Abstract
The processes of flow, mass and heat refer to extensive quantities (such as mass, momentum, energy and entropy), cf. Sect. 2.2.2, which are transported through a spatial domain of interest. This spatial domain is said to behave as a continuum which is occupied by matter for which a continuous distribution can be postulated. The matter may take a number of M aggregate forms or phases α, particularly: solid s, liquid l and gaseous g. It retains their continuity regardless how small volume elements the matter is subdivided in and interior material interfaces or surfaces exist. Any mathematical point we select can be assigned to matter as a physical point of given finite size.
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Fundamental Concept
The processes of flow, mass and heat refer to extensive quantities (such as mass, momentum, energy and entropy), cf. Sect. 2.2.2, which are transported through a spatial domain of interest. This spatial domain is said to behave as a continuum which is occupied by matter for which a continuous distribution can be postulated. The matter may take a number of M aggregate forms or phases α, particularly: solid s, liquid l and gaseous g. It retains their continuity regardless how small volume elements the matter is subdivided in and interior material interfaces or surfaces exist. Any mathematical point we select can be assigned to matter as a physical point of given finite size. In accordance with the assigned size of the physical point we can find, at least, two levels of a continuum description:
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1.
Microscopic level, where every point in the domain is occupied by only one phase (solid or liquid or gaseous).
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2.
Macroscopic level, where properties are defined at every point in the domain consisting of all phases (solid and liquid and gaseous).
At the microscopic level, the basic principles of fluid and solid mechanics can be used to solve the processes in the single phase domain, subject to BC’s on the interfaces of phases (e.g., liquid-solid, liquid-gaseous) that bounds this domain. However, at this level the complex interface geometry is neither observable nor describable. Accordingly, the solution of mass and transport processes in porous and fractured media at the microscopic level is impractical and widely impossible to obtain.
At the macroscopic level we can circumvent the difficulties associated with the geometric complexity of coexisting phases, at which measurable and continuous quantities may be determined and BC’s can be easily formulated. The continuum approach at such a macroscopic level is obtained via spatial averaging of the phase behaviors over a certain elementary volume. For each point within this macroscopic space, average values for variables and material properties result. The advantage of the macroscopic continuum approach is that (1) there is no need anymore for specifying exact configurations of interphase boundaries, (2) continuous and differentiable quantities result which can be employed by standard mathematical methods, and (3) the macroscopic quantities are measurable and applicable to practical problems.
Representative Elementary Volume (REV)
The transformation of variables and quantities from the microscopic to the macroscopic level needs spatial averaging referred to a certain elementary volume. It represents an appropriate transition, often termed as macroscopization, from a single phase to a multiphase level of description applied to a volume composed of all relevant phases (α = s, l, g) (or materials, M = 3) of interest. Formally, a porous (or fractured) medium can be defined as a multiphase material body characterized by the following features, e.g., [37]:
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1.
The averaging volume, denoted by dV, for a porous medium refers to a representative elementary volume (REV) which is occupied by a persistent solid phase s (Fig. 3.1). The remaining volumetric part, called void space, is occupied by one or more fluid phases (f = l, g). If such a REV cannot be found for a given domain, that domain cannot qualify as a porous medium.
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2.
The size of the REV is such that parameters that represent the distribution of the solid phase and void space within it are statistically meaningful.
The porous medium can be naturally formed (e.g., sand beds, rocks, soils) or engineered (e.g., tissues, concretes, polymer composites). Each phase (solid s, liquid l, gaseous g) is regarded as a continuum with smoothly varying properties, overlooking its molecular structure.
Representation of porous and fractured porous medium
The REV dV has to be sufficiently large for fluctuations of spatially averaged properties to be negligible. Phases α = s, f are regarded as material subdomains dV α separated by phase interfaces (e.g., solid-solid, fluid-solid, fluid-fluid). Each phase α is composed of N α miscible chemical species. It represents a molecular mixture of several identifiable chemical components k. By definition a chemical species k exists in only one phase α. Species that pass through different phases are regarded as separate, phase-pertinent constituents, accordingly the total number of chemical species \(N =\sum _{\alpha }{N}^{\alpha }\) holds. The fundamental assumption of continuum mechanics states that the resulting average quantities have to be independent of the size of the averaging volume dV and have to be continuous over time and space. Thus, the REV region dV is required to possess certain characteristics. Consider, for example, the void space, also termed as porosity for a porous medium, \(\varepsilon _{f} = \mathit{dV }_{f}/\mathit{dV }\). As the size of dV varies, the porosity \(\varepsilon _{f}\) varies as shown in Fig. 3.2. If dV is very small, erratic porosity results depending on whether the dV happens to cover voids or solids. Then as dV increases, fluctuations will appear in \(\varepsilon _{f}\) because relatively large portions of the one phase or other phases may become part of the averaging domain. As dV further increases within some interval of domain, there is a region when the porosity \(\varepsilon _{f}\) remains fairly constant. Within such an interval, in general, all average quantities become independent of the average domain dV. Further increase of dV may cause gross inhomogeneities of the medium that affect the stability of the average (macroscopic) quantities.
Porosity \(\varepsilon _{f}\) as function of averaging volume
In order to maintain meaningful average quantities the characteristic length of the averaging volume dV, denoted by D ∼dV 1∕3, must satisfy the inequality
where δ is the microscopic scale of the medium and L is the scale of the gross inhomogeneities. If for a particular medium these characteristic lengths cannot be identified, if inequality (3.1) does not hold, or if the scale of the problem of interest is of order D, there is no REV and accordingly the averaging technique is not applicable. With other words, the size of a REV must be much larger than the scale of microscopic heterogeneity due to the presence of solid and void space, and much smaller than the scale of the domain of interest (e.g., an aquifer system or a layered domain of soil) having macroscopic heterogeneity.
The features possessing for a REV of a porous medium can also be applied to a REV for the fracture media. In some cases we can recognize that an overlapping REV for the porous media and fractures exists (Fig. 3.3). Then, the problem can be treated as an equivalent continuum.
REVs for fractured media with overlapping continuum
Unlikely, if such an overlapping REV cannot be found (Fig. 3.4), fractures and porous media must be solved in a separate scale and have to be coupled via macroscopic interface conditions by using a discrete fracture approach. This is typical, for example, when fractures’ apertures are large, while the voids in the porous blocks are very small, practically all the flow takes place through the fractures. The discrete fracture approach requires information (e.g., aperture, length, orientation, spacing etc.) on every individual fracture. For solving large-scale problems of this type with hundreds or thousands of fractures a huge amount of detailed input data demands and a high computational effort can result.
REVs for fractured media without overlapping continuum
Average Operators and Average Quantities
The validity of the following averaging procedures is subject to the existence of a REV, the averaging volume dV. If valid, we can define an average over dV at every mathematical point, denoted by its position vector \(\boldsymbol{x}\), independent of whether or not \(\boldsymbol{x}\) falls inside the phase. This position vector \(\boldsymbol{x}\) serves as the centroid of the REV. On the other hand, let us identify the position of a particle within the REV by the vector \(\boldsymbol{r}\) and the position with respect to the centroid of the REV by the vector \(\boldsymbol{y}\) as shown in Fig. 3.5:
Hypothetical averaging volume dV with three phases present
For each phase a phase distribution function γ α may be defined as
The volume volume fraction of the α−phase, \(\varepsilon _{\alpha }\), is the fraction of dV occupied by the α−phase:
where \(dv(\boldsymbol{y})\) is the miscroscopic differential volume and dV is the macroscopic REV volume. Clearly, \(\varepsilon _{\alpha }\) is constrained by
For the macroscopization process we need three different averaging operators:
Volume average operator
Intrinsic volume average operator
Intrinsic mass average (Boltzmann) operator
From (3.6) and (3.7), it follows that the volume average and the intrinsic volume average of a scalar quantity ψ are related to each other by
where \(\varepsilon _{\alpha }\) is defined by (3.4). Furthermore, it results from (3.8)
The deviation of a microscopic quantity ψ at the point \(\boldsymbol{r}\) from its mass average of α−phase at the point \(\boldsymbol{x}\) is denoted by the fluctuation \({\tilde{\psi }}^{\alpha }\):
or
Since \({\overline{\psi }}^{\alpha }(\boldsymbol{x},t)\) is constant in the REV, dV, the following holds:
where \(\phi =\phi (\boldsymbol{r},t)\) is another scalar quantity.
Averaging Theorems
Averaging differential expressions within the REV, we have to consider terms providing averages of derivations with respect to space and time. The following theorems relate the average of a gradient and a time derivative to the gradient and time derivative of an average, respectively. For an extensive quantity \(\mathcal{F}\), cf. (2.116), it is valid:
Averaging theorem
Transport theorem
where dA α β is the macroscopic differential interface between α−phase and β−phase within dV, \(\mathit{da}(\boldsymbol{y})\) is an elemental portion of this area, \(\boldsymbol{{n}}^{\alpha \beta } = -\boldsymbol{{n}}^{\beta \alpha }\) is a normal direction vector on this surface pointing from the α−phase toward the β−phase, \(\boldsymbol{w}\) is the α β−interface velocity and \(\nabla \) is regarded as the macroscopic gradient operator with respect to the macroscopic coordinates \(\boldsymbol{x}\). We note that \(\mathcal{F}\) can be either vectorial or scalar quantities.
Aquifer Averaging
Flow and transport process modeling in aquifers possesses important special cases, in which the horizontal extent of a regional flow field can be much bigger than the thickness B of an aquifer. For such conditions vertical variations can often be neglected to reduce the full 3D equations to two-dimensional (2D) essentially horizontal relationships. Regarding groundwater hydraulic processes this procedure is associated with the well-known Dupuit assumption [33].
There are two distinctly different approaches to develop the macroscopic, 2D relations for aquifers. The standard two-step averaging procedure takes in a first step the above REV averaging technique to derive the general 3D macroscopic equations. In a second step, these equations have to be vertically integrated or vertically averaged. The difficulty with the two-step averaging procedure is that a number of terms whose physical meaning is not readily apparent arise involving derivations from averages which must be taken into account. In contrast, a more general and physically rigorous averaging technique has been proposed by Gray [202], which represents a one-step averaging procedure. It allows a direct transformation from 3D microscopic equations to 2D macroscopic aquifer-related equations. The procedure is termed as aquifer averaging and represents an extension to the REV concept.
Aquifer averaging is based on an aquifer REV, termed as AREV, as shown in Fig. 3.6.
Aquifer averaging volume (AREV) of height H and diameter D penetrating an aquifer of thickness B
Within the AREV the following constraint must be satisfied in addition
where B is the thickness of the aquifer, which may vary in space and time, and H is the total length of the cylinder, which is constant. In Fig. 3.6, dS denotes the projected circular planar area of the averaging volume, π D 2∕4. In the AREV the position vector \(\boldsymbol{r}\) only lies in the horizontal plane. The vertical direction is treated explicitly and referred to as the x 3−direction. Thus, a point in the AREV may be located by specification of its \((\boldsymbol{r},x_{3})\) coordinates.
In modification of the REV procedures, the AREV conception uses the following modified averaging operators, quantities and theorems:
Volume fraction
Volume average operator
Intrinsic volume average operator
Intrinsic mass average (Boltzmann) operator
Averaging theorem
Transport theorem
in which \(\boldsymbol{{n}}^{\mathrm{TB}}\) is the outward-directed unit normal at the top and bottom of the aquifer. We note that the gradient operator \(\nabla \) in (3.21) is only 2D, as there is no vertical gradient of B or \(\left \langle \mathcal{F}\right \rangle _{\alpha }\).
Fundamental Microscopic Balance Laws and Conservation Principles
The core of the mathematical modeling is formed by the four fundamental physical principles of
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\(\mathcal{M}_{k}\), mass balance,
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\(\mathcal{V}_{k}\), momentum balance,
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\(\mathcal{E}_{k} + \mathcal{K}_{k}\), total energy balance (first law of thermodynamics), and
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\(\mathcal{S}_{k}\), entropy balance
associated with species k. Mass, motion, energy and entropy-related quantities can be defined in a ‘microscopic’ (single phase) volume element (continuum), for which balance laws are postulated. Mass, momentum, internal (thermal) energy, kinetic energy and entropy, respectively, represent extensive quantities \(\mathcal{F}_{k} \in (\mathcal{M}_{k},\mathcal{V}_{k},\mathcal{E}_{k},\mathcal{K}_{k},\mathcal{S}_{k})\) of species k (i.e., those quantities are additive over volumes), cf. (2.116). Intensive quantities f k concern densities of these extensive properties being independent of the balance volume in form of mass densities, momentum densities, energy densities and entropy densities. In this context ρ k is introduced as a mass density function and ψ k as an intensive balance quantity. In accordance with (2.116) and Table 2.1, for an arbitrary volume Ω it is
where the intensive balance quantities ψ k are specified in Table 3.1 for the different extensive quantities.
