SOILSOILSOILSOIL2199-398XCopernicus GmbHGöttingen, Germany10.5194/soil-1-103-2015Coupled cellular automata for frozen soil processesNagareR. M.ranjeet.nagare@worleyparsons.comBhattacharyaP.KhannaJ.https://orcid.org/0000-0002-2279-0301SchincariolR. A.Department of Earth Sciences, The University of Western
Ontario, London, CanadaDepartment of Geosciences, Princeton University,
Princeton, USAAtmospheric and Oceanic Sciences, Princeton University,
Princeton, USAnow at: WorleyParsons Canada Services Ltd., Edmonton,
CanadaThese authors contributed equally to this work.R. M. Nagare (ranjeet.nagare@worleyparsons.com)14January2015111031164May201421May2014–26August2014This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://soil.copernicus.org/articles/1/103/2015/soil-1-103-2015.htmlThe full text article is available as a PDF file from https://soil.copernicus.org/articles/1/103/2015/soil-1-103-2015.pdf
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.
Introduction
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 ∘C for long
periods of time. The water and energy redistribution has significant
implications for regional hydrology, infrastructure and agriculture.
Understanding the physics behind this complex coupling remains an active area
of research. Field studies have been widely used to better understand the
mechanism of these thermohydraulic cycles (e.g. Hayashi et al., 2007).
Innovative column studies under controlled laboratory settings have allowed
for further insights by isolating the effects of factors that drive soil freezing
and thawing, a separation impossible to achieve in the field (e.g. Nagare et
al., 2012). Mathematical models, describing the mechanism of water and heat
movement in variably saturated freezing soils, have been developed to
complement these observational studies. Analytical solutions of freezing and
thawing front movements have been developed and applied (e.g. Stefan, 1889;
Hayashi et al., 2007) and numerical models have replicated the freezing-induced water redistribution with reasonable success (e.g. Hansson et al.,
2004). The optimization of existing numerical modelling approaches also remains
an active area of research. For example, improvements to numerical solving
techniques to address sharp changes in soil properties, especially behind
freezing and thawing fronts, and during special conditions such as
infiltration into frozen soils have been reported recently (e.g. Dall'Amico
et al., 2011).
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
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.
Mathematical description
Let Sit be a discrete state variable which describes the state of the
ith cell at time step t. If one assumes that an order of N elementary
repetitions of the unit cell describe the system spatially, then the complete
macroscopic state of the system is described by the ordered Cartesian product
S1t⊗S2t⊗…⊗Sit⊗…⊗SNt at time t. Let a local transition rule ϕ
be defined on a neighbourhood of spatial indicial radius r,ϕ:
Si-rt⊗Si-r+1t⊗…⊗Si+rt→Sit+1, where iϵ [1 +r, N-r]. The global
state of the system is defined by some global mapping, χ: S1t⊗S2t⊗…⊗Sit⊗…⊗SNt→Gt, where Gt is the global state variable of the system
defining the physical state of the system at time t. Given this algebra of
the system, Gt+1 is given by
Gt+1=χφω1t⊗φω2t⊗…⊗φωit⊗…⊗φωNt,
where ωit=S1-rt⊗Si-r+1t⊗…⊗Si+rt. The quantity r is generally called the radius of
interaction and defines the spatial extent on which interactions occur on the
local scale. In the case of the 1-D CA, the only choice of neighbourhood
which is physically viable is the standard von Neumann neighbourhood
(Fig. 1).
Physical description based on a heat flow problem in a hypothetical soil
column
Let us consider the CA simulation of heat flow in a soil column of length
Lc and a constant cross-sectional area. The temperature change in
the column is driven by a time varying temperature boundary condition applied
at the top. It is assumed that no physical variation in the soil properties
exist in the column at length intervals smaller than Δx. Each cell in
the 1-D CA model can therefore be assumed to be of length Δx.
