Heat and water movement in variably saturated freezing soils is a strongly coupled phenomenon. The coupling is a result of the effects of sub-zero temperature on soil water potential, heat carried by water moving under pressure gradients, and dependency of soil thermal and hydraulic properties on soil water content. This study presents a one-dimensional cellular automata (direct solving) model to simulate coupled heat and water transport with phase change in variably saturated soils. The model is based on first-order mass and energy conservation principles. The water and energy fluxes are calculated using first-order empirical forms of Buckingham–Darcy's law and Fourier's heat law respectively. The liquid–ice phase change is handled by integrating along an experimentally determined soil freezing curve (unfrozen water content and temperature relationship) obviating the use of the apparent heat capacity term. This approach highlights a further subtle form of coupling in which heat carried by water perturbs the water content–temperature equilibrium and exchange energy flux is used to maintain the equilibrium rather than affect the temperature change. The model is successfully tested against analytical and experimental solutions. Setting up a highly non-linear coupled soil physics problem with a physically based approach provides intuitive insights into an otherwise complex phenomenon.

Variably saturated soils in northern latitudes undergo repeated freeze–thaw
cycles. Freezing reduces soil water potential considerably because soil
retains unfrozen water (Dash et al., 1995). The resulting steep hydraulic
gradients move considerable amounts of water upward from deeper warmer soil
layers which accumulates behind the freezing front. The resulting
redistribution of water alters soil thermal and hydraulic properties, and
transports heat from one soil zone to another. As water freezes into ice, the
latent heat maintains soil temperatures close to 0

Although the coupling of heat and water movement in variably saturated freezing soils is complex, fundamental laws of heat and water movement coupled with principles of energy and mass conservation are able to explain the physics to a large extent. There is a paradigm shift in modelling of water movement in variably saturated soils using physically based approaches. For example, HydroGeoSphere and Parflow (Brunner and Simmons, 2012; Kollet and Maxwell, 2006) are examples of codes that explicitly use Richard's equation to model subsurface flow. Thus, the use of derived terms such as specific yield is not required. Mendicino et al. (2006) reported a three-dimensional (3-D) CA (cellular automata; direct solving) model to simulate moisture transfer in the unsaturated zone. Cervarolo et al. (2010) extended the application of this CA model by coupling it with a surface–vegetation–atmosphere-transfer scheme to simulate water and energy flow dynamics. Direct solving allows for unstructured grids while describing the coupled processes based on first-order equations. Use of discrete first-order formulations allow one to relax the smoothness requirements for the numerical solutions being sought. This has advantages, particularly in large-scale models, wherein use of relatively coarse spatial discretization may be feasible. Therefore, it is important to expand the application of direct solving to further complicated unsaturated soil processes.

This study presents a coupled CA model to simulate heat and water transfer in variably saturated freezing soils. The system is modelled in terms of the empirically observed heat and mass balance equations (Fourier's heat law and Buckingham–Darcy equation) and using energy and mass conservation principles. The liquid–ice phase change is handled with a total energy balance including sensible and latent heat components. In a two-step approach similar to that of Engelmark and Svensson (1993), the phase change is brought about by the residual energy after sensible heat removal has dropped the temperature of the system below freezing point. Knowing the amount of water that can freeze, the change in soil temperature is then modelled by integrating along the soil freezing curve. To our knowledge, coupled cellular automata have not yet been used to explore simultaneous heat and water transport in frozen variably saturated porous media. The model was validated against the analytical solutions of (1) the heat conduction problem (Churchill, 1972), (2) steady state convective and conductive heat transport in unfrozen soils (Stallman, 1965), (3) unilateral freezing of a semi-infinite region (Lunardini, 1985), and (4) the experimental results of freezing-induced water redistribution in soils (Mizoguchi, 1990).