In referring to a spatially fixed Eulerian coordinate system, the postulate of balance of the extensive quantity \(\mathcal{F}_{k}\) is stated as:
where F k corresponds to an external production (supply) and G k corresponds to a net rate of production of \(\mathcal{F}_{k}\). The material derivative (2.38) in (3.24) for the Eulerian description is given by
where \(\boldsymbol{v}_{k}^{\mathcal{F}}\) represents the velocity vector of the considered particle associated with the quantity \(\mathcal{F}_{k}\). The general balance statement (3.24) can be expressed by using the Reynolds’ transport theorem (2.83) as follows:
Thus, (3.26) can be simply written as
because the balance expression becomes independent of the volume Ω in a microscopic description. The particle velocity \(\boldsymbol{v}_{k}^{\mathcal{F}}\) can be further expressed via a diffusive law defined as
where \(\boldsymbol{j}_{k}^{\mathcal{F}}\) corresponds to a diffusive flux of species k associated with the extensive quantity \(\mathcal{F}_{k}\). It assumes a linear continuous relation to the particle velocity \(\boldsymbol{v}_{k}\) of species k. By using (3.28) the balance expression (3.27) takes the form
The diffusive fluxes \(\boldsymbol{j}_{k}^{\mathcal{F}}\) are summarized in Table 3.2 for the different extensive quantities. Since the particle velocity \(\boldsymbol{v}_{k}\) of species k is generally immeasurable, a diffusive flux
is introduced, which directly relates \(\boldsymbol{v}_{k}\) of species k to the mass-weighted (barycentric) velocity \(\boldsymbol{v}\) defined as
with
Accordingly, inserting (3.30) into (3.29) we obtain an appropriate balance expression, viz.,
where the particle velocity \(\boldsymbol{v}_{k}\) is eliminated, however, in account of specifying the diffusive fluxes \(\boldsymbol{j}_{k}^{\mathcal{F}}\) and \(\boldsymbol{j}_{k}\).
The balance equation (3.33) is still rather general because it implies expressions for each species k. Mostly, however, it is sufficient to specify only balance statements for mass-weighted (barycentric) quantities. In doing so, we sum (3.33) over all species k and obtain
where
with \(\boldsymbol{j}_{k}^{\mathcal{M}} = \mathbf{0}\ \ (\boldsymbol{v}_{k}^{\mathcal{M}}\equiv \boldsymbol{ v}_{k})\) according to the definition (3.28) and finding from (3.30) the identity
The barycentric variables ψ, \(\boldsymbol{j}\), F and G of the general microscopic balance equation (3.34) are listed in Table 3.3 for the different extensive quantities \(\mathcal{F}\) that need to be considered,
where E is the barycentric internal energy, S is the barycentric entropy, \(\boldsymbol{\sigma }\) is the barycentric stress tensor, \(\boldsymbol{j}_{T}\) is the barycentric thermal flux, \(\boldsymbol{j}_{S}\) is the barycentric entropy flux, Q is the barycentric supply of mass, \(\boldsymbol{g}\) is the barycentric supply of momentum, H is the barycentric supply of thermal energy, W is the barycentric supply of entropy and Υ is the barycentric entropy production. As seen from Table 3.3 the balance expression (3.34) can be considered as a general microscopic balance equation, where even the balance of species mass (3.33) (at \(\boldsymbol{j}_{k}^{\mathcal{M}} = \mathbf{0}\) and ψ k = 1) can be recognized if we formally set ψ →ω k , where ω k =ρ k ∕ρ is the mass fraction of species k (2.123) and in Table 3.3, \(\boldsymbol{j}_{k}\) accounts for the diffusive flux of species k and r k accounts for the production rate of species k.
Note that the net rate of production G for mass, momentum and internal energy is zero because these quantities are conserved, i.e., the balance statements for mass, momentum and energy represent conservation equations. On the other hand, however, entropy is a non-conservative quantity. The axiom of the second law of thermodynamics postulates that the entropy production is always non-negative, i.e.,
Macroscopization of Balance Equations
General Balance Equation
The transformation of the microscopic balance equation (3.34) to the macroscopic level uses the averaging procedures of (3.6)–(3.13) in combination with the averaging theorems (3.14) and (3.15) and finally leads to the following general formulation of the macroscopic balance equation written for the α−phase:
where
represents the macroscopic non-advective (dispersive) flux vector for \({\overline{\psi }}^{\alpha }\) consisting of the first part of macroscopic diffusion \(\left \langle \boldsymbol{j}\right \rangle _{\alpha }\) and the second part of macroscopic mechanical dispersion \(\left \langle \rho \right \rangle _{\alpha }{\overline{\tilde{\boldsymbol{{v}}{}^{\alpha }\ \tilde{\psi }}^{\alpha }}}^{\alpha }\). Furthermore, the term e α(ρ ψ) in (3.38) is
which describes the exchange of \({\overline{\psi }}^{\alpha }\) with other phases through phase changes caused by relative motion of phase boundaries. The term J α in (3.38) reads
which describes the diffusion of \({\overline{\psi }}^{\alpha }\) across the phase interfaces.
A constraint upon (3.38) may be obtained by summing over all phases α to obtain [226]
assuming that no properties are stored at a phase interface.
For the sake of simplicity the general balance equation (3.38) will be rewritten by omitting the averaging symbols in form of angular brackets and overbars indicating macroscopic quantities. In doing so, we replace the mass density \(\left \langle \rho \right \rangle _{\alpha }\) by its intrinsic average (3.9), i.e., \(\left \langle \rho \right \rangle _{\alpha } =\rho _{\alpha } =\varepsilon _{\alpha }{\left \langle \rho \right \rangle }^{\alpha } =\varepsilon { _{\alpha }\rho }^{\alpha }\). It is important to note that we shall designate always throughout the book an intrinsic quantity by a phase superscript and a bulk quantity by a phase subscript, e.g., \(\psi _{\alpha } =\varepsilon { _{\alpha }\psi }^{\alpha }\), \(Q_{\alpha } =\varepsilon _{\alpha }{Q}^{\alpha }\) and so forth. Using this convention, we can rewrite (3.38) in the following divergence formFootnote 1
where
represents a macroscopic exchange term occurring due to phase change and phase interaction, respectively. The definitions for ψ α, \(\boldsymbol{{j}}^{\alpha }\), F α, F ex α and G α in the general macroscopic balance equation (3.43) are listed in Table 3.4 for mass, momentum, energy and entropy balance. If we substitute the balance equation of the barycentric mass \({\mathcal{M}}^{\alpha }\) with
in (3.43), we find the convective formFootnote 2 of the balance equation as
where
defines the material derivative of ψ α of the α−phase. The divergence form (3.43) and the convective form (3.46) represent equivalent expressions for the same balance quantity ψ α. In the following the specific formulations of the macroscopic balance laws are described. Macroscopic conservation equations result for mass, momentum and energy with G α = 0.
Conservation of Mass
For conservation of mass, (3.43) becomes
where \(\boldsymbol{{v}}^{\alpha }\) is the (barycentric) velocity of α−phase, Q α represents the phase-internal supply of mass and \(Q_{\mathrm{ex}}^{\alpha }\) accounts for phase change of mass (e.g., ice melting). The conservation of mass requires that the total mass created over all phases must be identical to zero, i.e.,
Conservation of Species Mass
For a chemical species k in the α−phase, the mass conservation equation results from (3.43) as
written in the divergence form and from (3.46) with Table 3.4 as
written in the convective form, where \(\omega _{k}^{\alpha } =\rho _{ k}^{\alpha }{/\rho }^{\alpha }\) is the mass fraction of species k, \(\boldsymbol{j}_{k}^{\alpha }\) is the dispersive flux of species k, r k α is the homogeneous reaction rate of species k and R k α is the heterogeneous reaction rate of species k. Equations (3.50) and (3.51) are subject to the restrictions to insure a global conservation of mass:
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1.
The sum of mass fluxes of all k into phase α must be identical to the total mass change in the α−phase, i.e.,
$$\displaystyle{ \sum _{k}^{{N}^{\alpha }}(r_{ k}^{\alpha } + R_{ k}^{\alpha }) {=\rho }^{\alpha }({Q}^{\alpha } + Q_{\mathrm{ ex}}^{\alpha }) }$$(3.52) -
2.
The sum of dispersive fluxes of all k vanishes in the α−phase, i.e.,
$$\displaystyle{ \sum _{k}^{{N}^{\alpha }}\boldsymbol{j}_{ k}^{\alpha } = \mathbf{0} }$$(3.53)
Taking into account (3.52) and (3.53) and noting that \(\sum _{k}^{{N}^{\alpha } }\omega _{k}^{\alpha } = 1\), the mass balance equation (3.48) for the phase α is obtained by summing (3.50) over all species k.
The balance laws (3.50) and (3.51), respectively, for species k can be alternatively expressed if introducing the mass concentration \(C_{k}^{\alpha } =\rho _{ k}^{\alpha } =\omega _{ k}^{\alpha }{\rho }^{\alpha }\), cf. (2.117), according to
written in the divergence form and
written in the convective form.
Conservation of Momentum
The momentum equation may be obtained from (3.43) and (3.46) with Table 3.4 in its divergence form
and in its convective form
where \({\boldsymbol{\sigma }}^{\alpha }\) is the stress tensor of the α−phase, \(\boldsymbol{{g}}^{\alpha }\) is the α−phase external supply of momentum due to gravity (and electric or magnetic force fields) and \(\boldsymbol{f}_{\sigma }^{\alpha }\) is the interfacial drag term, which accounts for the exchange of momentum between the α−phase and all other phases due to mechanical interaction and exchange of mass. Note that the material derivative for \(\boldsymbol{{v}}^{\alpha }\) is
Conservation of Energy: First Law of Thermodynamics
The conservation of the total energy (the first law of thermodynamics) is obtained from (3.43) and (3.46) with Table 3.4 after subtraction of \(\boldsymbol{{v}}^{\alpha }\) dotted with (3.56) and (3.57), respectively, in the divergence form
and in the convective form
where \(\boldsymbol{j}_{T}^{\alpha }\) is the α−phase heat flux, H α is the α−phase external supply of energy and H ex α accounts for the exchange of energy between the α−phase and all other phases due to mechanical interaction and exchange of mass. The term \(\varepsilon {_{\alpha }\boldsymbol{\sigma }}^{\alpha }\boldsymbol{:} \nabla \boldsymbol{{v}}^{\alpha }\) in (3.59) and (3.60) represents a dissipation term of energy (for fluids it is termed as viscous dissipation) as an irreversible heat source due to internal forces (friction)
which is always negative (at the given definition) and produces thermal energy. For porous and fractured media, however, the energy dissipation is usually very small and is often negligible.
Entropy Balance
The entropy balance law is often neglected in derivations of equations for porous and fractured flow simulation. However, it is an important law when developing and proving constitutive relations for material properties. From (3.43) and (3.46) with Table 3.4 the entropy balance is in the divergence form
and in the convective form
where \(\boldsymbol{j}_{S}^{\alpha }\) is the α−phase entropy flux, W α is the α−phase external supply of entropy, W ex α accounts for the exchange of entropy between the α−phase and all other phases due to mechanical interaction and exchange of mass and Υ α is the α−phase net production of entropy. Usually, the entropy flux \(\boldsymbol{j}_{S}^{\alpha }\) is assumed proportional to the heat flux and the dispersive mass flux of chemical species such that [116]:
where (0 < T α < ∞) represents the absolute temperature of the α−phase and μ k α is the chemical potential of the kth-species in the α−phase. Furthermore, it may be assumed that the entropy supply term W α is proportional to the heat supply term according to
Second Law of Thermodynamics
The second law of thermodynamics dictates the sign of net entropy production. According to this axiom, the rate of net entropy production for the multiphase system must be always positive, i.e.,
Substitution of (3.64) and (3.65) into (3.63) and introduction of the Helmholtz free energy of the α−phase
into (3.60), replacement of the dispersive species flux \(\nabla \cdot (\varepsilon _{\alpha }\boldsymbol{j}_{k}^{\alpha })\) by (3.51), followed by elimination of H α between (3.59) and (3.63) yields:
Now division by T α and summation over all phases yield the Clausius-Duhem inequality of the total entropy production for the multiphase system:
expressing the second law of thermodynamics for porous media. Note that in (3.69) the interface condition (3.42) for the energy, \(\sum _{\alpha }\varepsilon {_{\alpha }\rho }^{\alpha }(H_{\mathrm{ex}}^{\alpha } +\boldsymbol{ f}_{\sigma }^{\alpha } \cdot \boldsymbol{ {v}}^{\alpha }) = 0\), with (3.44) and Table 3.4, has been used to replace the energy exchange term H ex α.