Therefore, the column can be discretized using Lc/Δx
elementary cells. To simulate the spatio-temporal evolution of soil
temperature in the column, an initial temperature for each elementary cell
has to be set. To study the behaviour of the soil column under external
driving (time varying temperature), a fictitious cell is introduced at the
top and/or the bottom of the soil column and subjected to time varying
temperatures. The transition rules need to be defined now. Once the
transition rules of heat exchange between neighbours are defined, the
fictitious boundary cells interact with the top and/or bottom cells of the
soil column as any other internal cell based on the prescribed rules and the
predefined temperature time series. Although the same set of rules govern
interaction among all cells of the column, heat exchange cannot affect the
temperature of the fictitious cells as that would corrupt the boundary
conditions. This is handled by assigning infinite specific heats to the
fictitious cells. This allows evolution of the internal cells and the
boundary cells according to the same mathematical rules/empirical equations.
The preceding mathematical description of the CA algebra is based on the
assumption that the state variable defining each cell is discrete in space
and time. But soil temperatures are considered to be continuous in space and
time. The continuous description of the soil temperature can be adapted to
the CA scheme by considering small time intervals over which the temperature
variations are not of interest and hence for all practical purposes can be
assumed constant. Conditions for convergence of the numerical temperature
profile set an upper limit on the size of this time interval for a given
value of Δx. Therefore, once the length scale of homogeneity Δx in the system and the local update rules have been ascertained, the CA is
ready for simulation under the given initial and boundary conditions.
Equations (2) and (3) (Sect. 3), applied sequentially, would be the local
update rules for this simple case of heat flow in a soil column (without
phase change) driven by time varying temperatures at the top.
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 (r).
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: Sit is the temperature of the ith cell at time t,
r=1, ϕ is a sequential application of Eqs. (2) and (3) describing heat
loss/gain by a cell due to temperature gradients with its two nearest
neighbours and temperature change due to the heat loss/gain respectively,
and χ is the identity mapping.
Coupled heat and water transport in variably saturated soils
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. r=1. The
one-dimensional conductive heat transport in variably saturated soils can be
given by the heat balance equation
qh=∑ζ=i-1i+1λi,ζ⋅Tζ-Tili,ζ,ζ≠i,
where subscripts i and ζ refer to the cell and its active
neighbours, qh is the net heat flux (J s-1 m-2) for the
ith cell, T is cell temperature (∘C), λi,ζ is
average effective thermal conductivity of the region between the ith and
the ζth cells (J s-1 m-1∘C-1), and
li,ζ is the distance between the centres of the ith and the
ζth cells (m). Effective thermal conductivity can be calculated using
one of the popular mixing models (e.g. Johansen, 1975; Campbell, 1985). The
empirical relationship between heat flux from Eq. (2) and the resulting change in
cell temperature (ΔTi=Tit+Δt-Tit) is
given as
Qh,i=qh⋅Δtli=Ci⋅ΔTi,
where li is the length of the cell (m) and Ci (J m-3∘C-1)
is the effective volumetric heat capacity of
the cell such that
Ci=Cwθw+Ciceθice+Csθs+Caθa,
where θ is volumetric fraction (m3 m-3) and subscripts w,
ice, s, and a represent water, ice, soil solids and air fractions.
The mass conservation equation in 1-D can be written as
ρw⋅ΔΘΔt+ρw⋅qwli+ρw⋅Ss=0,Θ=θw+ρiceρwθice,
where ρ is density (kg m-3), Θ is the total volumetric water
content (m3 m-3), qw is the Buckingham–Darcy flux
(m s-1), and Ss is sink/source term. In unfrozen soils,
θice=0 and Θ=θw.