Cellular automata were first described by von Neumann in 1948 (see von Neumann and Burks, 1966). The CA describe the global evolution of a system in space and time based on a predefined set of local rules (transition rules). Cellular automata are able to capture the essential features of complex self-organizing cooperative behaviour observed in real systems (Ilachinski, 2001). The basic premise involved in CA modelling of natural systems is the assumption that any heterogeneity in the material properties of a physical system is scale dependent and there exists a length scale for any system at which material properties become homogeneous (Hutt and Neff, 2001). This length scale characterizes the construction of the spatial grid cells (elementary cells) or units of the system. There is no restriction on the shape or size of the cell with the only requirement being internal homogeneity in material properties in each cell. One can then recreate the spatial description of the entire system by simple repetitions of the elementary cells. The local transition rules are results of empirical observations and are not dependant on the scale of homogeneity in space and time. The basic assumption in traditional differential equation solutions is of continuity in space and time. The discretization in models based on traditional numerical methods needs to be over grid spacing much smaller than the smallest length scale of the heterogeneous properties making solutions computationally very expensive. The CA approach is not limited by this requirement and is better suited to simulate spatially large systems at any resolution, if the homogeneity criteria at elementary cell level are satisfied (Ilachinski, 2001; Parsons and Fonstad, 2007). In fact, in many highly non-linear physical systems such as those describing critical phase transitions in thermodynamics and the statistical mechanical theory of ferromagnetism, discrete schemes such as cellular automata are the only simulation procedures (Hoekstra et al., 2010).

On the contrary, explicit schemes like CA are not unconditionally convergent and hence given a fixed space discretization, the time discretization cannot be arbitrarily chosen. Another limitation of the CA approach was thought to be the need for synchronous updating of all cells for accurate simulations. However, CA models can be made asynchronous and can be more robust and error resistant than a synchronous equivalent (Hoekstra et al., 2010).

The following section (Sect. 2.1) describes a 1-D CA in simplified, but precise mathematical terms. It is then explained with an example of heat flow (without phase change) in a hypothetical soil column subjected to a time varying temperature boundary condition.

Let

Let us consider the CA simulation of heat flow in a soil column of length

One-dimensional cellular automata grids based on the von
Neumann neighbourhood concept. How many neighbours (grey cells) interact
with an active cell (black) is controlled by the indicial radius (

The meaning of the terms used in the mathematical description of CA can now
be explained with respect to the heat flow simulation for the hypothetical
soil column:

The algorithm developed for this study simultaneously solves the heat and
water mass conservation in the same time step. The implementation is based on
the assumption of nearest neighbour interactions, i.e.

The mass conservation equation in 1-D can be written as

Buckingham–Darcy's equation is used to describe the flow of water under
hydraulic head gradients wherein it is recognized that the soil matric
potential (

Figure 2 shows a flow chart describing the algorithm driving the coupled CA
code. The code was written in MATLAB^{®}. Let
the superscript

Flow chart describing the algorithm driving the coupled CA code. Subscripts TC, HC and FT refer to changes in physical quantities due to thermal conduction, hydraulic conduction and freeze–thaw processes respectively. Hydraulic conduction and thermal conduction are two different CA codes coupled through updating of volumetric heat capacity and the freeze–thaw module to simulate the simultaneous heat and water movement in soils. Corresponding equations or sections containing module description are shown in red text in squared brackets.

The energy balance module is explained using an example of a system wherein
the soil temperature is dropping and phase change may take place if cell
temperature drops below the freezing point of pure water (

Graphical description of the phase change approach used in
this study. The curve is a soil freezing curve for a hypothetical soil. The
change in water content (d

If

The CA scheme described here is not unconditionally convergent. Hence, the size of the time step cannot be arbitrarily chosen. In our implementation of the CA model, adaptive time stepping has been achieved following the convergence analysis reported in Appendix A.

The ability of the CA model to simulate pure conduction under hydrostatic
conditions was tested by comparison to the analytical solution of
one-dimensional heat conduction in a finite domain given by Churchill (1972).
A soil column with total length (

Simulation parameters for heat conduction problems. Analytical solution for this example is given by Eq. (13) as per Churchill (1972).

Stallman's analytical solution (1965) to the subsurface temperature profile in a semi-infinite porous medium in response to a sinusoidal surface temperature provides a test of the CA model's ability to simulate one-dimensional heat convection and conduction in response to a time varying Dirichlet boundary.

Given the temperature variation at the ground surface described by

The parameters used in analytical examples for Stallman (1965), and the CA code are given in Table 2. The coupled CA code is able to simulate the temperature evolution due to conductive and convective heat transfer as seen from the excellent agreement with the analytical solution (Fig. 5).