Vertically Averaged Aquifer Balance Equations
The aquifer macroscopization of the microscopic balance equation (3.34) uses the specific averaging procedures and theorems (3.17)–(3.22). The following aquifer-averaged balance equation results in the general form written for the α−phase:
where
describing exchange of \({\overline{\psi }}^{\alpha }\) due to phase change and interphase transport, respectively. The new term a α(ρ ψ) in (3.70) and (3.71) contains the averages of the microscopic production plus apparent production in the 2D plane due to the addition of ψ at the upper and lower boundaries of the aquifer. The dispersion flux vector \(\boldsymbol{{j}}^{\alpha }\) corresponds to expression (3.39) introduced before. We note that the thickness of the aquifer B can vary in space and time. The gradient operator \(\nabla \) in (3.70) is only 2D in the horizontal extent of the aquifer. In summing over all phases a constraint is obtained similar to (3.42) which indicates that properties must be conserved when considering interface transport:
Introducing again the simplified notation the aquifer-average balance equation (3.70) can be written in the general divergence form
and in the convective form
in whichFootnote 3
where \(\boldsymbol{W}\) is the velocity of the macroscopic interface forming the upper and lower boundary of the aquifer. The interface condition for a α(ρ ψ) appearing in (3.75) will be subsequently used to specify BC’s at the top and bottom of the aquifer. The definitions for ψ α, \(\boldsymbol{{j}}^{\alpha }\), F α, F ex α and G α in the aquifer-average balance equations (3.73) and (3.74) are listed in Table 3.4 for mass, momentum, energy and entropy balance. The following expressions result:
Conservation of mass
Conservation of species mass
or
Conservation of momentum
or
Conservation of energy
or
Balance of entropy
or
Second law of thermodynamics
Constitutive Theory
Kinematics
As discussed in Sect. 3.2, a porous medium can be viewed as a body which consists of a number of coexistent continua (phases), one solid phase s and two (or more) fluid phases f = l, g. Each phase possesses a reference configuration at time t = 0, which will be altered by its motion. The motions of the phases are independent. For a solid phase s in particular, a typical particle which occupies a position \(\boldsymbol{{X}}^{s}\) at time t = 0 may be carried to a new position \(\boldsymbol{x}\) at time t. Then, the solid phase motion is given by a displacement function \(\boldsymbol{{u}}^{s}(\boldsymbol{{X}}^{s},t)\) such that (cf. Sect. 2.1.4):
Note that the lower case latin index i refers to the deformed position (i.e., spatial coordinates) and the upper case latin index I refers to the reference position (i.e., material coordinates). It is assumed that the inverse of (3.86) exists such that:
To have this mapping continuous and bijective at all times, the Jacobian J s of this motion must be non-zero and strictly positive (cf. Sect. 2.1.4), i.e.,
where \(\boldsymbol{{J}}^{s}\) represents the deformation tensor of the solid phase s.
With the deformation of the solid phase there is a differential change of the volume dV s occupied by the particle of the porous solid. This can be expressed by the Jacobian of the deformed and the reference (non-deformed) solid volumes:
Due to mass conservation the following must be valid
and accordingly
and finally
Because \((\varepsilon {_{s}\rho }^{s})_{0}\) does not depend on time or spatial coordinate \(\boldsymbol{x}\), substitution of (3.92) into (3.48) yields
where
is the symmetric rate of deformation tensor of the solid phase s, \({D}^{s}/\mathit{Dt} = \partial /\partial t + (\boldsymbol{{v}}^{s} \cdot \nabla )\) is the material derivative for the solid phase and \(\boldsymbol{\delta }\) is the Kronecker symbol (2.7). The second-order tensor \(\boldsymbol{{d}}^{s}\) is sometimes called rate of strain tensor appropriate for small deformations. The velocity of the solid phase is defined as the material time rate of change of solid phase motion (2.39):
where \(\vert _{\boldsymbol{{X}}^{s}}\) indicates that \(\boldsymbol{{X}}^{s}\) is held constant. The strain tensor of the solid phase is commonly defined as
where the relation \({D}^{s}\boldsymbol{{e}}^{s}/\mathit{Dt} =\boldsymbol{ {d}}^{s}\) holds. Note that the second-order strain tensor (3.96) is symmetric, i.e., \(\boldsymbol{{e}}^{s} =\boldsymbol{ {e}}^{{s}^{T} }\). This symmetry means that there are six rather than nine independent strains, as might be expected in a 3 × 3 matrix. For convenience it is conventional to arrange the strain components in a vector form termed as strain pseudovector \({\boldsymbol{\epsilon }}^{s}\) of the solid phase by using the so-called Voigt notation. This strain pseudovector \({\boldsymbol{\epsilon }}^{s}\) is related to the displacement \(\boldsymbol{{u}}^{s}\) of the solid phase by the following relationship with denoted matrix operations written in the Euclidean space ℜ 3:
introducing the symmetric gradient operator
with the strain pseudovector components
where \(\nabla _{1} = \partial /\partial x_{1}\), \(\nabla _{2} = \partial /\partial x_{2}\), \(\nabla _{3} = \partial /\partial x_{3}\) and \(\varepsilon _{i}^{s}\) and \(\gamma _{\mathit{ij}}^{s}\;(i,j = 1,2,3)\) denote the normal strain components and the shear strain components of the solid phase, respectively. Accordingly, the divergence of the solid velocity \(\nabla \cdot \boldsymbol{ {v}}^{s}(=\boldsymbol{\delta }\boldsymbol{:}\boldsymbol{ {d}}^{s})\) can be expressed by displacements as follows
where \(\boldsymbol{m}\) is a specific unit vector defined as
For the subsequent derivations it will be convenient to use the solid phase velocity \(\boldsymbol{{v}}^{s}\) as a reference velocity. We define the relative velocity of the α−phase as
and the material derivative of the α−phase can be written according to
Constitutive Equations
The balance laws given by Eqs. (3.48), (3.51), (3.56), (3.59), and (3.62) with the relations (3.64) and (3.67) constitute N + (3 + D)M equations with the N(3 + D) + M(5 + 3D + D 2) unknowns which are enumerated below:
Quantities which are not listed in (3.104) are considered as known or directly related to these variables. Therefore, to close the systems of balance equations N(2 + D) + M(2 + 2D + D 2) additional constitutive equations are needed, which must account for the material properties of the system and their interrelation. The development of these constitutive equations will be done in a more general way, where we choose a set of independent variables to express the unknowns. The entropy inequality, the objectivity principle and the material symmetries will be utilized in order to restrict the general relationships. The remaining N(2 + D) + M(2 + 2D + D 2) unknowns, chosen as dependent variables are members of the set {Ψ j } given below:
These variables are not directly measurable and they have to be determined as functions of directly measurable variables, hereby termed independent variables. The choice of independent variables is made in accordance with the following axioms [157, 521]:
-
1.
Principle of equipresence. A variable present as an independent variable in one constitutive equation should be so present in all.
-
2.
Principle of coordinate invariance – objectivity principle. Constitutive equations must be stated by a rule which holds equally in all inertial coordinate systems at any fixed time.
-
3.
Principle of admissibility. The constitutive relations do not violate the balance laws or the second law of thermodynamics.
The principle of objectivity requires that a constitutive equation must be unchanged under an orthonormal transformation of the spatial reference frame. As shown in [132, 228] this requirement implies that velocity \(\boldsymbol{{v}}^{\alpha }\) and velocity gradient \(\nabla \boldsymbol{{v}}^{\alpha }\) have to be replaced by the relative velocity \(\boldsymbol{{v}}^{fs} =\boldsymbol{ {v}}^{f} -\boldsymbol{ {v}}^{s}\) (3.102) and the rate of deformation tensor of the fluid phase \(\boldsymbol{{d}}^{f} = \tfrac{1} {2}[\nabla \boldsymbol{{v}}^{f} + {(\nabla \boldsymbol{{v}}^{f})}^{T}]\), respectively. Finally, the constitutive equations for the dependent variables (3.105) are postulated in terms of the following set {Ξ j } of independent variables:
Note that T 0 represents a reference temperature. In (3.106) \(\varepsilon _{s}\) is not chosen as independent variables because the volume fraction must sum to unity (3.5), knowledge of \(\varepsilon _{f}\) provides \(\varepsilon _{s}\).
The constitutive equations (3.106) are subject to the principle of admissibility. The Coleman and Noll method [94] is used in Appendix B to restrict the functional form of the constitutive variables. For the sake of simplicity the following assumptions are made:
-
The phases are considered ideal in so far as constitutive variables which account for intraphase processes \({(\boldsymbol{\sigma }}^{\alpha },{A}^{\alpha }{,\mu }^{\alpha },{S}^{\alpha }\quad \mbox{ for}\quad \alpha = s,f)\) depend only on the properties of that phase.
Accordingly, the restrictions obtained via the Coleman and Noll method in Appendix B yields:
Note that the dependence of A s and S s on \(\varepsilon _{f}\) must remain explicit because \(\varepsilon _{s}\) is not adopted as an independent constitutive variable, however, related directly via the unity \(\varepsilon _{s} = 1 -\varepsilon _{f}\), (3.5). Taking into account (3.107) and introducing the thermodynamic pressure of the fluid phases f and of the solid phase s, respectively, according to
the entropy inequality (B.5) takes the form:
Equilibrium Restrictions
Thermodynamic equilibrium is the state where the following independent variables of (3.106) controlling directly the entropy production (3.109)
are all zero and the constitutive functions satisfy:
where \(\bigr |_{e}\) denotes evaluation at the equilibrium and \({G}^{\alpha } = {A}^{\alpha } + {p}^{\alpha }{/\rho }^{\alpha }\) is the Gibbs free energy of the α−phase.
At the thermodynamic equilibrium the entropy production ρ Υ goes to zero, i.e., it attains its minimum value. The necessary and sufficient conditions to ensure that ρ Υ is a minimum at equilibrium are:
Application of restriction (3.112) to (3.109) yields:
It shows that \(\boldsymbol{f}_{\sigma }^{f}\) and \({\boldsymbol{\sigma }}^{f}\) are composed of an equilibrium part and a non-equilibrium (deviatoric) part, the latter being zero at equilibrium. Accordingly, it is useful to split \({\boldsymbol{\sigma }}^{f}\) and \(\boldsymbol{f}_{\sigma }^{f}\) in such a form
where \({\boldsymbol{\tau }}^{f}\) and \(\boldsymbol{f}_{\tau }^{f}\) represent the deviatoric fluid stress tensor and the deviatoric fluid momentum exchange vector, respectively. With (3.113) and (3.114) these flux and stress variables dependent on (3.106) ensure at equilibrium:
As a consequence, if \(\boldsymbol{f}_{\tau }^{f}\), \({\boldsymbol{\tau }}^{f}\), \(\boldsymbol{j}_{k}^{\alpha }\) and \(\boldsymbol{j}_{T}^{\alpha }\) will be developed subsequently for the chosen independent variables (3.106) in form of phenomenological equations,Footnote 4 the equilibrium condition (3.115) requires that dependency can only be allowed for the driving thermodynamic ‘forces’ \(\boldsymbol{{v}}^{fs}\), \(\boldsymbol{{d}}^{f}\), \(\nabla \omega _{k}^{\alpha }\) and \(\nabla {T}^{\alpha }\), i.e.,
Basic Balance Equations and Entropy Inequality
The combination of the above constitutive equations with the general balance equations (3.48), (3.51), (3.57), and (3.60) as well as with the entropy inequality (3.109) yields the following relations by using the substitution of bulk source/sink terms \(Q_{\alpha } =\varepsilon _{\alpha }({Q}^{\alpha } + Q_{\mathrm{ex}}^{\alpha }),H_{\alpha } =\varepsilon _{\alpha }({H}^{\alpha } + H_{\mathrm{ex}}^{\alpha }),\alpha = f,s\):
Mass conservation of fluid phases
Mass conservation of solid phase
Mass conservation of species k of fluid phases
Mass conservation of species k of solid phase
Momentum conservation of fluid phases
Momentum conservation of solid phase
Energy conservation of fluid phases
Energy conservation of solid phase
Entropy of the fluid-solid phase system
In the above equations, the following useful assumptions are made:
-
The stress tensors \({\boldsymbol{\tau }}^{f}\) and \({\boldsymbol{\sigma }}^{s}\) are symmetric.
