k = 5.0e‐14
k 7 5 14
0
50Tim T k = 7.5e‐14
k = 1.0e‐130
0 2 4 6 8 10 12
Fracture Spacing (m)
Figure 4.10: Sensitivity of treatment zoneboiling time to bulk medium properties.
CHAPTER 4. NUMERICAL MODELLING 77
When the fracture spacing is large, it is more difficult for the heated water in the
matrix to drain into a fracture. Consequently, the pressure rise is larger and the boiling
point more elevated. There appears to be a limit to this behaviour, however; as seen in
Figure 4.10, incremental increases in fracture spacing have a progressively smaller impact
on the time needed to reach complete steam saturation. Also apparent in Figure 4.10 is
the significant difference in boiling times between the EPM and 1 m fracture spacing cases.
Although there is no significant difference in computed temperatures between an EPM and
1 m-spaced fractures in any of the simulations conducted (Figure 4.8), the difference in
time needed to reach complete steam saturation may be over 50%. The large variability in
the time needed to reach complete steam saturation for a single value of bulk permeability
suggests a difficulty in predicting TCH treatment times in fractured bedrock, if complete
steam saturation is the goal of treatment.
4.5 Conclusions
The performance of thermal conductive heating in fractured rock environments may be
strongly dependent on the hydraulic properties of the rock matrix (permeability, porosity)
and the aperture and spacing of fractures. If complete steam saturation is the goal of thermal
treatment, treatment time will be strongly governed by the magnitude of the pressure spike
that occurs in the rock matrix during heating. When the rock matrix has a low permeability,
high porosity, or sparse fracturing, this pressure rise may be enough to significantly raise the
boiling point of water in the matrix, thus delaying treatment. Because a clear temperature
plateau may not be observed in the matrix during boiling, it may be difficult to determine
if boiling has occurred throughout a treatment area from temperature measurements alone.
Due to the importance of fracture spacing in determining the pressure rise in the matrix,
a discrete fracture model is most appropriate for modelling boiling in this context. However,
treatment zone temperatures are only moderately affected by the location of fractures,
CHAPTER 4. NUMERICAL MODELLING 78
for a given value of bulk permeability. An equivalent porous medium (EPM) model may
provide an adequate estimation of treatment zone temperatures, especially when the bulk
permeability is low or heating times are long.
4.6 Acknowledgements
The authors would like to thank the U.S. Department of Defense Environmental Security
Technology Certification Program (Contract ER-0715), the Natural Sciences and Engineer-
ing Research Council of Canada (Discovery Grant), and Queen’s University for financial
support of this work. Ron Falta is thanked for sharing his expertise with TOUGH2.
Chapter 5
Conclusions
5.1 Semi-Analytical Modelling
Results from semi-analytical modelling indicate that treatment zone target temperatures
may not be reached using thermal conductive heating if the cooling effect of groundwater
influx is not considered in the design of the treatment system. These results are in good
agreement with Elliott et al. (2003) and Elliott et al. (2004), who used numerical modelling
to study the cooling problem in porous media.
It was determined using a sensitivity analysis that, for the prediction of treatment
zone temperatures, the characterization of hydrogeological parameters (fracture aperture,
fracture spacing, hydraulic gradient) is more important than the characterization of rock
material properties (density, specific heat capacity, thermal conductivity). However, when
influx is low to moderate, predicted temperatures have only a minor dependence on the
particular combination of hydrogeological parameters used to arrive at a given bulk influx.
Two design options were shown to partially offset the cooling effect: increasing the power
delivered to the thermal wells or installing upgradient preheating wells. It was shown to be
more effective to increase the power of existing thermal wells before installing preheating
wells.
79
CHAPTER 5. CONCLUSIONS 80
5.2 Numerical Modelling
The treatment zone temperatures predicted using numerical modelling results were in agree-
ment with those of the semi-analytical solution. As with the results of the semi-analytical
modelling, a minor dependence of temperature on hydrogeological parameters was observed
for cases of low to moderate influx. Unlike the semi-analytical solution, the numerical model
was able to simulate boiling processes within the treatment zone.
Predictions of boiling in the treatment zone suggest that fracture spacing is important
in predicting the time needed to reach complete steam saturation, even in cases where the
temperature distribution is not affected by fracture spacing. It was shown that a pressure
spike can occur when a low-permeability rock matrix is heated, elevating the boiling point of
water. The magnitude of this pressure spike determines the time needed to reach complete
steam saturation in the rock matrix, which is typically the last part of the treatment zone
to boil. Matrix permeability, matrix porosity, and fracture spacing were all shown to play a
role in determining the magnitude of the pressure spike. For the ranges of properties tested,
the magnitude of the pressure spike was most sensitive to changes in matrix permeability.
Consequently, knowledge of these properties is important in the prediction of treatment
times.
Because boiling in the rock matrix does not occur at constant pressure, a temperature
plateau is not observed during boiling. Consequently, it may be difficult to determine
when the entire rock matrix has boiled, if temperature measurements are the only source
of information available.
5.3 Recommendations
Boiling in the a fractured rock environment is a very complex process, and a number of
simplifications were made in the modelling presented in this thesis. The removal of some of
these simplifications presents a number of opportunities for future work. In particular:
CHAPTER 5. CONCLUSIONS 81
• In this study, no consideration was given to the behaviour of NAPL in fractured rock
during heating. It would be useful to apply the TMVOC code (Pruess and Battistelli,
2002) to observe how NAPL boiling occurs in these environments.
• The thermal diffusivity of rocks tends to decline as temperature is increased (Clauser
and Huenges, 1995). In addition, anisotropy in thermal conductivity (Cermak and
Rybach, 1982) may affect temperature distributions during heating in fractured rock.
Investigation of these effects could provide useful insights into the design of thermal
wells. The TOUGH2 code would need to be modified in order to permit changes in
these properties (Pruess et al., 1999).
• When the magnitude of the pressure spike is large, it is possible that pressure-induced
fracturing may occur. In addition, thermally-induced expansion of the rock matrix and
pressure-induced compression may have an impact on the magnitude of the pressure
spike. In order to quantify the pressure response in the rock matrix when matrix
permeability is very low (e.g. 10−22 m2), it would be useful to apply a fully-coupled
thermohydromechanical (THM) model.
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Appendix A
Semi-analytical Solutions
A.1 Background
Heat transport in fractured rock can be described by a series of coupled partial differential
equations. When heat conduction in the rock is considered to be one-dimensional, the gov-
erning equations are reduced to a more easily solvable set of ordinary differential equations.
While this is a reasonable approximation for many applications, such as the analysis of
heat extraction from a geothermal reservoir, the one-dimensional approximation can not be
used when heat is generated within the rock matrix. In order to model thermal conductive
heating, multidimensional conduction must be included.
To the author’s knowledge, the semi-analytical solution of Cheng et al. (2001) is the only
solution to the problem that includes two-dimensional conduction in the rock without using
a lumping procedure. In this section, the Cheng et al. (2001) solution will be extended to
include a heat source from a thermal well, as well as a set of parallel fractures.
97
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 98
A.1.1 Governing Equations
Transient heat conduction in a two-dimensional block of rock is described by the heat
conduction equation:
∇2T =∂2T
∂x2+∂2T
∂y2= − g
Kr+
1αr
∂T
∂t(A.1)
An appropriate differential equation for heat transfer in the fracture can be developed
by considering a small control volume of a fracture. As shown in Figure A.1, the upper half
of a fracture element is considered, having x, y, z dimensions of ∆x, e2 , 1. The midpoint of
the fracture (y = − e2), which forms the lower boundary of the control volume, is a symmetry
boundary, across which no heat is transferred.
