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Thermodynamics and chemical transport through a deforming porous medium Exploration Geodynamics Lecture David Smith, Glen Peters The University of Newcastle Callaghan, Australia

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At thermodynamic equilibrium and so Chemical reaction Where K = equilibrium constant

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Principle of Minimum Potential Energy Structural Engineering Principle of Minimum Complementary Potential Energy (Castigliano’s Theorem)

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Deformation of a Truss

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Consilience (Edward Wilson) Physics (mechanics) traditionally been separate from chemistry. But they are not really separate. Two forces: gravity and electromagnetic. Electrical and chemical forces (actually merge one into another) Inorganic Chemistry: Shriver and Atkins

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Thermodynamics is required to understand: Solid mechanics ( e.g. fully coupled thermoelasticity ) Material science Geochemistry Interfacial phenomena

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OUTLINE What is thermodynamics? Concept of thermodynamic potentials Briefly discuss 2 applications Darcy’s law and why water flows through soil Brief history of thermodynamics Thermodynamics of dissolution processes Getting the governing differential equations right: Transport through a deforming porous media Coupling between (ir)reversible processes

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Rumford 1782 Carnot 1820’s Joule Kelvin Clausius 1840’s -60’s Helmholtz 1882 Gibbs 1880’s-90’s Caratheodory 1909 Onsager 1930’s Slater 1939 Prigogine 1950’s Katchalsky& Curran Broecker& Oversby 1960’s-70’s Mitchell Collins Houlsby 1970’s,80’s,90’s Truesdell Coleman Noll 1950’s-60’s Bowen 1980

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What is Thermodynamics? One approach to thermodynamics is through the atomic theory of matter (statistical mechanics) -1 gram MW of a substance contains 6.023 x 10 23 atoms or molecules 602,300,000,000,000,000,000,000

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To completely define 1 litre of water, the position and velocity of every nuclei and every electron in the litre of water would have to be specified 6.6 x 10 26 co-ordinates However, the litre of water can be characterized by the temperature, pressure and strength of the electro- magnetic field surrounding the water 3 co-ordinates 6.6 x 10 26 3 Statistical averaging

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Ludwig Boltzmann came up with a way of getting a statistical measure of the likelihood of a particular configurations of nuclei and electrons

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The Second Law of Thermodynamics This equation is crucial because it allows us to concentrate on the system alone Clausius inequality Corollary: At equilibrium, S is maximised

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The Diffusion Equation: Time’s Arrow Time’s Arrow (Arthur Eddington) Leads to counter-intuitive solutions If t

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The Wave Equation The wave equation is unchanged if time is reversed If t

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The First Law of Thermodynamics Change in internal energy Heat flow across the system boundary Work done on the system Definition: Internal energy of the system (U) is the sum of the total potential and the kinetic energies of the atoms in the system

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Thermodynamic Potentials For adiabatic systems, the amount of work required to change the internal energy of the system is independent of how the work is performed The system is dependent on its initial and final states but independent of how it got there Hence the internal energy is a state function (or potential) [A state function (or potential) has a path independent integral between two points in the same state space]

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‘pv’ is also a state function (or potential) Addition (or subtraction) of two potentials gives another potential v p

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‘TS’ is another potential, and this may be subtracted from U (Helmholtz free energy) (Gibbs free energy) ‘TS’ may be subtracted and ‘pv’ may be added to U

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And substituting dU shows (S,p indep. variables) (T,v indep. variables) (T,p indep. variables) (S,v indep. variables) Legendre Transformations

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Irreversible Processes Irreversible Processes where = thermodynamic flux For many slow processes of interest to engineers = thermodynamic force

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Well known thermodynamic fluxes: (Darcy’s Law) (Fick’s Law) (Ohm’s Law) (Fourier’s Law)

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Rates of Entropy Production (Darcy’s Law) (Fick’s Law) (Ohm’s Law) (Fourier’s Law)

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Lord Kelvin postulated existence of a ‘dissipation potential’ (D) A potential implies the general reciprocal relationships between thermodynamics forces and fluxes; These are known as the Onsager reciprocal relationship (Onsager (1931)) Coupled flows (and the Onsager relationships) are important in two phase materials e.g. clays (Mitchell, 1991).

