Kinetic approach for modeling salt precipitation in porous-media

Contents

1 Introduction
1.1 Motivation
1.2 Description of the process
1.3 Structure of study

2 Fundamentals
2.1 Porous medium:
2.1.1 Flow in the porous medium:
2.1.2 Multiphase flow in the porous medium:
2.2 Multiphase fluid velocity:
2.2.1 Hydrodynamic dispersion:
2.3 General mass balance for multiphase flow systems:
2.4 Chemistry
2.4.1 Description of Brine:
2.4.2 Important chemical definitions:

3 Model Concept
3.1 Main aspects:
3.2 Thermodynamic equilibrium
3.3 Mass transfer between the phases:
3.3.1 Fluid - Fluid Phases:
3.3.2 Fluid Phase - Solid Phase:
3.4 Transport Equations
3.4.1 Energy balance equation
3.4.2 Mass balance equations
3.4.3 Mass balance for the precipitated NaCl
3.4.4 Source, Sink terms for the model
3.5 Supplementary equations
3.6 Constitutive relationships
3.7 Primary variables
3.8 Numerical Discretization
3.8.1 Spatial Discretization:
3.8.2 Time Discretization:
3.8.3 Solution of the discretized equations:

4 Results
4.1 Scenario setup:
4.2 Scenario 1: Decoupled model - Isothermal case
4.2.1 Results and Discussions:
4.3 Scenario 2: Decoupled model - Non-isothermal case
4.3.1 Scenario setup
4.3.2 Results and Discussions:
4.4 Scenario 3: Comparison of the kinetic approach with the equilibrium approach

5 Summary
5.1 Summary
5.2 Future work

A Values of the parameters used for simulation
A.1 Brooks and Corey parameters
A.2 Debye Huckel constant values
A.2.1 Debye Huckel constant values
A.2.2 Truesdell and Jones constants

List of Figures

1.1 Saline and sodic soils in EU Zone

1.2 World map representing countries with salinity problems. See¹

1.3 Effect of Tsunami over land salinization

1.4 Representation of stages of salt precipitation in the porous media

2.1 Figure depicting transformation from pore scale to REV scale

2.2 pc,S α ,kr relations by Brooks-Corey and Van-Genuchten

2.3 Solubility of NaCl in water variation with temperature

3.1 Exchange of components in different phases

3.2 Unstrctured 2D BOX mesh, from Bielinski-⁸

4.1 Developed model for analysis

4.2 Temperature along the domain

4.3 Saturation of liquid initially and after 17 days

4.4 Mass fractions of Na and Cl after 10.5 days

4.5 Solidity after 10.5 days

4.6 Mass fractions of Na and Cl after 16.5 days

4.7 Solidity developed after 16.5 days

4.8 Mass fractions of Na and Cl after 6.5 days

4.9 Solidity developed after 6.5 days

4.10 Mass fractions of Na and Cl after 13 days

4.11 Solidity developed after 13 days

4.12 Mass fraction of components after 6.5 days in the wetting pahse

4.13 Mass fraction of components at a constant solidity of

4.14 Effect of salt precipitation on the porosity (13 days)

4.15 Effect of salt precipitation on the permeability (13 days)

4.16 Temperature variation at intial and final stages

4.17 Mass fractions of Na and Cl after 11.5 days

4.18 Solidity developed at the surface of the medium after 11.5 days

4.19 Mass fractions of Na and Cl after 21 days

4.20 Solidity developed at the surface of the medium after 13 days

LIST OF FIGURES

4.21 Solidity developed at the surface of the medium after 13 days by equi- librium approach

4.22 Comparison between the kinetic and equilibrium approaches over the solidity formation

A.1 Constant values for finding the activity coefficients. Debye- Huckel con- stants

A.2 Constant values for fing the activity coefficients. Truesdell and Jones constants

List of Tables

1.1 Saline soils in the world

2.1 Solubility of NaCl in water variation with temperature

3.1 Adaptive choice for the primary variables

4.1 Model parameters used for the approach

4.2 Component analysis

4.3 Precipitation rate constants

A.1 Brooks and Corey parameters

Nomenclature

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Chapter 1 Introduction

1.1 Motivation

Soil salinization is one of the major challenges faced by the world these days. Salinization is defined as the process of increasing the salt content in the soil, mainly caused due to the evaporation of the water used for irrigation. The accumulation of salt in the porous medium which poses a serious threat to agriculture, is seen as a common problem caused by the salt accumulation. All around the world, the salinity affected areas are estimated to be as given in Table 1.1.