V
Δ
2
x
y
Figure A.1: Heat balance
Because energy is conserved within the control volume,
ρwcw∆x(e
2
) ∂T∂t
= rate of heat accumulation− rate of heat loss (A.2)
For simplicity, axial conduction within the fracture is neglected, and transverse conduc-
tion is assumed to occur instantaneously, due to the small scale of e2 . Therefore, only two
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 99
sources contribute to heat accumulation in the control volume:
rate of heat accumulation = convection into CV + conduction from matrix into CV
= ρwcwv(e
2
)T (x) +Kr∆x
∂T
∂y
∣∣∣∣y=e/2
(A.3)
Similarly,
rate of heat loss = convection from CV
= ρwcwv(e
2
)(A.4)
Substituting (A.3) and (A.4) into the original statement of conservation (A.2) gives,
upon rearranging:
∂T
∂t= ρwcwv
T (x)− T (x+ ∆x)∆x
+2Kr
e
∂T
∂y
∣∣∣∣y=e/2
(A.5)
Taking the limit as ∆x→ 0:
∂T
∂t= −ρwcwv
∂T
∂x+
2Kr
e
∂T
∂y
∣∣∣∣y=e/2
(A.6)
Equation (A.6) is equivalent to the heat balance developed by Gringarten et al. (1975),
among others. Cheng et al. (2001) showed that the time partial derivative term may be
neglected from (A.6) without large error. A verification of this conclusion for the conditions
of thermal conductive heating may be found in Appendix B. Eliminating this term and
rearranging gives:∂T
∂x=
2Kr
ρwcwve
∂T
∂y
∣∣∣∣y=e/2
(A.7)
It should be noted that, although the time partial derivative has been eliminated, the
temporal dependence of the problem is retained through ∂T∂x and ∂T
∂y , both of which vary
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 100
with time.
A.1.2 Method of Solution
All of the following solutions are developed using Green’s functions. Green’s functions
provide a modular solution framework through which problem parameters such as boundary
conditions and heat generation rates may be easily modified without changing the structure
of the solution. A general form for a solution to (A.1) using Green’s functions is given by
Beck et al. (1992, p. 55):
T (x, y, t) =∫ a2
x′=a1
∫ b2
y′=b1
GXIJ(x, t|x′, 0)GYMN (y, t|y′, 0)F (x′, y′)dx′dy′
+α
K
∫ t
τ=0
∫ a2
x′=a1
∫ b2
y′=b1
GXIJ(x, t|x′, τ)GYMN (y, t|y′, τ)g(x′, y′, τ)dy′dx′dτ
+ Ix′=a1 + Ix′=a2 + Iy′=b1 + Iy′=b2 (A.8)
where GXIJ and GYMN are one-dimensional Green’s functions determined by the choice
of boundary conditions in the x and y directions, F (x′, y′) is a spatially-variable initial
condition, g(x′, y′, τ) is a point heat source, and the I-terms represent boundary condition
effects. To simplify the solution, we consider the case of homogeneous boundary conditions
(zero temperature or zero flux) and a uniform initial temperature of zero. Under these
conditions, the solution will have the form:
T (x, y, t) =α
K
∫ t
τ=0
∫ a2
x′=a1
∫ b2
y′=b1
GXIJ(x, t|x′, τ)GYMN (y, t|y′, τ)g(x′, y′, τ)dy′dx′dτ (A.9)
The time integral may be removed by recognizing that (A.9) is the convolution of the
Green’s function G(x, y, t|x′, y′, τ) = GXIJ(x, t|x′, τ)GYMN (y, t|y′, τ) and the source func-
tion g(x′, y′, τ) with respect to time. The convolution property of the Laplace transform
states that:
F (s)G(s) =∫ t
0F (τ)G(t− τ)dτ (A.10)
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 101
where s is the Laplace variable and the overbar denotes a Laplace-transformed quantity.
Applying this rule to (A.9) gives:
T (x, y, s) =α
K
∫ a2
x′=a1
∫ b2
y′=b1
L[GXIJ(x, t|x′, τ)GYMN (y, t|y′, τ)]L[g(x′, y′)]dy′dx′ (A.11)
A.2 Single Fracture Solution
A.2.1 Problem Formulation
Using the Green’s function solution framework, one simple solution that can be constructed
is that of a single infinite-length fracture at y = 0 intersected by a heater well at x = W
in an infinite two-dimensional body of rock. In this case, the Green’s functions GXIJ and
GYMN in are equivalent:
GXIJ = GYMN =
(1
2√παr(t− τ)
)exp
(− (z − z′)2
4αr(t− τ)
), z = x, y (A.12)
The product GXIJGYMN is then given in the Laplace domain by:
Gs(x, y|x′, y′) = L [GXIJGYMN ] =1
2παrsK0
(√s
αr[(x− x′)2 + (y − y′)2]
)(A.13)
where αr is the thermal diffusivity of the rock, s is the Laplace variable, and K0 is the
zero-order modified Bessel function of the second kind. The effect the fracture can be math-
ematically introduced using the source/sink function g(x′, y′, s). If q(x′, s) is the strength
of the source/sink along the fracture, the source function g(x′, y′, s) can be written as:
g(x′, y′, s) = q(x′, s)δ(y′) (A.14)
where q(x′, s) is the strength of the fracture source/sink. Substituting the Laplace-space
source function (A.14) and Green’s function (A.13) into the general solution form (A.11)
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 102
gives:
T (x, y, s) =1
2πKr
∫ L
0q(x′, s)K0
(√s
αr[(x− x′)2 + y2]
)dx′ (A.15)
This is the formulation used by Cheng et al. (2001) to describe heat extraction from
a hot dry rock geothermal reservoir. Although the domain in x extends from −∞ to ∞,
integration limits of 0 and L have been used to facilitate evaluation of the integral. Note that
the sources are modelled as a line and not a strip; consequently, the fracture is effectively
of zero aperture and occupies no physical space. However, the influence of the fracture
aperture on heat transfer is retained through the function q(x′, s).