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 (ψ) and hydraulic conductivity (k) are functions of liquid
water content (θw). The dependency of ψ and k on
θw can be expressed as a constitutive relationship. The
constitutive relationships proposed by Mualem–van Genuchten (van Genuchten,
1980) defining ψ(θw) and k(θw) are used
in this study:
ψ(θw)=Se-1m-11nα,k(θw)=Ks⋅Se0.5⋅1-1-Se1mm2,Se=θw-θresη-θres,
where θres (m3 m-3) is the residual liquid water
content, η (m3 m-3) is total porosity, Ks
(m s-1) is the saturated hydraulic conductivity, and α
(m-1), n and m are equation constants such that m=1-1/n. For an
elementary cell in a 1-D CA model, the Buckingham–Darcy flux in its simplest
form can be written as
qw=∑ζ=i-1i+1ki,ζ⋅ψ+zζ-ψ+zili,ζ,ζ≠i,
where all subscripts have the same meaning as introduced so far, z is the
cell elevation and k represents the average hydraulic conductivity of the
region between the ith and the ζth cells. In this study, phase change
and associated temperature change is brought about by integrating along a
soil freezing curve (SFC). SFCs can be defined because the liquid water
content in frozen soils must have a fixed value for each temperature at which
the liquid and ice phases are in equilibrium, regardless of the amount of ice
present (Low et al., 1968). Soil freezing curves for different types of soils
developed from field and laboratory observations between liquid water content
and soil temperature have been widely reported (e.g. Anderson and
Morgenstern, 1973; Stähli and Stadler, 1997). Van Genuchten's model can be
used to define a SFC (Eq. 7), wherein ψ(θw) is replaced
with T(θw), and n, m and α (∘C-1)
are equation constants.
The coupled CA model
Figure 2 shows a flow chart describing the algorithm driving the coupled CA
code. The code was written in MATLAB®. Let
the superscript t denote the present time step and subscript i be the
spatial index across the grid where each node represents centres of the cell.
The thermal conduction and hydraulic conduction modules represent two
different algorithms that calculate the net heat (qh,i) and
water (qw,i) fluxes respectively across the ith cell. In
essence, the thermal conduction and hydraulic conduction codes run
simultaneously and are not affected by each other in the same time step.
However, the processes are not independent and are coupled through updating
of model parameters and state variables at the end of each time step. Hydraulic
conduction is achieved by applying Eq. (10) to each elementary cell using the
hydraulic gradients between it and its immediate neighbours (r=1).
Similarly, Eq. (2) is used to calculate the heat flux between each elementary
cell and its immediate neighbours using the corresponding thermal gradients.
The change in mass due to the flux qw,i is used to obtain
change in pressure head (Δψi=ψit+Δt-ψit) from the ψ(θw) relationship. The updated value of
total water content is then used to update the volumetric heat capacity
Ci (Eq. 4). The updated value of Ci is used as an input to the
energy balance module along with computed heat flux qh,i. This
represents the first stage of coupling between hydraulic and thermal
processes. The energy balance module computes the total change in ice and
water content due to phase change, and the total temperature change (ΔTi) due to a combination of thermal conduction and phase change.
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 (Tfw=0∘C). Inside the energy balance module, the change in temperature
(ΔTi) is calculated using Eq. (3) and values of Ci and
Qh,i assuming that only thermal conduction takes place. If the
computed ΔTi for a given cell is such that Tit+Δt≥Tfw, then water cannot freeze; cell temperatures are updated
without phase change and the code moves into the next time step. In the
approach of this study, phase change and associated temperature change can
occur if and only if the present cell temperature (Ti) and water content
(θw,i) represent a point on the SFC. This point along the
SFC (Fig. 3) is defined here as the critical state point (Tcrit,
θwcrit). If ΔTi gives Tit+Δt < Tfw for any cell, then freezing point
depression along the SFC accounts for change in temperature due to
freeze–thaw. The freezing point depression or Tcrit is defined
for the cell by comparing the cell θw,i with the SFC.
However, the coupled nature of heat and water transport in soils perturbs the
critical state from time to time, e.g. when freezing induces water movement
towards the freezing front or infiltration into frozen soil leads to
accumulation or removal of extra water from any cell. In such a case,
Qh,i needs to be used to bring the cell to the critical state.
