Drainage subsidence associated with Arctic permafrost degradation



[1] Arctic sources of greenhouse gas associated with permafrost degradation constitute a large uncertainty in existing climate models. Greenhouse gas release from the Arctic subsurface is mediated by numerous interconnected physical processes; one facet of these is the interplay between surface deformation and melting of subsurface ice. First, we construct analytic solutions describing fluid drainage and soil subsidence subsequent to thawing of a 1-D permafrost column. These solutions lead to formulas giving the total amount of subsidence as well as the time over which subsidence occurs. We give an example application of the analytic model to peat plateau degradation in the Canadian Hudson Bay Lowland and show that the degree of subsidence predicted from our model is consistent with recorded subsidence of peat in western Norway that was drained for cultivation purposes. Second, we numerically model an initially frozen, fluid-saturated, 2-D soil matrix with a thaw zone advancing from the surface downward. With the surface temperature fixed at 5°C, a thaw front propagates to ∼10 m depth within 20 years, and due primarily to drainage of fluid from the pore space, a region of soil depressed by ∼3 m forms above an initially ice-rich subsurface zone. Soil underlying this depressed zone may have its permeability reduced by between 1 and 2 orders of magnitude; this reduction in permeability can act as a negative feedback to thawing.

1. Introduction

[2] Polygonal patterned ground, ice wedges, pingos, thermokarst depressions, subsidence and cryoturbation are some of the dramatic features and processes observed in the Arctic [e.g., Williams and Smith, 1989]. They occur in part because soils contract at cold temperatures, soils will compact under stress, and a volume change occurs when water freezes or ice melts. These patterns shape the surface topography and affect the flow of water across Arctic landscapes and even impact the integrity and characteristics of the active layer, indirectly influencing the production of greenhouse gases focused therein. Mathematical and computational models have had success in explaining many cold climate processes, but in order to extend their predictive ability, the mechanics of soil deformation and subsidence must be integrated with the thermal and hydrologic forces at work.

[3] There are many causes of soil subsidence, including peat oxidation [e.g., Leifeld et al., 2011; Gebhardt et al., 2010], settlement due to thawing of subsurface ice [Morgenstern and Nixon, 1971], placement of man-made structures on the surface [Terzaghi, 1943; Gibson et al., 1967], differential frost heave [Mackay, 1999], and settlement due to drainage of subsurface fluids [e.g., Lamoreaux and Newton, 1986], hereafter referred to as “drainage settlement.” All but the first can be understood as mechanical stress effects in porous media, the study of which dates back at least to M. A. Biot, who is credited with developing the concept of a poroelastic medium. In a series of papers published between 1935 and 1962, Biot developed his theory of poroelasticity, providing a mathematical description of the mechanical behavior of a poroelastic medium [see Biot, 1941, 1962]. Biot's equations of poroelasticity are derived from linear elasticity for the solid matrix, Navier-Stokes equations for a viscous fluid, and Darcy's law for flow of fluid through the porous matrix.Fowler and Noon [1999]corrected conceptual errors in Biot's theory, but found that Biot's equations nevertheless did not change for small deformations. They extended the theory to large-amplitude elastic-plastic deformation, and applied it to an approximate analysis of groundwater flow induced by the compaction of sediments by an overriding ice sheet. The theory was also applied to frost heave associated with ice sheet advance and retreat, using a Dupuit approximation to simplify the theory. To our knowledge, the above theory has not yet been applied to the problem of drainage settlement associated with permafrost degradation.

[4] Below we present novel analytic results describing the evolution of the porosity, permeability, and subsidence fields in an idealized 1D column of earth material evolving from a frozen to a thawed state. In order to understand the specific features of drainage settlement, we neglect settlement due to other causes. In addition to analytic results, we present 2D numerical simulations of soil subsidence employing rectangular geometry for the subsurface ice distribution. For these simulations, we employ the numerical modeling framework FEHM (Finite Element Heat and Mass Transfer), described by Zyvoloski [2007] and used in many previous studies involving subsurface fluid flow.

[5] Our key assumptions throughout this study are that (1) subsequent to thawing, all deformation results from fluid drainage in the fully saturated layer, (2) soil is partially supported by interstitial fluids, (3) the soil is sufficiently weak to subside continuously as fluid drains from the pore space, and (4) there is a drainage route located at depth, through which fluids may escape once the permafrost above this route has been thawed.

[6] Assumption 1 is partly based on convenience for this initial study. We note, however, that our results below would not be significantly altered by including a thin unsaturated zone near the surface, because the deformation our models assume occurs as the result of thawing and subsequent drainage of an ice-rich zone (IRZ) at depth, and because the soils to which our models apply have little internal cohesion, as is the case for unconsolidated, loose soils (seesection 5.1for more details). Henceforth, when we refer to “the surface” we mean the location of the surface separating the fully saturated zone from any overlying partially saturated soils. Assumption 3 obviates the need to consider stresses within the soil matrix because the soil immediately deforms instead of having stresses build up within it. While simplifying the analysis, this assumption does limit the applicability of our study to continuous soil subsidence as opposed to, e.g., catastrophic collapse associated with thermokarst events. We consider initially ice-rich permafrost, so that postthaw soil grains can be considered loosely packed. The adequacy of assumption 3 is discussed further insection 5.1.

