## 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.