-
The only external supply of momentum is provided by the gravity, i.e., \(\boldsymbol{g} =\boldsymbol{ {g}}^{\alpha }\).
-
Diffusive (dispersive) flux of chemical species in the solid phase does not exist, i.e., \(\boldsymbol{j}_{k}^{s} = \mathbf{0}\).
Furthermore, the entropy balance (3.63) is not included anymore because we need not to know explicitly the entropy variable S α in the subsequent analysis. Accordingly, the conservation laws (3.117)–(3.124) provide now N + M(2 + D) equations, which are available for solving:
To close this system of equations the following list of dependent variables remains
which must be determined by appropriate constitutive functions depending suitably on the independent variables (3.106). Furthermore, equations of state (EOS) have to be supplemented in order to determine the needed explicit information about the fluid density ρ f and the internal energy E α (α = s, f)
taking into account (3.67) and the restrictions (3.107).
Development of Phenomenological Equations and Constitutive Relations
The remaining dependent variables as listed in (3.127) have to be expressed by phenomenological and constitutive functions in terms of the independent variables (3.106):
where the obtained restrictions (3.107), (3.111), and (3.116) have been taken into account. Polynomial expansions are usually employed up to the desired degree of approximation. A truncated Taylor series around the state Ξ = 0 leads to expressions as exemplified for \(\boldsymbol{j}_{T}^{\alpha }\):
where \(\bigr |_{0}\) denotes evaluation at Ξ = 0 and HOT represents higher-order terms:
in which for instance
or written in index notation (i, j, m, n = 1, …, D)
In the above Taylor series the derivative terms \((.)\bigr |_{0}\) represent tensorial quantities, which account for material properties and have to be known (or to be determined) as material coefficients. Note that in (3.130) a negative sign is used for the development. This is required by the entropy inequality (3.125), where \(\boldsymbol{j}_{T}^{\alpha }\) has to be negative while the material coefficients remain positive. The higher order terms of the material coefficients lead to tensorial coefficients of higher order so as indicated in (3.133), where 3rd-order and 4th-order tensors \(A_{\mathit{ijm}}^{\alpha \gamma \beta },B_{\mathit{ijmn}}^{\alpha \gamma \beta }\) appear. If we assume
-
Higher order tensors for the α−phase are isotropic and symmetric.
-
Material coefficients related to the α−phase depend only on the properties of that phase, i.e., for instance \(\sum _{\gamma }\sum _{\beta }A_{\mathit{ijm}}^{\alpha \gamma \beta }(v_{j}^{\gamma s} \frac{\partial {T}^{\beta }} {\partial x_{m}}) = A_{\mathit{ijm}}^{\alpha }(v_{j}^{\alpha s} \frac{\partial {T}^{\alpha }} {\partial x_{m}})\).
then any 3rd-order tensor A ijm α γ β and any 4th-order tensor B ijmn α γ β simplify, cf. [12, 132]
where b 0 α and b 1 α correspond to material coefficients of the α−phases. The following derivations will take into account these assumptions.
Deviatoric Fluid Stress Tensor \({\boldsymbol{\tau }}^{f}\)
The Taylor series expansion of \({\boldsymbol{\tau }}^{f} {=\boldsymbol{\tau } }^{f}(\boldsymbol{{v}}^{fs},\boldsymbol{{d}}^{f},\nabla \omega _{k}^{f},\nabla {T}^{\alpha })\) for the 1st-order terms yields
where HOT are neglected and a negative sign is used due to the entropy restriction \({\boldsymbol{\tau }}^{f} \leq 0\) in (3.125). Considering isotropic conditions we find for the 3rd-order tensors:
and for the 4th-order symmetric tensor:
where λ f is denoted as dilatational (or bulk) viscosity and μ f is denoted as dynamic (or shear) viscosity of the f−phase. With (3.136) and (3.137) we obtain from (3.135)
In fluid mechanics the mechanical pressure \(p_{\mathrm{mech}}^{f}\) is defined as the average of the normal stress
The difference between the thermodynamic pressure p f defined by (3.108) and the mechanical pressure p mech f defined by (3.139) is obtained from (3.114) by contracting on the index i and dividing by 3. It results
The assumption that the two pressures are equal is known as Stokes’ assumption, and it means that
Then the deviatoric stress tensor \({\boldsymbol{\tau }}^{f}\) reaches its final form
which represents the Newton’s viscosity law of fluids. A consequence of Stokes’ assumption is that the average normal viscous stress is always zero, cf. [409], and the deviatoric stress tensor implies primarily viscous shear stress effects. We note that for pure fluid flow the momentum equation (3.121) with Newton’s viscosity law (3.142) is commonly referred to as the Navier-Stokes equation.
For further needs the divergence of the deviatoric stress tensor (3.142) gives
SinceFootnote 5
we can simplify (3.143) to
under the specific assumptions:
-
The spatial variability of the fluid viscosity is negligible, i.e., \(\Vert \varepsilon _{f}\boldsymbol{{d}}^{f} \cdot {\nabla \mu }^{f}\Vert \approx 0\).
-
Applied to the stress tensor the vector field \(\boldsymbol{{v}}^{f}\) is considered solenoidal (2.84) having \(\nabla \cdot \boldsymbol{ {v}}^{f} =\boldsymbol{\delta }\boldsymbol{:}\boldsymbol{ {d}}^{f} = 0\), which corresponds to the assumption of incompressibility usually made in classic fluid mechanics.
-
For the stress tensor the second derivative of volume fraction is negligible \({\nabla }^{2}\varepsilon _{f} \approx 0\) and the antisymmetric rate of deformation tensor associated with the gradient of volume fraction vanishes: \(\bigl (\nabla \boldsymbol{{v}}^{f} - {(\nabla \boldsymbol{{v}}^{f})}^{T}\bigr ) \cdot \nabla \varepsilon _{f} \approx \mathbf{0}\).
The following restriction on the dynamic viscosity μ f results from the entropy inequality (3.125):
The dynamic viscosity μ f can be considered as a thermodynamic function
which will be described as EOS further below.
Deviatoric Drag of Fluid Momentum Exchange \(\boldsymbol{f}_{\tau }^{f}\)
The truncated Taylor series expansion of \(\boldsymbol{f}_{\tau }^{f} =\boldsymbol{ f}_{\tau }^{f}(\boldsymbol{{v}}^{fs},\boldsymbol{{d}}^{f},\nabla \omega _{k}^{f},\nabla {T}^{\alpha })\) for 1st order, however, extended by terms in \(\boldsymbol{{v}}^{fs}\) up to third order gives
where other HOT are neglected and a negative sign is used due to the entropy restriction \(\boldsymbol{f}_{\tau }^{f} \leq 0\) in (3.125). We define for the second-order tensors:
Considering isotropic conditions we find for the 3rd-order tensors:
For the 4th-order symmetric tensor we prefer the following approximation:
where \(\boldsymbol{R}_{2}^{f}\) is a second-order tensor and ℑ F is an inertial coefficient. Using (3.149), (3.150), and (3.151) the drag of momentum exchange (3.148) reads
The third and fourth term of the RHS of (3.152) represent actions on the drag of momentum exchange which are controlled by the mass fraction and temperature gradients. Commonly, those cross effects on the momentum exchange are very small. For the sake of simplicity we shall assume:
-
Dependency of mass fraction and temperature gradients on the drag of momentum exchange in form of cross effects are negligible, i.e., \(\boldsymbol{A}_{k}^{f} \approx \mathbf{0}\), \(\boldsymbol{{B}}^{f} \approx \mathbf{0}\).
Typically in porous-media problems the drag parameter for the fluid momentum exchange is related to the dynamic viscosity of the fluid μ f and the permeability of the fluid phase, which is defined by
where \(\boldsymbol{{k}}^{f}\) is the intrinsic permeability tensor of the f−phase. With (3.153) the final form of the drag of momentum exchange of the fluid phase reads
where the first term of the RHS of (3.154) describes the Darcy flow effect [392] and the second term is recognized as Forchheimer flow effect on the momentum drag of porous-media flow in which ℑ F represent the Forchheimer coefficient, where there are different formulations and derivations in the literature, cf. [296, 534, 562]. Nield and Bejan [389] use a dimensionless form-drag constant c F , which is related to the Forchheimer coefficient \(\mathfrak{I}_{F}\) as
The above introduced material parameters for the fluid momentum exchange \(\boldsymbol{f}_{\tau }^{f}\) are restricted by the entropy inequality (3.125) according to
Solid Stress Tensor \({\boldsymbol{\sigma }}^{s}\)
The relation between the solid stress tensor \({\boldsymbol{\sigma }}^{s}\) and the solid-phase free energy A s given by
results from the Coleman and Noll method’s evaluation (3.107), where p s is the thermodynamic pressure of the solid phase s. Furthermore, the dependence of A s (and accordingly \({\boldsymbol{\sigma }}^{s}\) and p s) is restricted by
The term \({\rho }^{s}\partial {A}^{s}/{\partial \boldsymbol{\epsilon }}^{s}\) in (3.157) can be identified as the non-equilibrium solid stress
At the thermodynamic equilibrium we have to require
To satisfy this equilibrium constraint the non-equilibrium stress of the solid phase \({\boldsymbol{\tau }}^{s}\) must be independent of \(\varepsilon _{f}{,\rho }^{s},\omega _{k}^{s}\) and T s, i.e.,
and the truncated Taylor series expansion of \({\boldsymbol{\tau }}^{s}\) yields
In (3.162) a 4th-order deviatoric stress tensor appears
which can be simplified if we assume
-
The solid phase s is isotropic. The deviatoric stress tensor \(\boldsymbol{{t}}^{s}\) is symmetric. The solid phase can be considered as an elastic material.
Then
where λ s and μ s are the \(\mathrm{Lam\acute{e}}\) constants. The constitutive expression for \({\boldsymbol{\tau }}^{s}\) in (3.164) represents the Hook’s law for isotropic linear-elastic continua. The elastic material constants λ s and μ s are usually expressed by the shear modulus G, Young’s (or elastic) modulus E and Poisson’s ratio ν as follows:
Since strain \({\boldsymbol{\epsilon }}^{s}\) and displacement \(\boldsymbol{{u}}^{s}\) are related according to (3.97), the deviatoric stress tensor \({\boldsymbol{\tau }}^{s}\) can be expressed as
with the elasticity matrix
For the material coefficients the thermodynamic restrictions require
where with \(\nu = \tfrac{1} {2}\) the solid material is incompressible.