[7] First, we derive and solve analytically the differential equations describing evolution of the postthaw pore pressure field in a 1D draining soil column. From the pressure field the porosity and permeability fields are determined. Furthermore, the analytic solutions yield characteristic timescales that govern the rate of soil subsidence, and the maximum degree of soil subsidence. Second, we construct, verify, and employ a numerical scheme to study both the thaw and drainage stages of a 2D degrading permafrost system. The numerical simulations reveal edge effects not obtainable from 1D analysis, as well as interplay between degradation and flow in regions of strongly varying ice content and permeability.

2. Analytical Model

[8] Previous studies that involve analytic models of drainage subsidence have investigated the effects of subsurface fluid extraction via pumping. Most of these studies model the extraction inlet as an idealized point sink [e.g., Booker and Carter, 1986, 1987; Kanok-Nukulchai and Chau, 1990; Bear and Corapcioglu, 1981; Tarn and Lu, 1991], or as a circular area/ring line [Chen, 2003]. While these studies could in principle be adapted to study subsidence associated with Arctic drainage, some consider only the steady state [e.g., Booker and Carter, 1986; Chen, 2003] and all employ equations describing instantaneous force balances within the soil, which considerably complicates the analysis and resulting solutions. The approach of this study includes time evolution, is more easily adapted to consider different sink geometries and, due to assumption 3 discussed in section 1, avoids the complications of employing equations governing the soil stresses.

[9] As a simplified model of permafrost, consider a 10 m high 1-D column of fully saturated frozen soil. Suppose that at the base there is a drainage route, which would allow liquid water to escape the system. We study the pore pressure field and soil matrix subsidence immediately following complete thawing of the column. The initial pressure profile is hydrostatic, starting at 0.1 MPa at the surface and increasing to 0.2 MPa at depth. The drainage route can be taken to connect either to a shallow aquifer of lower pressure or to the surface at a different location. Assuming the latter, we fix the drainage route pressure at 0.1 MPa.

[10] This prescribed bottom boundary condition creates a sharp discontinuity in the initial pressure profile. Because our analytic solution method will depend on the Fourier series of this initial distribution, it might be thought that such a discontinuity would introduce inaccuracy into the solution, particularly near the bottom boundary. We therefore formulated the initial distribution so as to have an adjustable position for the maximum pressure and, after finding that our final solutions are not sensitive to the sharpness of the discontinuity in initial pressure, we reverted to the simpler initial conditions specification. Indeed, our solution turns out to be especially accurate near the bottom boundary of the system.

[11] The pressure difference between the drainage route and the fluid above will drive fluid from the bottom of the system. As the pressures equalize, a pulse of pressure low relative to hydrostatic values will propagate upward from the drainage route. Consider the case such that this pulse has traveled a short distance ΔZ from the drainage face during a time Δt (see Figure 1). Soil within a layer of height ΔZ will have been drained during this time, releasing a volume ΔV of water given by

display math

where k is the permeability, μ is the dynamic viscosity, ρ is the density of water, g is the gravitational acceleration constant, Ais the cross-sectional area of the column,p is the pore pressure, z is the depth, and angle brackets represent averaging across the layer. As implied by assumption 3 above, the volume of released fluid gives rise to an equivalent decrease in the void space volume of the soil matrix:

display math

where ϕ is the porosity, V is the volume of the layer, and the subscript i refers to the initial value (equation (2) can be derived from equation (29) by making the identifications Δh = ΔV, hi = ΔZ, and ϕf = ϕ). Combining equations (1) and (2) yields

display math

where pmin is the fluid pressure at the bottom of the layer, ΔZt has been replaced by dZ/dt, and we have used 〈∂p/∂z〉 = (pmin − p)/ΔZ; dZ/dt is the velocity of the pressure pulse, negative because z is positive downward. If ΔZ ≪ (p − pmin)/ρg, then first term may be neglected. An initially hydrostatic pressure profile with pmin equal to atmospheric pressure gives this condition as ΔZL, where L is the height of the column. Neglecting the first term in (3) leads to

display math

where ϕmin is the postdrainage value of the porosity and

display math

where ϕ in has been replaced by its average value. Equation (4) embodies the expectation that when pressures have reached steady state values, the fluid will have drained from the soil as completely as possible and the soil porosity will have reached some minimum allowed value. Due to (4), the initial porosity is then simply a rescaling of the initial pressures, with ϕ = ϕmin at the surface and ϕ = ϕmin + αρgL at the drainage route depth. We assume a similar relationship for the permeability k:

display math

Conservation of fluid mass is governed by

display math

We take ρ and μ as constants; we nondimensionalize the pressure and depth via the transformations

