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A Rigorous Homogenization for a Two-Scale Convergence Approach to Piping Flow Erosion with Deposition in a Spatially Heterogeneous Soil

DOI: 10.9734/JAMCS/2020/v35i330256 Editor(s): (1) Dr. Dragoş Pǎtru Covei, Bucharest University of Economic Studies, Romania. Reviewers: (1) Jackson Akpojaro, University Of Africa, Toru-Orua, Nigeria. (2) Niculescu Mihai Leonida, INCAS National Institute for Aerospace Research Elie Carafoli, Romania. (3) Sie Long Kek, Universiti Tun Hussein Onn Malaysia, Malaysia. Complete Peer review History: http://www.sdiarticle4.com/review-history/56987


Introduction
Several studies on porous media over the decades have been conducted and a lot of papers have been published, addressing the basics of flow and transport through porous media. In most cases averaging and the notion of representative elementary volume is used [1]. A number of studies [2,3] on conceptual models and the theory of mixtures have also been proposed.
Normally, the heterogeneous medium is described as a medium with local parameters that can be described by functions rapidly varying with respect to space variables and time. Homogenization is an approach that allows us to study the macro behaviour of a medium by its micro properties, it therefore seeks to replace a heterogeneous material by an equivalent homogeneous one.
One of the early studies was conducted by [4] and [5]. They respectively studied effective conductivity of a media with small concentrations of randomly and periodically arranged inclusions. Effective viscosity of suspensions in compressible viscous fluids was also investigated by [6]. A general approach based on asymptotic tools which can deal with a variety of different physical problems was also introduced later by [7].
The homogenization procedure is in two forms, namely the formal homogenization and rigorous homogenization. The former is based on construction of asymptotic expansions using multiple scales. At least two natural spatial length scales are introduced. One measuring variations within one period cell (the fast scale) and the other measuring variations within the domain of interest (the slow scale). The effectiveness of the use of multiple space scales to treat systematically boundary value problems with rapidly varying periodic structure was established by [8].
The latter is also based on energy estimates. Since the coefficients of the equations involved are rapidly oscillating and the derivatives are multiplied by the characteristic length scale ϵ which is the ratio of the microscopic length scale to the macroscopic length scale, obtaining estimates independent of ϵ is very difficult. To achieve this however, one must pass to the limit in weak sense by using integration by parts and suitable test functions [9,10]. This process acts as a rigorous justification of results normally obtained using the formal homogenization [11,12].
In this paper we model the piping flow erosion phenomena in a spatially heterogeneous soil and justify mathematically by rigorous use of periodic homogenization thus showing that the results obtained from formal homogenization can be verified through this rigorous homogenization process via two scale convergence.
In what follows we give a few mathematical theorems and lemma [12] useful in the quest to rigorously obtain a homogenized model for the piping flow erosion via two-scale convergence.

Important theories on homogenization
For a bounded sequence u ϵ ∈ L 2 (ΛT ), there exist a subsequence and a function u0 ∈ L 2 (ΛT × Y ) such that u ϵ two scale converges to u0. However, u ϵ converges weakly in L 2 (ΛT ) to the average of the two scale limit over the unit cell : (., ., y)dy, weakly in L 2 (ΛT ) 2 Rigorous Homogenization of the Microscopic Problem via Two-scale Convergence

Choice of the micro structure and the microscopic problem
A bounded heterogeneous soil structure Λ in R 2 of coordinates x = (x1, x2) with periodic positioning of pores through which a water/soil particles mixture of volume Λ f flowing through a soil matrix Λs with a purely geometrical fluid/soil interface ∂Λs of no thickness. The spatial variable x is a macroscale (global) variable. We take in the space R 2 a unit cell Y with a microscale (local) variable y = (y1, y2) and define a characteristic length scale ϵ = l L with yi = x i ϵ for i ∈ {1, 2}, l and L denotes the characteristic length of the unit cell Y and the soil domain Λ. The reference unit cell Y has two pairwise disjoint connected domains Y s and Y f with smooth fluid/soil boundary ∂Y s . A repeating arrangement of copies ϵY occupying the entire region Λ as shown in Fig. 1 was created.

Function spaces and unknowns of the microscopic problem P ϵ
We define a parameterized domain Λ ϵ and discuss briefly the function spaces and few technical assumptions on the unknown objects in the microscopic model.

Function spaces
For ϵ = l L > 0 as in Fig. 1, x ϵ ) = Eu(t, y). The notations used in this section are listed as follows:

Unknowns and parameters of the microscopic model
The unknowns in the microscopic model are: Concentration of soil particles in water/soil mixture C ϵ The parameters in the model are: As a working assumption we assume the following: We thus have the dimensionless coupled equations at the microscopic level for the piping flow erosion as

The weak formulation of the microscopic problem
On the weak formulation of the microscopic problem we postulate definition 2.1

The ϵ−independent prior estimate (Energy Estimates)
In this section and the next, the well-posedness of the microscopic problem is discussed. We prove the existence of solution to (2.9) and (2.10) thus postulate Lemma 2.1.