This may require thermal conduction without phase change
(Tcrit > Ti) or freezing of water without
temperature change (Tcrit < Ti). This process
gives us an additional change in temperature or water content which is purely
due to the fact that the additional water accumulation disturbs the critical
state. This is another and a subtle form of coupling between heat and water
flow. Because of the above consideration to perturbation of critical state
caused by additional water added/removed from a cell, infiltration into
frozen soils during the over-winter or spring melt events need no further
modifications to the process of water and heat balance.
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 (dθw) due to Qres,i is used to
determine Tnew by integrating along the SFC (Eq. 11).
If Qh,i is such that a cell can reach critical state and still
additional heat needs to be removed, then this additional heat
(Qres,i) removal leads to the freezing of water. The freezing of water
leads to change in the temperature of the cell such that
minθw,i,Qres,iLf=∫TcritTnew,idθw,i,
where Lf is the latent heat of fusion (334 000 J kg-1)
and Tnew,i is the new temperature of the cell (Fig. 3). If the
change in water content due to freezing is such that θw,i=θres, then no further freezing of water can take place and
Qres,i is used to decrease the temperature of the cell using
Eq. (3) and the updated value of Ci (i.e. after accounting for change
in Ci due to phase change). The soil thawing case is exactly similar as
described above; the only dissimilarity is that a different SFC may be used
if hysteric effects are observed in SFC paths as observed in studies by
Quinton and Hayashi (2008), and Smerdon and Mendoza (2010). If the cell
temperature is above freezing, then the matric potential is calculated using
Eq. (7). For cell temperatures below freezing point, the water pressure
(matric potential) can be determined by the generalized Clausis–Clapeyron
equation by assuming zero ice gauge pressure:
Lf⋅ΔTiTi+273.15=g⋅ΔΨi,
where g is acceleration due to gravity (9.81 m s-2). At the end of
the energy balance calculations, temperatures of all the cells are updated
using the ΔTi computed in the energy balance module. Water content for
each cell is updated by considering the change due to freeze/thaw inside the
energy balance module and qw,i. Hydraulic conductivity of each
cell is updated (Eq. 8) using the final updated values of water content.
Pressure and total heads in each cell are updated considering water movement
(Eq. 7) and freezing/thawing (Eq. 12). The volumetric heat capacity of each
cell is updated one more time (Eq. 4) to incorporate the changes due to
freeze/thaw inside the energy balance module. Thermal conductivity of each
cell is updated using a mixing model (e.g. Johansen, 1975). This completes
all the necessary updates and the model is ready for computations of the next
time step.
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.
Comparison with analytical solutionsHeat transfer by pure conduction
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 (Lc) of 4 m was assumed to have
different initial temperatures in its upper (Tu=10∘C)
and lower (Tl=20∘C) halves (Fig. 4). The system is
hydrostatic at all times and there is no flow. At the interface, heat
conduction due to the temperature gradient will occur until the entire domain
reaches an average steady state temperature of 15 ∘C. The analytical
solution given by Churchill (1972) can be expressed as
T(z,t)=Tu⋅0.5+2π∑n=1∞(-1)n-12n-1⋅cos(2n-1)⋅π⋅zLc⋅exp-(2n-1)⋅πLc2⋅λC⋅t+Tl⋅0.5-2π∑n=1∞(-1)n-12n-1⋅cos(2n-1)⋅π⋅zLc⋅exp-(2n-1)⋅πLc2⋅λC⋅t.
The parameters used in analytical examples for Churchill (1972), and the CA code
are given in Table 1. There is excellent agreement between the analytical
solution and the CA simulation (Fig. 4).
Simulation parameters for heat conduction problems.
Analytical solution for this example is given by Eq. (13) as per Churchill (1972).