The effect of the thermal well is introduced by adding a term to the source function, a
line source of magnitude gw located at x = W and of infinite length in y:
g(x′, y′, s) = q(x′, s)δ(y′) +gwsδ(W − x) (A.16)
Substituting the revised source function (A.16) into the general solution (A.11) gives:
T (x, y, s) =1
2πKr
∫ L
0q(x′, s)K0
(√s
αr[(x− x′)2 + y2]
)dx′
+g
2πKrs
∫ ∞−∞
K0
(√s
αr[(W − x)2 + (y − y′)]
)dy′ (A.17)
Using this formulation, several thermal wells may be incorporated simply by adding
additional terms to (A.17). The unknown source function q(x, s) in the above equations
can be written in terms of the heat flux into the fracture:
q(x, s) = −2Kr∂T (x, y, s)
∂y
∣∣∣∣y=0
(A.18)
Making use of the heat balance in the fracture (A.7), the above can be written in terms
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 103
of the gradient in the x-direction:
q(x, s) = −veρwcw∂T (x, 0, s)
∂x(A.19)
Substituting this into (A.17) gives:
T (x, y, s) =−veρwcw
2πKr
∫ L
0
∂T (x′, 0, s)∂x′
K0
(√s
αr[(x− x′)2 + y2]
)dx′
+gw
2πKrs
∫ ∞−∞
K0
(√s
αr[(W − x)2 + (y − y′)]
)dy′ (A.20)
A.2.2 Solution Along Fracture/Matrix Interface
The first integral can be expanded by parts:
∫ L
0
∂T (x′, 0, s)∂x′
K0
(√s
αr[(x− x′)2 + y2]
)dx′ =[
T (x′, 0, s)K0
(√s
αr[(x− x′)2 + y2]
)]L0
−√
s
αr
∫ L
0T (x′, 0, s)
x− x′√y2 + (x− x′)2
K1
(√s
αr[(x− x′)2 + y2]
)dx′
(A.21)
Substitution of the above into (A.17) gives:
T (x, y, s) = −veρwcw2πKr
[T (L, 0, s)K0
(√s
αr[(L− x)2 + y2]
)− T (0, 0, s)K0
(√s
αr[x2 + y2]
)−√
s
αr
∫ L
0T (x′, 0, s)
x− x′√y2 + (x− x′)2
K1
(√s
αr[(x− x′)2 + y2]
)dx′
]
+gw
2πKrs
∫ ∞−∞
K0
(√s
αr[(W − x)2 + (y − y′)]
)dy′ (A.22)
The temperature in the fracture (y = 0) is thus described by the following integral
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 104
equation:
T (x, 0, s) = −veρwcw2πKr
T (L, 0, s)K0
(√s
αr(L− x)
)+veρwcw2πKr
T (0, 0, s)K0
(√s
αrx
)+
veρwcw2πKr
√s
αr
∫ L
0T (x′, 0, s)
x− x′
|x− x′|K1
(√s
αr|x− x′|
)dx′
+gw
2πKrs
∫ ∞−∞
K0
(√s
αr[(W − x)2 + (y′)2]
)dy′ (A.23)
To remove the singularity in the third term, a subtraction method is used (e.g. Delves
and Mohamed, 1985, p. 268). The following similar integral is evaluated analytically:
√s
αr
∫ L
0T (x, 0, s)
x− x′
|x− x′|K1
(√s
αr|x− x′|
)dx′ =
T (x, 0, s)[K0
(√s
αr(L− x)
)−K0
(√s
αrx
)](A.24)
Multiplying (A.23) through by 2πKrveρwcw
and subtracting (A.24) gives:
gwveρwcws
∫ ∞−∞
K0
(√s
αr[(W − x)2 + (y′)2
)dy′ =
T (L, 0, s)K0
(√s
αr(L− x)
)− T (0, 0, s)K0
(√s
αrx
)+[
2πKr
veρwcw−K0
(√s
αr(L− x)
)+K0
(√s
αrx
)]T (x, 0, s)
−√
s
αr
∫ L
0[T (x′, 0, s)− T (x, 0, s)]
x− x′
|x− x′|K1
(√s
αr|x− x′|
)dx′ (A.25)
Equation (A.25) can be numerically solved to determine the Laplace-space temperature
along the fracture-matrix boundary. When the final integral in (A.25) is approximated by a
numerical quadrature rule, a n× n system of nonhomogeneous linear equations will result,
with the Laplace temperatures at n discrete points along the fracture as the unknowns.
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 105
The system of equations is described by:
gwveρwcws
∫ ∞−∞
K0
(√s
αr[(W − xi)2 + (y′)2
)dy′ = TnK0
(√s
αr(L− xi)
)− T1K0
(√s
αrxi
)+[
2πKr
veρwcw−K0
(√s
αr(L− xi)
)+K0
(√s
αrxi
)]T (x, 0, s)
−√
s
αr
n∑j=1
[Tj − Ti]wjxi − xj|xi − xj
K1
(√s
αr|xi − xj |
), i = 0..n (A.26)
Once the Laplace-space temperature is known, a numerical Laplace inversion algorithm
may be used to calculate the real-space temperature at any time.
A.2.3 Calculation of Temperature Within the Rock Matrix
Once the Laplace-space temperature along the fracture/matrix boundary is known from
solution of equation (A.26), temperatures within the rock may be calculated directly using
a discretized version of equation (A.22):
T (x, y, s) = −veρwcw2πKr
[TnK0
(√s
αr[(L− x)2 + y2]
)− T1K0
(√s
αr[x2 + y2]
)
−√
s
αr
n∑j=1
Tjwjx− xj√
y2 + (x− xj)2K1
(√s
αr[(x− xj)2 + y2]
)+
gw2πKrs
∫ ∞−∞
K0
(√s
αr[(W − x)2 + (y − y′)]
)dy′ (A.27)
A.2.4 Implementation Details
The reader is referred to Section A.3.4 for a discussion of the influence of time units on the
accuracy of calculated temperatures within the rock matrix near the fracture plane.
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 106
A.3 Fracture Set Solution (Bessel Function Solution)
A.3.1 Problem Formulation
A useful modification of the preceding solution would allow for an infinite set of evenly-
spaced parallel, discrete fractures. One approach is to consider a large finite number of
fractures and explicitly model the effect of each fracture: 2m+ 1 fracture pairs are assumed
to exist, located at y = ±kH, n = 0...m. As m is increased, the computed temperatures
near the central fracture approach those predicted by an infinite set of fractures. Using this
approach, a solution can be constructed from:
T (x, y, s) =1
2πKr
m∑k=−m
∫ L
0qk(x′, s)K0
(√s
αr
√(x− x′)2 + (y − kH)2
)dx′
+gw
2πKrs
∫ ∞−∞
K0
(√s
αr[(W − x)2 + (y − y′)2]
)dy′ (A.28)
where qk(x′, s) represent the heat sink created by each of the 2m+1 fractures. The distances
from the heat sinks are given by rk =√
(x− x′)2 + (y − kH)2.
A.3.2 Solution Along Fracture/Matrix Interface
The solution follows the same procedure as was used for the single fracture. Substituting
in the fracture heat balances (A.7) and integrating by parts gives:
T (x, y, s) = −veρwcw2πKr
m∑k=−m
T (L, kH, s)K0
(√s
αr[(L− x)2 + (kH − y)2]
)
+√
s
αr
∫ L
0T (x′, kH, s)
x− x′√(kH − y)2 + (x− x′)2
K1
(√s
αr[(kH − y)2 + (x− x′)2]
)dx′
− T (0, kH, s)K0
(√s
αr[x2 + (kH − y)2]
)+
gw2πKrs
∫ ∞−∞
K0
(√s
αr[(W − x)2 + (y − y′)2]
)dy′ (A.29)
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 107
If an adequate number of fractures is chosen, it can be assumed that the temperature
in all fractures is equal (T (x′, kH, s) = T (x′, 0, s)). Making this substitution and evaluating
at y = 0 gives the following:
T (x, 0, s) = −veρwcw2πKr
m∑k=−m
T (L, 0, s)K0
(√s
αr[(L− x)2 + (kH)2]
)
+√
s
αr
∫ L
0T (x′, 0, s)
x− x′√(kH)2 + (x− x′)2
K1
(√s
αr[(x− x′)2 + (kH)2]
)dx′
− T (0, kH, s)K0
(√s
αr[x2 + (kH)2]
)+
gw2πKrs
∫ ∞−∞
K0
(√s
αr[(W − x)2 + (y′)2]
)dy′ (A.30)
As in the single fracture solution, regularization needs to be performed on the integral
on the second line of (A.30). When k = 0, the integral will reach a singularity at x′ = x.