display math


display math

where L is the height of the column. Equations (8) and (9) ensure that the nondimensionalized pressure, math formula, and depth, χ, vary between zero and unity. When k is constant, equations (4) and (7) yield the diffusion equation in p with a diffusive timescale proportional to μαL2/k. By analogy, we rescale the time as

display math

for the case of nonconstant k. With the above rescalings, equations (4), (6), and (7) yield the nondimensionalized equation

display math


display math

Equation (11)has nonlinearities in the third and fourth terms on the left-hand side, which prevent solution by classical methods of integration. However, ifk only depends weakly on p; that is, if β is small, and if kmin is not too small, then ϵ can be treated as a small parameter. We use the parameters given in Table 1, for which ϵ ≈ 0.5. We write math formula as a power series in ϵ as

display math

where each math formula is considered as a function of χ and τ. Substituting (13) into (11), setting the coefficients of equal powers of ϵ to zero, and ignoring all powers of ϵ above the first order yields

display math


display math

Equation (14) is the classical diffusion equation and has the solution

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where the bn are the Fourier coefficients of the sin expansion of the initial pressure profile, given by

display math

We attempt a solution of (15) in the form of

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Using (16)to evaluate the right-hand side of(15) and substituting (18) into (15) yields

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where Sn abbreviates sin(nπχ), Cn abbreviates cos(nπχ), and the prime denotes differentiation with respect to τ. Multiplying both sides of (19) by sin(mπχ) and integrating from zero to one results in

display math


display math

Equation (20) is an ordinary differential equation in cm(τ) and has the solution

display math

where q is an abbreviation for π2(l2 + p2 − m2) and

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The trigonometric integrals in Glpm can be evaluated analytically and are given as

display math


display math

where, if ⊙ represents + or −,

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We therefore have a solution for math formula in the form of (18) with coefficients given by (22)(26). Figure 2shows the pressure after 1, 4, and 8 days according to the zeroth-order solution (constant permeability case, shown as the green curve), the perturbed solution (temporally and spatially varying permeability, shown as the red curve), a finite difference solution of the problem (black curve), and the solution obtained from the code FEHM (blue curve). There is a closer match between the perturbed solution and the computational solutions than between the zeroth-order solution and the latter, particularly toward the drainage route. The values of ΔZ and dZ/dt from equation (5) can be estimated from the position of the peak pressure of the nonperturbed solution at day one in Figure 2. The values ΔZ ≈ 4 m and dZ/dt ≈ 5 × 10−5 m s−1 yield α ≈ 0.5 × 10−6, consistent with the value of α used in Table 1 to derive the analytic solution.

Figure 1.

A 1-D conceptual model of a column of permafrost.

Table 1. Parameters Used in the Perturbed Analytic Solution
g9.8 m s−2L10 m
ρ1000 kg m−3kmin10−13 m2
μ10−3 Pa sϕmin0.3
α3 × 10−6 Pa−1pmin105 Pa
β5 × 10−19 m2 Pa−1pmax2 × 105 Pa
Figure 2.

Comparison between the zeroth-order (green), perturbed (red), finite difference (black), and FEHM-derived (blue) solutions.

[12] We now derive a formula to calculate the maximum degree of surface subsidence, starting from

display math

where V is the volume, the subscript tot refers to the total volume, s refers to the soil phase, l refers to the liquid phase, and f refers to some final time. Because the volume of the soil phase does not change, Vsf = Vsi, where the subscript i refers to initial time. Introducing this relation into (27), and writing the volumes in terms of porosity, we obtain

display math

where Ais the cross-sectional area of the column andh is the height of the column element. We solve (28) in terms of hf and calculate the subsidence of a volume element as

display math

If the distance from the bottom is defined as ζ = L − z, then the subsidence for the entire column can be calculated by letting hi →  in (29) and integrating:

display math

Substituting (4) along with the initial pressures into (30) and evaluating the integral yields

display math

where math formula is the fractional degree of subsidence. Inserting the parameter values from Table 1 into (31) gives math formula or 2.2 m of surface subsidence.

[13] The assumption that α is constant in equation (4) leads to an initial porosity profile that is merely a rescaling of the initial pressure. Here we show how this limitation may be at least partially overcome for the case of constant permeability; because the solution method employs an eigenfunction expansion similar to equation (14), the regular perturbative solution for nonconstant k can in principle be carried over to the present case.

[14] We now assume that the initial porosity in the column is equal to a constant, ϕmax, so that α varies with depth as

display math

where pi(z) is the initial pressure profile (excluding the bottom boundary) and Δϕ is defined as ϕmax − ϕmin. Although α(z) is infinite at z = 0, the porosity is always finite because (p − pmin) → 0 as 1/z → . Substituting (32) and (4) into (7) yields, after simplification,

display math

Again we nondimensionalize the pressure and depth according to (8) and(9), but we now rescale the time as

display math

The nondimensionalized mass conservation equation then becomes

display math

Equation (35) is separable; we write the separated solution as math formula = X(χ)T(τ) and take the separation constant as −λ2. With this choice, the time-dependent part can be solved readily as

display math

The remaining spatial part satisfies

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Under the transformations

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display math

equation (37) becomes

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Equation (40) is a version of Bessel's equation and has solutions