SymbolParameterValueηporosity0.35λbulk thermal conductivity2.0 J s-1 m-1∘C-1Cwvolumetric heat capacity of water4 174 000 J m-3∘C-1Csvolumetric heat capacity of soil solids2 104 000 J m-3∘C-1ρwdensity of water1000 kg m-3ρsdensity of soil solids2630 kg m-3llength of cell0.01 mtlength of time step in CA solution1 sHeat transfer by conduction and convection
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
T(z0,t)=Tsurf+A⋅sin2⋅π⋅tτ,
the temperature variation with depth is given by
T(z,t)=Ae-a⋅z⋅sin2⋅π⋅tτ-b⋅z+T∞,a=πCρλτ2+14qfCwρw2λ40.5+12qfCwρw2λ20.5-qfCwρw2λ,b=πCρλτ2+14qfCwρw2λ40.5+12qfCwρw2λ20.5,
where A is the amplitude of temperature variation (∘C),
Tsurf is the average surface temperature over a period of τ
(s), T∞ is the initial temperature of the soil column and temperature
at infinite depth, and qf is the specific flux through the column top.
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).
Heat transfer with phase change
Lunardini (1985) presented an exact analytical solution for propagation of
subfreezing temperatures in a semi-infinite, initially unfrozen soil column
with time t. The soil column is divided into three zones (Fig. 6a) where
zone 1 is fully frozen with no unfrozen water; zone 2 is “mushy” with both
ice and water; and zone 3 is fully thawed. The Lunardini (1985) solution as
described by McKenzie et al. (2007) is given by following set of
equations:
T1=(Tm-Ts)⋅erfx2D1terfϑ+Ts,T2=(Tf-Tm)⋅erfx2D4t-erfγerfγ-erfϑD1D4+Tf,
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).
SymbolParameterValueηporosity0.40λbulk thermal conductivity2.0 J s-1 m-1∘C-1Cwvolumetric heat capacity of water4 174 000 J m-3∘C-1Csvolumetric heat capacity of soil solids2 104 000 J m-3∘C-1ρwdensity of water1000 kg m-3ρsdensity of soil solids2630 kg m-3llength of cell0.01 mtlength of time step in CA solution1 sqfspecific flux4×10-7 m s-1 downwardτperiod of oscillation of temperature at the ground surface24 hAamplitude of the temperature variation at the ground surface5 ∘CTsurfaverage ambient temperature at the ground surface20 ∘CT∞ambient temperature at depth20 ∘C
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 ∘C,
and the upper surface is subjected to a sinusoidal temperature
with amplitude of 5 ∘C and period of 24 h.
T3=(T0-Tf)⋅-erfcx2D3terfγD4D3+T0,
where T1, T2 and T3 are the temperatures at distance x from the
temperature boundary for zones 1, 2, and 3 respectively; T0,
Tm, Tf, and Ts are the temperatures of
the initial conditions, the solidus, the liquidus, and the boundary
respectively; D1 and D3 are the thermal diffusivities for zones 1 and
3, defined as λ1/C1 and λ3/C3 where C1 and C3, and
λ1 and λ4 are the volumetric bulk-heat capacities
(J m-3∘C-1) and bulk thermal conductivities
(J s-1 m-1∘C-1) respectively of the two zones. The
thermal diffusivity of zone 2 is assumed to be constant across the transition
region, and the thermal diffusivity with latent heat, D4, is defined as
D4=λ2C2+γdLfΔξTf-Tm,
where γd is the dry unit density of soil solids, and Δξ=ξ1-ξ3 where ξ1 and ξ3 are the ratio of unfrozen
water to soil solids in zones 1 and 3 respectively. For a time t in the
region from 0≤x≤X1(t) the temperature is T1 and X1(t) is
given by
X1(t)=2ϑD1t;
and from X1(t)≤x≤X(t) the temperature is T2 where X(t) is
given by
(a) Diagram showing the setting of Lunardini's (1985)
three-zone problem. Equations (18), (19), and (20) are used to predict temperatures in
the completely frozen zone (no phase change and sensible heat only), mushy zone
(phase change and latent heat + sensible heat), and unfrozen zone
(sensible heat only) respectively. (b) Linear freezing function used to
predict unfrozen water contents for two cases used in this study (Tm=-1∘C and Tm=-4∘C).