Equation (A.24) is multiplied by veρwcw/2πKr and added to (A.30), giving:
2πKr
veρwcwT (x, 0, s) =
T (0, 0, s)K0
(√s
αrx
)− T (L, 0, s)K0
(√s
αr(L− x)
)− T (x, 0, s)
[K0
(√s
αr(L− x)
)−K0
(√s
αrx
)]−√
s
αr
∫ L
0[T (x′, 0, s)− T (x, 0, s)]
x− x′
|x− x′|K1
(√s
αr‖(x− x′)2‖
)dx′
−m∑
k=−m,k 6=0
T (L, 0, s)K0
(√s
αr[(L− x)2 + (kH)2]
)
+√
s
αr
∫ L
0T (x′, 0, s)
x− x′√(kH)2 + (x− x′)2
K1
(√s
αr[(kH)2 + (x− x′)2]
)dx′
− T (0, kH, s)K0
(√s
αr[x2 + (kH)2]
)+
gwveρwcws
∫ ∞−∞
K0
(√s
αr[(W − x)2 + (y′)2]
)dy′ (A.31)
A numerical quadrature is used to express the boundary integrals in (A.31) in terms of
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 108
summations. The following is a generalized discretization of the fracture-rock interface:
Ti
[2πKr
veρwcw+K0
(√s
αr(L− xi)
)−K0
(√s
αrxi
)]=
T1
m∑k=−m
K0
(√s
αr[x2i + (kH)2]
)− Tn
m∑k=−m
K0
(√s
αr[(L− xi)2 + (kH)2]
)
+√
s
αr
n∑j=1
wj [Tj − Ti]xi − xj|xi − xj |
K1
(√s
αr|xi − xj |
)
+√
s
αr
m∑k=−m,k 6=0
n∑j=1
wjTjxi − xj√
(kH)2 + (xi − xj)2K1
(√(kH)2 + (xi − xj)2
)
+gw
veρwcws
∫ ∞−∞
K0
(√s
αr
√(W − x)2 + (y′)2
)dy′ (A.32)
Values for the abscissas (xi) and weights (wj) may be found in a handbook of quadrature
tables.
A.3.3 Calculation of Temperature Within the Rock Matrix
Once the temperature along the discretized fracture-rock boundary has been calculated, the
temperature at any point within the rock matrix may be rapidly calculated. Since y > 0,
the singularity in (A.29) is avoided, although error stemming from the evaluation of K0(x)
near its singuarities at y = ±kH may be unacceptable. Applying the same quadrature rule
to (A.29) gives:
T (x, y, s) = −veρwcw2πKr
m∑k=−m
TnK0
(√s
αr[(L− x)2 + (kH − y)2]
)
+√
s
αr
n∑j=1
Tjwjx− xj√
(kH − y)2 + (x− xj)2K1
(√s
αr[(kH − y)2 + (x− xj)2]
)
− T0K0
(√s
αr[x2 + (kH − y)2]
)+
gw2πKrs
∫ ∞−∞
K0
(√s
αr[(W − x)2 + (y − y′)2]
)dy′ (A.33)
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 109
The temperature at points along the fracture plane should be evaluated by interpolating
between the previously calculated Ti rather then using the above equation.
A.3.4 Implementation Details
Number of Modelled Fracture Pairs
An important parameter of the solution is the number of fracture pairs to explicitly model.
If too few pairs are modelled, temperatures in the area of interest will be erroneously high.
However, the modelling of too many pairs will slow computations needlessly. In order to
determine an appropriate number of fracture pairs, a sensitivity analysis was conducted on
the number of fracture pairs. The temperature after heating at five points along the fracture
plane was modelled, using an increasing number N of fracture pairs (up to N = 40).
The normalized temperatures TNTN=40
at each of the five points are plotted in Figure A.2.
From these data, it is clear that at least 15 pairs of fractures should be explicitly modelled
in order to represent an infinite system of fractures.
Choice of Units
Although the regularization allows (A.32) to be evaluated exactly along y = 0, interior
temperatures calculated from evaluating (A.33) near y = ±kH will have an unacceptable
error. This error is caused by the evaluation of the modified Bessel functions K0(x) and
K1(x) near their singularity at x = 0. A judicious choice of units will increase the magnitude
of the arguments, thereby reducing the error.
The following is a typical call of K0 as y approaches 0 or kH:
K0
(√s
α(x− xj)2
)(A.34)
It can be shown that the dimensions of the argument are [T]. Therefore, a change
of time units can increase the magnitude of the function arguments, thus increasing the
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 110
0 5 10 15 20 25 30 35 40 45 500
0.2
0.4
0.6
0.8
1
1.2
1.4
Number of Fracture Sets Modelled
Normalized
Tem
perature
x = 100x = 105x = 110x = 115x = 120
Figure A.2: Determining an appropriate number of fracture sets
distance from the singularity. In the De Hoog et al. (1982) algorithm, values of s are
inversely proportional to log t. When years or centuries are used as the time units, values
of t are small and values of s become large, thus avoiding the singularity and improving
accuracy. Even with these corrections, however, temperatures may be inaccurate near the
fracture-matrix boundary.
A.4 Fracture Set Solution (Improved Solution)
A.4.1 Problem Formulation
As noted in Section (A.3.3), the accuracy of the free space (infinite domain) Green’s function
solution may be unacceptable for interior points close to the fracture. By avoiding the use of
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 111
the Bessel functions, the following solution improves the accuracy of points calculated near
a singularity of the Green’s function. Further, since nonelementary functions are avoided,
computational time is reduced. The following Green’s functions are used Beck et al. (1992):
GX00(x, t|x′, τ) =
(1
2√πα(t− τ)
)exp
(− (x− x′)2
4α(t− τ)
)(A.35)
GY 22(y, t|y′, τ) =1H
[1 + 2
∞∑m=1
exp(−m2π2α(t− τ)
H2
)cos(mπyH
)cos(mπy′
H
)](A.36)
where the subscript X00 indicates that the x-domain is bounded by two zeroth-type bound-
ary conditions, and Y 22 indicates that the y-domain is bounded by two second-type condi-
tions. Beck et al. (1992) recommend that (A.36) be used for “large” values of dimensionless
time, when the Fourier number Fo = αtH2 exceeds 0.25. For the times to be considered in
thermal heating for fractured rock, Fo should always be large enough to used the above
function; however, this should be verified prior to computation. If Fo is small, the solution
may still be used, but a large number of terms should be evaluated to ensure convergence.
The Laplace transform of the product of these two functions is given by:
Gs(x, y|x′, y′) =1
2H√sα
exp(−|x− x′|
√s
α
)
+∞∑m=1
exp(−|x− x′|
√sα + m2π2
H
)cos(mπyH
)cos(mπy′
H
)√H2s+m2π2α
(A.37)
whose derivative with respect to x′ is equal to:
∂Gs∂x′
(x, y|x′, y′) =sgn(x− x′)
2Hαexp
(−|x− x′|
√s
α
)
+∞∑m=1
sgn(x− x′) exp(−|x− x′|
√sα + m2π2
H
)cos(mπyH
)cos(mπy′
H
)Hα
(A.38)
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 112
A.4.2 Solution Along Fracture/Matrix Interface
A solution can be constructed using (A.37) in exactly the same manner as the Bessel function
solution of Section (A.3). The form of the Green’s function solution from which the integral
equation is constructed is nearly identical (to A.17); only the limits of integration and the
Green’s function have been changed.