Compactness results (Two-scale)
From the boundedness results obtained and theorem 1.3, we have the following results: there exist ϵ ⇀û0(t, x, y) weakly, p ϵ ⇀p0(t, x, y) weakly, C ϵ s ⇀Ĉs0(t, x, y) weakly and (iv) u ϵ ⇀ũ0(t, x) strongly, p ϵ ⇀p0(t, x) strongly, C ϵ s ⇀Cs0(t, x) strongly consequently uniqueness implies 2.6 Passing the microscopic problem to ϵ → 0 via two scale convergence which is the weak form of the limit two scale problem. Next we expand and shift the derivatives from the test functions: Now we let Π0 = 0, Π1 = 0 within F ×Λ, F ×Λ×Y and their boundaries we obtain the homogenized equation: It must be noted that (2.14) and (2.15) are respectively the macro problem and cell problem.
Similarly from the weak formulation in (2.9) we have We choose test function ) and obtain: this further expands to Next we expand and shift the derivatives from the test functions Choosing ω0 = 0, ω1 = 0 within F × Λ, F × Λ × Y and their boundaries we get Equations (2.16) and (2.17) coincides with the equations obtained through the formal homogenization, these are respectively the homogenized macroscopic model and the equation at the unit cell level from which the cell problem can be obtained. We have thus shown that the results obtained from the formal homogenization can be verified through this rigorous homogenization process via two scale convergence.

Numerical Computation of the Homogenized Macroscopic Problem
The model is used to simulate the piping flow erosion with deposition in a highly erodable soil under a tangential flow instigated by seepage of water through the embarkment. We imposed a constant pressure drop between inlet and outlet with a constant flux at the inlet, the tangential velocities are assumed continuous across ∂Λ. The cell problems and macroscopic equations were discretize using the finite element method (Galerkin).The momentum equation was decoupled using the Incremental Pressure Correction Scheme (IPCS) [17,18].
The numerical task performed include: (i) Computing the cell problem: .n(y) = 0 for i, k = 1, 2,j = 1, 2 and χ c j , χ µ j − Y periodic (ii) Using the results in (i) to compute the Homogenized effective characteristic coefficients: Eudy for i, k = 1, 2,j = 1, 2 (iii) We then solved 2D the Homogenized problem : With concluding equations for deposition and pore space dynamics Initial and boundary conditions imposed are: Numerical values of the parameters used are listed in Table 1 for a rectangular soil domain of dimensions 2m × 1m with compacity C soil = 1 − ϕ. The characteristic erosion time was computed as in [19], ter = 2Lρ p kerpin = 3000hrs. The density of the water/soil mixture ρ = ϕ(ρ p −ρ w )+ρ w , where ρ w and ρ p are water and soil particles density. The time-stepping was performed using Backward Euler Method [20] with a time step δt = 0.2 for 0 ≤ t ≤ 20

Numerical Results and Discussion
Using the Galerkin finite element on an unstructured triangular mesh we divided the soil domain into 10116 elements as in Fig. 2. Define an inflow at x1 = 0 and an outflow at x1 = 2. No slip conditions are imposed on the walls x2 = 0 and x2 = 1. The soil domain was subjected to a constant pressure drop of 0.1. Under a parabolic velocity profile realized from the momentum equation (Fig.  3), the results obtained in terms of percentage increase and decrease in soil particle deposition and entrainment are presented.

Fig. 2. Meshing of the soil domain
We observed a decrease in concentration of soil particles in the flow over the period simulated, to substantiate the extent of decrease we computed the percentage decrease in soil particle concentration in the flow. At the early stages of the simulation, eroded soil particles transforms the flow into a concentrated suspension causing enlargement of the pipe downstream. However these eroded soil particles are entrained towards the outflow, variations in levels of percentage decrease in concentration of particles in the flow as depicted by our model can be seen in Fig. 4. A rapid decrease in soil particle concentration was observed at the early stages of the process with higher levels (almost 50%) close to inflow, this is due to the higher pressure levels at the inflow thereby propagating the eroded particles from inflow towards outflow. It can be seen that, at the later stages of the process the decrease in soil particle is much slower as most of the particles entrained are deposited at the outlet.

Fig. 5. Percentage increase in concentration of soil particles deposited
Deposition of soil particles entrained occurred over the period. We observed further from the percentage increase in deposition computed that, the concentrated suspension at the early stages of the piping flow erosion causes a rapid increase in soil deposition. Though deposition increases, however we observed as in Fig. 5 that this increase is slower with time thus deposition increases at a decreasing rate.

Conclusion
The paper started off with a microscopic system of equations for modelling piping flow erosion with deposition in a spatially heterogeneous soil with periodic positioning of pores. The main aim is to provide a rigorous backing for the formal homogenization results obtained in the earlier paper, thus we proved the energy bound of the weak solution to the microscopic model thereby preparing the way for the rigorous passing of ϵ → 0 via two scale convergence. We have accomplished the task of rigorous homogenization of the microscopic model and have obtained a homogenized macroscopic model with effective coefficients capable of simulating the piping flow erosion phenomena in a spatially heterogeneous soil. A strong observation from the numerical simulation was that, soil particle concentration in the water/soil mixture decreases but at a decreasing rate whereas soil particle deposition increases at regions with increasing amount of particle concentration in the flow causing a reduction in bare pore spaces across the soil domain. These numerical results from the proposed model clearly shows the various trends [21,22] associated with soil particle concentration in the flow and deposition in the piping phenomena.