X(t)=2γD4t;
and for x≥X(t) the temperature is T3. The unknowns, ϑ and
γ, are obtained from the solution of the following two simultaneous
equations:
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).
SymbolParameterValueηporosity0.20λ1bulk thermal conductivity of frozen zone3.464352 J s-1 m-1∘C-1λ2bulk thermal conductivity of mushy zone2.941352 J s-1 m-1∘C-1λ3bulk thermal conductivity of unfrozen zone2.418352 J s-1 m-1∘C-1C1bulk-volumetric heat capacity of frozen zone690 360 J m-3∘C-1C2bulk-volumetric heat capacity of mushy zone690 360 J m-3∘C-1C3bulk-volumetric heat capacity of unfrozen zone690 360 J m-3∘C-1ξ1fraction of liquid water to soil solids in frozen zone0.0782ξ3fraction of liquid water to soil solids in unfrozen zone0.2llength of cell0.01 mtlength of time step in CA solution1 sLflatent heat of fusion334 720 J kg-1γddry unit density of soil solids1680 kg m-3Tssurface temperature at the cold end-6 ∘CTmtemperature at the boundary of frozen and mushy zones-1, -4 ∘Cγ∗equation parameter estimated using Eqs. (24) and (25)1.395, 2.062ϑ∗equation parameter estimated using Eqs. (24) and (25)0.0617, 0.1375T0initial temperature of the soil column4 ∘C
* values taken from McKenzie et al. (2007)
Tm-TsTm-Tf⋅e-ϑ21-D1D4=λ2λ1erfϑD1D4erfγ-erfϑD1D4,Tm-Tfλ2λ1T0-Tf⋅D3D4⋅e-γ21-D4D3=erfγ-erfϑD1D4erfcγD4D3.
The verification example based on Lunardini's (1985) analytical solution used
in this study is the same as that used by McKenzie et al. (2007). Lunardini (1985)
assumed the bulk-volumetric heat capacities of the three zones, and thermal
conductivities in each zone, to be constant. It was also assumed for the sake
of the analytical solution that the unfrozen water varies linearly with
temperature. As stated by Lunardini (1985), if unfrozen water varies linearly
with temperature then an exact solution may be found for a three-zone
problem. Although this will be a poor representation of a real soil system,
it will constitute a valuable check for approximate solution methods. The
linear freezing function used in this study is shown in Fig. 6b and the
parameters used in Lunardini's analytical solution are given in Table 3. The
excellent agreement between the analytical solution and coupled CA model
simulations (Fig. 7a, b) for two different cases of Tm shows
that the model is able to perfectly simulate the process of heat conduction
with phase change.
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: (a)Tm=-1∘C and
(b)Tm=-4∘C (Table 3, Fig. 6).
Comparison of total water content (ice + liquid) between
experimental (Mizoguchi, 1990, as cited by Hansson, 2004) and coupled CA
model results: (a) 12, (b) 24, and (c) 50 h.