Heat Flux Vector \(\boldsymbol{j}_{T}^{\alpha }\)
The Taylor series expansion for the heat flux vector \(\boldsymbol{j}_{T}^{\alpha } =\boldsymbol{ j}_{T}^{\alpha }(\boldsymbol{{v}}^{fs},\boldsymbol{{d}}^{f},\nabla \omega _{k}^{\alpha },\nabla {T}^{\alpha })\) of the α−phase (α = f, s) up to third order for ∇T α product terms becomes
where we again have assumed that the material coefficients of the α−phase depend only on the properties of that phase. In (3.169) a negative sign is used due to the entropy restriction \(\boldsymbol{j}_{T}^{\alpha } \leq 0\) in (3.125). For an isotropic medium the odd-order tensorial quantities vanish in (3.169). Furthermore, it is assumed that only first-order approximation with respect to \(\nabla {T}^{\alpha }\) is considered. Using the following definitions for the remaining second and fourth tensors in (3.169) as
the heat flux vector \(\boldsymbol{j}_{T}^{\alpha }\) becomes
in which the 2nd-order tensor of hydrodynamic thermodispersion is introduced as
consisting of two parts: (1) the tensor of thermal conductivity \(\boldsymbol{\varLambda }_{0}^{\alpha } {=\varLambda }^{\alpha }\boldsymbol{\delta }\) and (2) the tensor of mechanical thermodispersion \(\boldsymbol{\varLambda }_{\mathrm{mech}}^{\alpha }\) given by
where \(\bar{\alpha }_{L}^{\alpha }\) and \(\bar{\alpha }_{T}^{\alpha }\) represent the specific longitudinal and transverse thermodispersivity, respectively. In contrast to the form (3.173) the classic dispersion models developed by Scheidegger [460] and Bear [33] postulate only a linear velocity dependence for the mechanical dispersion \(\boldsymbol{\varLambda }_{\mathrm{mech}}^{\alpha }\) according to
where with \(\alpha _{L}^{\alpha } =\bar{\alpha }_{ L}^{\alpha }\Vert \boldsymbol{{v}}^{fs}\Vert\) and \(\alpha _{T}^{\alpha } =\bar{\alpha }_{ T}^{\alpha }\Vert \boldsymbol{{v}}^{fs}\Vert\) new longitudinal and transverse thermodispersivity coefficients appear, respectively. The Scheidegger-Bear dispersion model (3.174) is commonly used in practice. However, it is important to note that in an isotropic medium the first-order approximation of \(\boldsymbol{j}_{T}^{\alpha }\) explicitly contains only \(\boldsymbol{{v}}^{fs}\boldsymbol{{v}}^{fs}\) terms and not the \(\boldsymbol{{v}}^{fs}\) terms. The material parameters for the heat flux \(\boldsymbol{j}_{T}^{\alpha }\) appearing in (3.171), (3.172), and (3.174) are restricted by the entropy inequality (3.125) according to
In (3.171) the heat flux is also affected by cross effects driven by the flow velocity \(\boldsymbol{{v}}^{fs}\) and the mass fraction gradient \(\nabla \omega _{k}^{\alpha }\) of species k. The influence of the concentration (mass) gradient on the heat flux is known as Dufour effect, where N k α corresponds to the Dufour coefficient. It is apparent that if mechanical and Dufour effects are neglected, we recover the conventional form of the heat flux as
known as the Fourier heat flux, where the tensor of hydrodynamic thermodispersion \({\boldsymbol{\varLambda }}^{\alpha }\) is used in the form
Species Mass Flux Vector \(\boldsymbol{j}_{k}^{f}\)
Similarly to the heat flux, the mass flux vector \(\boldsymbol{j}_{k}^{f} =\boldsymbol{ j}_{k}^{f}(\boldsymbol{{v}}^{fs},\boldsymbol{{d}}^{f},\nabla \omega _{k}^{f},\nabla {T}^{\alpha })\) of the species (k = 1, …, N f) in the fluid phase (f = l, g) is developed via a Taylor series expansion up to third order now for \(\nabla \omega _{k}^{f}\) product terms. It yields
where a negative sign is used due to the entropy restriction \(\boldsymbol{j}_{k}^{f} \leq 0\) in (3.125). In a direct analogy to the heat flux we assume that the medium is isotropic and that only a first-order approximation with respect to \(\nabla \omega _{k}^{f}\) is considered. Finally, we find for the species mass flux \(\boldsymbol{j}_{k}^{f}\) the following expression:
with the 2nd-order tensor of hydrodynamic dispersion \(\boldsymbol{D}_{k}^{f}\) of species k
consisting of the tensor of diffusion
where D k f is the coefficient of molecular diffusion of species k of the fluid phase f in the porous medium, and the tensor of mechanical dispersion Footnote 6 of the porous medium
written for the Scheidegger-Bear dispersion model, where β L f and β T f are the longitudinal and transverse dispersivities, respectively. In (3.179) cross effects for the mass flux are incorporated due to \(\boldsymbol{{v}}^{fs}\) and \(\nabla {T}^{f}\). The temperature influence is known as Soret effect (or thermodiffusion), where M f describes the Soret coefficient. Mechanical and Soret effects are commonly negligible and the species mass flux (3.179) reduces to the well-know linear Fick’s law of macroscopic hydrodynamic dispersion
where the tensor of hydrodynamic dispersion \(\boldsymbol{D}_{k}^{f}\) is used in the form
We find for the hydrodynamic dispersion \(\boldsymbol{D}_{k}^{f}\) that the dependency on the species k is only associated with the coefficient of molecular diffusion D k f. It is important to note that the molecular diffusion coefficient D k f of the species k in the fluid phase f of the porous medium is usually smaller than the corresponding diffusion coefficient \(\breve{D}_{k}^{f}\) in an open fluid body due to geometric effects of the porous medium [37, 38]. They are related by
with the tortuosity T ∗ f, which ranges between zero and unity and can be approximated as [38]
For the linear Fick’s law (3.183) the dispersive mass flux \(\boldsymbol{j}_{k}^{f}\) of a species k is proportional to the mass fraction gradient. However, it has been shown [232, 464] that if high concentrations of solutes occur, typically arising in concentrated brine transport, nonlinear effects become important and \(\boldsymbol{j}_{k}^{f}\) should be replaced by an extended nonlinear non-Fickian dispersion law,
where \(\mathfrak{I}_{H}\) represents an additional high-concentration (HC) dispersion coefficient and \(\boldsymbol{D}_{k}^{f}\) is the Scheidegger-Bear dispersion tensor according to (3.184). It has been found [464] that \(\mathfrak{I}_{H}\) varies inversely with the flow velocity, i.e., \(\mathfrak{I}_{H} = \mathfrak{I}_{H}(\boldsymbol{{v}}^{fs})\).
The material parameters for the species mass flux \(\boldsymbol{j}_{k}^{f}\) introduced in (3.179), (3.181), (3.182), and (3.187) are restricted by the entropy inequality (3.125) according to
Species Reaction Rate \(r_{k}^{\alpha } + R_{k}^{\alpha }\)
The reaction rates r k α and R k α differ between homogeneous and heterogeneous reactions of species k in the multiphase system, respectively, where r k α concerns intraphase reactions and R k α covers interphase reactions. If a species k exists in different phases the mass conservation has to be related to the overall (summed) balance of mass, (3.119) plus (3.120), and a bulk reaction rate of species k for the multiphase system has to be taken into account. This bulk reaction rate can be defined as
For the constitutive representations of the rates the following functionals hold
where the dependency on ρ f can be discarded from (3.129) since the knowledge of \(\omega _{k}^{f} \subset \omega _{k}^{\alpha }\) provides the fluid density according to (2.117) and (2.123). Since r k α, R k α and R k possess the same functional structure, a polynomial representation of (3.190), exemplified for R k , may be written as
with
where n k ≥ 0 corresponds to an exponent of species k and the coefficients \(b_{k}^{p}\ (p = 0,1,\ldots,N)\) are rate constants of species k depending on the overall reaction mechanism, which can be dependent on the temperature T α
applicable to a nonisothermal reaction mechanism. There are many reactive systems which can be broadly classified into simple and complex kinetic reactions. According to the mechanism of a reaction, the functional form of r k α, R k α or R k can be very complicated and may not be representable as a polynomial in a form of (3.191) for all cases. Reaction mechanisms for irreversible (kinetic) and reversible (equilibrium) reactions will be discussed in more detail in Chap. 5.
Equations of State (EOS)
Fluid Density ρ f
The fluid density ρ f is composed of N f miscible chemical species k with a partial fluid density \(\rho _{k}^{f} = C_{k}^{f} {=\rho }^{f}\omega _{k}^{f}\) (mass of species k per unit volume of fluid), cf. (2.117) and (2.123), so that
holding for a mixture, where ω k f (and C k f) stands for all species present in the fluid phase f. However, it is important to note that only N f − 1 of the mass fractions ω k f can be specified independently because the sum of the mass fractions must be unity. Let us for convenience designate the N fth species as the one that is dependent, the constitutive relation is:
It simply states that if we know the mass fractions of species 1 through N f − 1, we know the mass fraction of species N f. A typical example refers to a diluted aqueous phase, where water (species k: = N f = H2O) is referred to as a solvent \(\omega _{{N}^{f}}^{f}\) because it is the predominant species in a liquid phase, while the N f − 1 species as solutes constitute only a small portion of the phase. In this context we define the special case of a single-species solute, where only one dissolved component exists and the aqueous phase is composed of two miscible species (one solute and one solvent), i.e., N f = 2.
The density ρ f is regarded as a dependent thermodynamic variable for which the following constitutive relationship, or EOS, (3.128) holds
It is to be noted that dependence is indicated on only N f − 1 of the species mass fractions as shown in (3.195). We can differentiate (3.196) to obtain:
where γ f is the fluid compressibility, and α k f and β f are the specific solutal and thermal expansion coefficients, respectively. A negative sign is introduced for the thermal expansion coefficient β f to take into account that the fluid density decreases when temperature increases. Regarding γ f and α k f it implies that the density ρ f increases when the pressure p f and/or the mass fractions ω k f increase, respectively. If (and only if) we assume that γ f, α k f and β f are constant, the integration of (3.197) immediately leads to the EOS for the fluid density ρ f in the common form:
where suitable reference values appear for the density \(\rho _{0}^{f} {=\rho }^{f}(p_{0}^{f},\omega _{k0}^{f},T_{0}^{f})\) at reference pressure p 0 f, reference mass fraction ω k0 f and reference temperature T 0 f. The EOS for the fluid density (3.198) is often linearly approximated in the form:
and commonly γ f, α k f and β f are considered constant [389]. While for the most practical applications this assumption is valid for compressibility γ f and specific solutal expansion α k f, a constant thermal expansion β f may become inappropriate for geothermal applications where a larger temperature range has to be considered and thermal anomalies in ρ f (such as the 4 ∘C anomaly for water) can also play a role (Fig. 3.7). For temperatures within the range from 0 to 100 ∘C, the thermal expansion of freshwater \((\omega _{k}^{f} =\omega _{ k0}^{f} = 0,\,k = 1,\ldots,{N}^{f} - 1,{p}^{f} = p_{0}^{f})\) actually varies from − 0. 68 ⋅ 10−4 up to 7. 5 ⋅ 10−4 K−1, and is zero at 4∘C [120]. To improve the relationship (3.199), a more accurate 6th-order polynomial \({\rho }^{f} {=\rho }^{f}({T}^{f})\) can be introduced. As shown in Appendix C a Taylor series expansion of the polynomial up to the 6th-order term results in a nonlinear variable thermal expansion \({\beta }^{f} {=\beta }^{f}({T}^{f})\), which is applied to the EOS in form of (3.199).
Density of freshwater ρ f (at \(\omega _{k}^{f} =\omega _{ k0}^{f},{p}^{f} = p_{0}^{f}\)) as a function of temperature T f in a range of 0 and 100 ∘C. Close-up view indicates density anomaly at 4 ∘C
Internal Energy E α
For the internal energy of the fluid phase E f and the solid phase E s the following dependencies exist according to (3.128):
Using the chain rule of differentiation it follows that
and
where for (3.202) we have assumed that E s depends only on properties of the solid phase s. The task here is to find
Taking into account from (3.67), (3.107), (3.108), (3.164), and (3.166)
we obtain
where with c α the specific heat capacity of the α−phase is introduced which is usually positive c α > 0. Note that c α need not be constant. We substitute (3.205) into (3.201) and (3.202) to find the expression for the material derivatives,Footnote 7 viz.,
and
In general, the chemical potential \(\mu _{k}^{\alpha } =\mu _{ k}^{\alpha }{(\rho }^{\alpha },\omega _{k}^{\alpha },{T}^{\alpha })\), cf. (3.107), is a dependent variable and further constitutive relations are required. However, in most applications
-
The density, solid strain and chemical effects on the internal energy are negligible,
so that E α becomes only dependent on the temperature T α
and the material derivatives (3.206) and (3.207) simplify in
If the specific heat capacity c α is independent of the temperature T α, the internal energy \({E}^{\alpha }({T}^{\alpha }) = {E}^{\alpha }(T_{0}^{\alpha }) +\int _{ T_{0}^{\alpha }}^{{T}^{\alpha } }{c}^{\alpha }{\mathit{dT}}^{\alpha }\) can be given explicitly
where \(E_{0}^{\alpha } = {E}^{\alpha }(T_{0}^{\alpha })\) is a constant reference value of internal energy.
