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

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

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

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

The basis for the CVPM model is the conservation of mass and enthalpy over
time within any finite volume

In permafrost, the enthalpy at temperature

Below 500

Unlike most materials, experimental data for liquid water show an anomalous
increase in specific heat (

Coefficients

For water ice (

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

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,

Sensitivity of the volume fraction of liquid water

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

The undercooling

When solutes remain dilute, the freezing-point depression due to impurities
can be approximated using simple relationships such as Blagden's law

Figure

Volume fraction of liquid water

As previously mentioned, the interfacial melting parameter is best determined
experimentally for natural Earth materials. Inversion of unfrozen water data
(shown in Fig.

Given that permafrost occurs at depths in excess of 1

Sensitivity of the volume fraction of liquid water

Several mixing models are available for estimating the bulk thermal
conductivity of multi-component systems. Of these, CVPM uses the

For matrix minerals, the thermal conductivity depends primarily on the
temperature and mineral composition. Using thermal conductivity data obtained
by

Coefficients

To find the thermal conductivity of the pores (

Variation of the pore thermal conductivity

For the thermal conductivity of liquid water

Variation of the thermal conductivity with temperature for liquid
water, ice, air (terrestrial atmosphere),

Experimental data for the thermal conductivity of ice

For the terrestrial environment, the thermal conductivity of air

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

In sedimentary basins, overburden pressure causes the porosity

CVPM assumes the enthalpy-production rate

Schematic showing the nomenclature associated with a control volume
centered on grid point P for 2-D Cartesian (

The CVPM modeling system implements the governing equations in 1-D, 2-D, and
3-D Cartesian coordinates (

Substituting Eqs. (

Geometric factors

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

Discretization coefficient

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

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.

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

Vertical sequence of sedimentary rocks used for the example
simulations in Sect.

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.

Thermophysical parameters for the sedimentary rock units in
Fig.

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

With the above setup, CVPM was run forward in its 1-D vertical mode from
255

Upper boundary condition (

As the simulation confirms, the volume fraction of ice

Ice content

Volumetric heat capacity

Bulk thermal conductivity

We now return to the state of the sedimentary sequence simulated in
Sect.

Temperature change

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

Volumetric ice content

Temperatures in the sedimentary sequence (Fig.

Bulk thermal conductivity

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
(

Here, we briefly explore the permafrost response over a 35-

Simulated permafrost temperatures over a 35-

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.

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 (

Volume fraction of ice (

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.

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

The CVPM source code, test cases, examples, and a user
guide are publicly available at the Community Surface Dynamics Modeling
System repository at

The author declares that he has no conflict of interest.

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