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where J1(x) is a Bessel function of order 1. In terms of X(χ) we have

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X(χ) already satisfies the boundary condition X(0) = 0, and the other boundary condition requires that

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Because the roots of J1 are given approximately by π(n + 1/4) for integers n ≥ 4, λ must satisfy

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The first three values of λn are given approximately by 1.92, 3.51, and 5.09. The initial condition in terms of X(χ) is simply X(χ) = χ, which we satisfy via the sum

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Multiplying both sides of (45) by math formula and integrating from zero to one gives the expansion coefficients as

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where J2 is a Bessel function of order 2. The complete solution is now given by

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Figure 3shows the zeroth-order solution from the previous section together with solution(47) with ϕmax = 0.45. Although the evolution timescales given by equations (10) and (34) are different, the pressure decays at approximately the same rate in both systems as long as the initial porosities and permeabilities have the same average values. This agreement justifies use of the simpler model with average parameter values if it is the overall timescale that is of primary interest.

Figure 3.

Comparison between pressure solutions for constant α (green) and for α ∝ 1/z (red).

[15] The decay timescales for solutions of equations (14) and (15) are identical; similarly, if the permeability does not vary too strongly in time or space, the timescale implied by equation (47) will also apply to the case of nonconstant k. According to this equation and equation (34), the time required for the pressure to decay by a factor 1/e, and hence the subsidence time, satisfies

display math

where K is the hydraulic conductivity and math formula. This timescale is much shorter than the time required for the thaw front to reach the drainage face. The latter may be calculated exactly by solving the so-called Stefan problem [Carslaw and Jaeger, 1959] for the temperature field inside and outside a retreating sheet of ice. Alternatively, because we are not interested in the evolution of the entire temperature field, we may estimate the propagation time of the thaw front by considering the energy balance over a control volume (CV) such as that shown in Figure 4. In Figure 4, TH is the temperature of the surface heat source, TM is the melting temperature of ice, h(t) is the distance from the surface to the boundary separating thawed from unthawed material, and u(x) is the horizontal Darcy velocity. Because of the additional energy required to melt the ice, we expect the time required for the thaw front to propagate to a given depth to be much larger than the single-phase pure conductive timescale given by

display math

where κ is the thermal diffusivity. Therefore, suppose that the temperature in the thaw zone assumes approximately the linear steady state profile for all times. Then the heat flux into the top boundary of the CV is given by

display math

where λ is the effective medium thermal conductivity. The heat flux out the bottom of the CV is given by

display math

where Lf is the latent heat of fusion, cp is the specific heat, and the subscript i refers to the ice phase. The total heat flux into the left and right sides is

display math

where 〈T〉 is the average melt zone temperature and the subscript l denotes the liquid phase. Because the average temperature of the thaw zone remains constant, the total heat flow into the CV must vanish for all times. By performing a mass balance on the CV, it can be shown that

display math

Substituting (53) into (52), summing the resulting term with equations (50) and (51), and setting the sum equal to zero leads to

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display math


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Equation (54) is separable and can therefore be integrated to yield

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Hence, the time τthw taken for the thaw front to reach the depth L is given as

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Equation (58) has the form of proportionality (49) with κ replaced by the effective diffusivity (55). Figure 5 shows a comparison of the position of the thaw front given by (57) and that computed by FEHM from the first simulation of section 4.1. In the associated single-phase problem, the high-temperature front would reach the drainage face depth in only ∼3 years.

Figure 4.

Energy balance for determining the advance rate of a thaw front. The term h(t) is the distance from the surface to the liquid-ice interface,TH is the temperature of the heat source, and TM is the melt temperature, 0°C. The terms u(x) and u(x + δx) are the horizontal Darcy velocities at the left and right faces of the CV.

Figure 5.

Comparison between the thaw front depths computed from FEHM and those calculated analytically.

3. Numerical Model

[16] Numerical modeling of subsurface processes in the Arctic permafrost presents additional challenges compared to more temperate regions. A number of models have tackled the extra complexity of freezing and thawing processes in cold soils. There has been a progression of models of mass and energy transport in freezing soils, ranging from those, for example, compared by Romanovsky et al. [1997], to ones such as that by Hansson et al. [2004], which solves a 1-D model of flow and heat transport with freezing/thawing in variably saturated media, to recent models, for example, byRowland et al. [2011], Painter [2011] and Frampton et al. [2011]. These most recent models solve equations for 3-D, three phase, unsaturated nonisothermal flow and heat transport with freezing/thawing, and are being used to examine long-term subsurface response in the Arctic permafrost to climate change. They include detailed thermohydrologic dynamics, but they do not consider mechanical processes.