T (x, y, s) =1
2πKr
∫ L
0q(x′, s)Gs(x, y|x′, 0)dx′ +
g
2πKrs
∫ H
0Gs(x, y|W, y′)dy′ (A.39)
Like the Bessel function-based Green’s function, the derivative (A.38) is singular as
x′ → x; again, this singularity can be removed by subtraction. The solution is calculated
by solving the n× n system of equations described by:
Ti = −veρwcwαrKr
TnGs(xi, 0|L, 0)− T1Gs(xi, 0|0, 0)−n∑j=1
[Ti − Tj ]wj∂Gs∂x′
(xi, 0|xj , 0)
− Ti[Gs(xi, 0|L, 0) +Gs(x, y|0, 0)]+gwαrKrs
∫ H
0Gs(xi, y|W, y′)dy′ (A.40)
A.4.3 Calculation of Temperature Within the Rock Matrix
Interior points may then be calculated using:
T (x, y, s) = −veρwcwαrKr
TnGs(x, y|L, 0)− T1Gs(x, y|0, 0)
−n∑j=1
[Tj − T (x, 0, s)]wj∂Gs∂x′
(x, y|xj , 0)− T (x, 0, s)[Gs(x, y|L, 0) +Gs(x, y|0, 0)]
+gwαrKrs
∫ H
0Gs(x, y|W, y′)dy′ (A.41)
An excellent “perk” of this formulation is that the heater well terms can be evaluated
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 113
analytically:
gwαrsKr
∫ H
0Gs(x, y|W, y′)dy′ =
gwKr
√αrs
exp(−|W − x|
√s
αr
)(A.42)
A.4.4 Verifications
Disappearing Fractures
As the aperture of the fractures approaches zero, their importance should become negligible
and the computed temperature field should approach the solution for a continuous plane
source, as provided by Carslaw and Jaeger (1959, p. 263):
T (x) =gwρwcw
[(t
πα
) 12
exp(−(x− x′)2
4αt
)− |x− x
′|2α
erfc(|x− x′|2√αt
)](A.43)
A test case was prepared with two heater wells, located at x = 30 m and x = 33 m. Each
well produced 40 W/m of heat for a period of 1 year. Figure A.3 shows the temperature
profile along along x for six values of time.
No Heater Well; Incoming Water is Heated
An alternative scenario exists in which the rock is heated not by a thermal well, but rather
by the injection of hot fluid. In this case, the solution should behave similarly to that of
Gringarten et al. (1975), whose analytical solution models one-dimensional heat conduction
from a set of equally-spaced parallel fractures. Previous numerical modelling work (Kolditz,
1995; Cheng et al., 2001) suggests that the solutions should differ at large time, where the
multidimensional solution should predict a lower temperature.
In notation consistent with this solution, Gringarten et al. (1975)’s solution can be
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 114
50
60
70
80
90
100
erature (°C)
t = 33 dayst = 100 dayst = 166 dayst = 232 dayst = 365 daysAnalytical Solution
0
10
20
30
40
20 22 24 26 28 30 32 34 36 38 40
Tempe
Distance (m)
Figure A.3: Comparison of new solution with that of Carslaw and Jaeger (1959, p. 263) forthe case of zero fracture aperture.
written in Laplace space as:
T (x, 0, s) =1s
exp
(xs1/2 tanh
ρwcwbvHs1/2
Kr
)(A.44)
Equation (A.44) can be inverted numerically, with the values of s calculated from the
dimensionless time tD:
tD =(veρwcw)2
Krρrcr
(t− x
v
)(A.45)
Multidimensional conduction effects are far more pronounced for very low Peclet num-
bers, defined here as Pe = ve/αr. In order to facilitate comparison of the two solutions, the
hydrogeological parameters used by Cheng et al. (2001) are chosen, resulting in a Peclet
number of 19.18 (Table A.1).
After 100 days, the temperature profiles were calculated using the two solutions (Figure
A.4). It is important to note the sudden drop in temperature after x = 0 in the integral
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 115
Table A.1: Parameters used for verification against the Gringarten et al. (1975) solution
Parameter Value
v (m/day) 432e (mm) 3H (m) 10
ρr (kg/m3) 2750cr (J/kg·K) 1000Kr (W/m·K) 2.15
equation solution. Because this solution permits multidimensional conduction in the matrix,
heat may be lost to the area directly to the left of the injection point. This is not possible
with unidirectional conduction in the matrix, so it is to be expected that the solution of
Gringarten et al. (1975) should predict less heat loss from the fracture at small x.
0 4
0.5
0.6
0.7
0.8
0.9
1
perature (°C)
Semianalytical Solution
Gringarten et al. (1975)
0
0.1
0.2
0.3
0.4
0 25 50 75 100 125 150
Temp
Distance (m)
Figure A.4: Comparison of the present solution with that of Gringarten et al. (1975), t =100days
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 116
Energy Balance
Another simple verification can be performed by adding up the heat contained everywhere
in a domain and comparing it to the heat added to the domain through the heater wells.
As long as the parameters and time scale are chosen such that no heat has left the modelled
area at x = L, the total heat in the domain can be calculated by:
∆x∆yρrcrn−1∑i=1
m−1∑j=1
(Ti,j + Ti+1,j+1)/2 + ∆xρwcw(e
2
) n−1∑i=1
(Ti,1 + Ti+1,1) (A.46)
The parameters used for the energy balance are as follows. A single heater well at x = 30
provided heat a constant rate of 100 W/m. The modelled domain consisted of a set of 1
mm fractures spaced at 1 m; water flowed at a constant velocity of 20 m/day. After 1 year,
the heat content of the area x = 0 to x = 70 and y = 0 to y = H/2 − e/2 was calculated
using (A.46). Grid spacings were 0.2341 m in x and 0.0172 m in y. The heat balance ratio
was calculated to be:heat found in domain
heat provided to domain= 1.0004255. (A.47)
Laplace Inversion Algorithm
In order to verify the results of the numerical Laplace inversion, a fracture temperature
profile was calculated using two different Laplace inversion algorithm. Figure A.5 shows
the fracture temperature after 1 year as calculated using the algorithms proposed by De
Hoog et al. (1982) and by Weeks (1966).
Comparison to Numerical Solution
In order to further verify the analytical solution, a numerical model was used to run an
equivalent simulation. A Cartesian domain was created in TOUGH2 consisting of a 200 m
long strip, 1 m in depth and 0.5 m in height. In order to model the fracture, the domain
was divided into two materials, with properties given in Table A.2. The model also required
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 117
40
50
60
70
80
90
100
perature (°C)
De Hoog et al. (1980)
Weeks (1966)
0
10
20
30
40
0 10 20 30 40 50 60 70 80 90 100
Temp
Distance (m)
Figure A.5: Comparison of temperatures in fracture after one year of heating, as calculatedusing the De Hoog et al. (1982) and Weeks (1966) algorithms.
capillary pressure and relative permeability functions for both materials, but these have no
influence at the comparison time, before the onset of boiling.
Initially, the domain was discretized using 182 non-uniform blocks in the x-direction
(parallel to the direction of flow), 1 block in the y-direction, and 15 non-uniform blocks in
the z-direction (normal to the fracture plane). Improved agreement with the semi-analytical
solution was achieved by progressively refining the discretization in the direction of flow.
Figure A.6, created using 309 blocks in the x-direction, shows the predicted temperature
profile in the fracture after 4 months of heating.
