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**Model description paper**
06 Dec 2018

**Model description paper** | 06 Dec 2018

CVPM 1.1: a flexible heat-transfer modeling system for permafrost

- Institute of Arctic and Alpine Research, University of Colorado, Boulder, CO, USA

- Institute of Arctic and Alpine Research, University of Colorado, Boulder, CO, USA

**Correspondence**: Gary D. Clow (gary.clow@colorado.edu)

**Correspondence**: Gary D. Clow (gary.clow@colorado.edu)

Abstract

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The Control Volume
Permafrost Model (CVPM) is a modular heat-transfer modeling system designed
for scientific and engineering studies in permafrost terrain, and as an
educational tool. CVPM implements the nonlinear heat-transfer equations in
1-D, 2-D, and 3-D Cartesian coordinates, as well as in 1-D radial and 2-D
cylindrical coordinates. To accommodate a diversity of geologic settings, a
variety of materials can be specified within the model domain, including
organic-rich materials, sedimentary rocks and soils, igneous and metamorphic
rocks, ice bodies, borehole fluids, and other engineering materials. Porous
materials are treated as a matrix of mineral and organic particles with pore
spaces filled with liquid water, ice, and air. Liquid water concentrations at
temperatures below 0 ^{∘}C due to interfacial, grain-boundary, and
curvature effects are found using relationships from condensed matter
physics; pressure and pore-water solute effects are included. A radiogenic
heat-production term allows simulations to extend into deep permafrost and
underlying bedrock. CVPM can be used over a broad range of depth,
temperature, porosity, water saturation, and solute conditions on either the
Earth or Mars. The model is suitable for applications at spatial scales
ranging from centimeters to hundreds of kilometers and at timescales ranging
from seconds to thousands of years. CVPM can act as a stand-alone model or the
physics package of a geophysical inverse scheme, or serve as a component
within a larger Earth modeling system that may include vegetation, surface
water, snowpack, atmospheric, or other modules of varying complexity.

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How to cite.

Clow, G. D.: CVPM 1.1: a flexible heat-transfer modeling system for permafrost, Geosci. Model Dev., 11, 4889–4908, https://doi.org/10.5194/gmd-11-4889-2018, 2018.

1 Introduction

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Given the recent surge of interest in the cryosphere and its role in the
Earth's climate system, a large number of permafrost models have been
developed over the past few decades. An important characteristic of
permafrost, especially in its fine-grained form, is that significant amounts
of liquid water can coexist with ice within the pore spaces at temperatures
well below 0 ^{∘}C due to a combination of interfacial, grain-boundary,
curvature, solute, and pressure effects (Dash et al., 2006; Davis, 2001). Even at
standard atmospheric pressure, liquid water has been observed at temperatures
as low as −31 ^{∘}C in silty soils and −40 ^{∘}C in glass
powders (Watanabe and Mizoguchi, 2002). The existence of liquid water at such low
temperatures is of interest biologically (Rothschild and Mancinelli, 2001), particularly
in the case of Mars, where permafrost could serve as a microbial refuge from
high radiation levels if life ever existed there. From a geoscience
perspective, it is also the presence of liquid water that makes permafrost so
dynamic and interesting. Since the liquid water content of permafrost is
highly temperature dependent and the thermal properties of solid and liquid
water are so different (Anderson et al., 1973; Holten et al., 2012; Huber et al., 2012; Yen, 1981), the thermal
response of permafrost to any temperature change is complicated by
nonlinearities and feedbacks. As with the thermal properties, the mechanical
properties of permafrost can be highly sensitive to temperature and the
unfrozen water content, particularly within a few degrees of the freezing
point *T*_{F}. As temperatures approach *T*_{F}, the material
strength generally declines, increasing the likelihood of downslope creep,
slope failures, accelerated lakeshore and coastal erosion, and ultimately
thaw settlement (thermokarst) if temperatures become warm enough. If enough
liquid water is available and the permafrost is sufficiently permeable,
migration of liquid water towards colder temperatures can lead to significant
frost heave, damaging buildings, roadways, and other facilities. With a
warming climate, the dynamic response of permafrost is expected to be
amplified, leading to accelerated landscape changes, disruption of vulnerable
habitats and ecosystems, and damage to human infrastructure
(ACIA, 2005; USARC, 2003).

A wide range of models has been developed to better understand the
occurrence of permafrost and its dynamics in a warming world. These models
range from simple analytical models to sophisticated numerical codes with
integrated vegetation, snow, and atmospheric layers overlying permafrost
(Riseborough et al., 2008; Zhang et al., 2003). The vast majority of these are 1-D
vertical models. Although useful for simulating conditions beneath a uniform
surface, 1-D models ignore important lateral heat-transfer effects occurring
near large land-surface contrasts such as at the boundaries between tundra,
rivers, lakes, oceans, glaciers, and human infrastructure. In addition, these
models almost universally use empirical equations to predict the unfrozen
water content at temperatures below 0 ^{∘}C. A significant limitation
with this approach is that the coefficients appearing in the unfrozen water
equations must be “calibrated” using field data for every material type,
pressure, water saturation, and solute condition. Even with calibration, the
empirical equations remain valid only over a limited range of temperatures.
With an emphasis on simulating shallow permafrost and active-layer
conditions, most permafrost models currently neglect the freezing-point
depression due to pressure and the radiogenic heat-source term, both of which
are needed to simulate conditions in deep permafrost.

In this paper, we present the new Control Volume Permafrost Model (CVPM v1.1)
which is designed to relax several of the limitations imposed by previous
models. CVPM implements the nonlinear heat-transfer equations in 1-D, 2-D,
and 3-D Cartesian coordinates, as well as in 1-D radial and 2-D cylindrical
coordinates. A variety of materials can be specified within the modeling
domain, including organic-rich materials, sedimentary rocks and soils,
igneous and metamorphic rocks, ice bodies, borehole fluids, and other
engineering materials. Numerical implementation is based on the
control-volume method (Anderson et al., 1984; Minkowycz et al., 1988; Patankar, 1980), allowing
enthalpy fluxes to be exactly balanced at control-volume interfaces (e.g., at
the interface between an ice lens and a siltstone). The unfrozen water
content at temperatures below 0 ^{∘}C is found using relationships from
condensed matter physics that utilize physical quantities (e.g., particle
radii), rather than non-physical empirical coefficients requiring
calibration. Pore pressure and solute effects are included in the unfrozen
water equations. A radiogenic heat-production term is also included to allow
simulations to extend into deep permafrost and underlying bedrock. CVPM is
designed for use over a broad range of depth, temperature, rock and soil
types, porosity, water saturation, and solute conditions. These conditions
include the coldest temperatures experienced on Earth through the ice ages,
as well as conditions on Mars, where the upper crust of the planet consists
entirely of permafrost (Squyres et al., 1992). The model is suitable for
applications at spatial scales ranging from centimeters to hundreds of
kilometers and at timescales ranging from seconds to thousands of years. To
achieve the greatest flexibility, CVPM does not include heat-transfer
processes within a vegetation canopy, snowpack, or atmospheric boundary
layer. Rather, CVPM focuses on permafrost and the underlying Earth materials.
In this way, CVPM can act as a stand-alone model or the physics package of a
geophysical inverse scheme, or serve as a component within a larger Earth
modeling system that may include vegetation, surface water, snowpack,
atmospheric, or other modules of varying complexity.

2 Governing equations

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The basis for the CVPM model is the conservation of mass and enthalpy over
time within any finite volume *V*. In integral form, the conservation
equations take the form

$$\begin{array}{}\text{(1)}& {\displaystyle}& {\displaystyle}\underset{V}{\int}{\displaystyle \frac{\partial \mathit{\rho}}{\partial t}}\phantom{\rule{0.33em}{0ex}}\mathrm{d}V=-\underset{A}{\int}\mathit{\rho}\mathit{v}\cdot \phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{A}\left(\mathrm{mass}\right)\text{(2)}& {\displaystyle}& {\displaystyle}\underset{V}{\int}{\displaystyle \frac{\partial \left(\mathit{\rho}H\right)}{\partial t}}\phantom{\rule{0.33em}{0ex}}\mathrm{d}V=-\underset{A}{\int}\mathit{J}\cdot \phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{A}\phantom{\rule{0.125em}{0ex}}+\phantom{\rule{0.125em}{0ex}}\underset{V}{\int}S\phantom{\rule{0.33em}{0ex}}\mathrm{d}V\left(\mathrm{enthalpy}\right),\end{array}$$

where *ρ* is the bulk density, *ρ*** v** is the mass flux,

In permafrost, the enthalpy at temperature *T* consists of two components:
one associated with the vibrational modes of the molecular lattice and the
other due to the latent heat associated with the phase change of water:

$$\begin{array}{}\text{(3)}& \mathit{\rho}H\left(T\right)\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\mathit{\rho}\underset{\mathrm{0}}{\overset{T}{\int}}\phantom{\rule{-0.125em}{0ex}}{c}_{p}\left({T}^{\prime}\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}{T}^{\prime}\phantom{\rule{0.125em}{0ex}}+\phantom{\rule{0.125em}{0ex}}{\mathit{\rho}}_{\mathrm{\ell}}\phantom{\rule{0.125em}{0ex}}\mathrm{\Delta}{H}_{\mathrm{fus}}\phantom{\rule{0.125em}{0ex}}{\mathit{\varphi}}_{\mathrm{\ell}}\left(T\right).\end{array}$$

Here, *c*_{p} is the specific heat of the bulk material, *ρ*_{ℓ} is the
density of liquid water, and Δ*H*_{fus} is the specific enthalpy
of fusion for water. Differentiating Eq. (3), the volumetric heat
capacity at constant pressure, defined by $C\equiv \mathit{\rho}(\partial H/\partial T{)}_{\mathrm{P}}$, is given by the sum of the lattice-vibration and latent-heat
terms:

$$\begin{array}{}\text{(4)}& C=\mathit{\rho}{c}_{p}+{\mathit{\rho}}_{\mathrm{\ell}}\phantom{\rule{0.125em}{0ex}}\mathrm{\Delta}{H}_{\mathrm{fus}}{\displaystyle \frac{\partial {\mathit{\varphi}}_{\mathrm{\ell}}}{\partial T}}.\end{array}$$

Since the density of air is much less than that of the other permafrost constituents, the lattice-vibration term is well approximated by the volume-weighted sum of the specific heats of the matrix materials, liquid water, and ice:

$$\begin{array}{}\text{(5)}& \mathit{\rho}{c}_{p}=(\mathrm{1}-\mathit{\varphi}){\mathit{\rho}}_{\mathrm{m}}{c}_{p\mathrm{m}}+{\mathit{\varphi}}_{\mathrm{\ell}}{\mathit{\rho}}_{\mathrm{\ell}}{c}_{p\mathrm{\ell}}+{\mathit{\varphi}}_{\mathrm{i}}{\mathit{\rho}}_{\mathrm{i}}{c}_{p\mathrm{i}},\end{array}$$

assuming *ϕ*_{a}<1.