[17] Mechanical effects of freezing and thawing in permafrost soils are significant, and challenging to numerical approaches because of the coupled nature of the dynamics, but numerical models are catching up to the theory. Recently, Nicolsky et al. [2008]developed a 2-D finite element thermomechanical model that couples heat diffusion with phase change and a stress equation plus an equation for soil skeleton movement. The soil is assumed saturated with ice or water or a combination of both. Their governing set of equations is essentially the same as in quasi-static poroelasticity theory. They applied their model to explain the dynamics of nonsorted circle features seen in Arctic landscapes, due to frost heave. They analyzed frost heave and changes in porosity for several simplifying cases, and obtained good quantitative to qualitative agreement with frost heave measurements at different field sites.Thomas et al. [2009]presented a similar finite element model and applied it to active layer freezing, ice lens formation, and also to solifluction (a kind of thaw-related slow mass movement).

[18] Similar to the analytical model, our numerical model uses equation (4) to calculate the change in porosity due to fluid drainage. Then the surface subsidence is given by computing (29) for each computational node and summing.

3.1. Ice Model

[19] The equations of FEHM governing fluid flow require expressions for the density, enthalpy, and viscosity of each thermodynamic phase present in a mixture. Because the freezing curve for water has a steep slope in the thermodynamic regime of interest (i.e., −10°C < T < 5°C, 0.1 MPa < P < 0.5 MPa), we model the ice-liquid phase transition as a step function inTlocated at 0°C. In order to keep first derivatives of the fluid properties well defined everywhere, we approximate the step function as a hyperbolic tangent located between −0.25 and 0.25°C. Fluid properties in the single-phase regions are treated linearly as

display math

where ξ is density, enthalpy, or viscosity, ξ0 is a reference value, and γ1 and γ2 are empirical constants representing the rate of increase of ξ with increasing temperature and pressure. For example, when ξ is the liquid density γ1 is the liquid thermal expansion coefficient and γ2 is the liquid compressibility. For temperatures between −0.25 and 0.25°C, we interpolate between the endpoints by using the hyperbolic tangent approximated step function.

3.2. Frozen Wall Bench Mark

[20] To check our numerical scheme's ability to model fluid flow in systems with ice, we employ the “frozen wall” bench mark as suggested by McKenzie et al. [2007]. Figure 6shows the system geometry, initial, and boundary conditions for this simulation. No heat or mass flow is allowed through the top or bottom boundaries. The temperature and pressure are both fixed at the left boundary. The pressure is fixed at the right boundary, but a flow-through temperature condition is enforced there, i.e., such that the temperature at a node is set equal to that of the nearest “upstream” node as determined by the direction of fluid flow. The solution domain contains 2244 nodes. The horizontal grid resolution is 0.5 m in the domain interior and 0.25 m for two columns of nodes, each one at a vertical boundary; the vertical resolution is 0.5 m in the interior and 0.25 m for two rows of nodes, each one at a horizontal boundary.

Figure 6.

Boundary and initial conditions for the freezing wall bench mark.

[21] McKenzie et al.'s solution to the frozen wall problem incorporates the effects of longitudinal and transverse thermal dispersivity. Because FEHM does not include thermal dispersivity effects, our results are not directly comparable with those of McKenzie et al. [2007]; therefore, we compare the output of FEHM with output from the code ARCHY [Rowland et al., 2011], which is also equipped to simulate the dynamics of systems involving the liquid-ice phase transition.Figure 7shows simulation results at times of 4, 40, and 400 days for FEHM and ARCHY. The extent of the ice wall is shown by the blue regions. Results from FEHM and ARCHY compare quite well. Our ice model implies an abrupt liquid-ice transition; however, ARCHY solves the full governing equation for ice saturation, so that the strong agreement between FEHM and ARCHY argues that the step function approximation to the liquid-ice transition is adequate in this thermodynamic regime, for a fully saturated soil.

Figure 7.

Comparison between (left) FEHM and (right) ARCHY temperature solutions for the frozen wall bench mark.

4. Subsidence Simulations

[22] The simulations in the following sections employ the reference model setup depicted in Figure 8. The initial temperature is −3°C everywhere. The system is fully saturated and all water is initially in the ice phase. The initial pore pressure varies from 0.1 MPa at the top to ∼0.2 MPa at depth. We model a drainage route at depth by fixing the pressure there to 0.1 MPa and imposing a flow-through temperature condition. The initial porosity is 0.4 every where. In order to minimize the effects of the side boundaries on the internal system dynamics, we extend the simulation domain by an additional 100 m on each side of the 100 m wide central region. The side boundaries are set to allow zero mass or heat flux. Our grid resolution is 0.1 m in the vertical direction and 1 m in the horizontal. For each of the simulations in the following sections, we insert a region of ice-rich soil inside the system and mimic conditions of talik (i.e., permanently thawed ground) formation by imposing a constant temperature of 5°C at the top boundary to induce soil thawing from the top down. The parameterα in equation (4) is made to vary spatially according to

display math

where the subscript i denotes values of the initial profiles, and ϕmin is taken as 0.3 for all simulations; α(z) is set to zero for z such that pi(z) = pmin. Formula (31) was derived assuming constant α, but this formula may still be employed to calculate subsidences for comparison with numerical results by using an average value of α, i.e.,

display math

where the angle brackets indicate spatial averaging. In addition to the decrease in porosity due to drainage implied by equation (4); for the simulations below we also include the relatively small change due to the decrease in specific volume of ice upon melting.