APPENDIX A. SEMI-ANALYTICAL SOLUTIONS 118
Table A.2: Numerical model material propertiesMaterial
Property Units Variable Name ROCK FRACT
ρ kg/m3 DROK 2757 1000φ - POR 0.01 0.99kx m2 PER(1) 10−14 2.0833× 10−8
ky m2 PER(2) 10−14 2.0833× 10−8
kz m2 PER(3) 10−14 2.0833× 10−8
Kwet W/m·K CWET 2.98 0.616c J/kg·K SPHT 1180 4184
60
70
80
90
100
ure (°C)
Semianalytical
TOUGH2 Discrete Fracture
0
10
20
30
40
50
20 25 30 35 40 45 50 55 60
Tempe
ratu
Distance (m)
Figure A.6: Fracture temperature after 4 months of heating, computed using semi-analyticalsolution and TOUGH2.
Appendix B
Verification of Simplified Heat
Balance
In the fracture heat balance described in Section A.1.1, the heat storage term is omitted,
following the approach of Cheng et al. (2001). Although these authors verified the validity
of this simplification using numerical modelling, it is prudent to independently verify its
appropriateness for the conditions of subsurface heating using thermal conductive heating.
If the heat storage term is retained in the heat balance, a solution may be derived
using the method of Section A.4 without any modification. It will be shown here that the
temperature difference introduced by neglecting the heat storage term is negligible (on the
order of 10−4 degrees); yet, the increase in computing time is significant.
A heat balance for a control volume of a fracture is given by equation (A.6). Applying
the Laplace transformation to (A.6) gives:
sT = −ρwcwv∂T
∂x+
2Kr
e
∂T
∂y
∣∣∣∣y=e/2
(B.1)
Equation (A.39) is again used as the basis for the solution. When the heat storage
term is retained, the fracture-matrix heat exchange term q(x′, s) takes on a somewhat more
119
APPENDIX B. VERIFICATION OF SIMPLIFIED HEAT BALANCE 120
complex form:
q(x′, s) = −s(e
2
)T − ρwcwv
e
2∂T
∂x′(B.2)
Substitution of (B.2) into (A.39) gives:
T (x, y, s) =−se
2
∫ L
0T (x′, y, s)Gs(x, y|x′, 0)dx′
− ρwcwve
2
∫ H
0
∂T
∂x′Gs(x, y|x′, 0)dx′
+gwαrKrs
∫ H
0Gs(x, y|W, y′)dy′ (B.3)
At this point we set y = 0 to solve for the fracture temperature. The second integral is
then integrated by parts to remove the spatial derivative from its kernel:
T (x, y, s) = −αrse2Kr
∫ L
0T (x′, 0, s)Gs(x, 0|x′, 0)dx′
− ρwcwveαr2Kr
−∫ L
0T (x′, 0, s)
∂Gs∂x′
(x, 0|x′, 0)dx′
+ T (L, 0, s)Gs(L, 0|x′, 0)− T (0, 0, s)Gs(0, 0|x′, 0)
+gwαrKrs
∫ H
0Gs(x, y|W, y′)dy′ (B.4)
Subtracting the singularity at |x′ − x| → 0 gives:
T (x, y, s) = −αrse2Kr
∫ L
0[T (x′, 0, s)− T (x, 0, s)]Gs(x, 0|x′, 0)dx′
+ T (x, 0, s)∫ L
0Gs(x, 0|x′, 0)dx′
− ρwcwveαr
2Kr
T (L, 0, s)Gs(L, 0|x′, 0)− T (0, 0, s)Gs(0, 0|x′, 0)
+∫ L
0[T (x′, 0, s)− T (x, 0, s)]
∂Gs∂x′
(x, 0|x′, 0)dx′
− T (x′, 0, s) [Gs(x, 0|L, 0)−Gs(x, 0|0, 0)]
+gwαrKrs
∫ H
0Gs(x, y|W, y′)dy′ (B.5)
APPENDIX B. VERIFICATION OF SIMPLIFIED HEAT BALANCE 121
In discretized form, the above equation is given as:
Ti =−αrse2Kr
n∑j=1
[Tj − Ti]wjGs(xi, 0|xj , 0) + Ti
∫ L
0Gs(xi, 0|x′, 0)dx′
− veρwcwαr
Kr
TnG(xi, 0|L, 0)− T1GF (xi, 0|0, 0)−n∑j=1
[Ti − Tj ]wj∂G
∂x′(xi, 0|xj , 0)
− Ti[G(xi, 0|L, 0) +G(x, y|0, 0)]
+gwαrKrs
∫ H
0G(xi, y|W, y′)dy′ (B.6)
Equations (A.40) and (B.6) differ only by the first line of terms, which result from the
inclusion of the heat storage term. The code used to solve the simplified version (A.40)
was modified to include the extra terms. The parameters for the base case, shown in Table
B.1 were used to determine the temperature in the fracture after 1 year. The maximum
computed temperature difference between the two solutions at any point during the model
period was 0.000349 C. Runtime is nearly doubled when the heat storage term is included
(see Table B.2). It is thus concluded that inclusion of the heat storage term produces a
negligible difference in predicted temperatures, while significantly increasing computational
time.
Table B.1: Parameters used for verification of heat storage term omissionParameter Value Parameter Value Parameter Value
e (µm) 500 Kr (W/m·K) 2.59 gw (W/m) 100H (m) 1 ρr (kg/m3) 2650 Wi (m) x = 30, 33∇h -0.005 cr 1046 J/kg-K ∆x 0.2 m
Table B.2: Comparison of solution times when heat storage is included and neglectedSimulation Runtime
Heat storage term included (equation B.6) 6.16 hoursHeat storage term neglected (equation A.40) 3.27 hours
Appendix C
Numerical Discretization
C.1 Discretization of Fracture
C.1.1 Two-Dimensional Simulations
Although it is desirable to model fractures using a discrete fracture approach, the very small
gridblocks make numerical solution especially challenging. For most of the simulations in
this thesis, a “fracture zone” approach is used, whereby a high-permeability zone is used to
approximate the effect of a single fracture. Further discussion of this approach is found in
Section 2.6.1.
Using a 1 m fracture spacing, the base case from the semi-analytical parameter sensitivity
analysis (Table 3.1) is reproduced in TOUGH2 using both a discrete fracture and 10 cm
fracture zone discretization. Using these three solution methods, the temperature in the
fracture is computed.
After 122 days of heating, prior to the onset of boiling, the maximum percent difference
between the discrete fracture numerical model and the semi-analytical solution is 6.02%;
between the fracture zone numerical model and the semi-analytical solution it is 5.75%
(Figure C.1(a)). After 214 days of heating, the maximum percent diference between the
122
APPENDIX C. NUMERICAL DISCRETIZATION 123
fracture zone numerical model and the discrete fracture numerical model is 1.62 % (Figure
C.1(b)).
60708090
100110120
rature (°C)
Semianalytical
TOUGH2 Discrete Fracture
TOUGH2 10 cm Fracture Zone
01020304050
20 25 30 35 40 45 50 55 60 65 70 75 80
Tempe
Distance (m)
(a) t = 122 days
60708090
100110120
erature (°C)
TOUGH2 Discrete FractureTOUGH2 10 cm Fracture Zone
01020304050
20 25 30 35 40 45 50 55 60 65 70 75 80
Tempe
Distance (m)
(b) t = 214 days
Figure C.1: Comparison of semi-analytical solution, discrete fracture numerical solution,and “fracture zone” numerical solution before and after boiling.
Accurate representation of fractures is believed to be more difficult for larger values of
fracture spacing, where flow heterogeneity is higher and the effect of an individual fracture
is more significant. Using a 5 m fracture spacing, the discrete fracture numerical model
failed to converge at the onset of boiling. Just prior to this point, the maximum percent
difference between the fracture zone model and the discrete fracture model was calculated
to be 6.27%.