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Figure 1.1 describes the saline and sodic soils in the EU zone and Figure 1.2 repre- sents the nations having problem with saline soils all over the world. It is assumed that around 77 million hactares of soil is salinized by the human activities. Figure 1.3 shows the natural case of a Tsunami responsible for soil salinization along the coasts.

Though the process of soil salinization had been taking place over decades, the research studies have been increased over in the recent times. This is due to the increased practices in the agricultural sector which are supplemented by the irrigational efforts that are inturn dependent on saline ground waters, taken from⁴. The extent of the

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Table 1.1: Saline soils in the world.

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Figure 1.1: Saline and sodic soils in EU Zone.

problem is magnified, when poor drainage conditions lead to an increase in the salt content of the soil because the accumulated salt above the soil surface is not flushed out regularly and piles up further. A group of studies conducted by Sharma, Prihar (1973) have shown that excess irrigation water, when subjected to higher rates of evaporation would result in salt precipitation. See⁵.

On the porous medium side, the salt is transported under the influence of the viscous, capillary, gravitational forces and by the advective, diffusive fluxes. However, to depict reality, the influence of the free flow needs to be considered, with effects such as the wind velocities, air temperature which can have a direct impact on the evaporation rates.

The salt can be precipitated in two forms: Efflorescence which describes the salt precipitation over the surface of the soil, Subflorescence which describes the salt precipitation within the porous medium, see¹². Salt precipitation weakens the porous medium as the amount of expansion and contraction of the salts due to the temperature variations are comparitively higher than the silicate minerals thereby leading to erosion. Furthermore, the developed salt crystals excert an additional pressure over the surface of the silicate minerals that could lead to erosion. See².

In this independent study, we attempt to describe and determine the chemical parame- ters that have an influence on the salt precipitation. Furthermore, we make a compari- sion of the developed kinetic model with the available previously developed equilibrium

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Figure 1.2: World map representing countries with salinity problems. See¹.

numerical model. The model is implemented on DuMux, in the numerical framework of an open source simulator for the description of flow and transport in porous medium. It is built on top of DUNE (Distributed and Unified Numerics Environment). DuMux also includes a variety of non-linear solvers including the management of sophesticated time steps and thereby increasing the computational efficiency. Modularity is an important aspect of DuMux where various parameters can be changed in the input according to a specific case, see⁶. Additionally for any further work, the code implemented can be effectively made use of or developed.

1.2 Description of the process

Salinity is the degree to which water contains dissolved salts. Usually it is expressed in terms of ’parts per thousand’.

Salinization is the accumulation of soluble minerals like NaCl, CaCO3, CaSO4, KCl in the soil.

According to the studies performed by Fisher (1923), Coussot (2000), Lehmann (2008) and Nachshon (2011), the phenomenon of pure water evaporation from a homogeneous porous medium in the absence of constant water supply was described. The entire process was classified into three stages:

Stage 1: In the initial stage the porous medium is fully water saturated. The water is subjected to high evaporation rates which are capillary-driven. The evaporation continues until there exists a hydraulic conductivity between the receding drying front and the free flow - porous medium interface.

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Figure 1.3: Effect of Tsunami over land salinization.

Stage 2: As the progression of the Stage 1 continues, there comes a point where the hydraulic conductivity gets disturbed due to the falling water levels. There is a restricted evaporation rate which is governed by the vapor diffusion through the porous medium.