Below 500 K, the specific heat of most matrix minerals is strongly
temperature dependent, primarily due to the energies associated with the
acoustic and optical modes of vibration (Kieffer, 1979, 1980). Due to
the wide variety of crystalline mineral structures, simple analytic
expressions for the temperature dependence of *c*_{pm} encompassing
all minerals are unavailable. However, the relatively smooth temperature
dependence predicted by detailed models indicates the specific heat for a
given mineral can be adequately represented by a three-term Taylor expansion over
the temperature range of interest for permafrost (Fig. 1). For
most minerals, the specific heat falls within the range
630–870 $\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ at 300 K and tends towards zero
as *T*→0 K (Kieffer, 1979; Kittel, 1967; Robertson, 1988). Taylor
expansions describing the temperature dependence *ω*(*T*) of
*c*_{pm} for most common mineral groups are incorporated into CVPM.
The matrix specific heat is then given by ${c}_{p\mathrm{m}}={c}_{p\mathrm{m}}^{\circ}\phantom{\rule{0.125em}{0ex}}\mathit{\omega}\left(T\right)$, where ${c}_{p\mathrm{m}}^{\circ}$ is the
specific heat of the dominant minerals at a standard temperature
of 293.15 K.

Unlike most materials, experimental data for liquid water show an anomalous
increase in specific heat (*c*_{pℓ}) with decreasing temperature.
Holten et al. (2012) explained this and other peculiar behaviors of supercooled
water with a thermodynamic model that assumes the existence of a
liquid–liquid critical point at low temperatures. Based on their
interpretation of available thermodynamic data, the liquid–liquid critical
temperature *T*_{c} is near 227 K. A least-squares fit to a
composite of data reported by Angell et al. (1982) below 273 K and the
International Association for the Properties of Water and Steam (IAPWS) 2008
values above 273 K provide the following relationships:

$$\begin{array}{}\text{(6)}& {c}_{p\mathrm{\ell}}\left(T\right)=\phantom{\rule{-0.125em}{0ex}}\left[\begin{array}{l}{a}_{\mathrm{1}}+{a}_{\mathrm{2}}{\left({\displaystyle \frac{T}{{T}_{\mathrm{c}}}}\phantom{\rule{0.125em}{0ex}}-\mathrm{1}\right)}^{-\mathrm{1}},\mathrm{235}\phantom{\rule{0.125em}{0ex}}\mathrm{K}<T\le \mathrm{265}\phantom{\rule{0.125em}{0ex}}\mathrm{K}\\ \sum _{i=\mathrm{1}}^{\mathrm{5}}{b}_{\mathrm{i}}\left({\displaystyle \frac{T}{\mathrm{310}\phantom{\rule{0.25em}{0ex}}\mathrm{K}}}-\mathrm{1}\right)\phantom{\rule{0.125em}{0ex}},\mathrm{265}\phantom{\rule{0.125em}{0ex}}\mathrm{K}<T\le \mathrm{360}\phantom{\rule{0.125em}{0ex}}\mathrm{K}.\end{array}\right.\end{array}$$

Values for the coefficients *a*_{1}, *a*_{2}, and *b*_{i} are listed in
Table 1.

For water ice (*I*_{h}), lattice vibrations lead to a simple linear
relationship between the specific heat *c*_{pi} and temperature for
*T*>150 K (Yen, 1981). Based on Yen's empirical relationship,
*c*_{pi} is well represented by

$$\begin{array}{ll}{\displaystyle}{c}_{p\mathrm{i}}\left(T\right)=& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{-0.125em}{0ex}}{a}_{\mathrm{1}}+{a}_{\mathrm{2}}\left({\displaystyle \frac{T}{\mathrm{273.15}\phantom{\rule{0.125em}{0ex}}\mathrm{K}}}-\mathrm{1}\right),\\ \text{(7)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\mathrm{150}\phantom{\rule{0.125em}{0ex}}\mathrm{K}<T<\mathrm{273.15}\phantom{\rule{0.125em}{0ex}}\mathrm{K},\end{array}$$

with *a*_{1}=2096.1 and *a*_{2}=1943.8 ($\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$).

Studies dating back to the mid-1800s show that a melt layer can stably exist
at the interface between ice and a foreign substrate (e.g., a mineral grain),
even at temperatures well below the bulk freezing temperature of water
*T*_{f} (Dash, 2002; Dash et al., 1995). A melt layer can similarly exist at the
grain boundaries within polycrystalline ice. For both interfacial
and grain-boundary melting, the liquid phase exists because it
reduces the system's total free energy. Electrostatic interactions in
molecular substances such as ice tend to be dominated by non-retarded van der
Waals forces. In this case, the thickness of the liquid layer adjacent to a
planar substrate is $L=\mathit{\lambda}\phantom{\rule{0.125em}{0ex}}\mathrm{\Delta}{T}^{-\mathrm{1}/\mathrm{3}}$, where $\mathrm{\Delta}T=({T}_{\mathrm{f}}-T)$ is the temperature below the bulk freezing point
(Dash et al., 2006; Wettlaufer and Worster, 1995). A ramification of this behavior is that the
melting and freezing of ice in contact with mineral grains occur over a
range of temperatures, rather than at a distinct temperature. Depending on
the value for the interfacial melting parameter *λ*, substantial
amounts of liquid water can exist at temperatures well below *T*_{f}.
For a planar substrate, the interfacial melting parameter is given
approximately by

$$\begin{array}{}\text{(8)}& \mathit{\lambda}={\left({\displaystyle \frac{\mathrm{2}{\mathit{\sigma}}^{\mathrm{2}}\mathrm{\Delta}\mathit{\gamma}\phantom{\rule{0.125em}{0ex}}{T}_{\mathrm{f}}}{{\mathit{\rho}}_{\mathrm{i}}\mathrm{\Delta}{H}_{\mathrm{fus}}}}\right)}^{\mathrm{1}/\mathrm{3}},\end{array}$$

where Δ*γ* is the difference in the interfacial free energy with
and without the melt layer, and *σ* is a constant on the order of a
molecular diameter (Wettlaufer and Worster, 1995). Imperfections due to internal
disorder (polycrystallinity, point defects, dislocations) within the ice and
irregularities (pits, scratches, steps) in the substrate's surface can
greatly increase the magnitude of Δ*γ* and thereby the effective
interfacial melting parameter *λ*. Due to the irregular nature of
mineral surfaces and the likely disorder within interstitial ice, reliable
expressions for parameter *λ* are currently lacking for most Earth
materials. Thus, *λ* is best determined experimentally for frozen
ground.

Surface curvature also affects the interfacial free energy and hence the
thickness of liquid water films surrounding mineral grains. By considering
the detailed effects of curvature along with interfacial and grain-boundary
melting, Cahn et al. (1992) found that the volume fraction of liquid water in a
porous medium consisting of spheres with radius *r* can be described by the
sum of two temperature-dependent terms:

$$\begin{array}{}\text{(9)}& {\mathit{\varphi}}_{\mathrm{\ell}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}{a}_{\mathrm{1}}\left({\displaystyle \frac{\mathit{\lambda}}{r\mathrm{\Delta}{T}^{\mathrm{1}/\mathrm{3}}}}\right)+{a}_{\mathrm{2}}{\left({\displaystyle \frac{\mathit{\xi}}{r\mathrm{\Delta}T}}\right)}^{\mathrm{2}}.\end{array}$$

The first term is due to the combined effects of interfacial melting at the
surface of the spherical particles and grain-boundary melting within
polycrystalline pore ice. The second term is associated with high-curvature
areas on the ice–liquid interface (e.g., near mineral grain-contact points
and where ice-grain boundaries approach mineral surfaces). Coefficients *a*_{1}
and *a*_{2} depend on how the particles are packed. For simple cubic packing,
*a*_{1}=1.893 and *a*_{2}=3.367, while for cubic close packing, *a*_{1}=2.450
and *a*_{2}=8.572 (Cahn et al., 1992). Earth materials are likely to have
intermediate values for the packing coefficients. In addition to its
dependence on particle size and temperature, the second term depends on the
interfacial free energy at the ice–water interface (*γ*_{sℓ}).
The curvature coefficient at this interface, $\mathit{\xi}={\mathit{\gamma}}_{\mathrm{s}\mathrm{\ell}}{T}_{\mathrm{f}}/\left({\mathit{\rho}}_{\mathrm{i}}\mathrm{\Delta}{H}_{\mathrm{fus}}\right)$, numerically
evaluates to 0.0259 µm K (Cahn et al., 1992). Given the temperature
dependencies, the interfacial/grain-boundary term ($\mathrm{\Delta}{T}^{-\mathrm{1}/\mathrm{3}}$)
dominates for strong undercooling (Δ*T*), while the second term (Δ*T*^{−2}) dominates as temperatures approach the bulk freezing point
(*T*_{f}). As the particle size *r* decreases, the transition between
the two behaviors shifts to larger Δ*T* values (colder temperatures).
While the first term is inversely proportional to *r*, the second term has an
even stronger dependence on particle size (${\mathit{\varphi}}_{\mathrm{\ell}}\propto {r}^{-\mathrm{2}}$). Both
terms result in greater amounts of unfrozen water in fine-grained materials
(Fig. 2). Experimental data with graphitized carbon black and
polystyrene powders confirm the form of Eq. (9) for monosized
particles (Cahn et al., 1992).

