CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 4, Number 1, Winter 1996 PRIMARY FROST HEAVE: A QUASI-STEADY APPROACH ZHENZHONG DING AND FEDERICO TALAMUCCI 1. Introduction. When a soil saturated with water is subject to a freezing process, three different zones are usually observed: (i) a completely frozen zone, i.e., a mixture of soil grains and ice, possibly including "lenses" of pure ice, (ii) an unfrozen zone, where. water fills the pores and migrates through them, and (iii) an intermediate region where ice and water coexist with soil grains. This last part is the so-called "frozen fringe." From a mathematical point of view, models for freezing in saturated porous media can be grouped in two main categories: models in which the fringe is neglected and a sharp interface is assumed to exist between the frozen and unfrozen zones (class A, according to [5]) and models in which the change of phase is assumed to occur over a fringe of finite width (class B). These two classes of models correspond to "primary frost heave," and "secondary frost heave," models, respectively. Indeed, suction of water towards the freezing front often causes the soil swelling, a phenomenon that is relevant in many practical cases. In this paper we present a comprehensive discussion of models of class A, starting from the basic assumption. As a matter of fact, some of the models of primary frost heave used in the literature are not fully justified, or are simply incorrect. For the sake of simplicity, we restrict our attention to one-dimensional cases that cover many of the large scale applications. The models are summarized in Section 4. In the second part of the paper (Sections 5 and 6) we present explicit solutions to special problems in the quasi-steady approximation, i.e., when each phase can be considered at thermal equilibrium at any time. Primary frost heave does not represent a realistic and complete description of the phenomenon, but it represents a simple approach to the problem and also a reasonable approximation, at least when the Received by the editors in revised form on September 21, 1995. Work partially supported by the USAERO/ECMI research contract DAJA 45- 934-0046. Copyright 01996 Rocky Mountain Mathematics Consortium 31 32 Z. DING AND F. TALAMUCCI thickness of the fringe is negligible with respect to the length scale of the process studied. We believe that models of class A can be regarded as a first stage in the study of ground freezing. Indeed, when models of class B are concerned, there is no consensus on the laws governing the evolution of the frozen fringe, and an accurate discussion of primary frost heave could allow for discriminating among different models for secondary frost heave, checking whether or not the corresponding model of class A can be recovered as a limiting case. 2. Mass and energy conservation. The one-dimensional model of class A examined in this paper is based on the following assumptions: i) the porous medium is saturated with water and chemically inactive; ii) the porous matrix in the unfrozen region is undeformable and at rest. It represents the frame of reference with respect to which the equations are written; iii) all water that reaches the frost line freezes instantaneously. Taking an upwards-directed vertical coordinate z, let z = z~ and z = zs denote the bottom and the top (soil surface) of a soil sample. While z~ is a constant, as will be an unknown function of time, because of the heaving process. The position of the freezing front is given by the unknown function z = z~(t).The region z~ < z < zF(t) is the unfrozen part of the soil, while z~(t)< z < zs(t) is totally frozen. At z = zg and z = zs the temperature will be prescribed as well as a boundary condition of mechanical type as, for instance, the pressure of water at E = ZB and the force per unit area acting on the soil surface (e.g., equal to the atmospheric pressure). In the following, suffixes u and f refer to the unfrozen and frozen soil, respectively, while we will use suffixes W, I, S to denote water, ice and soil. The density p and the porosity E are related in the following way: If q represents the volumetric velocity of water (i.e., the volume passing through a unit surface in a unit time), conservation of mass PRIMARY FROST HEAVE 59 pores and reaches the lens, which will increase in thickness. Thus, we can have a situation (described by a primary frost heave model) where a lens can grow "behind" the freezing front, and not "at" the freezing front. 7. Conclusions. Recalling conditions (5.11) and (5.19) and taking a frame of reference with zs = 0, ZF = b, zs(0) = c (the same considered in ii)), we have that, when the boundary temperatures ho and go are constant, the conditions for lens formation are -ho < -- < -ho + (lens formation), b C-b b where (we take 7 = 1for simplicity; hence, pi = pw = p). Equation (7.1) can be read as a double condition for the temperature go, once ho and pw(O, t) has been prescribed; indeed, go must be lower than an upper threshold in order to allow the freezing process to take place, and greater than the critical value in order to avoid frost penetration. On the other hand, condition (5.24), valid for frost penetration, gives (frost penetration). If (7.3) is satisfied, i.e., if go < gb, frost penetration occurs and the water pressure ~w(zF(~),t) takes the critical value -p,. 60 2. DING AND F. TALAMUCCI Note that in this case (go > gl), if one prescribes the critical pressure -pc at the front (and then the freezing temperature TF), melting would occur since the contribution of convection at the separating front is too high to allow freezing. However, in the same last case go > gb lens formation is possible, but obviously pw at the front cannot be prescribed, and it will be PW(%F(~),~)> -PC. Let us summarize the analysis in the following statement: Once ho and pw at the bottom of the soil are prescribed, we have i) go < 91 and pw (ZF(t) ,t) = -pc (hence T (ZF(t), t) = TF)implies frost penetration; ii) gu > go > gl implies lens formation; iii) go > gu implies melting. The example of transition from frost penetration to lens formation that has been analyzed in Section 6 has to be framed in the previous scheme in the following way (although go is not constant): a) the temperature go(t) is lower than gl as long as t < T and condition (7.3) is fulfilled, so that frost penetration can take place for t E [O,7); b) when t = T, go reaches the value gl so that (7.3) is no longer valid, but inequality (7.1) is true for t > T; therefore, ice grows at the freezing front and a lens will form. Remark 1. If we prescribe the thermal fluxes at the bottom and on the top of the soil, we can assign, for instance, where fl and f2 are two (positive) constants. In the case of lens formation, water flux and upper boundary of the .