Comparison with experimental data
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-3
resulting into a total porosity of 0.535 m3 m-3. The columns were
thermally insulated from all sides except the tops and brought to uniform
temperature (6.7 ∘C) and volumetric water content
(0.33 m3 m-3). The tops of three cylinders were exposed to a
circulating fluid at -6 ∘C. One cylinder at a time was removed from
the freezing apparatus and sliced into 1 cm thick slices after 12, 24, and
50 h. Each slice was oven dried to obtain the total water content (liquid
water + ice). The fourth cylinder was used to precisely determine the
initial condition. The freezing-induced water redistribution observed in
these experiments was simulated using the coupled CA code. Parameters used
were saturated hydraulic conductivity of 3.2×10-6 m s-1
and van Genuchten parameters α=1.11 m-1 and n=1.48. The
hydraulic conductivity of the cells with ice was reduced using an impedance
factor of 2. The thermal conductivity formulation of Campbell (1985) as modified
and applied by Hansson et al. (2004) was used. In their simulations of the
Mizoguchi (1990) experiments, Hansson et al. (2004) calibrated the model
using a heat flux boundary at the top and bottom of the columns. The heat
flux at the surface and bottom was controlled by heat conductance terms
multiplied by the difference between the surface and ambient, and bottom and
ambient temperatures respectively. Similar boundary conditions were used in
the CA simulations. The value of heat conductance at the surface was allowed
to decrease non-linearly as a function of the surface temperature squared
using the values reported by Hansson et al. (2004). The heat conductance
coefficient of 1.5 J s-1m-2∘C-1 was used to
simulate heat loss through the bottom. Hansson and Lundin (2006) observed
that the four soil cores used in the experiment performed by Mizoguchi (1990)
were quite similar in terms of saturated hydraulic conductivity, but probably
less so in terms of the water-holding properties where more significant
differences were to be expected. Such differences in water-holding capacity
would result in significant differences in unsaturated hydraulic
conductivities of the columns at different times during the freezing
experiments. The simulated values of total water content agree very well with
the experimental values (Fig. 8). The region with a sharp drop in the water
content indicates the position of the freezing front. There is clear freezing-induced water redistribution, which is one of the principal phenomena for
freezing porous media and is well represented in the coupled CA simulations.
Mizoguchi's experiments have been used by a number of researchers for validation
of numerical codes (e.g. Hansson et al., 2004; Painter, 2011; Daanen et al.,
2007). The CA simulation shows a comparable or improved simulation for total
water content as well as for the sharp transition at the freezing front.
Conclusions
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.
Convergence analysis
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 r=1, this can be
written as
CiliT̃it+Δt-T̃itΔt+ρwLfliθ̃wit+Δt-θ̃witΔt=λi,i+1T̃i+1t-T̃itli+λi,i-1T̃i-1t-T̃itli,
where li is the uniform cell size and λi,i+1 and
λi,i-1 are the average effective thermal conductivity of the region
between the ith, and the i+1th and i-1th cells respectively. The second
term on the left-hand side of equation is the contribution of freeze–thaw to
the thermal energy conservation. T̃it=Tit+eit
is some approximation of the exact solution for temperature Tit at
time t and cell index i given an approximation error eit.
Similarly, θ̃wit=θwit+ei′t is an approximation, subject to the discretization error
ei′t, of the exact solution for the volumetric fraction of water
θwit. It is useful to rewrite the equation in the
following simpler form in order to decouple the thermal and hydraulic
processes in terms of known parameters:
CiliT̃it+Δt-T̃itΔt+ρwLfθ̃wit+Δt-θ̃witT̃it+Δt-T̃it⋅liT̃it+Δt-T̃itΔt=λi,i+1T̃i+1t-T̃itli+λi,i-1T̃i-1t-T̃itli.
The quantity in parentheses in the second term of the equation can be
approximated as ρwLfdθw/dT∣T=Tit, where
dθw/dT∣T=Tit is the slope of the
soil freezing curve at T=Tit, a known quantity. Finally, we
introduce the term Ci′=Ci+ρwLfdθw/dT∣T=Tit as apparent heat
capacity, and rewrite Eq. (A1) as
Ci′liT̃it+Δt-T̃itΔt=λi,i+1T̃i+1t-T̃itli+λi,i-1T̃i-1t-T̃itli.
Rearranging the terms, we obtain
eit+Δt=eit1-λ¯iΔtli2Ci′+λi,i+1Δtli2Ci′ei+1t+λi,i-1Δtli2Ci′ei-1t+λi,i+1Δtli2Ci′Ti+1t+λi,i-1Δtli2Ci′Ti-1t+Tit1-λ¯iΔtli2Ci′-Tit+Δt,
where λ¯i=λi,i-1+λi,i+1. Replacing all
the error terms by the maximum absolute error term, defined as Et=max∣eit∣, we obtain
Δt≤li2Ci′λ¯i.