Although the particle and associated pore-size distributions in sandstones, limestones, and other rocks are often unimodal, those in mudrocks and soils typically are not (Kuila and Prasad, 2013). To accommodate a multimodal pore-size distribution, CVPM finds the total liquid water content by summing the contributions from the dominant modes. This is implemented in CVPM using a variant of Eq. (9):

$$\begin{array}{}\text{(10)}& {\mathit{\varphi}}_{\mathrm{\ell}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\sum _{j}{\mathrm{\Psi}}_{j}\left[{a}_{\mathrm{1}}\left({\displaystyle \frac{\mathit{\lambda}}{{r}_{j}\mathrm{\Delta}{T}^{\mathrm{1}/\mathrm{3}}}}\right)+{a}_{\mathrm{2}}{\left({\displaystyle \frac{\mathit{\xi}}{{r}_{j}\mathrm{\Delta}T}}\right)}^{\mathrm{2}}\right],\end{array}$$

where Ψ_{j} is the relative volume fraction of pores associated with the
mode whose mean particle size is *r*_{j}. Tests with sample pore-size
distributions show the larger pore sizes contribute by far the most to the
unfrozen water content. Thus, CVPM currently limits the number of modes
contributing to *ϕ*_{ℓ} to two (*j*≤2). In this case,

$$\begin{array}{}\text{(11)}& {\mathrm{\Psi}}_{\mathrm{1}}={\displaystyle \frac{{\left(\frac{{r}_{\mathrm{1}}}{{r}_{\mathrm{2}}}\right)}^{\mathrm{3}}}{{\left(\frac{{r}_{\mathrm{1}}}{{r}_{\mathrm{2}}}\right)}^{\mathrm{3}}+\left(\frac{{n}_{\mathrm{2}}}{{n}_{\mathrm{1}}}\right)}},{\mathrm{\Psi}}_{\mathrm{2}}={\displaystyle \frac{\left(\frac{{n}_{\mathrm{2}}}{{n}_{\mathrm{1}}}\right)}{{\left(\frac{{r}_{\mathrm{1}}}{{r}_{\mathrm{2}}}\right)}^{\mathrm{3}}+\left(\frac{{n}_{\mathrm{2}}}{{n}_{\mathrm{1}}}\right)}},\end{array}$$

where (*n*_{2}∕*n*_{1}) is the ratio of the number density of pores with radius
*r*_{2} to those with radius *r*_{1}. As an example, suppose there are 100 times
as many small pores (with a mean modal radius of 0.1 µm) as large
pores (mean modal radius of 2 µm). Despite the greater number of
small pores, the relative volume fraction of large pores is Ψ_{1}=0.988,
while that of the smaller pores is only Ψ_{2}=0.012.

The undercooling Δ*T* used to evaluate the interfacial, grain-boundary,
and curvature effects is measured relative to the bulk freezing temperature
(*T*_{f}). For permafrost, pore pressures and dissolved solutes can
significantly reduce *T*_{f} below the point at which pure water
freezes (${T}_{\mathrm{f}}^{*}$): 273.16 K at the triple point pressure
*P*_{tp}=611.66 Pa (Guildner et al., 1976; Kittel, 1969; Nicholas and White, 2001). If *θ*_{P} and *θ*_{s} are the
freezing-point depressions due to pressure and solutes, respectively, CVPM predicts
the bulk freezing temperature using

$$\begin{array}{}\text{(12)}& {T}_{\mathrm{f}}={T}_{\mathrm{f}}^{*}-{\mathit{\theta}}_{\mathrm{s}}-{\mathit{\theta}}_{\mathrm{P}}.\end{array}$$

When solutes remain dilute, the freezing-point depression due to impurities can be approximated using simple relationships such as Blagden's law (Delapaz, 2015). However, due to the insolubility of most solutes in ice, impurities become strongly enriched in the liquid pore water as permafrost freezes. As a result, solute–solute interactions become increasingly important, leading to significant deviations from the ideal behavior exhibited by dilute solutions. To account for the non-ideal behavior of aqueous electrolyte solutions at higher solute concentrations, CVPM uses the relationship

$$\begin{array}{}\text{(13)}& -\mathrm{ln}\left({a}_{\mathrm{w}}\right)={\displaystyle \frac{\mathrm{\Delta}{H}_{\mathrm{fus}}^{\circ}}{R\phantom{\rule{0.125em}{0ex}}{T}^{\circ}}}\left({\displaystyle \frac{{\mathit{\theta}}_{\mathrm{s}}}{{T}^{\circ}}}\right)+\left({\displaystyle \frac{\mathrm{\Delta}{H}_{\mathrm{fus}}^{\circ}}{R\phantom{\rule{0.125em}{0ex}}{T}^{\circ}}}-{\displaystyle \frac{\mathrm{\Delta}{c}_{p}^{\circ}}{\mathrm{2}R}}\right){\left({\displaystyle \frac{{\mathit{\theta}}_{\mathrm{s}}}{{T}^{\circ}}}\right)}^{\mathrm{2}}\end{array}$$

between the water activity *a*_{w} and the solute freezing-point
depression *θ*_{s} (Robinson and Stokes, 1959). Here, $\mathrm{\Delta}{H}_{\mathrm{fus}}^{\circ}$ is the molar enthalpy of fusion at a standard
temperature *T*^{∘}, $\mathrm{\Delta}{c}_{p}^{\circ}$ is the difference between the molar
heat capacities of liquid water and ice, and *R* is the gas constant. Using
established values for $\mathrm{\Delta}{H}_{\mathrm{fus}}^{\circ}$, $\mathrm{\Delta}{c}_{p}^{\circ}$,
*R*, and *T*^{∘}, Eq. (13) simplifies to

$$\begin{array}{}\text{(14)}& -\mathrm{ln}{a}_{\mathrm{w}}=(\mathrm{9.687}\times {\mathrm{10}}^{-\mathrm{3}})\phantom{\rule{0.125em}{0ex}}{\mathit{\theta}}_{\mathrm{s}}+(\mathrm{4.76}\times {\mathrm{10}}^{-\mathrm{6}})\phantom{\rule{0.125em}{0ex}}{\mathit{\theta}}_{\mathrm{s}}^{\mathrm{2}},\end{array}$$

where *θ*_{s} is in Kelvin. The extent to which aqueous
electrolyte solutions deviate from ideal behavior varies greatly, depending
on the composition of the solute. As a result, the water activity
*a*_{w} depends on the particular solute and its mole fraction
*x*_{s}. Several expressions have been proposed for the water activity
of non-ideal electrolyte solutions. CVPM uses the following equation proposed by
Miyawaki et al. (1997):

$$\begin{array}{}\text{(15)}& {a}_{\mathrm{w}}=(\mathrm{1}-{x}_{\mathrm{s}})\mathrm{exp}\left(\mathit{\alpha}{x}_{\mathrm{s}}^{\mathrm{2}}+\mathit{\beta}{x}_{\mathrm{s}}^{\mathrm{3}}\right),\end{array}$$

where the coefficients *α* and *β* are solute dependent. For
example, *α*=1.825, 4.754, and 11.859 for NaCl, KCl, and MgCl_{2},
respectively, while $\mathit{\beta}=-\mathrm{20.78}$, −49.37, and −404.5 (Miyawaki et al., 1997).
Since essentially all the solutes are concentrated in the aqueous solution
upon freezing, the solute mole fraction at any stage during freezing or
thawing is given by ${x}_{\mathrm{s}}={x}_{\mathrm{s}}^{\star}\left[\right({\mathit{\varphi}}_{\mathrm{\ell}}+{\mathit{\varphi}}_{\mathrm{i}})/{\mathit{\varphi}}_{\mathrm{\ell}}]$, where ${x}_{\mathrm{s}}^{\star}$ is the solute
concentration in the fully melted system (*ϕ*_{i}=0). Once
*x*_{s} and *a*_{w} have been established, the freezing-point
depression *θ*_{s} can be found by solving Eq. (14).

Figure 3 shows the sensitivity of the unfrozen water content
*ϕ*_{ℓ} to solutes predicted by CVPM for a medium-grained silt along with
the temperature sensitivity $\partial {\mathit{\varphi}}_{\mathrm{\ell}}/\partial T$ needed to find
the latent-heat component of the volumetric heat capacity *C*. The results
show that even small amounts of solute can significantly affect *ϕ*_{ℓ}
and $\partial {\mathit{\varphi}}_{\mathrm{\ell}}/\partial T$. As noted by Dash et al. (2006), the great
sensitivity of *ϕ*_{ℓ} to impurities is a likely cause for the
considerable disagreement between the results of various unfrozen water
experiments. An additional sensitivity can occur at cold temperatures if
solute concentrations are sufficiently high. This occurs when the solution
reaches its saturation limit, beyond which the solute begins to precipitate
upon further cooling. This leads to the spike in $\partial {\mathit{\varphi}}_{\mathrm{\ell}}/\partial T$ values near −21 ^{∘}C seen in Fig. 3b when
aqueous NaCl concentrations ${x}_{\mathrm{s}}^{\star}$ exceed 0.005. The
temperature at which the solute-saturation spike occurs varies, depending on
the particular solute. Outside of the saturation limit, the largest $\partial {\mathit{\varphi}}_{\mathrm{\ell}}/\partial T$ values occur as the last bits of pore ice melt upon
warming (*ϕ*_{i}→0). The volumetric heat capacity
mirrors the temperature sensitivity $\partial {\mathit{\varphi}}_{\mathrm{\ell}}/\partial T$ but with
minimum values established by the lattice-vibration term in Eq. (4).

As previously mentioned, the interfacial melting parameter is best determined
experimentally for natural Earth materials. Inversion of unfrozen water data
(shown in Fig. 3a) from Yuanlin and Carbee (1987) using CVPM yields a
value of *λ*=0.36 µm K^{1∕3} for Fairbanks silt. Other
parameters determined by the inversion are Ψ_{1}≃1 and
${x}_{\mathrm{s}}^{\star}=\mathrm{0.0008}$. Thus, the pore-size distribution for this
material is approximately unimodal. In addition, trace amounts of impurities
appear to have been present during the unfrozen water experiments despite
efforts to eliminate them. The *λ* value determined for Fairbanks silt
is about 100 times that determined for liquid films adjacent to smooth metal
wires (Cahn et al., 1992), testifying to the importance of mineral surface
irregularities and imperfections on the interfacial free energy in natural
Earth materials. While more work needs to be done to quantify *λ* for
the range of materials expected in permafrost, preliminary inversions for
sedimentary materials (e.g., Suffield silty clay and kaolinite) yield values
within ±10 % of that found for Fairbanks silt. At this point, the
interfacial melting parameter *λ* does not appear to vary substantially
amongst natural Earth materials.