Dynamic Viscosity μ f
The dynamic viscosity μ f of the fluid phase f = l, g is regarded as a thermodynamic function of mass fraction ω k f and temperature T f, cf. (3.147):
A truncated Taylor series expansion for μ f around reference mass fraction ω k0 f and reference temperature T 0 f up to the 3rd order gives
where the terms \(\ldots \bigr |{}_{T_{0}^{f},\omega _{k0}^{f}}\) are constant coefficients of fluid viscosity at reference temperature and reference mass fraction, and μ 0 f is the reference fluid viscosity at reference temperature and reference mass fraction.
Furthermore, viscosity dependencies have been developed by using empirical polynomial relationships in the literature. Regarding the mass fraction dependency, particularly for high-concentration saltwater, Lever and Jackson [343] and Hassanizadeh [224] proposed the following relationship:
with
where ω l is the overall mass fraction of the total dissolved solids (TDS) in the liquid ( = water) phase l and \(\bar{\mu }_{0}^{l}\) is a specific reference viscosity valid for ω 0 l = 0 at, however, unspecified temperatures. On the other hand, an empirical relation for the temperature dependence of the dynamic viscosity μ l of the liquid ( = water) phase l has been proposed by Mercer and Pinder [371] in the form:
with
where the specific reference viscosity \(\bar{\mu }_{0}^{l}\) is related to a reference temperature of 150 ∘C (or \(\varsigma = 0\)) at, however, unspecified mass fraction of solutes. A combination of both influences yields the following expression:
where \(\bar{\mu }_{0}^{l} {=\mu }^{l}(0,15{0}^{\circ }\mathrm{C})\). Employing an arbitrary reference mass fraction ω 0 l and reference temperature T 0 l, a viscosity relation function f μ l can be obtained, which is related to these proper reference conditions:
where \(\mu _{0}^{l} {=\mu }^{l}(\omega _{0}^{l},T_{0}^{l})\) is the reference viscosity with respect to the reference mass fraction ω 0 l and the reference temperature T 0 l. For example, typical viscosity relations of water l are displayed in Figs. 3.8 and 3.9 for the temperature range T l between 0 and 300 ∘C and mass concentrations \({C}^{l} {=\omega { }^{l}\rho }^{l}\) between 0 and 200 g/l for the chosen reference concentration \(\omega _{0}^{l} = C_{0}^{l} = 0\) (freshwater) and reference temperature \(T_{0}^{l} = 1{0\,}^{\circ }\mathrm{C}\).
Viscosity relation \(\mu _{0}^{l}{/\mu }^{l}\) of water l as function of mass concentration \({C}^{l} {=\omega { }^{l}\rho }^{l}\) [g/l] and temperature T l [∘C] using \(C_{0}^{l} =\omega _{ 0}^{l} = 0\) (freshwater) and \(T_{0}^{l} = 1{0\,}^{\circ }\mathrm{C}\)
Viscosity relation \({\mu }^{l}/\mu _{0}^{l}\) of water l as function of mass concentration \({C}^{l} {=\omega { }^{l}\rho }^{l}\) [g/l] and temperature T l [∘C] using \(C_{0}^{l} =\omega _{ 0}^{l} = 0\) (freshwater) and \(T_{0}^{l} = 1{0\,}^{\circ }\mathrm{C}\)
Additional Closure Relations
The volume fraction \(\varepsilon _{\alpha }\,(\alpha = f,s)\) (3.4) appears as a geometry-dependent variable resulting from the volume averaging process over a REV. It is convenient to express \(\varepsilon _{\alpha }\) for fluid phases f = l, g and the solid phase s of a porous medium as
with
and \(\sum _{\alpha }\varepsilon _{\alpha } =\varepsilon _{l} +\varepsilon _{g} +\varepsilon _{s} \equiv 1\) (3.5), where \(\varepsilon\) is the porosity (void space) and s is the fluid saturation referring to the dynamic liquid l and gas g phases of the porous medium. For the following considerations we assume that the liquid phase l represents the wetting phase and the gas phase g represents the nonwetting phase of the two coexisting fluids filled in the void space \(\varepsilon\) of the porous medium.
Capillary Pressure p c
The macroscopic representation of the equilibrium with the pressure difference between adjacent nonwetting and wetting fluid phases at the interface of the two fluids in the void space of a porous medium is recorded by the macroscopic capillary pressure p c , defined as
for which constitutive relationships are required, e.g.,
We have to note that the dependency of p c on the liquid saturation s l does not follow consistently from the thermodynamic dependence of the liquid pressure as stated above in (3.108). The conflict results from the difference between the pressure variable p α of α−phase, which is a volume average over the REV, and the capillary pressure p c , which is basically an interface variable and accordingly should refer to as a surface average. A comprehensive discussion on this matter and an extended alternative theoretical approach can be found in Gray and Hassanizadeh [205, 206], Hassanizadeh and Gray [231], and Hassanizadeh et al. [233]. Temperature effects on capillary pressure are presented by Grant [200]. Commonly, the capillary pressure p c is considered to be dependent on the wetted liquid phase l only, viz.,
where numerous empirical relations exist to express p c (s l). The explicit functional form for p c (s l) must be considered to be specific to the combination of the pair of fluids and the porous medium, basically also dependent on the medium temperature and the chemical composition of the fluids. The function of p c is also known to exhibit hysteresis in that the equilibrium value of p c as a function of s l is found to be dependent on the direction of the process (i.e., drainage (drying) or imbibition (wetting)). A schematic depiction of the p c versus s l curves is given in Fig. 3.10.
Schematic plot of capillary pressure p c versus liquid saturation s l possessing hysteresis
In soil science, the p c (s l)−relationship is called retention curve as it shows how much water is retained in a soil by the capillary pressure. Numerous empirical parametric models exist to describe retention curves based on fitted analytical expressions. The most common empirical relations are summarized in Appendix D. Alternatively, spline approximations of the retention curve can be useful in cases where analytical functions do not fit suitably to the experimental data, for more see also Appendix D.
Relative Permeability k r f
The presence of more than one fluid phase (f = l, g) in a porous medium has consequences on the interfacial momentum exchange \(\boldsymbol{f}_{\tau }^{f}\) too. For the intrinsic permeability tensor \(\boldsymbol{{k}}^{f}\) of the fluid phase f appearing in the drag term of fluid momentum exchange (3.154) a saturation-dependency is postulated in the following form:
where with k r f the saturation-dependent relative permeability, sometimes called relative conductivity, is introduced via a variable separation and the split \(\boldsymbol{k}\) is the fluid-independent permeability tensor of the porous medium, which is anisotropic in general. The permeability tensor \(\boldsymbol{k}\) is also termed as saturated permeability equivalent to the intrinsic permeability at full saturation s f = 1. Various empirical relationships for \(k_{r}^{f} = k_{r}^{f}({s}^{f})\) exist, where the most useful parametric models are summarized in Appendix D. A typical curve of \(k_{r}^{l}({s}^{l})\) for the liquid phase l is exhibited in Fig. 3.11. Note that the relative permeability k r f can also imply hysteretic effects [34, 38, 422]. In case of need spline approximation for \(k_{r}^{l}({s}^{l})\) can be beneficial to get better fits to experimental data, see Appendix D.
Schematic plot of relative permeability k r l versus liquid saturation s l
Complete Equations of Multiphase Flow and Transport in Deforming Porous Media
General Formulation
The formulation of a complete mathematical model for solving multiphase flow, mass and heat transport in deforming porous media is based on the balance laws of Sects. 3.7 and 3.8.4 in combination with the phenomenological equations of Sect. 3.8.5, the constitutive relations of Sect. 3.8.6 and the additional closure relations of Sect. 3.8.7 including the assumptions as stated in these sections. It results in a rather general set of equations as follows:
Mass conservation \({\mathcal{M}}^{f}\) of fluid phases f = l,g
Mass conservation \({\mathcal{M}}^{s}\) of solid phase s Footnote 8
Mass conservation \(\mathcal{M}_{k}^{f}\) of species k of fluid phases f = l,g
divergence form
convective form
Mass conservation \(\mathcal{M}_{k}^{s}\) of species k of solid phase s
divergence form
convective form
Momentum conservation \({\mathcal{V}}^{f}\) of fluid phases f = l,g
divergence form
convective form
Momentum conservation \({\mathcal{V}}^{s}\) of solid phase s
convective form
Energy conservation \({\mathcal{E}}^{f} + {\mathcal{K}}^{f}\) of fluid phases f = l,g
divergence form
convective form
Energy conservation \({\mathcal{E}}^{s} + {\mathcal{K}}^{s}\) of solid phase s
divergence form
convective form
Constitutive relations
with \(\boldsymbol{{v}}^{fs} =\boldsymbol{ {v}}^{f} -\boldsymbol{ {v}}^{s}\), \(\boldsymbol{{d}}^{f} = \tfrac{1} {2}[\nabla \boldsymbol{{v}}^{f} + {(\nabla \boldsymbol{{v}}^{f})}^{T}]\), \(\varepsilon _{f} =\varepsilon {s}^{f}\), \(\varepsilon _{s} = 1-\varepsilon\), \(\boldsymbol{{v}}^{s} = \partial \boldsymbol{{u}}^{s}/\partial t\) and \({\boldsymbol{\epsilon }}^{s} =\boldsymbol{ L} \cdot \boldsymbol{ {u}}^{s}\). We introduced above appropriate bulk quantities denoted by phase subscripts as follows: \(\boldsymbol{j}_{fk} =\varepsilon _{f}\boldsymbol{j}_{k}^{f},\boldsymbol{D}_{fk} =\varepsilon _{f}\boldsymbol{D}_{k}^{f},\boldsymbol{D}_{f\mathrm{mech}} =\varepsilon _{f}\boldsymbol{D}_{\mathrm{mech}}^{f},\boldsymbol{\varLambda }_{\alpha } =\varepsilon { _{\alpha }\boldsymbol{\varLambda }}^{\alpha }\). In (3.238) we made use of the fact that the mechanical dispersion is a property of the porous medium and independent of the actual transport quantity; accordingly, we substituted \(\boldsymbol{\varLambda }_{\mathrm{mech}}^{f} {=\rho }^{f}{c}^{f}\boldsymbol{D}_{\mathrm{mech}}^{f}\). We use both divergence and convective forms of the balance statements if necessary due to mathematical reasons as discussed further below. For the energy conservation it is obvious that the convective form naturally results from replacing the internal energy by the temperture variable, cf. (3.206) and (3.207). A divergence form of energy conservation with the temperature variable results from the basic energy balance equation (3.59) by inserting (3.210). That means, their expressions (3.234) and (3.236) in terms of temperatures T f and T s, respectively, are possible if (and only if) the specific heat capacities c f and c s are assumed independent of temperatures. Such an assumption is not needed for the convective forms (3.235) and (3.237) of energy conservation.
The conservation laws (3.225)–(3.237) for the three (l − g − s) phases form a closed equation system consisting of 6 + N + 3D equations, which can be solved for the following independent primary variables:
The complexity of the governing equations is very high and a further reduction is useful and really possible in many applications. The reduction will be done in three levels in a top-down manner:
-
1.
First level reduction: Multiphase variable-density flow, mass and heat transport in porous media based on the general Darcy-Brinkman-Forchheimer (DBF) flow equation.
-
2.
Second level reduction: Single liquid phase variable-density flow, mass and heat transport in variably saturated porous media based on the Darcy flow equation [59].
-
3.
Third level reduction: Variable-density Darcy-type flow, mass and heat transport in groundwater (fully saturated porous media), including vertically integrated formulations for aquifers.
Proper Reduction of Governing Equations for Multiphase Variable-Density Flow, Mass and Heat Transport in Porous Media: First Level Reduction
Introducing the volumetric flux density (Darcy velocity)Footnote 9 for the fluid phases f = l, g
the general model Eqs. (3.225)–(3.238) can be significantly reduced if the following assumptions are made:
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Due to the generally slow motion of fluid flow in porous media the inertial effects appearing in the momentum conservation (3.231) or (3.232) in form of local acceleration \(\partial (\varepsilon {_{f}\rho }^{f}\boldsymbol{{v}}^{f})/\partial t\) and of convective acceleration \(\nabla \cdot (\varepsilon {_{f}\rho }^{f}(\boldsymbol{{v}}^{f}\boldsymbol{{v}}^{f}))\) are negligible, cf. [389].