Stage 3: This stage is marked by further decrease in the evaporation rates gov- erned by vapor diffusion. The fall of the evaporation rates is attributed to the fall of the water levels to greater depths and the cut down in the hydraulic conduc- tivities between the receding water front and the interface of free flow - porous medium.

There had been observations related to extended S1 stage by Lehmann et al.(2008) due to higher capillary forces in the finer parts of the porous medium and other heterogeneous conditions.

But under normal conditions there excists a certain portion of salt in the form of dissolved minerals in water. Studies of Shimojimaa (1996), Horton (1999), Fujimaki (2006) have made an analysis of salt precipitation under homogeneous conditions. Nachshon et al. (2011) made a study of salt precipitation under the heterogeneous conditions and classified the process into three stages:

Salt Stage 1: This stage is assumed to be initially salt water saturated. There are higher rates of evaporation with minor decrease in the evaporation rates due to the osmotic pressure variations caused due to the difference in the concentration of solutes between the different fluid systems in contact (i.e Air and Water).

Salt Stage 2: As the salt water starts to evaporate, there is a considerable quan- tity of salt that gets precipitated at the porous medium - free flow interface and thereby blocking the free evaporation of the water and thus resulting in decreased rates of salt water evaporation, compared to the stage 2 of evaporation rates with pure water.

Salt Stage 3: This stage exhibits a further drop in the rates of evaporation which are governed by the diffusion across the accumulated salt crust.

The various stages are represented in Figure 1.4 taken from⁹. The colors represent the following phases, grey - soil, blue - water and white - air. The green areas show the precipitated salt. The solid line above the porous medium represents the water trans- ferred to the free flow domain by diffusive flux through soil and the dashed line shows the water brought to the free flow domain by the diffusive flux through precipitated salt.

Numerical modeling is a powerful tool to make complex analysis and transfer the data to an application scale. The application of numerical modeling for the above discussed is a challenging task. There have been many studies by scientists: Clement (2001), Woods (2005), Lehmann and Or (2009), Bechtold et al. (2011) to develop mathematical models and perform the numerical simulations for the above discussed. Developing a numerical model for the above described has been the main scope of this study.

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Figure 1.4: Representation of stages of salt precipitation in the porous media.

1.3 Structure of study

The content of this report is divided into five chapters:

Chapter 1, gives the motivation part which includes introduction, the process of the salt precipitation in the porous medium, scope of study, objectives of the study.

The second chapter of the study deals with review of literature, relevant laws, definitions and various processes concerning the porous medium.

Chapter 3 of this report, comprises the brief description of the conceptual model used in the study along with the assumptions involved in the study, transfer of mass between phases, transport equations and supplementary equations.

Chapter 4 deals with data analysis and results for various scenarios.

The final chapter, gives the summary of the study, conclusions and scope for further research.

Chapter 2 Fundamentals

This chapter gives a brief description of the porous medium, flow in porous medium, and information related to the precipitation of salt in the porous medium. Additionally, a brief description of the chemistry driven approach used in the course of this work is specified in this chapter.

2.1 Porous medium:

Porous medium is a skeletal solid matrix with pores within. Important parameters related to the porous medium are discussed below. In the model there are two main phases which are liquid and gas.

2.1.1 Flow in the porous medium:

The flow in the sub-surface is through the interconnected pores. The pores are either occupied by liquid or gas phase. The pore scale is difficult to be resolved and work upon. So, the porous medium is transferred to a parameter averaged Representative Elementary Volume (REV). The REV scale introduces a few extra features of description such as the porosity, permeability etc. These parameters are essential to describe the interactions in the porous medium on the REV scale. Figure 2.1 depicts the tranformation from the pore scale to a REV scale.

Porosity: The porous medium is comprised of the solid matrix and the pores. The ratio of the volume of pores to that of the entire volume of the REV is called the porosity.

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(2.1)

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Figure 2.1: Figure depicting transformation from pore scale to REV scale.