Given that permafrost occurs at depths in excess of 1 km in some
high-latitude areas and at 3–4 km beneath the polar ice sheets
(Davis, 2001; MacGregor et al., 2016), the effect of pressure can be substantial on
the bulk freezing temperature *T*_{f} and thereby the unfrozen water
content *ϕ*_{ℓ}. If the interstitial pores are freely connected to the
planet's surface, the pore pressure is equal to the hydrostatic
pressure (*P*=*ρ*_{ℓ}*g**z*), where *g* is the acceleration of gravity and *z*
is the depth below the surface. However, if the pore water is trapped, the
pore pressure can be nearly equal to or exceed the lithostatic
pressure (*P*=*ρ**g**z*) (Turcotte and Schubert, 1982). In either case, the freezing-point depression due to pore pressure is

$$\begin{array}{}\text{(16)}& {\mathit{\theta}}_{\mathrm{P}}=a(P-{P}_{\mathrm{tp}}).\end{array}$$

As water in permafrost is likely to be saturated with air, the appropriate
value for coefficient *a* is $\mathrm{9.8}\times {\mathrm{10}}^{-\mathrm{8}}$ K Pa^{−1}
(Cuffey and Paterson, 2010). Since both pressure situations are known to occur in
sedimentary basins, both are implemented in CVPM. The lithostatic effect is
generally 2–3 times that of the hydrostatic effect. Not only does the
pressure effect increase the unfrozen water content with depth, it also
increases the temperature sensitivity ($\partial {\mathit{\varphi}}_{\mathrm{\ell}}/\partial T$) and
therefore the volumetric heat capacity *C* (Fig. 4).

Several mixing models are available for estimating the bulk thermal conductivity of multi-component systems. Of these, CVPM uses the Brailsford and Major (1964) two-phase random mixture (BM2) and three-phase (BM3) models, which are recommended as being the best for use with in situ Earth materials (Roy et al., 1981). Assuming a random mixture of pores and matrix material, the bulk thermal conductivity of permafrost can be described by the BM2 model:

$$\begin{array}{ll}{\displaystyle}k=& {\displaystyle}\phantom{\rule{0.25em}{0ex}}{k}_{\mathrm{m}}[{\displaystyle \frac{(\mathrm{2}\mathit{\chi}-\mathrm{1})-\mathrm{3}\mathit{\varphi}(\mathit{\chi}-\mathrm{1})}{\mathrm{4}\mathit{\chi}}}\\ \text{(17)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}+\phantom{\rule{0.125em}{0ex}}{\displaystyle \frac{{\left\{{\left[(\mathrm{2}\mathit{\chi}-\mathrm{1})-\mathrm{3}\mathit{\varphi}(\mathit{\chi}-\mathrm{1})\right]}^{\mathrm{2}}+\mathrm{8}\mathit{\chi}\right\}}^{\frac{\mathrm{1}}{\mathrm{2}}}}{\mathrm{4}\mathit{\chi}}}],\end{array}$$

where *k*_{m} is the conductivity of the matrix material,
*k*_{p} is the conductivity of the pores, and *χ* is their ratio
(*k*_{m}∕*k*_{p}).

For matrix minerals, the thermal conductivity depends primarily on the temperature and mineral composition. Using thermal conductivity data obtained by Birch and Clark (1940a, b) over the temperature range 273–473 K, Sass et al. (1992) found the temperature dependence could be separated from the compositional dependence using a function of the form

$$\begin{array}{ll}{\displaystyle}{k}_{\mathrm{m}}\left(T\right)=& {\displaystyle}\phantom{\rule{0.25em}{0ex}}{k}_{\mathrm{m}}^{\circ}{\left[{a}_{\mathrm{1}}+\left(T-\mathrm{273.15}\phantom{\rule{0.125em}{0ex}}\mathrm{K}\right)\left({a}_{\mathrm{2}}-{\displaystyle \frac{{a}_{\mathrm{3}}}{{k}_{\mathrm{m}}^{\circ}}}\right)\right]}^{-\mathrm{1}},\\ \text{(18)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\mathrm{150}\phantom{\rule{0.125em}{0ex}}\mathrm{K}<T<\mathrm{570}\phantom{\rule{0.125em}{0ex}}\mathrm{K},\end{array}$$

where ${k}_{\mathrm{m}}^{\circ}$ is the value of the matrix conductivity at a
standard temperature of 273.15 K. As the *a*_{i} coefficients are
fairly insensitive to rock type, the effects of mineralogy and texture are
almost entirely encapsulated in ${k}_{\mathrm{m}}^{\circ}$. A more recent analysis
indicates the *a*_{i} coefficients (Table 2) are
slightly different for the mineral assemblages that dominate sedimentary
rocks from those that occur in magmatic and metamorphic rocks
(Vosteen and Schellschmidt, 2003). The upper temperature limit for Eq. (18) is set
below the temperatures at which metamorphosis occurs in sedimentary rocks and
well below the point where radiative heat transfer within crystal lattices
becomes important (Clauser and Huenges, 1995). As very little thermal conductivity data
exist for rocks and minerals below 273 K, the validity of
Eq. (18) has yet to be tested at lower temperatures. The little data
that do exist suggest that the transition from the intermediate-temperature
behavior (Eq. 18) to the low-temperature behavior (*k*_{m}∝*T*^{3}) (Parrott and Stuckes, 1975) generally occurs below 100 K. For
example, in garnets, the transition occurs at 20–30 K
(Slack and Oliver, 1971). We tentatively set the lower limit of validity for
Eq. (18) at 150 K.

To find the thermal conductivity of the pores (*k*_{p}), CVPM utilizes
the three-phase BM3 model:

$$\begin{array}{}\text{(19)}& {k}_{\mathrm{c}x}={k}_{\mathrm{1}}\left[{\displaystyle \frac{{\mathit{\psi}}_{\mathrm{1}}+\mathrm{3}{\displaystyle}\left({\displaystyle \frac{{\mathit{\psi}}_{\mathrm{2}}}{\mathrm{2}{\mathit{\chi}}_{\mathrm{2}}+\mathrm{1}}}+{\displaystyle \frac{{\mathit{\psi}}_{\mathrm{3}}}{\mathrm{2}{\mathit{\chi}}_{\mathrm{3}}+\mathrm{1}}}\right)}{{\mathit{\psi}}_{\mathrm{1}}+\mathrm{3}{\displaystyle}\left({\displaystyle \frac{{\mathit{\psi}}_{\mathrm{2}}{\mathit{\chi}}_{\mathrm{2}}}{\mathrm{2}{\mathit{\chi}}_{\mathrm{2}}+\mathrm{1}}}+{\displaystyle \frac{{\mathit{\psi}}_{\mathrm{3}}{\mathit{\chi}}_{\mathrm{3}}}{\mathrm{2}{\mathit{\chi}}_{\mathrm{3}}+\mathrm{1}}}\right)}}\right],\end{array}$$

where the three phases (*x*) are liquid water, ice, and air. Here, ${\mathit{\chi}}_{\mathrm{2}}=({k}_{\mathrm{1}}/{k}_{\mathrm{2}})$, ${\mathit{\chi}}_{\mathrm{3}}=({k}_{\mathrm{1}}/{k}_{\mathrm{3}})$, and the *ψ*_{x} are the relative volume
fractions of the pore's constituents (${\mathit{\psi}}_{x}={\mathit{\varphi}}_{x}/\mathit{\varphi}$). Similar to
other three-phase models, BM3 assumes phases 2 and 3 are randomly distributed
within a continuous phase 1. If the relative volume fraction of any of the
three constituents exceeds a threshold (*ψ*_{x}≥*α*), it is assumed that
component is the continuous phase and *k*_{p} is calculated directly
from Eq. (19). A comparison of the results from the BM3 model in
the limit *ψ*_{2}→0 or *ψ*_{3}→0 with those of the
BM2 model suggests a reasonable choice for *α* is ∼0.75. If none
of the relative volume fractions exceed *α*, Eq. (19) is used
to calculate the conductivity of the pore space assuming each of the three
components, in turn, is the continuous phase to produce values
*k*_{cℓ} (continuous liquid water phase), *k*_{ci}
(continuous ice phase), and *k*_{ca} (continuous air phase). The pore
conductivity is then found from a simple weighted average:

$$\begin{array}{}\text{(20)}& {k}_{\mathrm{p}}={w}_{\mathrm{\ell}}{k}_{\mathrm{c}\mathrm{\ell}}+{w}_{i}{k}_{\mathrm{ci}}+{w}_{\mathrm{a}}{k}_{\mathrm{ca}},\end{array}$$

where the weights *w*_{x} are based on the relative volume fractions *ψ*_{x}
and the requirement that *k*_{p} be continuous across the lines
*ψ*_{ℓ}=*α*, *ψ*_{i}=*α*, and *ψ*_{a}=*α*
in three-phase space (Fig. 5).

For the thermal conductivity of liquid water *k*_{ℓ}, CVPM uses the
simplified correlating equation recommended by Huber et al. (2012) for use at
0.1 MPa:

$$\begin{array}{}\text{(21)}& {k}_{\mathrm{\ell}}\left(T\right)=\sum _{i=\mathrm{1}}^{\mathrm{4}}{a}_{\mathrm{i}}{\left({\displaystyle \frac{T}{\mathrm{300}\phantom{\rule{0.125em}{0ex}}\mathrm{K}}}\right)}^{{b}_{\mathrm{i}}}\phantom{\rule{-0.125em}{0ex}},\mathrm{250}\phantom{\rule{0.125em}{0ex}}\mathrm{K}<T\le \mathrm{383}\phantom{\rule{0.125em}{0ex}}\mathrm{K},\end{array}$$

with coefficients *a*_{i} and *b*_{i} (Table 2).
Although the formal lower limit for Eq. (21) is 273.15 K,
Huber et al. (2012) find that it extrapolates in a physically reasonable manner
down to ∼250 K (Fig. 6), producing results very close
to those of the new detailed IAPWS formulation for *k*_{ℓ}. Thermal
conductivity data for supercooled water do not appear to exist below
250 K at this time, preventing the development of accurate
correlating equations at lower temperatures. This is a minor limitation for
the thermal model as the relative amount of liquid water is expected to be
small at colder temperatures.