C.1.2 Simulations in Radial Geometry
The fracture zone approach was also used for numerical simulations in a radial geometry
(Chapter 4). Here, the importance of the fracture zone thickness is explored for the param-
eters corresponding to Run G5 (kb = 10−13 m2, km = 10−18 m2, fracture spacing = 2.5 m).
In this case, the fracture aperture was computed to be 144 µm. Therefore, for a discrete
fracture simulation, the first layer of cells should have a thickness of 72 µm.
Using a fracture layer cell thickness of 72 µm, the model failed to converge at the onset
APPENDIX C. NUMERICAL DISCRETIZATION 124
of boiling (Run G55). The model also failed to converge when the fracture layer thickness
was increased to 100 µm (Run G61). However, convergence was obtained using fracture
layer thicknesses of 500µm and above (Runs G56-G60).
Using a variety of fracture zone thicknesses, the temperature in the fracture at the along
the line at the upper edge of the domain (θ = 30). For each simulation, the maximum
percent temperature difference was calculated using Run G60 (the converging run with the
smallest fracture zone thickness) as the “true value”. These differences, along with the CPU
time for each run, are plotted in Figure C.2.
3 00%
4.00%
5.00%
6.00%
7.00%
1 5
2
2.5
3
3.5
perature Differen
ce
me (hou
rs)
CPU Time
% Difference
0.00%
1.00%
2.00%
3.00%
0
0.5
1
1.5
0 0.02 0.04 0.06 0.08 0.1 0.12
Maxim
um Tem
p
CPU Ti
Thickness of Fracture Zone (m)
Figure C.2: Comparison of CPU time and error in temperature for different values offracture zone thickness.
It can be seen that some error is incurred in the use of a 10 cm fracture zone. Therefore,
all simulation runs were conducted using the smallest fracture zone thickness possible.
Most runs were able to converge with a fracture zone thickness of 500 µm, although it was
necessary to use a 1 mm fracture zone in some cases.
APPENDIX C. NUMERICAL DISCRETIZATION 125
C.2 Radial Discretization
C.2.1 Size of Domain
In order to avoid the influence of the outer boundary condition on the temperature and
pressure in the treatment zone, a domain size dependence study was conducted. Using the
base case run from the numerical study (Table 4.1), the value of the primary variables in a
the fracture were compared for three different values of rmax, the domain size in the radial
direction (Figure C.3).
From these figures it appears that there is little difference in any of the output variables
between a radial grid size of 25 m and 50 m. Consequently, a 25 m grid was used in all
simulations.
C.2.2 Radial Grid Refinement
The effect of grid refinement in the radial direction is shown in Figure C.4, where the
base case parameters from the numerical study (Table 4.1) are used to compute pressure,
temperature, and steam saturation against time in two reference gridblocks: one near the
center of the rock matrix, and one located in the fracture. For all of the simulations discussed
in Chapter 4, where the steam saturation is of greatest interest, the medium-level radial
discretization (Nr = 90) is used.
If the radial discretization is too coarse, the pressure spike that occurs prior to boiling
will be relayed radially inward as each cell boils during outward movement of the boiling
front. This will cause pressure oscillations in cells within the boiled zone (Figure C.4(a)). A
similar effect was observed by (Falta et al., 1992) in their study of steam injection into the
subsurface. When the discretization in the R-direction is refined, the magnitude of further
pressure oscillations is reduced.
APPENDIX C. NUMERICAL DISCRETIZATION 126
250
300
350
400
450
500
erature (°C)
Rmax = 10
Rmax = 25
0
50
100
150
200
0 5 10 15 20 25 30 35 40 45 50
Tempe
Radial Distance (m)
Rmax = 50
(a) Temperature Profiles
2.44
2.46
2.48
2.5
sure (b
ar)
Rmax = 10
Rmax = 25
2.38
2.4
2.42
0 5 10 15 20 25 30 35 40 45 50
Press
Radial Distance (m)
Rmax = 50
(b) Pressure Profiles
0.6
0.8
1
1.2
Saturation
Rmax = 10
Rmax = 25
0
0.2
0.4
0 5 10 15 20 25 30 35 40 45 50
Steam
Radial Distance (m)
Rmax = 50
(c) Gas Saturation Profiles
Figure C.3: Fracture primary variable profiles for three domain sizes
APPENDIX C. NUMERICAL DISCRETIZATION 127
400
500
600
700
re (kPa
)
Nr = 50Nr = 90Nr = 180
0
100
200
300
0 100 200 300
Pressu
Heating Time (days)
(a) Pressure near matrix center
244
245
246
247
248
249
re (kPa
)
239
240
241
242
243
244
0 100 200 300
Pressu
Heating Time (days)
Nr = 50Nr = 90Nr = 180
(b) Pressure in fracture
0.6
0.8
1
1.2
aturation
0
0.2
0.4
0.6
0 100 200 300
Steam Sa
Heating Time (days)
Nr = 50Nr = 90Nr = 180
(c) Steam saturation near matrix center
0.6
0.8
1
1.2
aturation
0
0.2
0.4
0.6
0 100 200 300
Steam Sa
Heating Time (days)
Nr = 50Nr = 90Nr = 180
(d) Steam saturation in fracture
200
250
300
350
400
ature (°C)
0
50
100
150
200
0 100 200 300
Tempe
ra
Heating Time (days)
Nr = 50Nr = 90Nr = 180
(e) Temperature near matrix center
200
250
300
350
400
ature (°C)
0
50
100
150
200
0 100 200 300
Tempe
ra
Heating Time (days)
Nr = 50Nr = 90Nr = 180
(f) Temperature in fracture
Figure C.4: Effect of radial discretization on pressure, steam saturation, and temperatureat two reference points.
Appendix D
Calculated Numerical Model Input
Parameters
A number of model input parameters are not treated as independent variables, but rather
are derived from other input parameters. An outline of all calculated variables is provided
in this appendix.
D.1 Fracture Zone Physical Properties
In all simulations, a “fracture zone” approach has been used, whereby a layer of gridblocks
of thickness ∆zfz is assigned properties reflective of a fracture of aperture e and an amount
of matrix material of thickness (∆zfz − e/2). In this way, many of the benefits of a discrete
fracture model are retained, while avoiding the numerical difficulties associated with very
small gridblocks. Some calculation is necessary to derive appropriate properties for the
gridblocks in the fracture zone.
128
APPENDIX D. CALCULATED NUMERICAL MODEL INPUT PARAMETERS 129
kbulk kmatrix φmatrix
kfz
Equation (C.2)
e
Equation (C.6)
φfz
Equation (C.7)
Krock
Kwet‐mat
Kdry‐mat
Equations (C.10) & (C.11)
Kwet‐fz Kdry‐fz
Equations (C.10) & (C.11)
Pe‐frac Pd‐mat
Equation (C.16)
Equation (C.14)
λ
m
αmat
αfz
Equation (C.18)
Equation (C.19)
Equation (C.19)
Figure D.1: Outline of calculation of derived input parameters
D.1.1 Fracture Zone Permeability
For most simulations, the matrix permeability and bulk permeabilty are specified. However,
the fracture zone permeability must be calculated. The bulk permeability kb of the fractured
medium is equal to the weighted arithmetic mean of the fracture zone permeability kfz and
the matrix permeability km:
kb =kfz∆zfz + km∆zm
∆zfz + ∆zm(D.1)
where ∆zfz is the thickness of the fracture zone and ∆zmat is the thickness of the
modeled portion of the matrix. Equation (D.1) can be rearranged to solve for the fracture
zone permeability kfz:
kfz =kb (∆zm + ∆zfz)− km (∆zm)
∆zfz(D.2)
APPENDIX D. CALCULATED NUMERICAL MODEL INPUT PARAMETERS 130
D.1.2 Fracture Aperture
The fracture aperture e is not used as a model input. However, it is necessary to calculate
the fracture aperture in order to calculate several model inputs, such as the fracture zone
porosity and capillary pressure parameters.