Figure 8.

Reference model for permafrost degradation simulations.

[23] We have not included the change in vertical distance between control volume nodes as the top boundary moves downward during simulations, because neglecting this change introduces only a small error. The decrease in vertical distances between nodes results in decreased areas of vertical control volume faces, so that horizontal fluid flow rates would be less than otherwise. However, before the thaw zone reaches the drainage face all fluid velocities are near zero and the thermal regime is dominated by conduction; during fluid drainage, the fluid fluxes are primarily oriented downward. For simulations that include varying nodal permeabilities, the change in permeabilities at the control volume interfaces is included.

4.1. Rectangular Ice-Rich Zone

[24] We insert a rectangular ice-rich zone into the reference system (see above) by setting the initial porosity to 0.6 in a rectangular 20 × 6 m2 region. The permeability is 10−12 m2 everywhere and held constant, i.e., thawing does not affect the permeability for this first simulation. Figure 9 shows the geometry of the IRZ. The IRZ is both wider and shallower than it appears because of the skewed aspect ratio of Figure 9. Imposing a constant 5°C top boundary temperature condition and running the simulation for 20 years leads to the porosity and subsidence profiles shown in Figure 10. The solid black curve indicates the boundary between the liquid and ice phases, while the thick purple curve shows the approximate position of the surface after maximum subsidence has been achieved. Outside the IRZ, formula (60) gives 〈α〉 ≈ 2 × 10−6 Pa−1 and (31) yields a maximum subsidence of 1.4 m. Inside the IRZ, 〈α〉 ≈ 4 × 10−6 Pa−1 and the maximum subsidence is ∼2.9 m. Both of these estimates are consistent with the surface subsidence values shown in Figure 10. Figure 11 shows the detailed behavior of the system during the drainage stage, which starts at ∼16 years. Due to the larger amount of latent heat that must be supplied to initiate melting, the IRZ takes longer to thaw than the surrounding medium. Consequently, the residual ice below the IRZ (z = 0 to 2 m) slows subsidence above significantly relative to that of the surrounding medium, which is instead governed by the timescale given in (10) and is on the order of weeks instead of months. Because the edges of the IRZ are the first portions of the IRZ to thaw, subsidence is first most pronounced above these regions, increasing toward the middle as the IRZ continues to thaw inward.

Figure 9.

Porosity distribution representing a 20 × 6 m2subsurface ice-rich region.

Figure 10.

Porosity and subsidence (m) after 5, 10, 15, and 20 years (top to bottom) of simulation time for the setup shown in Figures 8 and 9.

Figure 11.

Porosity and subsidence (m) after 16, 16.1, 16.3, and 16.5 years (top to bottom) of simulation time for the setup shown in Figures 8 and 9. Subsidence begins at 16 years because the thaw front has taken this amount of time to reach the drainage route.

[25] In the simulation above, we assumed that the permeability is a constant; however, the permeability in actual systems may vary significantly both spatially and temporally. One way to include these effects is to employ an equation relating the porosity and permeability. Many such relations exist, such as the Kozeny-Carman equation [e.g.,Xu and Yu, 2008] and various fractal-based formulations [Gimenez et al., 1997]. In the next simulation we repeat previous simulation but employ the Kozeny-Carman equation in the form

display math

where C is determined from the initial conditions as

display math

with ϕi the initial porosity and ki the initial permeability. Figure 12 shows the porosity, subsidence and permeability after 20 years of simulation time. The permeability at the bottom of the IRZ has dropped from 10−12 m2 to ∼10−13.5 m2. This lower permeability slows fluid drainage from the IRZ even after the system has thawed completely. The trapped IRZ fluid slows the rate of subsidence above, particularly around x = 150 m.

Figure 12.

Porosity, subsidence (m), and permeability (m2) fields after 20 years of simulation time when the permeability is related to the porosity via equation (63).

[26] For the last simulation of this section, we repeat the first simulation, again holding the permeability constant in time, but with the permeability of the soil surrounding the IRZ set to 10−13 m2 while that of the IRZ remains at 10−12m2. Figure 13 shows the porosity and subsidence for this system during the drainage stage near 16.6 years of simulation time. This time sequence is comparable to that shown in Figure 11, with the former shifted forward in time by ∼7 months and expanded from a duration of half a year to a full year.

Figure 13.

Porosity and subsidence (m) after 16.6, 16.9, 17.3, and 17.6 years (top to bottom) of simulation time.

4.2. Hexagonal Ice-Rich Zone

[27] In order to investigate the sensitivity of our results to the geometry of the IRZ, we approximate a circular subsurface IRZ roughly as a hexagonal region of elevated porosity, as shown in Figure 14. Other parameters are the same as in the first simulation of section 4.1. Figure 15 shows the porosity and subsidence after 15.8, 16.0, 16.2, and 16.3 years of simulation time. Figure 16 shows the final subsidence profile. The maximum degree of subsidence is the same as in Figure 10, but now the region of maximum subsidence is concentrated toward the center of the IRZ.