Experimental data for the thermal conductivity of ice *k*_{i} exist
at temperatures as cold as 60 K. Based on these data, Yen (1981)
recommends the function

$$\begin{array}{}\text{(22)}& {k}_{\mathrm{i}}\left(T\right)={a}_{\mathrm{1}}\phantom{\rule{0.125em}{0ex}}\mathrm{exp}(-{a}_{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}T),\mathrm{60}\phantom{\rule{0.125em}{0ex}}\mathrm{K}<T\le \mathrm{273.15}\phantom{\rule{0.125em}{0ex}}\mathrm{K}\end{array}$$

for describing the temperature dependence of *k*_{i}, with *a*_{1}=9.828 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ and *a*_{2}=0.0057 K^{−1}.

For the terrestrial environment, the thermal conductivity of air
*k*_{a} can be separated into the sum of two terms: a “dilute gas”
term *k*_{o} that depends solely on temperature and a “residual”
term Δ*k* that depends on air density:

$$\begin{array}{}\text{(23)}& {k}_{\mathrm{a}}({\mathit{\rho}}_{\mathrm{a}},T)={k}_{\mathrm{o}}\left(T\right)+\mathrm{\Delta}k\left({\mathit{\rho}}_{\mathrm{a}}\right).\end{array}$$

For the dilute gas term, Stephan and Laesecke (1985) recommend the correlating equation:

$$\begin{array}{ll}{\displaystyle}{k}_{\mathrm{o}}\left(T\right)=& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\sum _{i=\mathrm{1}}^{\mathrm{9}}{a}_{\mathrm{i}}{\left({\displaystyle \frac{T}{\mathrm{132.52}\phantom{\rule{0.125em}{0ex}}\mathrm{K}}}\right)}^{\phantom{\rule{-0.125em}{0ex}}(i-\mathrm{4})/\mathrm{3}}\phantom{\rule{-0.125em}{0ex}},\\ \text{(24)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\mathrm{70}\phantom{\rule{0.125em}{0ex}}\mathrm{K}<T<{\mathrm{10}}^{\mathrm{3}}\phantom{\rule{0.125em}{0ex}}\mathrm{K},\end{array}$$

with *a*_{i} coefficients (Table 2). At typical
terrestrial surface pressures (∼0.1 MPa), the residual term Δ*k*
is $\mathrm{5.17}\times {\mathrm{10}}^{-\mathrm{5}}$ $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ (Stephan and Laesecke, 1985).

When considering permafrost on Mars, the thermal properties of a different
atmospheric gas must be used. The Martian atmosphere is currently 95 %
carbon dioxide, a gas that has a thermodynamic critical point at
304.107 K, 7.3721 MPa. At gas densities below
25 kg m^{−3}, the effects of the critical region are small enough
that the thermal conductivity can again be described by Eq. (23). In
the case of CO_{2}, the dilute gas contribution to *k*_{a} is

$$\begin{array}{ll}{\displaystyle}{k}_{\mathrm{o}}\left(T\right)=& {\displaystyle}\phantom{\rule{0.25em}{0ex}}{\displaystyle \frac{\mathrm{4.75598}\times {\mathrm{10}}^{-\mathrm{4}}\left(\mathrm{1}+{\displaystyle \frac{\mathrm{2}{c}_{\mathrm{int}}}{\mathrm{5}{k}_{\mathrm{B}}}}\right){T}^{\mathrm{1}/\mathrm{2}}}{{\mathcal{C}}_{\mathrm{R}}\left(T\right)}},\\ \text{(25)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\mathrm{200}\phantom{\rule{0.125em}{0ex}}\mathrm{K}<T<{\mathrm{10}}^{\mathrm{3}}\phantom{\rule{0.125em}{0ex}}\mathrm{K},\end{array}$$

where *c*_{int} is the ideal gas heat capacity, *k*_{B} is the
Boltzmann constant, and 𝒞_{R} is the reduced effective
cross-section (Vesovic et al., 1990). The correlating equations for
𝒞_{R} and *c*_{int} provided by Vesovic et al. (1990)
are

$$\begin{array}{}\text{(26)}& {\displaystyle}& {\displaystyle}{\mathcal{C}}_{\mathrm{R}}\left(T\right)=\sum _{i=\mathrm{0}}^{\mathrm{7}}{a}_{\mathrm{i}}{\left({\displaystyle \frac{T}{\mathrm{251.196}\phantom{\rule{0.125em}{0ex}}\mathrm{K}}}\right)}^{-i}\text{(27)}& {\displaystyle}& {\displaystyle}{c}_{\mathrm{int}}={k}_{\mathrm{B}}\phantom{\rule{-0.125em}{0ex}}\left[\mathrm{1}+\mathrm{exp}\left({\displaystyle \frac{-\mathrm{183.5}\phantom{\rule{0.125em}{0ex}}\mathrm{K}}{T}}\right)\sum _{i=\mathrm{1}}^{\mathrm{5}}{b}_{\mathrm{i}}\phantom{\rule{-0.125em}{0ex}}{\left({\displaystyle \frac{T}{\mathrm{100}\phantom{\rule{0.125em}{0ex}}\mathrm{K}}}\right)}^{\mathrm{2}-i}\right]\end{array}$$

(coefficients *a*_{i} and *b*_{i} given in
Table 2). At current Martian surface pressures
(0.6 kPa), the residual component Δ*k* is $\mathrm{3.53}\times {\mathrm{10}}^{-\mathrm{7}}$ $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$. Even when the Martian atmosphere was
denser, the contribution of Δ*k* to the overall thermal conductivity of
the CO_{2} gas would have been quite small.

In sedimentary basins, overburden pressure causes the porosity *ϕ* to
decrease with depth due to pressure solution and mechanical-compaction
processes (Revil et al., 2002). The former process changes the mineral shapes in
response to grain-contact stresses, while the latter results in the slippage
and rotation of the grains. With increasing overburden pressure, the porosity
ultimately reaches a residual (or critical) porosity *ϕ*_{c} that
depends on the grain-shape and grain-size distribution. Shales and mudstones
are much more easily compacted than sandstones due to the plate-like shape of
the mineral grains. Although fairly sophisticated compaction models now
exist, CVPM uses the simple frequently used compaction function attributed to
Athy (1930),

$$\begin{array}{}\text{(28)}& \mathit{\varphi}\left(z\right)={\mathit{\varphi}}_{\mathrm{0}}\phantom{\rule{0.125em}{0ex}}\mathrm{exp}\left(-z/{h}_{\mathrm{c}}\right),\mathit{\varphi}\ge {\mathit{\varphi}}_{\mathrm{c}},\end{array}$$

to account for overburden pressures. This function has been successfully used
in a large number of studies (Burns et al., 2005; Fjeldskaar et al., 2004). Here,
*ϕ*_{0} is the porosity extrapolated to the surface, while *h*_{c} is
the compaction length scale. Parameters *ϕ*_{0} and *h*_{c} depend on
both the lithology and the effective stress history.

CVPM assumes the enthalpy-production rate *S* is associated with the decay of
radionuclides. In this case,

$$\begin{array}{}\text{(29)}& S\left(z\right)={S}_{\mathrm{0}}\phantom{\rule{0.125em}{0ex}}\mathrm{exp}\left(-z/{h}_{\mathrm{s}}\right),\end{array}$$

where *S*_{0} is the radioactive heat-production rate extrapolated to the
surface and *h*_{s} is the heat-production length scale
(Turcotte and Schubert, 1982). Surface heat-production rates *S*_{0} can vary from 0.002
to 5.5 µW m^{−3}, depending on lithology (Rybach, 1988), while
*h*_{s} is typically on the order of 10 km.

3 Numerical implementation

Back to toptop
The CVPM modeling system implements the governing equations in 1-D, 2-D, and
3-D Cartesian coordinates (*X*, *X**Z*, *X**Y**Z*), as well as in 1-D radial (*R*)
and 2-D cylindrical (*R**Z*) coordinates. Discretization follows the
control-volume approach (Anderson et al., 1984; Minkowycz et al., 1988; Patankar, 1980) in which
the problem domain is divided into a set of contiguous control volumes
(CVs). Scalars such as temperature *T* and thermal conductivity *k* are
computed at grid points located in the center of the CVs, while the enthalpy
fluxes *J* are computed at control-volume interfaces
(Fig. 7). Development of the CVPM permafrost model begins by
integrating the two conservation equations
(Eqs. 1–2) over a time step Δ*t*. With
velocity ** v**≃0, the conservation equations become

$$\begin{array}{ll}\text{(30)}& {\displaystyle}& {\displaystyle}\underset{V}{\int}\left({\mathit{\rho}}^{n+\mathrm{1}}-{\mathit{\rho}}^{n}\right)\mathrm{d}V\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\mathrm{0}{\displaystyle}& {\displaystyle}\underset{V}{\int}\left[(\mathit{\rho}H{)}^{n+\mathrm{1}}-(\mathit{\rho}H{)}^{n}\right]\mathrm{d}V\\ \text{(31)}& {\displaystyle}& {\displaystyle}=-\underset{{t}^{n}}{\overset{{t}^{n+\mathrm{1}}}{\int}}\phantom{\rule{-0.125em}{0ex}}\phantom{\rule{-0.125em}{0ex}}\underset{A}{\int}\mathit{J}\cdot \mathrm{d}\mathit{A}\phantom{\rule{0.125em}{0ex}}\mathrm{d}t\phantom{\rule{0.125em}{0ex}}+\phantom{\rule{0.125em}{0ex}}\underset{{t}^{n}}{\overset{{t}^{n+\mathrm{1}}}{\int}}\phantom{\rule{-0.125em}{0ex}}\phantom{\rule{-0.125em}{0ex}}\underset{V}{\int}S\phantom{\rule{0.125em}{0ex}}\mathrm{d}V\mathrm{d}t,\end{array}$$

where superscript *n* refers to the time step following standard numerical
nomenclature (e.g., ${t}^{n+\mathrm{1}}={t}^{n}+\mathrm{\Delta}t$). Equation (30)
states the bulk density integrated over any control volume is time invariant.
The conservation of enthalpy equation (Eq. 31) can be put in a
more convenient form by noting