All coefficients of error terms on the right-hand side of Eq. (A4) are either
positive or zero. Given this, the upper bound on the error at time t+Δt, defined as Et+Δt=max∣eit+Δt∣, must be
Et+Δt≤Et+maxfT,λ=Et+F,
where f(T,λ) is the term in squared brackets in Eq. (A4) and F=maxf(T,λ). Therefore as long as Eq. (A5) is satisfied, the
error always has an upper bound controlled solely by the discretization
error. This is the condition for stability. But, because Ci′ is a
function of time, an adaptive time stepping scheme would be well suited to
solve the problem. The adaptive time stepper would need to satisfy Eq. (A5)
at each time step.
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
Ci′ to account for the freeze–thaw effect) in order to prove convergence
of our method. To do this we note the following recurrence relations:
Et+Δt≤Et+F≤Et-Δt+2F≤E0+(n+1)F,t=nΔt.
It is worth noting that here we have assumed a constant value of F
through all time steps. In the following we argue that this does not affect the
generality of the convergence analysis that follows next.
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. E0=0. Therefore,
Et+Δt≤(n+1)F.
Now, from the definition of f(T,λ) we have
f(T,λ)=1liλi,i+1Ti+1t-Titli+λi,i-1Ti-1t-Titli-Ci′Tit+Δt-TitΔtΔtCi′.
For limΔt, li→0, we have the cluster of terms
within the square brackets converge to the expression
∂∂zλ(z)∂H∂z-C′∂T∂t.
As T is an exact solution of the above diffusion-equation form, we must
have the terms within square bracket converge to 0 as limΔt, li→0. This argument for the boundedness of F as Δt,
li→0 holds at each time step and, hence, would have led to the same
conclusion if we would have used a time variable maximum value of
f(T,λ) in Eq. (A8). Therefore, in general, at any time tlimΔt,li→0Et→0.
This formally shows that our numerical algorithm, with time stepping
satisfying Eq. (A5), is consistent and hence follows the convergence of the
thermal module.
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)
∂θw∂t+ρiceρw∂θice∂t=∂∂zk(z)∂H∂z.
The left-hand side of Eq. (A12) follows from the continuum version of the
first term on the left-hand side of Eq. (5) where Eq. (6) has been used to
eliminate Θ. The term on the right-hand side is the Darcy flux,
introduced as the continuum version of Eq. (10) where the total head H=ψ+z. The effect of freeze–thaw on the total head can be accounted for
as a Clausis–Clapeyron process as given in Eq. (12). To make this clear, we
rewrite Eq. (A12) as follows:
∂θw∂H+ρiceρw∂θice∂T∂T∂H∂H∂t=∂∂zk(z)∂H∂z.
We can make use of the
following relations to eliminate temperature from Eq. (A13):
∂θice∂T=-LfgT∂θw∂H,∂T∂H=gTLf.
Therefore, we can rewrite Eq. (A12) finally as
1-ρiceρwCw∂H∂t=∂∂zk(z)∂H∂z,
where Cw=∂θw/∂H∣H=H(t) is
the local slope of the soil retention curve which can be derived from
Eq. (7). Equation (A16) now has the same form as the expression in Eq. (A10).
It is immediately clear that, if one would have followed the full formal
arguments as outlined for the thermal module, the condition for stability of
the variably saturated flow dynamics part of our algorithm is of the form
Δt≤li2Cwi′k¯i,
where Cw′=Cw(1-ρice/ρw)
and k¯i=ki,i-1+ki,i+1. We refer the reader to Mendicino et
al. (2006) for a thorough convergence analysis of the water flow problem. The formal
argument is exactly equivalent to that presented by us for the heat flow
problem. Combining Eqs. (A5) and (A17), the following condition gives
stability and convergence conditions for the overall CA problem:
Δt≤minli2Cwi′k¯i,li2Ci′λ¯i.
Acknowledgements
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
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