On unsteady flows of pore pressure-activated granular materials
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
On unsteady flows of pore pressure-activated granular materials Anna Abbatiello, Miroslav Bul´ıˇcek , Tom´aˇs Los , Josef M´ alek
and Ondˇrej Souˇcek
Abstract. We investigate mathematical properties of the system of nonlinear partial differential equations that describe, under certain simplifying assumptions, evolutionary processes in water-saturated granular materials. The unconsolidated solid matrix behaves as an ideal plastic material before the activation takes place and then it starts to flow as a Newtonian or a generalized Newtonian fluid. The plastic yield stress is non-constant and depends on the difference between the given lithostatic pressure and the pressure of the fluid in a pore space. We study unsteady three-dimensional flows in an impermeable container, subject to stick-slip boundary conditions. Under realistic assumptions on the data, we establish long-time and large-data existence theory. Mathematics Subject Classification. 76D03, 76D05, 35Q30, 35Q35. Keywords. Granular material, Plastic solid, Non-Newtonian fluid, Implicit constitutive equation, Long-time and large-data existence, Weak solution.
1. Introduction The purpose of this study is to investigate mathematical properties of a system of nonlinear partial differential equations (PDEs) developed in [2] to describe processes, such as static liquefaction or enhanced oil recovery, in water-saturated (geological) materials. Such materials can be viewed as two component mixtures consisting of a granular solid matrix and a fluid filling the interstitial pore space. More specifically, we investigate the following system of PDEs: div v = 0, m s
(∂t v + div (v⊗v)) = ∂t pf + v · ∇pf =
div S − ∇p + m s b , KΔpf − div(Km f b)
+ ∂t ps + v · ∇ps , 1 v f = v − φ(p − pf ) (∇pf − m f b) , α
(1.1a) (1.1b) (1.1c) (1.1d)
where S and Dv satisfy Dv = O ⇒ |S| ≤ τ (pf ), with τ (pf ) := q∗ (ps − pf )+ . Dv + Dv + 2ν∗ (|Dv| − δ∗ ) , Dv = O ⇒ S = τ (pf ) |Dv| |Dv|
(1.1e)
The system (1.1) coincides with equations (2.24)–(2.26) stated in [2] (see also [7]) provided that we set δ∗ = 0 in (1.1e) and we identify the symbols v and pf with v s and ptf used in [2]. In [2], Eq. (1.1) are summarized at the end of Sect. 2 as the outcome of derivation starting from the general principles of the theory of interacting continua, also using several well-motivated simplifying assumptions. In (1.1), v represents the velocity of the granular solid matrix, v f is the velocity of the interstitial fluid, p stands for the total pressure of the whole mixture and pf is the pressure of the fluid in a pore space. The 0123456789().: V,-vol
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Fig. 1. Representation of the material response described by (1.2)
vector and scalar fields v, v f , p and pf represent the unknowns, the other quantities are given material − pf ) is the porosity given as a function of the “effective” functions/parameters. More precisely, φ = φ(p m m pressure p − pf , s and f are the constant material densities of the
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