Lunardini (1985) presented an exact analytical solution for propagation of
subfreezing temperatures in a semi-infinite, initially unfrozen soil column
with time

Simulation parameters for predicting the subsurface temperature profile in a semi-infinite porous medium in response to a sinusoidal surface temperature. The analytical solution to this one-dimensional heat convection and conduction problem in response to a time varying Dirichlet boundary is given by Eqs. (14)–(17) as per Stallman (1965).

Comparison between the analytical solution given by Churchill (1972) and coupled cellular automata model simulation for a perfectly thermally insulated 4 m long soil column. Lines represent the analytical solution and symbols represent the CA solution for time points as shown in the legend. The initial temperature distribution is shown on the right.

Comparison between the analytical (Stallman, 1965) and coupled
CA model steady state solutions for conductive and convective heat transfer.
The soil column in this example is infinitely long, initially at 20

Simulation parameters for predicting the subsurface temperature profile with phase change in a three-zone semi-infinite porous medium. The analytical solution to this one-dimensional problem with sensible and latent heat zones is given by Eqs. (18)–(25) as per Lundardini (1985).

Comparison between analytical solution of heat flow with
phase change (Lunardini, 1985) and coupled CA model solutions for heat
transfer with phase change. Lunardini's (1985) solution is shown and compared
with the CA simulation for two cases:

Comparison of total water content (ice

Hansson et al. (2004) describe laboratory experiments of Mizoguchi (1990) in
which freezing-induced water redistribution in 20 cm long Kanagawa sandy
loam columns was observed. The coupled CA code was used to model the
experiment as a validation test for simulation of frost-induced water
redistribution in unsaturated soils. Four identical cylinders, 8 cm in
diameter and 20 cm long, were packed to a bulk density of 1300 kg m

The study provides an example of application of direct solving to simulate highly non-linear processes in variably saturated soils. The modelling used a one-dimensional cellular automata (CA) structure wherein two cellular automata models simulate water and heat flow separately and are coupled through an energy balance module. First-order empirical laws in conjunction with energy and mass conservation principles are shown to be successful in describing the tightly coupled nature of the heat and water transfer. In addition, use of an observed soil freezing curve (SFC) is shown to obliviate the use of non-physical terms such as apparent heat capacity and provide insights into a further subtle mode of coupling. This approach of coupling and use of SFC is easy to understand and follow from a physical point of view and straight forward to implement in a code. The results were successfully verified against analytical solutions of heat flow due to pure conduction, conduction with convection, and conduction with phase change. In addition, freezing-induced water redistribution was successfully verified against experimental data.

The CA scheme described in this paper is not unconditionally convergent. Hence, the size of the time step cannot be arbitrarily chosen. In this section we present a detailed evaluation of the convergence criteria of our code to address the choice of the time step.

The heat and flow convergence criteria are derived one after another. We
start with the heat balance portion. The local energy balance is the basic
principle used in our approach. This is imposed by ensuring flux continuity
of heat. The local heat balance is described by Eqs. (1) and (2) and
the freeze–thaw effect. For a 1-D CA application, assuming

As long as the thermal energy balance component of our CA algorithm obeys the
time stepping–spatial discretization relationship in Eq. (A5) it remains
stable. For such time step control, using the Lax–Richtmeyer equivalence
theorem, one only needs to show that the thermal module represents a
consistent numerical approximation to the full diffusion equation (including

Clearly, if the only source of error in our approximate solution is
the discretization of a continuous process, then our initial values must be
error-free; i.e.

We can construct a similar convergence analysis for the hydraulic module.
But we will approach this problem from the continuum version of the modified
Richard's equation for variably saturated flow for the sake of brevity. The
modified Richard's equation for variably saturated flow can be written as
(in the absence of a source term)

We wish to acknowledge the financial support of the Natural Science and Engineering Research Council (NSERC) and BioChambers Inc. (MB, Canada) through a NSERC-CRD award, NSERC Strategic Projects grant, and the Canadian Foundation for Climate and Atmospheric Sciences (CFCAS) through an IP3 Research Network grant. The authors want to thank the contributions of Lalu Mansinha and Kristy Tiampo, in helping to improve the manuscript, and Jalpa Pal during different stages of this work. Edited by: P. Hallett