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Zeigler (1983) assumed the existence of a dissipation potential in solid mechanics implying

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Performing a Legendre transform on D implying

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yield condition flow rule If D is a homogeneous function of degree one, then

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Thermodynamic Force (Gradient of Potential) Flow J Hydraulic head TemperatureElectricalChemical concentratio n Fluid Hydraulic conduction Darcy’s law Thermo- osmosis Electro- osmosis Chemical osmosis Heat Isothermal heat transfer Thermal conduction Fourier’s law Peltier effect Dufour effect Current Streaming current Thermoelectrici ty Seeback effect Electric conduction Ohm’s law Diffusion potential and membrane potential Ion Streaming current Thermal diffusion of electrolyte Soret effect Electrophores is Diffusion Fick’s law Couplings of Irreversible Processes

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Couplings through constitutive equations Young’s modulus = Viscosity = Yield surface = Permeability =Resistance = Diffusion coeff. =

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EXAMPLE 1 Darcy’s law and why water flows through soil

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Darcy’s law - flow of water through soil total head where, = Gibbs free energy of an incompressible pore fluid per unit volume

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For water,, the chemical potential is Standard state Pressure contribution Position contribution entropy component thermal component If Hubbert potential (1940)

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Why water flows through soil?

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EXAMPLE 2: Dissolution and precipitation

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supersaturated undersaturated, assume

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EXAMPLE: reactive transport (i.e. transport with precipitation)

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Soil Physical Chemistry 1999: Ed Sparks

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Ion Activity Product varies from soil to soil: Solid not pure Amorphous or crystalline (size of crystals important) Surface is charged (leads to concept of intrinsic and apparent IAP i.e. concentrations at surface of solid are critical, not those in the bulk solution). Stress induced change in IAP (IAP strong function of temperature)

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Nucleation (Stumm and Morgan 1996) Dissolution (Sparkes 1999) interfacial energy atoms in formulae unit

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Change in solubility product with pressure Consider reaction:

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Change in solubility product with pressure Standard partial molar compressibility For the reaction we have

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Change in solubility product with pressure (Langmuir 1997) Pressure generally increases the solubility of minerals Considering the reaction Change in pressure of 180 bar increases solubility by 50%.

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Soil Physical Chemistry 1999: Ed Sparks Effect of shear stress

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The Advection-Dispersion Equation: Boundary Conditions

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Solute Mass Flux Flux = Advection + Diffusion Two flows: 1) A mean flow (advection) 2) Perturbation about the mean (Mechanical Dispersion) Flow due to chemical potential gradients

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Advection and Mechanical Dispersion These processes cause perturbations of solute concentration and pore water velocity, hence,

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Mean Advection or Plug Flow Mechanical dispersion This represents a cross-correlation between concentration and velocity fluctuations. Fickian under “ideal” conditions

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The Advection-Dispersion Equation The solute mass flux is And leads to the standard ADE where D = D e +D md = D e +av

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Boundary Conditions At a boundary mass conservation requires the flux, f, to be continuous, that is, f(left of boundary,t) = f(right of boundary,t) f(0-,t)= f(0+,t) This holds for all times.

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A simple example Consider a porous medium between an upstream reservoir with a concentration of c=c0 and a downstream reservoir that allows solute to drip freely from the porous medium. c=c 0 v Porous Medium c=c e Inlet BoundaryOutlet Boundary

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The Inlet Boundary Condition This boundary condition ensures mass is conserved. Solute concentration is not continuous at boundary (solute concentration is continuous at the microscopic scale). Solute mass flux must be continuous,

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The outlet boundary However, this does not agree with experiment. Experiment agrees with the semi-infinite model evaluated at the point x=L. Both solute mass and solute mass flux must be continuous leading to the b/c

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Why? Consider solute advection Solution Solute advection is only affected by upstream boundary conditions. However, the ADE requires downstream boundary conditions (ADE is a parabolic equation).

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Mechanical dispersion is inherently an advective process and so should be described by a hyperbolic equation (i.e. the ADE is incorrect).

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Conclusions Thermodynamics is a `keystone theory’ in modern physics, underpinning theories in all the applied sciences and engineering. In some disciplines, the relation between thermodynamics and their discipline has become obscured by the continual telling and retelling by successive generations.

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Conclusions The contaminant transport equation requires some understanding of the underlying assumptions in order to use it properly. The transport of chemicals through a deforming porous media requires the derivation of a suitable transport equation from first principles. In much of engineering, thermodynamics is usually not taught in a systematic way, and first principles behind theories are skimmed over. This hampers fundamental research.

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Conclusions The great task that lies ahead of the engineering and the applied sciences this century, is consilience between the different disciplines.

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