Figure 14.

Porosity distribution representing a hexagonal subsurface ice-rich region.

Figure 15.

Porosity and subsidence (m) after 15.8, 16.0, 16.2, and 16.3 years (top to bottom) of simulation time.

Figure 16.

Final subsidence field when the IRZ is hexagonal.

5. Discussion

[28] We now discuss the implications of the analytic model and give an example of how it may be used in practice to calculate the degree of subsidence and the time to complete subsidence. Afterward we discuss the numerical results, with an emphasis on phenomena revealed by the simulations that is beyond the capabilities of the 1D analytical approach.

5.1. Analytic Model

[29] Terzaghi [1943] derived an equation describing consolidation of a 1D vertical soil column due to surface loading; the key parameter in his equation is the consolidation coefficient, cv. Due to the engineering significance of determining ground settlement under man-made structures, estimates ofcv are readily available for a variety of soils, including those of Arctic interest.

[30] A relationship between cv and α can be derived starting from equation (29). Setting hi = Vmin, Δh = ΔV, ϕf = ϕ, solving (29) for ϕ, and subtracting ϕmin from both sides yields

display math

Setting the right-hand side ofequation (64) equal to αΔp and solving for α gives

display math

where βs is the soil matrix compressibility. Using (29) to eliminate ΔV/Vmin from (65) gives

display math

When ϕβs is small compared to unity, αβs; however, for highly compressible materials one may replace ϕ with the average of the minimum and maximum porosities, 〈ϕ〉. The consolidation coefficient is then related to α via

display math

[31] Equation (67) may be combined with equation (31) to give

display math

Equation (68), together with equations (58) and (48), provides a basis for practical applications of the drainage subsidence model.

[32] Thawing of frozen soil alters its physical properties, and this alteration may need to be taken into account in applying equations (68) and (48). Freeze-thaw cycling of soils may increase hydraulic conductivity by 1 or 2 orders of magnitude and significantly raise the compressibility [Qi et al., 2006]. Hence, the subsidence times may be much shorter and total subsidences greater than expected based on prethaw soil properties. When the effects of thawing on soil properties cannot be quantified, prethaw soil properties in equations (68) and (48) will yield estimates of the maximum subsidence time and the minimum total degree of subsidence.

[33] According to (67) the values in Table 1 give cv ≈ 10−5 m2 s−1, which is within the range 10−5,6 m2 s−1 for those of fibrous peats [Mesri and Ajlouni, 2007]. Peat terrain is prevalent in many areas of the Arctic [Wright et al., 2008], for example in the Hudson Bay Lowland of central Canada, Rogovaya in northeastern European Russia, and Tavvavuoma in northern Sweden [Sannel and Kuhry, 2011]. We now give an example application of the drainage model, describing what future subsidence may look like as the climate continues to warm at locations featuring extensive raised peat plateaus, such as in the Canadian Hudson Bay Lowland. This location features soil for which assumptions 1–4 from section 1may be expected to hold to a reasonable degree of approximation, i.e., peat or peat-rich soil.

[34] In the Northern Hudson Bay Lowland, fens surround raised terrain composed of frozen peat bog plateaus, the plateaus having a relative relief of 1–2 m. Subsurface wedge and segregated ice inhibits drainage of fluid from the peat plateaus into the surrounding fens; however, snow accumulation and wave action at plateau edges act to degrade the ice, expanding the fen area [Dyke and Sladen, 2010]. Such degradation may enhance plateau drainage by providing subsurface fluid flow pathways leading into the fens. As the Arctic climate continues to warm, these pathways may lead to inland regions of isolated, depressed peat plateau. If the hydraulic conductivity is 10−6 m s−1 [Wong et al., 2009], L is 2 m, ϕmin is 0.7, 〈ϕ〉 is 0.8 [Parent and Ilnicki, 2003], and the consolidation coefficient is set to the value above for peat, then the plateau would be expected to subside approximately 0.1 m from drainage alone. The timescale governing the process depends on the initial peat plateau ice content, the duration of surface heating, and the location of subsurface drainage routes. For the case of no drainage previous to thawing, the postthaw plateau soil having an initial porosity of 0.9 and drainage occurring 2 m below ground, equation (48) gives τsub ≈ 17 days. For drainage occurring during thaw, equation (58) indicates that a thaw front would progress 2 m during the order of a year for a surface temperature fixed at 5°. For this example, subsidence pits that drain during the thaw process would therefore realistically be expected to take several years to form completely.

[35] In the years 1951–1966 and 1976–1983, commercial drainage of peatland in west Norway resulted in subsidence of the ground surface [Gronlund et al., 2008]. The mean depth of the peat was 3.68 m, and they argue that “compaction was likely to be the dominant component” in ∼60 cm of the resulting subsidence occurring between 1951 and 1966, the other component being soil loss due to conversion of soil carbon to CO2. Using the same parameter values as above except for L ≈ 3.7 gives S ≈ 45 cm, consistent with the Gronlund et al.'s conclusion.