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}(\mathit{\rho}H{)}^{n+\mathrm{1}}-(\mathit{\rho}H{)}^{n}=\left[\mathit{\rho}{c}_{p}+{\mathit{\rho}}_{\mathrm{\ell}}\phantom{\rule{0.125em}{0ex}}\mathrm{\Delta}{H}_{\mathrm{fus}}\phantom{\rule{-0.125em}{0ex}}{\left.{\displaystyle \frac{\partial {\mathit{\varphi}}_{\mathrm{\ell}}}{\partial T}}\right|}^{{T}^{n}}\right]\phantom{\rule{-0.125em}{0ex}}\left({T}^{n+\mathrm{1}}-{T}^{n}\right)\\ \text{(32)}& {\displaystyle}& {\displaystyle}={C}^{n}\left({T}^{n+\mathrm{1}}-{T}^{n}\right).\end{array}$$

This follows from Eq. (3) and a Taylor series expansion. To provide the flexibility of running the model in either explicit, implicit, or fully implicit modes, the net heat flux into a control volume is approximated by a linear combination of values at either end of a time step:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}-\underset{{t}^{n}}{\overset{{t}^{n+\mathrm{1}}}{\int}}\phantom{\rule{-0.125em}{0ex}}\phantom{\rule{-0.125em}{0ex}}\underset{A}{\int}\mathit{J}\cdot \mathrm{d}\mathit{A}\phantom{\rule{0.125em}{0ex}}\mathrm{d}t=-\mathrm{\Delta}t\left[f{\left(\underset{A}{\int}\mathit{J}\cdot \mathrm{d}\mathit{A}\right)}^{n+\mathrm{1}}\right.\\ \text{(33)}& {\displaystyle}& {\displaystyle}\left.+\phantom{\rule{0.125em}{0ex}}(\mathrm{1}-f){\left(\underset{A}{\int}\mathit{J}\cdot \mathrm{d}\mathit{A}\right)}^{n}\right].\end{array}$$

The explicit/implicit weighting factor *f* can take any value between 0 and
1. Following Patankar (1980), the heat fluxes across the *z*_{u}
and *z*_{d} interfaces in the vertical direction (see
Fig. 7) are

$$\begin{array}{}\text{(34)}& {J}_{\mathrm{u}}=-{\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{u}}\left({\displaystyle \frac{{T}_{\mathrm{P}}-{T}_{\mathrm{U}}}{{z}_{\mathrm{P}}-{z}_{\mathrm{U}}}}\right),{J}_{\mathrm{d}}=-{\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{d}}\left({\displaystyle \frac{{T}_{\mathrm{D}}-{T}_{\mathrm{P}}}{{z}_{\mathrm{D}}-{z}_{\mathrm{P}}}}\right),\end{array}$$

where ${\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{u}}$ and ${\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{d}}$ are the “effective” conductivities at the upper and lower interfaces, defined by

$$\begin{array}{}\text{(35)}& {\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{u}}=\phantom{\rule{-0.125em}{0ex}}\left[{\displaystyle \frac{\mathrm{1}}{{\displaystyle \frac{(\mathrm{1}-{\mathit{\epsilon}}_{\mathrm{u}})}{{k}_{\mathrm{U}}}}+{\displaystyle \frac{{\mathit{\epsilon}}_{\mathrm{u}}}{{k}_{\mathrm{P}}}}}}\right],{\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{d}}=\left[{\displaystyle \frac{\mathrm{1}}{{\displaystyle \frac{(\mathrm{1}-{\mathit{\epsilon}}_{\mathrm{d}})}{{k}_{\mathrm{P}}}}+{\displaystyle \frac{{\mathit{\epsilon}}_{\mathrm{d}}}{{k}_{\mathrm{D}}}}}}\right],\end{array}$$

with fractional distances

$$\begin{array}{}\text{(36)}& {\mathit{\epsilon}}_{\mathrm{u}}=\left({\displaystyle \frac{{z}_{\mathrm{P}}-{z}_{\mathrm{u}}}{{z}_{\mathrm{P}}-{z}_{\mathrm{U}}}}\right),{\mathit{\epsilon}}_{\mathrm{d}}=\left({\displaystyle \frac{{z}_{\mathrm{D}}-{z}_{\mathrm{d}}}{{z}_{\mathrm{D}}-{z}_{\mathrm{P}}}}\right).\end{array}$$

Subscripts used here indicate grid point and interface locations. For
example, *T*_{P} is the temperature at grid point P, while
*J*_{u} is the heat flux across the interface located at depth
*z*_{u}. Fluxes across the other interfaces are defined in a
completely analogous way. The use of effective conductivities guarantees that
the heat fluxes exactly balance at an interface between materials with very
different thermal properties (e.g., between a siltstone and an ice lens). The
source-term integral in Eq. (31) is left in a very general
form:

$$\begin{array}{}\text{(37)}& {\mathcal{S}}_{\mathrm{P}}=\mathrm{\Delta}t\underset{V}{\int}S\phantom{\rule{0.125em}{0ex}}\mathrm{d}V.\end{array}$$

Substituting Eqs. (32)–(34) and (37) into Eq. (31), the discrete form of the enthalpy balance for a control volume centered on grid point P can be written as

$$\begin{array}{}\text{(38)}& {a}_{\mathrm{P}}{T}_{\mathrm{P}}^{n+\mathrm{1}}=\sum {a}_{\mathrm{nb}}\phantom{\rule{0.125em}{0ex}}{T}_{\mathrm{nb}}^{n+\mathrm{1}}+\sum {a}_{\mathrm{nb}}^{\prime}\phantom{\rule{0.125em}{0ex}}{T}_{\mathrm{nb}}^{n}+b,\end{array}$$

where the sums are taken over the values at the neighboring (nb) grid points
(W, E, N, S, U, D). Putting all the geometric information into factors *A*_{x}
and *V*_{P} (Table 3), the discretization coefficients for
the internal control volumes are

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{a}_{\mathrm{W}}=f\mathrm{\Delta}t\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{w}}{\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{w}},{a}_{\mathrm{E}}=f\mathrm{\Delta}t\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{e}}{\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{e}}\\ {\displaystyle}& {\displaystyle}{a}_{\mathrm{S}}=f\mathrm{\Delta}t\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{s}}{\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{s}},{a}_{\mathrm{N}}=f\mathrm{\Delta}t\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{n}}{\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{n}}\\ \text{(39)}& {\displaystyle}& {\displaystyle}{a}_{\mathrm{U}}=f\mathrm{\Delta}t\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{u}}{\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{u}},{a}_{\mathrm{D}}=f\mathrm{\Delta}t\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{d}}{\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{d}}{\displaystyle}& {\displaystyle}{a}_{\mathrm{P}}={V}_{\mathrm{P}}\phantom{\rule{0.125em}{0ex}}{C}_{\mathrm{P}}^{n}\phantom{\rule{0.125em}{0ex}}+\phantom{\rule{0.125em}{0ex}}\sum {a}_{\mathrm{nb}},{a}_{\mathrm{P}}^{\prime}={V}_{\mathrm{P}}\phantom{\rule{0.125em}{0ex}}{C}_{\mathrm{P}}^{n}\phantom{\rule{0.125em}{0ex}}-\phantom{\rule{0.125em}{0ex}}\sum {a}_{\mathrm{nb}}^{\prime}\\ {\displaystyle}& {\displaystyle}b={\mathcal{S}}_{\mathrm{P}}\phantom{\rule{0.125em}{0ex}}.\end{array}$$

The primed counterparts of *a*_{W}, *a*_{E}, *a*_{S},
*a*_{N}, *a*_{U}, and *a*_{D} are identical, except that *f*
is replaced by (1−*f*).

Consideration of the enthalpy balance shows that the discretization
coefficients are slightly different for CVs adjacent to the boundaries of the
problem domain. CVPM can be “forced” at the boundaries using either a
prescribed temperature (Dirichlet) or heat-flux (Neumann) boundary condition
(a convective boundary condition will be introduced in a later version). For
a control volume adjacent to a boundary with a Dirichlet boundary condition,
a factor of (4∕3) is introduced into the discretization coefficients
(Eq. 39) associated with the boundary and the opposing
interface. When the heat flux is prescribed on a boundary (Neumann BC), the
coefficients associated with the boundary are zero and the specified heat
flux appears in discretization coefficient *b* (Table 4).
Boundary conditions along the edges of the problem domain are allowed to vary
both spatially and temporally in CVPM.

To complete the setup of the discretization coefficients, the material
properties must be specified at every grid point within the model domain.
Parameters controlling these properties include material type, mean density
of matrix particles *ρ*_{m}, mineral-grain thermal conductivity
${k}_{\mathrm{m}}^{\circ}$ at 273.15 K, mineral-grain specific heat
${c}_{p}^{\circ}$ at 293.15 K, heat-production rate extrapolated to the
surface *S*_{0}, heat-production length scale *h*_{s}, porosity
extrapolated to the surface *ϕ*_{0}, critical porosity *ϕ*_{c},
compaction length scale *h*_{c}, degree of pore saturation *S*_{r},
dominant solute type, solute mole fraction in the fully melted system
${x}_{\mathrm{s}}^{\star}$, interfacial melting parameter *λ*, mean diameter
of larger mode pores *d*_{1}, mean diameter of smaller mode pores *d*_{2}, and
the ratio of the number density of small pores to large pores (*n*_{2}∕*n*_{1}).
The material “type” specifies which governing equations to utilize when
finding the heat capacity and thermal conductivity (Sect. 2);
types include pure ice and a variety of rocks, soils, organic-rich materials,
fluids, and metals. Only a subset of these properties needs to be specified
for non-porous layers (e.g., an ice lens or a borehole casing). Examples of
the required thermophysical parameters are provided in
Sect. 4 and in the CVPM version 1.1 modeling system user's
guide (Clow, 2018).

Any temperature field can be used to set the initial temperature condition, including a user-supplied field (e.g., a measured temperature field), a CVPM-determined steady-state field consistent with the boundary conditions and material properties, or a field generated by a previous CVPM modeling experiment.

With the initial condition, boundary conditions, and discretization
coefficients specified, the enthalpy-balance equation (Eq. 38) is
solved recursively at each time step using the tridiagonal matrix algorithm
(TDMA) for 1-D models and the line-by-line method with TDMA for 2-D and 3-D
models. Given their temperature sensitivities, the thermophysical properties
(*ϕ*_{ℓ}, *ϕ*_{i}, *C*, *k*, etc.) are updated at every time
step. In order for the numerical scheme to remain unconditionally stable, all
of the discretization coefficients must be non-negative. This consideration
leads to the numerical stability condition:

$$\begin{array}{}\text{(40)}& \mathrm{\Delta}t\phantom{\rule{0.25em}{0ex}}<\phantom{\rule{0.25em}{0ex}}{\displaystyle \frac{{V}_{\mathrm{P}}\phantom{\rule{0.125em}{0ex}}{C}_{\mathrm{P}}^{n}}{{\displaystyle}(\mathrm{1}-f)\sum {A}_{\mathrm{nb}}{\stackrel{\mathrm{\u0303}}{k}}_{\mathrm{nb}}}}\phantom{\rule{0.125em}{0ex}},\end{array}$$

which must be satisfied in all the CVs at each time step for the scheme to remain unconditionally stable.