-
Energy dissipation terms in the energy conservations equations (3.234)–(3.237) can be neglected: \(\varepsilon _{f}{p}^{f}\nabla \cdot \boldsymbol{ {v}}^{f} \approx 0\), \(\tfrac{2} {3}\varepsilon {_{f}\mu }^{f}{(\nabla \cdot \boldsymbol{ {v}}^{f})}^{2} \approx 0\), \(2\varepsilon {_{f}\mu }^{f}\boldsymbol{{d}}^{f}\boldsymbol{:}\boldsymbol{ {d}}^{f} \approx 0\), \(\varepsilon _{s}{p}^{s}\nabla \cdot \boldsymbol{ {v}}^{s} \approx 0\), \(\varepsilon _{s}(\boldsymbol{{t}}^{s} {\cdot \boldsymbol{\epsilon }}^{s})\boldsymbol{ {:}\boldsymbol{\epsilon } }^{s} \approx 0\).
-
It is assumed that the phases of the porous medium are locally in a state of thermodynamic equilibrium. That means that the REV-averaged temperatures of all phases l, g, s are assumed to be equal at each point in the multiphase system:
$$\displaystyle{ {T}^{l} = {T}^{g} = {T}^{s} = T }$$(3.241)where T represents the system temperature. As the consequence of (3.241) the energy conservation equations (3.234)–(3.237) can be summed up over all phases and only one energy equation for the multiphase systems finally results. Additionally, for the gas phase g the thermal capacity c g and thermal hydrodynamic conductivity \({\boldsymbol{\varLambda }}^{g}\) can be neglected with respect to the solid and liquid phases. Another direct consequence of (3.241) is that the overall thermal conductivity \(\boldsymbol{\varLambda }=\boldsymbol{\varLambda } _{f} +\boldsymbol{\varLambda } _{s} =\boldsymbol{\varLambda } _{0} {+\rho }^{f}{c}^{f}\boldsymbol{D}_{f\mathrm{mech}}\) leads to a weighted arithmetic mean of the thermal conductivities of the fluid and solid phases in the form of \(\boldsymbol{\varLambda }_{0} = [\varepsilon {s{}^{f}\varLambda }^{f} + {(1-\varepsilon )\varLambda }^{s}]\boldsymbol{\delta }\) as a natural result in which the thermal conductivities of the fluid and solid phases occur in parallel Footnote 10.
-
The solid phase s is assumed deformable, but solid grains are incompressible. Inserting (3.240) into (3.225) and using the definition (3.219) the mass conservation equation for the fluid phase f reads
$$\displaystyle{ \varepsilon {s}^{f}\frac{{\partial \rho }^{f}} {\partial t} {+\rho }^{f}{s}^{f} \frac{\partial \varepsilon } {\partial t} {+\rho }^{f}\varepsilon \frac{\partial {s}^{f}} {\partial t} + \nabla \cdot {(\rho }^{f}\boldsymbol{q}_{ f}) + \nabla \cdot (\varepsilon {s{}^{f}\rho }^{f}\boldsymbol{{v}}^{s}) {=\rho }^{f}Q_{ f}\quad }$$(3.242)Assuming slowly deformable media and slightly compressible fluids the following approximation holds [37]
$$\displaystyle{ \nabla \cdot (\varepsilon {s{}^{f}\rho }^{f}\boldsymbol{{v}}^{s}) \approx \varepsilon {s{}^{f}\rho }^{f}(\nabla \cdot \boldsymbol{ {v}}^{s}) }$$(3.243)The expression \(\nabla \cdot \boldsymbol{ {v}}^{s}\) is obtained from the solid mass balance (3.226)
$$\displaystyle{ \frac{\partial [{(1-\varepsilon )\rho }^{s}} {\partial t} + \nabla \cdot [{(1-\varepsilon )\rho }^{s}\boldsymbol{{v}}^{s}] = 0 }$$(3.244)where Q s = 0 is assumed. For incompressible solid grains, (3.244) becomes
$$\displaystyle{ \nabla \cdot \boldsymbol{ {v}}^{s} \approx \biggl ( \frac{1} {1-\varepsilon }\biggr )\frac{\partial \varepsilon } {\partial t} }$$(3.245)In changing the porosity \(\varepsilon\) of the porous-medium compression work of the skeleton is taken into account. Let us consider the porosity as a function of fluid pressure and let mass fraction and temperature effects be disregarded, we have the differential
$$\displaystyle{ d\varepsilon = \frac{\partial \varepsilon } {\partial {p}^{f}}d{p}^{f} =\biggl (\underbrace{\mathop{ \frac{1} {1-\varepsilon } \frac{\partial \varepsilon } {\partial {p}^{f}}}}\limits _{\upsilon }\biggr )(1-\varepsilon )d{p}^{f} =\upsilon (1-\varepsilon )d{p}^{f} }$$(3.246)where \(\upsilon\) represents the coefficient of skeleton compressibility. It takes into account a vertical deformation of the porous medium. The relations (3.245) and (3.246) decouple the fluid equations from the solid equations and there is no need anymore to solve explicitly the momentum conservation equation (3.233) and mass conservation equation (3.226) for the solid s. This approach is a common practice in subsurface modeling, where the movement of the solid phase is modeled only implicitly. Rather than trying to obtain detailed information about movement of the solid phase, only its compression is considered. This assumption can be inappropriate for problems of land subsidence, slope or embankment stability in a geotechnical context [344] or for large deformations in absorbent swelling industrial porous material [144, 147, 375].
-
For the species mass, momentum and energy conservation equations the velocity term \(\varepsilon _{f}\boldsymbol{{v}}^{f}\) can be replaced by the volumetric flux density \(\boldsymbol{q}_{f}\) (3.240), assuming that the terms associated with the solid movement \(\varepsilon _{f}\boldsymbol{{v}}^{s}\) are negligible.
-
Let us consider a species k, which occurs both in the fluid phase f and the solid phase s. Let us assume that k is sorbed at the solid phase, which can be expressed by the sorption isotherm (for more details see Chap. 5):
$$\displaystyle{{ \rho }^{s}\omega _{ k}^{s} {=\rho }^{f}\varphi _{ k}\,\omega _{k}^{f} }$$(3.247)where \(\varphi _{k} =\varphi _{k}(\omega _{k}^{f})\) is the dimensionless adsorption function, which can be dependent on ω k f. In such an adsorption process a fluid phase f can occupy only part of the void space \(\varepsilon\) and therefore only part of the total area of the solid can be exposed to adsorption [39]. Sometimes, it is assumed [38, 422] that the wetting phase completely coats the solid such that no other (nonwetting) fluid phase is in contact with the solid. To take into account this solid-fluid contact phenomenon for the adsorption of chemical species k occurring both in the wetting fluid phase f and in the absorbing solid phase s, we subdivide the solid volume fraction \(\varepsilon _{s}\) into chemically active and inactive parts of solid mass \(\varepsilon _{s} =\varepsilon _{{s}^{\mathrm{active}}} +\varepsilon _{{s}^{\mathrm{inactive}}}\) (cf. Sect. 5.2.2). Accordingly, the mass balances (3.227) and (3.229) (assuming \(\nabla \cdot (\varepsilon {_{s}\rho }^{s}\boldsymbol{{v}}^{s}\omega _{k}^{s}) \approx 0\)) of species k in both phases f and s can be written as
$$\displaystyle{ \begin{array}{rcl} \frac{\partial } {\partial t}(\varepsilon {s{}^{f}\rho }^{f}\omega _{ k}^{f}) + \nabla \cdot {(\rho }^{f}\boldsymbol{q}_{ f}\omega _{k}^{f}) + \nabla \cdot \boldsymbol{ j}_{ k}^{f}& =&\varepsilon {s}^{f}(r_{ k}^{f} -\vartheta _{ k}\omega _{k}^{f} +\tilde{ R}_{ k}^{f}) \\ \frac{\partial } {\partial t}(\varepsilon {_{{s}^{\mathrm{active}}}\rho }^{s}\omega _{k}^{s})& =&\varepsilon _{{s}^{\mathrm{active}}}(r_{k}^{s} -\vartheta _{k}\omega _{k}^{s} +\tilde{ R}_{k}^{s})\qquad \end{array} }$$(3.248)where in the heterogeneous reaction rates R k f and R k s the linear reaction parts of decay are split off according to \(R_{k}^{f} = -\vartheta {_{k}\rho }^{f}\omega _{k}^{f} +\tilde{ R}_{k}^{f}\), \(R_{k}^{s} = -\vartheta {_{k}\rho }^{s}\omega _{k}^{s} +\tilde{ R}_{k}^{s}\), introducing a joint linear decay rate constant \(\vartheta _{k}\) of species k (see Sect. 5.4.2). The ratio of the area of the adsorbing solid-fluid interface to the total area of the solid, which can be assumed equal to the ratio of active (adsorbing) solid mass to the total mass of solid, and accordingly assumed equal to the ratio of the active (adsorbing) solid mass fraction \(\varepsilon _{{s}^{\mathrm{active}}}\) to the total solid mass fraction \(\varepsilon _{s}\), is apparently a function of the saturation s f of the wetting fluid phase: \(\varepsilon _{{s}^{\mathrm{active}}}/\varepsilon _{s} = f({s}^{f})\), where \({s}^{f} \leq f({s}^{f}) \leq 1\) is a surface contact ratio function, which has to be specified. In many applications a suited approximation is f(s f) ≈ s f and we use
$$\displaystyle{ \varepsilon _{{s}^{\mathrm{active}}} = f({s}^{f})\varepsilon _{ s} \approx {s}^{f}(1-\varepsilon ) }$$(3.249)Inserting (3.247) and (3.249) into (3.248), we can add up the mass conservation equations (3.248) to obtain:
$$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial t}(\varepsilon {s{}^{f}\rho }^{f}\mathfrak{R}_{ k}\omega _{k}^{f}) + \nabla \cdot {(\rho }^{f}\boldsymbol{q}_{ f}\omega _{k}^{f}) + \nabla \cdot \boldsymbol{ j}_{ k}^{f} +\varepsilon {s{}^{f}\rho }^{f}\vartheta _{ k}\mathfrak{R}_{k}\omega _{k}^{f}& & \\ =\underbrace{\mathop{ {s}^{f}{\Bigl [\varepsilon (r_{k}^{f} +\tilde{ R}_{k}^{f}) + (1-\varepsilon )(r_{k}^{s} +\tilde{ R}_{k}^{s})\Bigr ]}}}\limits _{\tilde{R}_{k}}\quad & & {}\end{array}$$(3.250)written in the divergence form and
$$\displaystyle\begin{array}{rcl} \varepsilon {s{}^{f}\rho }^{f}\acute{\mathfrak{R}}_{ k}\frac{\partial \omega _{k}^{f}} {\partial t} {+\rho }^{f}\boldsymbol{q}_{ f} \cdot \nabla \omega _{k}^{f} + \nabla \cdot \boldsymbol{ j}_{ k}^{f} +\varepsilon {s{}^{f}\rho }^{f}\vartheta _{ k}\mathfrak{R}_{k}\omega _{k}^{f}& & \\ =\tilde{ R}_{k} {-\rho }^{f}\omega _{ k}^{f}Q_{ f}& & {}\end{array}$$(3.251)written in the convective form, where ℜ k is the retardation factor
$$\displaystyle{ \mathfrak{R}_{k} = 1 +{\Bigl ( \frac{1-\varepsilon } {\varepsilon } \Bigr )}\varphi _{k} }$$(3.252)and \(\acute{\mathfrak{R}}_{k}\) is the derivative term of retardation
$$\displaystyle{ \acute{\mathfrak{R}}_{k} = 1 +{\Bigl ({ \frac{1-\varepsilon } {\varepsilon \rho }^{f}} \Bigr )}\,\frac{\partial {(\rho }^{f}\varphi _{k}\omega _{k}^{f})} {\partial \omega _{k}^{f}} }$$(3.253) -
As stated in Sect. 3.7.3, the summation of the species mass balance equations over all \(N =\sum _{\alpha }{N}^{\alpha }\) species must give the total mass balance of the phase(s), so that only \({N}^{{\ast}} =\sum _{\alpha }({N}^{\alpha } - 1)\) of the species mass fractions are independent, because if N α − 1 are known, the N αth may be computed directly from \(\omega _{{N}^{\alpha }}^{\alpha } = 1 -\sum _{k=1}^{{N}^{\alpha }-1 }\omega _{k}^{\alpha }\). Accordingly, only N ∗ species mass transport equations are needed to be solved, where N ∗ denotes the essential number of species.