Permeability: The intrinsic permeability, K relates to the potential gradients of pressure, ▽ p with the respective Darcy flow velocity, v for a given fluid with a viscosity, μ . This parameter is dependent over the porosity, spatial distribution and the connectivity of pores in the porous medium.

(2.2)

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Where p is the pressure gradient, ρ is the fluid density and g is the gravity vector.

2.1.2 Multiphase flow in the porous medium:

The concept of multiphase flow analysis is brought into light, when there are more number of immiscible components that form distinctive phases. The analysis of the multiphase flow systems requires additional equations describing the interactions between the different fluid phases.

Saturation: In a multiphase system, pores are occupied by two or more phases. Saturation is a parameter that describes the volume of a specific fluid to that of the total pore volume in the medium.

(2.3)

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The limiting case related to the saturation ( S α ) is that, the sum of the saturations of the fluids in the porous space is equal to one. This is because the pores would be filled in with at least any of the fluid.

(2.4)

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where: α refers to the specific phase saturation and the number of phases are represented by n.

Capillary pressure: For multiple immiscible phases in a medium, surface tension is observed at the sharp interface which is formed due to the cohesive forces within the perticular phase and adhesive forces at the interface between different phases. To balance the mechanical equilibrium, it is noted that the pressures for the wetting and the non-wetting phases are different. The difference between the pressures of the non-wetting phase to that of the wetting phase is called the capillary pressure.

(2.5)

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The relation of the S α , kr and pc are defined by the Brooks-Corey and Van-Genuchten graphs. See[7].

Following gives a brief description about the Brooks-Corey and Van-Genuchten rela- tions.

2.1.2.1 Brooks-Corey:

The emperical description for the capillary pressure is given as:

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(2.6)

2.1.2.2 Van-Genuchten:

The emperical relation for determing the capillary pressure by the Van-Genuchten method is:

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(2.7)

where, Swr is the residual wetting phase saturation, λ in the Brooks-Corey is the pore size distribution dependent function, pd is the entry pressure where the non-wetting phase enters the largest pore. In the Van Genuchten approach α , m, n are the fitting parameters. The comparision for both the analysis is depicted in Figure 2.2.

Relative Permeability: The relative permeability, kr, α is the ratio of the effective permeability of a particular phase to that of the absolute permeability. The relative permeability can be expressed as a function of saturation. ([illustration not visible in this excerpt]). This parameter also includes a host of other features such as the increase in the tortuosity, reduction in the cross sectional area of a fluid due to the presence of other fluid in the media and also the preference of the pore diameter for the fluid to flow according to the wetting phase fluid behaviour. The relative permeability can be computed by two methods. Brooks-Corey and Van-Genuchten approaches, taken from[7].

2.1.2.3 Brooks-Corey:

For the wetting phase, the relation between the wetting phase relative permeability and the saturation is as follows:

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(2.8)

and for the non-wetting phase, the relation between the non-wetting phase relative permeability and the saturation is as follows:

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(2.9)

where, kr,w and kr,n are the relative permeabilities of the wetting and the non-wetting fluids respectively and the effective saturation ( Se ) is given by:

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(2.10)

2.1.2.4 Van-Genuchten:

The other method to determine the relative permeability is by the Van-Genuchten formulation which is described as follows:

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(2.11)

and

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(2.12)

where, m and n are Van-Genuchten parameters. The relation for both the methods are graphically shown with respect to the saturation in Figure 2.2.

2.2 Multiphase fluid velocity:

On the REV scale, Darcy velocity is used for analysis. The Darcy velocity is the flux divided by the cross sectional area of flow. In the multiphase flow analysis, it

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Figure 2.2: pc, S α , kr relations by Brooks-Corey and Van-Genuchten

is required that the relative permeability needs to be considered into the original equation of the Darcy velocity due to the presence of different fluids. Thereby, the transformed equation for the Darcy velocity is given as:

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(2.13)

where, v α is the Darcy velocity in the REV scale, μα is the viscosity and ρα is the density of the phase α . The effect of the gravitation is denoted by g, ▽ p represents the pressure gradient. Gravity and the pressure gradients are the driving forces of the fluid.