Model verification was conducted in two phases. In the first phase, the general
model structure and numerical implementation were tested by comparing model
results with analytic solutions for a series of simple heat-transfer problems
without phase change. Test problems included steady-state and transient
boundary conditions, homogeneous and composite media with fixed thermal
properties, materials whose thermal properties vary linearly with
temperature, and materials with and without radiogenic heating. In all cases,
maximum model errors *ϵ* are on the order of 0.1 mK or less
under the test conditions (spatial resolution, time step, etc.). For most
cases, max(*ϵ*) ranges 1 µK to 0.01 pK. Since
analytic solutions are unavailable for simultaneously testing all of the
model physics, the second testing phase consisted of separately testing each
physics module to guarantee it properly simulates the appropriate governing
equations (Sect. 2).

4 Example simulations with a sedimentary sequence

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To illustrate the capabilities of the CVPM model, several examples are provided in this section based on the response of a thick vertical sequence of sedimentary rocks to changing boundary conditions. The sequence consists of flat-lying mudrock, carbonate, and sandstone units of various thicknesses (Fig. 8). Values of the parameters controlling the thermophysical properties (Sect. 2) are listed in Table 5. Heat-production rates are from Rybach (1988), while the compaction length scale is loosely derived from values found for a partially exhumed basin on the Arctic slope of Alaska (Burns et al., 2005). Pore spaces are assumed to be fully saturated with water throughout the geologic section [${S}_{r}\equiv (\mathrm{1}-{\mathit{\varphi}}_{\mathrm{a}}/\mathit{\varphi})=\mathrm{1}$]. Sodium chloride, present at relatively low levels when the sediments are completely thawed (${x}_{\mathrm{s}}^{\star}\simeq \mathrm{0.003}$), is the dominant pore-water solute.

The first simulation explores the response of the sedimentary sequence to
surface-temperature changes over the last ice-age cycle. The upper boundary
condition is based on the surface-temperature history determined for the
Greenland Ice Sheet during the Holocene and Wisconsin glacial period by
Cuffey and Clow (1997) and the Eemian interglacial by the NEEM Community Members (2013). To
construct the upper BC for the permafrost simulation, the Greenlandic
temperature history was rescaled and shifted to yield an 8 K warming
between the Last Glacial Maximum (LGM) and the early Holocene and a mean
surface temperature of −11 ^{∘}C during the 1800s
(Fig. 9). For the most recent period, upper boundary
temperatures warm 2.5 K between the mid-1800s and 1980 and an
additional 2.5 K by the present time, similar to the record for the
Arctic Coastal Plain of Alaska (Clow, 2017). The temperature history was
replicated back several ice-age cycles to allow for model spin-up. At the
lower boundary (depth 2 km), the conductive heat-flux *q*_{b}
was fixed at 60 mW m^{−2}, slightly above the continental average.
To initialize the model, the subsurface-temperature profile was assumed to be
in a steady-state condition at 255 ka with the mean surface
temperature over an ice-age cycle. For the spatial grid, 2 m thick
control volumes were used above the 550 m depth. Below this, the grid
spacing Δ*z* increased progressively to 50 m near the lower
boundary. To explore the response of permafrost well below the surface, a
20-year computational time step (Δ*t*) was deemed sufficient.

With the above setup, CVPM was run forward in its 1-D vertical mode from
255 ka to the present. During the last ice-age cycle, the base of
permafrost *P*_{d} (defined by the 0 ^{∘}C isotherm) is found to
vary by 90 m in this example from 435 to 525 m
(Fig. 9). Of greater physical significance is the maximum
depth where interstitial ice is present in permafrost. Due to the
freezing-point depression caused by interfacial, curvature, pressure, and
solute effects, the base of ice-bearing permafrost (B-IBPF) is located
20–27 m above the *P*_{d} throughout the simulation. During
the most recent ice-age cycle, the B-IBPF and *P*_{d} both reached
their greatest depths at ∼14 ka, a delay of 10 kyr from
the Last Glacial Maximum. Since then, these interfaces have been steadily
rising. With the conditions of this simulation, the B-IBPF is currently
located at 431 m in a silty claystone, about 22 m below the
shallowest depth projected to have occurred following the last interglacial,
while the base of permafrost *P*_{d} is currently at 467 m in
the underlying sandstone unit. Both interfaces are predicted to be rising
about 1 cm yr^{−1} at present, a rate that may be detectable with a
carefully designed experiment.

As the simulation confirms, the volume fraction of ice *ϕ*_{i}
depends strongly on lithology (Fig. 10). This leads to zones
with relatively high ice contents in coarse-grained sediments as is often
observed in electrical resistivity geophysical logs
(Hnatiuk and Randall, 1977; Osterkamp and Payne, 1981). An interesting facet of this behavior is that
a pocket of ice-rich sandstone can occur below a fine-grained unit that has
little or no ice (e.g., the silty claystone above the sandstone at
450 m in Fig. 10). Except near the B-IBPF, the ice
content in coarse-grained materials appears to be relatively stable. In
contrast, the volume fractions of ice *ϕ*_{i} and unfrozen water
*ϕ*_{ℓ} are much more dynamic in fine-grained materials over ice-age
cycles due to their greater temperature sensitivity. In the upper two-thirds
of the permafrost zone, up to 15 % of the water within silty claystones and
limestones converts between ice and unfrozen water over ice-age cycles; the
percentages are much higher in the lower third of the permafrost zone. Due to
the phase change of water, a high heat capacity zone occurs just above the
B-IBPF (Fig. 11). This zone, which tracks the B-IBPF over time,
is roughly 100 m thick. Pockets of elevated heat capacity also occur
in fine-grained materials closer to the surface, especially during periods
affected by the warm interglacials. Thermal conductivity variations over an
ice-age cycle are also primarily driven by the melting and refreezing of pore
ice. Hence, conductivity variations are greatest near the B-IBPF, where
conductivity changes of ±15 % occur (Fig. 12). In the
upper two-thirds of the permafrost zone, thermal conductivity variations are
greater in the fine-grained materials, at least on a percentage basis.

We now return to the state of the sedimentary sequence simulated in
Sect. 4.1 during the year 1980. By this time, temperatures
in the upper 100 m of the sedimentary sequence have warmed in
response to the 2.5 K surface warming since the termination of the
Little Ice Age (∼1850). At greater depths, temperatures still reflect
conditions earlier during the Holocene and the Wisconsin glacial period. With
this initial condition, we consider the drilling of a 3 km deep
borehole through the example sedimentary sequence (Fig. 8,
Table 5) over a 60-day period. Drilling fluids pumped into
the proposed 30 cm diameter hole at 30 ^{∘}C interact thermally
with the drill pipe and surrounding rock as they circulate to the bottom of
the hole and back to the surface. As a result of the drilling processes,
temperatures within the borehole warm within 700 m of the surface.
The degree of warming depends on both depth and time as the drill bit
advances into the warmer rocks below (Clow, 2015).
Figure 13 shows the evolution of temperature changes (Δ*T*_{a}) at the borehole wall during drilling based on the
Szarka and Bobok (2012) wellbore model. This thermal drilling disturbance, when
added to the initial 1980 temperature field, establishes the boundary
condition at the borehole wall over the 60-day drilling period. To
complete the CVPM setup for a cylindrical 2-D simulation of the permafrost
surrounding the borehole, the problem domain was extended radially far enough
(40 m) from the hole that the radial heat flux at the outer boundary
could be set to zero. The heat flux on the lower boundary was again
60 mW m^{−2}, while the upper boundary remained at its initial 1980
temperature (−8.5 ^{∘}C). The vertical grid was identical to that
used in Sect. 4.1, while the radial grid spacing increased
progressively from 2.5 cm near the borehole to 2 m near the
outer boundary. To resolve the rapid temperature changes that occur near the
advancing drill bit, the computational time step was set to 0.2 days.
The initial condition was provided by the values of all the state variables
from the previous example (Sect. 4.1) during 1980.

Running CVPM with the described initial and boundary conditions, the drilling
disturbance is found to be great enough in this simulation to melt all of the
permafrost ice within 1–2 m of the well by the time the borehole is
completed on day 60 (Fig. 14). In this case, borehole
electrical resistivity measurements used to detect ice in permafrost must be
able to penetrate at least 1.5 m of rock to successfully detect ice.
The exact location of the melting front is controlled partially by lithology,
with it advancing further from the borehole in fine-grained high-conductivity
sediments (e.g., limestones) than in coarser grained low-conductivity
mudrocks such as siltstones. Sediment texture is a factor because it affects
the volumetric ice content of a material in its undisturbed state. Although
the thermal disturbance due to drilling is greatest inboard of the melting
front, a substantial disturbance also occurs beyond the front, particularly
in the upper couple hundred meters of permafrost, where the initial
undisturbed temperatures were −7 to −9 ^{∘}C
(Fig. 15). Outboard of the melting front, the drilling
disturbance extends further into the higher conductivity limestone and
sandstone units than in the low-conductivity mudrocks. Because of the effect
of temperature on the thermal conductivity of minerals, ice, and water, and
because of the conversion of ice to liquid water, the bulk thermal
conductivity drops about 30 % in the sedimentary units near the wellbore
during drilling (Fig. 16). Attempts to infer the
thermophysical properties of the sedimentary units from borehole temperature
measurements using geophysical inverse methods must carefully consider these
changes (Nicolsky et al., 2007).