The permeability of the fracture zone is given by the weighted arithmetic mean of the
fracture permeability (kfrac) and the permeability of the matrix material in the fracture
zone gridblocks (km):
kfz =kfrac
(e2
)+ km
(∆zfz − e
2
)∆zfz
(D.3)
The permeability of the fracture itself is given by:
kfrac =e2
12(D.4)
Substituting (D.4) into (D.3) gives:
kfz =
(e3
24
)+ km
(∆zfz − e
2
)∆zfz
(D.5)
The term kme2 is relatively small and can be ignored. The above equation can then be
simplified to solve explicitly for e:
e = 3
√24∆zfz (kfz − km) (D.6)
D.1.3 Fracture Zone Porosity
The porosity of a fracture is 1.0, while the porosity the rock matrix is given by φmat. The
porosity of the fracture zone is calculated from the weighted arithmetic mean of the fracture
porosity and the porosity of the matrix material in the fracture zone gridblocks:
APPENDIX D. CALCULATED NUMERICAL MODEL INPUT PARAMETERS 131
φfz =1(e2
)+ φm
(∆zfz − e
2
)∆zfz
(D.7)
Since the fracture aperture is not an input to the model, it is back-calculated from the
fracture zone permeability, as shown in Section D.1.2. The fracture zone porosity is only
appreciably larger than the matrix porosity when the fracture aperture is large, i.e. e > 100
µm.
D.1.4 Thermal Conductivity
Thermal conductivity in TOUGH2 is calculated as a function of liquid saturation, using
(Pruess et al., 1999):
K(Sl) = Kdry +√Sl (Kwet −Kdry) (D.8)
The wet and dry thermal conductivities Kwet and Kdry can be calculated using the geo-
metric mean thermal conductivity (Woodside and Messmer, 1961):
Kgeo = KφfK
1−φs (D.9)
where Kf is the thermal conductivity of the pore fluid, and Ks is the thermal conductiv-
ity of the matrix material. When the pores are fully saturated with water (K = Kwet), Kf
is the thermal conductivity of water, 0.587 W/m·K (Washburn, 2003). When the pores are
fully saturated with steam, Kf is taken to be 2.17×10−4 W/m·K (Washburn, 2003). Using
these values, the wet and dry thermal conductivities were calculated for all simulations as:
Kwet = (0.587)φK1−φr (D.10)
Kdry =(2.17× 10−4
)φK1−φr (D.11)
APPENDIX D. CALCULATED NUMERICAL MODEL INPUT PARAMETERS 132
D.2 Capillary Pressure
In both the rock matrix and the fractures, the capillary pressure-saturation relationship
of van Genuchten (1980) is used. In TOUGH2, the setting ICP = 7 implements this
relationship as (Pruess et al., 1999):
Pcap = −P0
(S−1/me − 1
)1−m(D.12)
where m is a parameter related to the pore-size distribution, P0 = αvg/ρwg, and the
effective saturation Se is defined as:
Se =Sl − SlrSls − Slr
(D.13)
where Sl is the water saturation, Slr is the residual (irreducible) water saturation (assumed
to be 0.18 in the matrix and 0.08 in the fracture zone), and Sls is the maximum liquid
saturation (assumed to be 1.00). In this implementation, the capillary pressure is allowed
only to be within the range:
−Pmax ≤ Pcap ≤ 0
Pmax is assumed to be equal to 107 Pa, following the example problems in Pruess et al.
(1999).
The fracture entry pressure Pe is calculated from the fracture aperture as (Kueper and
McWhorter, 1991, eq. 2):
Pe =2σ cos θ
e(D.14)
where the contact angle θ is assumed to be 0, and the interfacial tension between
water and steam (σws) is taken to be 0.055 N/m (Washburn, 2003). In the matrix, the
displacement pressure Pd is estimated from the rock matrix permeability (Reynolds and
APPENDIX D. CALCULATED NUMERICAL MODEL INPUT PARAMETERS 133
Kueper, 2003, eq. 2):
Pd = 10(−0.227 log(km)+0.889) (D.15)
This regression was created for an interfacial tension of 0.04 N/m. Again taking the
interfacial tension between water and steam to be 0.055 N/m, the entry pressure can be
scaled using the Leverett function (Leverett, 1941):
Pd−mat = Pd
( σws0.04
)=(
0.0550.04
)10(−0.227 log(km)+0.889) (D.16)
Assuming a Brooks-Corey pore size distribution index (λ) of 2.5, the Brooks-Corey p is
calculated as (Morel-Seytoux et al., 1996, eq. 8a):
p =2 + 3λλ
= 3.8 (D.17)
The corresponding value of m for the (van Genuchten, 1980) capillary pressure equation
(D.12) is (Morel-Seytoux et al., 1996, eq. 16a):
m =2
p− 1= 0.714 (D.18)
The value of αvg corresponding to the entry pressure calculated in (D.14) is given by
(Morel-Seytoux et al., 1996, eq. 18):
1αvg
=Peρg
[2p (p− 1)]p+ 3
(55.6 + 7.4p+ p2
147.8 + 8.1p+ 0.092p2
)(D.19)
A summary of the capillary pressure parameters is given in Table D.1.
APPENDIX D. CALCULATED NUMERICAL MODEL INPUT PARAMETERS 134
Table D.1: Capillary pressure parameters used in TOUGH2 simulationTOUGH2 Parameter Symbol Value in Fracture Zone Value in Matrix
ICP - 7 7CP(1) m computed from (D.18) computed from (D.18)CP(2) Slr 0.18 (assumed) 0.08 (assumed)CP(3) 1/P0 = ρwg/αvg computed from (D.19) computed from (D.19)CP(4) Pmax 107 Pa (assumed) 107 Pa (assumed)CP(5) Sls 1.00 (assumed) 1.00 (assumed)
D.3 Relative Permeability
In both the rock matrix and the fractures, the relative permeability-saturation relationship
of van Genuchten (1980) is used. In TOUGH2, the setting IRP = 7 implements this
relationship (Pruess et al., 1999). The liquid relative permeability is calculated as:
krl =
Se
[1−
(1− S1/m
e
)m]2if Sl < Sls
1 if Sl ≥ Sls(D.20)
where Se is defined by equation (D.13). Assuming the residual gas phase saturation (Sgr)
to be zero, the gas relative permeability is calculated as:
krg = 1− krl (D.21)
A summary of the relative permeability parameters used is given in Table D.2.
Table D.2: Relative permeability parameters used in TOUGH2 simulationsTOUGH2 Parameter Symbol Value in Fracture Zone Value in Matrix
IRP - 7 7RP(1) m computed from (D.18) computed from (D.18)RP(2) Slr 0.20 (assumed) 0.10 (assumed)RP(3) Sls 1.00 (assumed) 1.00 (assumed)RP(4) Sgr 0.00 (assumed) 0.00 (assumed)