5.2. Numerical Model

[36] The analytic solutions discussed above are limited to 1D conditions, and to scenarios in which the permeability varies only weakly with drainage. The numerical simulations are not bound by either of these constraints, and contain the additional effect of subsidence associated with the ice-liquid phase change. Two-dimensional effects are evident at the interface between the IRZ and the outside medium. In all simulations, soil subsidence is first greatest at the edges of the IRZ, as this zone thaws from the edges inward. However, we note that by concentrating more of the ice toward the interior of the IRZ, such as in the hexagonal IRZ simulation, the increased edge subsidence can be delayed and possibly eliminated. It is important to note that the thaw time (equation (58)) and subsidence time (equation (48)) can be calculated analytically far outside or far inside of the IRZ, because the problem is approximately 1D in such regions; however, these formulas do not apply to areas that show lateral inhomogeneity of material properties, such as the area of significant extent between the interior and exterior domains. The simulation shown in Figure 12 includes effects of strongly varying permeability; the decrease in permeability of the IRZ significantly decreases the rate of fluid drainage from that area.

[37] Although the IRZ geometries employed in the numerical model were chosen partly for numerical convenience as well as to study the case of large subsidence, there are some areas where Arctic soils with large amounts of excess pore and segregated ice cover an area comparable to or greater than that used in the numerical simulations [Pollard and French, 1980].

[38] Another feature of the numerical model that was chosen partly for convenience is the constant temperature top boundary condition. Land surface temperatures in the Arctic show large seasonal variability, and therefore this boundary condition is unrealistic. There are two reasons why using different boundary conditions may not necessarily affect the subsidence timescales and degree of subsidence predicted by our models. First, when the drainage route is located at depth, no drainage occurs until the thaw zone reaches that depth. Once drainage does begin, the timescale and final subsidence are still governed by equations (48) and (31) as long as freezing does not reinitiate during drainage. Such may often be the case, because the drainage timescales can be as short as weeks, and prolonged above freezing surface temperatures seem likely given the current Arctic warming trend. Second, for the case of seasonally varying but gradually warming surface temperatures, equation (58) still provides a minimum estimate of the time for a thaw front to reach the drainage face; if fluid subsidence occurs incrementally during the thaw process, formula (68) can still be used to estimate the maximum possible subsidence.

6. Conclusion

[39] We have introduced analytical and numerical models that capture different aspects of permafrost degradation and subsequent soil subsidence. Analytic models can provide characteristic thaw and subsidence timescales, as well as subsurface porosity and permeability profiles for approximately 1D scenarios such that the permeability does not vary strongly spatially or temporally. The analytic results obtained with varying permeability can be reproduced by a model using constant permeability for an appropriately chosen average value of this parameter. Hence, the 1-D formulas may be applicable in the case of strongly varying permeabilities if appropriate average values are chosen. An example application of these formulas to subsidence in the Hudson Bay Lowland has been given. Also, subsidence predicted from our model has been compared favorably with observed subsidence from commercially drained peat. Numerical models overcome the limitations of the analytical model and compare favorably with analytical results when such a comparison is meaningful. By incorporating such a model into a regional-scale numerical landscape evolution scheme, the part of subsidence due to peatland drainage may be used to calculate subsurface heat flow more accurately and thus to improve estimates of total greenhouse gas release.






liquid dynamic viscosity.


pore pressure.


depth in the analytic model; distance from drainage face in the numerical model.


fluid density.


gravitational acceleration constant.


cross-sectional area.






distance from the drainage face of the drained zone boundary.


constant of proportionality between Δϕ and Δp.


constant of proportionality between Δk and Δp.


soil matrix compressibility.


length of soil column.

math formula

dimensionless pressure.


dimensionless depth.


dimensionless time.


perturbation parameter.


Fourier coefficients of the initial pressure profile.


Fourier coefficients of the perturbed solution.


shorthand for sin (nπχ).


shorthand for cos (nπχ).


height of a column element.


amount of soil subsidence.

math formula

dimensionless subsidence.


space part of the analytic solution with variable α.


time part of the analytic solution with variable α; temperature in the calculation of τthw.


Bessel function of order n.


(dimensional) time to total subsidence.


(dimensional) pure conductive timescale.


(dimensional) thaw timescale.


heat flux.


latent heat of fusion.


horizontal volumetric fluid flux.


thermal diffusivity.


consolidation coefficient.


hydraulic conductivity.


[40] This work was supported by LANL Laboratory Directed Research and Development Project 20120068DR: Predicting Climate Impacts and Feedbacks in the Terrestrial Arctic. We thank Sai Rapaka for his helpful suggestions regarding the analytic solutions and for setting up the finite difference solution shown by the black line in Figure 2. This project was also funded by the NGEE Arctic project. The Next-Generation Ecosystem Experiments (NGEE Arctic) project is supported by the Office of Biological and Environmental Research in the DOE Office of Science.