Shallow lakes are ubiquitous on the Arctic Coastal Plain. In thermokarst areas, these lakes are constantly in transition, shrinking, enlarging, draining, and filling new depressions in response to changing temperatures and stream flows. The seasonal ice that forms on these lakes is categorized as “bedfast” ice if it freezes solid to the bottom of the lake, “floating” ice if some liquid remains beneath the ice throughout the winter, and “intermittent” if it is bedfast some years and floating during others. Whether a lake is a bedfast-ice lake or a floating-ice lake depends on whether the maximum seasonal ice-cover thickness (${Z}_{\mathrm{ice}}^{\mathrm{max}}$) exceeds the depth of the lake. During the 1970s and 1980s, ${Z}_{\mathrm{ice}}^{\mathrm{max}}$ for lakes along the Beaufort Sea coast of Alaska was 2.0±0.2 m (Arp et al., 2012; Weeks et al., 1981). By 2010, the maximum seasonal ice thickness had decreased to about 1.5 m due to the warming climate in the region over the last few decades (Bieniek et al., 2014; Wang et al., 2017). Thus, 1.5 m deep lakes that would have been bedfast-ice lakes during the 1970s and 1980s would have transitioned to intermittent-ice lakes by 2010. Arp et al. (2012) recently provided lake–bed temperature data for seasonally ice-covered lakes near the Beaufort Sea coast. Based on these data, lakes whose depth is 0.5–1.0 m less than ${Z}_{\mathrm{ice}}^{\mathrm{max}}$ have a mean-annual temperature ∼4 K warmer than the surrounding tundra, while intermittent-ice lakes are about 9 K warmer.

Here, we briefly explore the permafrost response over a 35-year period
to the instantaneous creation of a 200 m wide lake on the Arctic
Coastal Plain of Alaska during 1980. The lake is assumed to be 1.5 m
deep with a 100 m wide shallow (1.0 m deep) shelf and is
assumed to occur in the same geologic terrain as in the previous two
examples. Thus, the lake lies on a 50 m thick silty claystone unit
which overlies deeper sedimentary rocks (Fig. 8,
Table 5). To investigate the permafrost response, CVPM was used to
simulate conditions along a 2-D vertical cross-section passing beneath the
lake and surrounding tundra. Initial conditions were identical to those used
in the previous example (Sect. 4.2). The upper boundary was
set at the 1.5 m depth, i.e., at the bottom of the deeper portion of
the lake. Temperatures on this boundary, which were used to force the model,
varied depending on location. Beneath the tundra, temperatures were assumed
to warm from the initial 1980 surface temperature (−8.5 ^{∘}C) at
0.75 K decade^{−1}, similar to recent trends observed along
the Beaufort coast of Alaska (Clow, 2017; Wang et al., 2017). With these temperatures,
the seasonal ice cover is projected to have been bedfast across the entire
lake when it was first created and to have transitioned to intermittent over
the deeper lake section towards the end of the simulation. Thus, lake–bed
temperatures beneath the deeper portion of the lake, assumed to have
initially been 4 K warmer than the surrounding tundra, were prescribed to
have warmed until they were 9 K warmer than the tundra by 2015. Temperatures
beneath the shelf were prescribed to be 4 K warmer than the tundra
throughout the simulation. Temperatures beneath the tundra, shelf, and deeper
portion of the lake provided the upper BC for the simulation. A heat-flux BC
was prescribed on the lower boundary (*q*_{b}=60 mW m^{−2}), taken to occur at the 300 m depth in this
example. The problem domain was extended laterally far enough beyond the lake
that the heat flux across the outer boundaries could be set to zero. Vertical
grid spacing was Δ*z*=0.1 m in the upper silty claystone
unit. Beneath this, Δ*z* increased progressively to 10 m near
the lower boundary. Horizontal grid spacing was 1 m beneath the lake
and 5 m beneath the surrounding tundra. Although the prescribed upper
BC does not include seasonal effects, the computational time step Δ*t*
was set to 0.1 years to satisfy the numerical stability condition
(Eq. 40).

Running CVPM in its 2-D Cartesian mode for 35 years with the described
boundary conditions, temperatures beneath the deeper portion of the lake are
found to become warm enough to melt all of the pore ice at the lake–bed
interface 19 years after the lake is created (Fig. 17).
Thereafter, the melting front propagates downward in the upper silty
claystone unit at ∼19 cm yr^{−1}, creating a thaw bulb in
its wake (Fig. 18). Lateral migration of the melting front
is much more modest: ∼4 cm yr^{−1} where the deep
section adjoins the tundra and ∼6 cm yr^{−1} where it
meets the shallow shelf. By the end of the simulation, a thaw bulb has not
yet begun to form beneath the shelf. There are two ramifications of the
growing thaw bulb beneath the deeper section of the lake. (1) Fine-grained
materials such as silty clay lose much of their mechanical strength once they
thaw. In this state, the sides of the lake are much more vulnerable to
erosion, which may lead to eventual drainage of the lake. (2) Old carbon
stocks stored in the previously frozen permafrost are likely to decompose in
the anaerobic thaw bulb and contribute greenhouse gases to the atmosphere
(Anthony et al., 2016).

5 Summary and conclusions

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This paper presents the governing equations and numerical methods underlying
the Control Volume Permafrost Model v1.1 which was designed to relax several
of the limitations imposed by previous models. CVPM implements the nonlinear
heat-transfer equations in 1-D, 2-D, and 3-D Cartesian coordinates, as well
as in 1-D radial and 2-D cylindrical coordinates. To accommodate a diversity
of geologic settings, a variety of materials can be specified within the
modeling domain, including organic-rich materials, sedimentary rocks and
soils, igneous and metamorphic rocks, ice bodies, borehole fluids, and other
engineering materials. Porous materials are treated as a matrix of mineral
and organic particles with pores spaces filled with liquid water, ice, and
air. Functions describing the temperature dependence of the specific heat and
thermal conductivity are built into CVPM for a wide variety of rocks and
minerals, liquid water, ice, air, and other substances. For porous materials,
the bulk thermal conductivity is found using a random two-phase (matrix
particles, pores) relationship, while the conductivity of the pores themselves
is found using a three-phase (liquid water, ice, air) mixing model. This scheme
allows the bulk thermal conductivity to be determined for a wide range of
porosities, water saturations ranging 0 %–100 %, and different planetary
atmospheres. In addition to the lattice-vibration term (*ρ**c*_{p}), the
volumetric heat capacity *C* depends on a latent-heat term proportional to
the change in liquid water content with temperature ($\partial {\mathit{\varphi}}_{\mathrm{\ell}}/\partial T$). At temperatures below 0 ^{∘}C, the unfrozen
water content is found using relationships from condensed matter physics that
utilize physical quantities (i.e., particle size) rather than non-physical
empirical coefficients requiring calibration. Solute and pore pressure
effects are included in the unfrozen water equations. Due to the insolubility
of most solutes in ice, impurities become strongly enriched in the liquid
pore water as permafrost freezes. To allow for the non-ideal behavior that
occurs at high solute concentrations, the water activity *a*_{w} is
used to find the effect of solutes on the bulk freezing temperature of water
*T*_{f}. With this approach, solute concentrations up to the eutectic
point are allowed. Pore pressure effects on *T*_{f} are found in CVPM
using either hydrostatic or lithostatic equations, whichever is more
appropriate geologically. A radiogenic heat-production term is also included
to allow simulations to extend into deep permafrost and underlying bedrock.
For the current version of CVPM, liquid water velocities are assumed to be
small enough that the associated advective heat flux is negligible compared
to the diffusive heat flux.

Numerical implementation of the governing equations is accomplished using the control-volume approach, allowing enthalpy fluxes to be exactly balanced at control-volume interfaces (e.g., at the interfaces between ice lenses, sedimentary units, bedrock, or a borehole casing). This approach was chosen because the expressions tend to be more accurate than with other methods near boundaries and where strong thermal or physical-property contrasts occur. Very large thermal-property contrasts generally occur near the water freezing point in permafrost. Despite the magnitude of the contrasts and the fact that the freezing front typically migrates over time, the numerical scheme used in CVPM remains stable as long as the stability criterium (Eq. 40) is satisfied. CVPM can be run in either explicit, implicit, or fully implicit modes.

CVPM has been designed for a wide range of scientific and engineering applications, and as an educational tool. The model is “forced” by changes in the boundary conditions at the edges of the problem domain. These conditions include user-prescribed temperatures and/or heat fluxes that are allowed to vary both spatially and temporally along the edges. The model is suitable for use at spatial scales ranging from centimeters to hundreds of kilometers and at timescales ranging from seconds to thousands of years. CVPM can be used over a broad range of depth, temperature, porosity, water saturation, and solute conditions on either the Earth or Mars. Through its modular design, CVPM can act as a stand-alone model or the physics package of a geophysical inverse scheme, or serve as a component within a larger Earth modeling system that may include vegetation, surface water, snowpack, atmospheric, or other models of varying complexity.

One of the goals of CVPM was to eliminate the empirical equations typically
used to predict the unfrozen water content at temperatures below
0 ^{∘}C, replacing them with condensed matter relationships containing
quantities that both have a physical meaning and can be determined using
relatively simple laboratory techniques or from geophysical logs. The
physical quantities in the resulting equations tend to occur within
reasonably narrow limits for the material categories defined in CVPM. Thus, if
measurements of the physical quantities are unavailable, it may be possible
to estimate the behavior of permafrost in a region from a geologic
description of the area.

Code availability

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Code availability.

The CVPM source code, test cases, examples, and a user guide are publicly available at the Community Surface Dynamics Modeling System repository at https://csdms.colorado.edu/wiki/Model:CVPM (last access: 3 December 2018) and in Clow (2018). CVPM v1.1 is implemented in the MATLAB programming language and is distributed under the GNU General Public License v3.0.

Competing interests

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Competing interests.

The author declares that he has no conflict of interest.

Acknowledgements

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Acknowledgements.

This work was supported by the U.S. Geological Survey through a grant from
the Climate and Land Use Change Program. Any use of trade, firm, or product
names is for descriptive purposes only and does not imply endorsement by the
U.S. Government. The author thanks the referees for their careful reviews
and constructive suggestions which helped to improve the
manuscript.

Edited by: David Lawrence

Reviewed by: two anonymous referees

References

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Short summary

CVPM is a modular heat-transfer modeling system designed for scientific and engineering studies in permafrost terrain, and as an educational tool. CVPM implements the heat-transfer equations in both Cartesian and cylindrical coordinates. To accommodate a diversity of geologic settings, a variety of materials can be specified within the model domain. CVPM can be used over a broad range of depth, temperature, porosity, water saturation, and solute conditions on either Earth or Mars.

CVPM is a modular heat-transfer modeling system designed for scientific and engineering studies...

Geoscientific Model Development

An interactive open-access journal of the European Geosciences Union