2.2.1 Hydrodynamic dispersion:

The effect of different velocities are pronounced due to the presence of heterogenities along with differences in the pore size distribution in the porous medium. For analysing the distribution of the components subjected to varied velocities, Hydro- dynamic dispersion is an effective tool to describe this behaviour. On an averaged REV scale, the phenomenon of dispersion can be analysed as diffusion and also has a dependency on the velocity of flow indicating the dependence of the advective portion. In general the hydrodynamic dispersion coefficient of the porous medium ( Dpm, α ) is expressed as:

(2.14)

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In the above described equation, the first part refers to the molecular diffusion whereas, the second and the third parts refer to the velocity defined mechanical dispersion in the porous medium. Parameters in the equation describe: Dk pm, α referstothedispersion coefficient of the component k in phase α , in the porous medium. Dk α referstothe molecular diffusion coefficient of the component k ; τ pm refers to the tortuosity of the component in the porous medium which is computed by the parameters of saturation and porosity as:

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(2.15)

where, φ is the porosity of the medium. The v e, α is the effective velocity that describes the flow through the porous medium. It is defined as the Darcy velocity divided by the porosity. v e, α = v αφ . As a result, the effective velocity is always greater than the Darcy velocity because the posity is lesser than one. Thereby, the effect of reduced cross section flow is accounted for in the equation.

2.3 General mass balance for multiphase flow sys- tems:

In general, a fluid phase in the porous domain can be described by a mass conservation equation, which consists of four major parts in the equation namely: Storage; Advection; Dispersion and the Source/Sink terms.

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(2.16)

where, k stands for the component under analysis in the phase α . S α stands for the saturation of the medium with the phase α .

2.4 Chemistry

For the developement of model by kinetic approach, various parameters that have an influence over the salt precipitation are considered under this section. A brief description about brine and other formulas that were used for the model are discussed.

2.4.1 Description of Brine:

Brine is a solution of salt in water. In the scope of this study, it is assumed that the dissolved solution is exclusively Brine. Brine consists of Sodium and Chloride ions as it’s main composition along with water. The solubility, defines the amount of salt that could be dissolved into a certain quantity of water. It is an important basic factor of description. For NaCl the value of solubility is around 360 g/L. Solubility is

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Table 2.1: Solubility of NaCl in water variation with temperature.

a paramter that depends on various other features such as the:

1. Forces between particles: The two main forces inbetween the solute and the solvant are the intermolecular and interionic forces. For the dissolution of the solute to take place, the molecules of the solute must overcome the forces of attraction of the solvant. Furthermore, the molecules of the solvant must have sufficient attraction for the solute particles to contian the solute particles within.
2. Temparature: The solubility of NaCl increases with the increase in temperature. Table 2.1 shows the change in values of solubility with respect to the temperature. Figure 2.3 shows the graphical variation of the solubility with respect to temperature.
3. Pressure: The effect on the solubilities of solids and liquids based over the appli- cation of pressure over the solution is less pronounced than that for the gases in liquids.

2.4.2 Important chemical definitions:

In this portion, some basic terminology in terms of chemistry and their overview is explained with the help of equations.

Mole: A mole is a chemical mass unit, defined to be 6 . 022X10²³ molecules (or) atoms. Molar mass: The mass of a substance in grams, divided by the quanity of the substance in number of moles gives the molar mass of a substance.

2.4.2.1 Mole Fraction:

The mole fraction ( xi ) is defined as the ratio of number of moles of a specific component to that of the total number of moles in the entire solution.

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Figure 2.3: Solubility of NaCl in water variation with temperature.

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(2.17)

2.4.2.2 Mass Fraction:

The mass fraction is defined as the ratio of the mass of the specific component to that of the total mass of the substance.

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(2.18)

The fractions are dimensionless as they are compared to the same parameter.

2.4.2.3 Concentrations:

The molar concentration (c) is defined as the ratio of the number of moles of a specie to that of the total volume.

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(2.19)

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