Taking into account the above assumptions we find the governing equations of the first level reduction as summarized in Table 3.5. In the momentum equations three terms are emphasized which are of specific concern.
First, the Brinkman term \(\tfrac{{\mu }^{f}} {\varepsilon _{f}}{\nabla }^{2}\boldsymbol{q}_{f}\) results from the viscous shear stresses of the fluid. Brinkman (see [389, 534] for references) has firstly described this term in the context of porous media, but, had set the term to \({\mu }^{f}{\nabla }^{2}\boldsymbol{q}_{f}\). However, the correct factor must be \(\tfrac{{\mu }^{f}} {\varepsilon _{f}}\) instead of μ f resulting directly from by the present volume averaging procedure, cf. [394]. It was pointed out by Tam [505] that whenever the length scale of the investigated problem is much greater than \({(\Vert \boldsymbol{k}\Vert /\varepsilon )}^{1/2}\) the Brinkman term becomes negligible in comparison to the Darcy term. Only for thin boundary layers with a thickness lower than \({(\Vert \boldsymbol{k}\Vert /\varepsilon )}^{1/2}\) the Brinkmann term could have effects for practical applications.
Second, the Darcy term \({\mu }^{f}{(k_{r}^{f}\boldsymbol{k})}^{-1} \cdot \boldsymbol{ q}_{f}\) represents a linear relationship to \(\boldsymbol{q}_{f}\) due to viscous drag by friction at the solid-fluid interfaces of the porous medium. This holds when \(\boldsymbol{q}_{f}\) is sufficiently small, which is valid for most of the porous-medium applications. The characteristic measure is provided by the pore Reynolds number Re p of the flow defined by
where d is the characteristic length dimension representing the elementary channels of the porous medium. It could be used as a mean grain diameter or estimated via \({(\Vert \boldsymbol{k}\Vert /\varepsilon )}^{1/2}\). The linear drag of the Darcy flow regime is valid as long as Re p does not exceed some values between 1 and 10
However, as \(\boldsymbol{q}_{f}\) (particularly Re p ) increases the viscous drag becomes nonlinear.
Third, a quadratic drag term is provided by the Forchheimer term \({\rho }^{f}{(k_{r}^{f}\boldsymbol{k})}^{-1/2}c_{F}\Vert \boldsymbol{q}_{f}\Vert \cdot \boldsymbol{ q}_{f}\), which takes into account that the transition from linear to nonlinear drag is smooth and does not mean that there is a sudden transition from laminar to turbulent flow. Even in laminar flow regimes the linearity is broken due to the fact that the form drag due to solid obstacles is now comparable with the surface drag due to friction. An upper limit of Re p at about 100 is suggested [33] for the nonlinear laminar flow regime. For higher Re p −numbers a turbulent flow regime occurs, which requires the extension to turbulent transport mechanisms based on the full momentum equations [118]. In the Forchheimer term we introduced the more common dimensionless form-drag constant c F (3.155), which is often approximated by 0.55. In dependence on the meaning of the different terms in the momentum equation \({\mathcal{V}}^{f}\) of Table 3.5 we differ between (1) the Darcy-Brinkman-Forchheimer (DBF) equation, (2) the Darcy-Forchheimer (DF) equation and (3) the Darcy equation as listed in Table 3.6.
Final Model Equations for Flow, Mass and Heat Transport
Single Liquid Phase Variable-Density Flow, Mass and Heat Transport in Variably Saturated Porous Media: Second Level Reduction
For the practical modeling of variable-density flow, mass and heat transport in porous media the above balance laws with their constitutive relations of the first level reduction as listed in Tables 3.5 and 3.6 are further simplified. The following assumptions are made as the second level reduction of the governing model equations:
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Fluid phase assumption: In the void space two fluid phases f coexist: a liquid phase l (e.g., water) and a gas phase g (e.g., air). In many applications, however, the gas phase g can be assumed stagnant, i.e.,
$$\displaystyle{ \boldsymbol{q}_{g} =\varepsilon _{g}(\boldsymbol{{v}}^{g} -\boldsymbol{ {v}}^{s}) \equiv \mathbf{0} }$$(3.256)Accordingly, there is no need to consider anymore the momentum balance for the gas phase. As a consequence of (3.256) a hydrostatic gas pressure condition with \(\nabla {p}^{g} {=\rho }^{g}\boldsymbol{g}\) results from the momentum balance equations for the gas phase (Table 3.6) and p g could be solved explicitly as a simple function of gas density ρ g and location \(\boldsymbol{x}\). This assumption reduces the problem to a single-phase flow, where the only dynamic fluid phase is the liquid phase l, however, under variable liquid saturation s l of the void space \(\varepsilon\), for which a further assumption is required to decouple finally the liquid phase from the gas phase.
-
Capillary pressure assumption: The liquid saturation s l is determined from the capillary pressure (3.223): \(p_{c}({s}^{l}) = {p}^{g} - {p}^{l}\). Taking into account that the density of liquid is much higher than of gas (e.g., note that the relation between water and air is \({\rho }^{l}{/\rho }^{g} \approx 800\)), with the hydrostatic gas phase condition (3.256) we find that \(\nabla {p}^{l} {\gg \rho }^{g}\boldsymbol{g}\) and conclude that gravitational effects on the gas pressure p g are negligible in comparison to the liquid pressure p l. This allows us to assume a constant gas pressure p g ≈ const. Practically, we refer to a constant atmospheric pressure and set p g = 0. It simplifies the capillary pressure relation according to
$$\displaystyle{ p_{c}({s}^{l}) = -{p}^{l} }$$(3.257)With this assumption the liquid phase is actually decoupled from the gas phase and the flow and transport process of the liquid phase may be modeled without need to explicitly model the gas phase. It represents the key assumption of flow and transport modeling in variably saturated porous media. Based on the relationships as described in Appendix D it is now easy to relate directly the liquid pressure to the liquid saturation \({p}^{l} = {p}^{l}({s}^{l})\) and, inversely, to express the liquid saturation as a function of the liquid pressure \({s}^{l} = {s}^{l}({p}^{l})\).
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Momentum equation assumption: It can be usually assumed that the liquid phase moves slowly in the porous medium and the condition (3.255) is satisfied. Accordingly, the momentum balance for the fluid phase l can be described by the Darcy equation (Table 3.6):
$$\displaystyle{ \boldsymbol{q}_{l} = -{\frac{k_{r}^{l}\boldsymbol{k}} {\mu }^{l}} \cdot (\nabla {p}^{l} {-\rho }^{l}\boldsymbol{g}) }$$(3.258)
Choice of Suited Variables
In groundwater hydraulics and subsurface hydrology it is common to measure pressures at a point P above a reference datum in an equivalence to a head of liquid (e.g., water) with given density in a vertical column (e.g., pipe, well) as shown in Fig. 3.12.
Pressure head \({\psi }^{l} = {p}^{l}/(\rho _{0}^{l}g)\) and hydraulic head \({h}^{l} {=\psi }^{l} + z\) measured in a piezometric pipe
We define the pressure head ψ l of the liquid l
and the hydraulic head (piezometric head) h l of the liquid l
which are related to the constant reference liquid density ρ 0 l, where the subscript j = 1, 2 or 3 indicates the direction of gravity aligned to a major coordinate direction of \(\boldsymbol{x}\). Typically, in a vertical direction it is x j = x 3 = z and \(\boldsymbol{{g}}^{T} = (0\;0\;g)\), where \(g =\Vert \boldsymbol{ g}\Vert\) is the gravitational acceleration. Introducing the gravitational unit vector
we can express the Darcy equation (3.258) by the variables of hydraulic head h l and pressure head ψ l, respectivelyFootnote 11
where
defines the hydraulic conductivity,
is the viscosity relation function (3.218) and
is the dimensionless buoyancy coefficient of the liquid phase l.
It is important to note that the hydraulic conductivity \(\boldsymbol{{K}}^{l}\) incorporates both porous-medium and liquid properties, however, the liquid parameters ρ 0 l and μ 0 l in (3.263) represent constant reference values and accordingly \(\boldsymbol{{K}}^{l}\) remains de facto a parameter of the porous medium scaled with constant liquid parameters ρ 0 l and μ 0 l and the gravitational constant g. Through the Darcy equation (3.262) formulated with the hydraulic conductivity \(\boldsymbol{{K}}^{l}\), the actual pressure, species concentration and temperature effects on the liquid density ρ l and liquid viscosity μ l are implied by the buoyancy coefficient χ l (3.265) with (3.199) and the viscosity relation function f μ l (3.218), respectively. Clearly, the h∕ψ−formulations (3.262) are fully physically equivalent to the basic p−formulation (3.258) of the Darcy equation.
In the above species mass balance equations of Table 3.5 the dimensionless mass fraction ω k α appears as the natural variable of mass conservation. In practice, however, the mass concentration C k α (2.117) is often preferred, which is related to the mass fraction ω k α according to (2.123)
The replacement of mass fraction by mass concentration in the species mass conservation equations requires for some specific terms a further consideration, which is part of the next subject.
Oberbeck-Boussinesq Approximation and Extension
The system of balance equations listed in Table 3.5 is coupled by the nonlinearity in the fluid density ρ l. Its analysis can be substantially simplified by the so-called Oberbeck-Boussinesq (OB) approximation, sometimes termed only as Boussinesq approximation. As pointed out in [255, 389] the term OB approximation seems more appropriate because Oberbeck [393] addressed this problem before Boussinesq [49].
The OB approximation consists in neglecting all density dependencies in the balance terms, except for the crucial buoyancy term \({\rho }^{l}\boldsymbol{g}\) (or \({\chi }^{l}\boldsymbol{e}\)) which is retained in the momentum equation of Table 3.6 (or (3.262)). For the buoyancy term the fluid density dependency (3.199) is incorporated as a function of mass fraction ω k l and temperature T, however, no pressure dependency is considered here. Pressure dependency remains a subject of the derivative term ∂ ρ l∕∂ t appearing in the LHS of the liquid mass balance equation as further discussed below. Referring to saturated and nondeformable porous media and considering liquid incompressibility as well as no sources/sinks, the liquid mass conservation \({\mathcal{M}}^{l}\) of Table 3.5 reduces then to the simple expression \(\nabla \cdot \boldsymbol{ q}_{l} = 0\) and the velocity becomes solenoidal, cf. Sect. 2.1.10. This incompressibility assumption is common in most analytical and stability analyses of convection phenomena.
The OB approximation is valid if density changes Δ ρ l remain small in comparison to the reference density ρ 0 l. Criteria for the validity of the OB approximation for liquids and gases were given by Gray and Giorgini [204]. Obviously, the OB approximation becomes invalid for large density variations, e.g., at high-concentration brines and/or high temperature gradients. However, it is often not clear what consequences practically result if the full dependencies are incorporated (so-called non-Boussinesq effects). Usually, extensions to non-Boussinesq formulations can be introduced by ‘correction’ terms written for the liquid mass conservation equation \({\mathcal{M}}^{l}\) of Table 3.5 in the following form
with the extended Boussinesq approximation term
where \(\vert _{T,\omega _{k}^{l}}\) and \(\vert _{{p}^{l}}\) indicate that T, ω k l and p l, respectively, are held constant. Inserting the EOS for the liquid density (3.199) into (3.268) we can approximate
introducing a generalized thermal expansion coefficient β l∗
where β l is a given constant, while β l(T) and \({\partial \beta }^{l}(T)/\partial T\) correspond to (C.8) and (C.10), respectively, derived in Appendix D.
Kolditz et al. [318] compared OB solutions and some extended forms exemplified for the Elder cellular convection problem (cf. Sect. 11.11.4). For this case, OB solutions were rather close to non-Boussinesq model results. Only slight differences in pressure and concentration distributions in some parts of the model domain were observed. Evans and Raffensperger [160] studied the limitation of the OB approximation for a problem which is similar to the Elder problem. They found differences in the concentration distributions up to 9 % comparing the results of the different formulations. Gartling and Hickox [186] studied adjustments for the variation of fluid properties in the heat transport equation, while assuming the constraint of incompressibility, \(\nabla \cdot \boldsymbol{ q}_{l} = 0\). They found that the OB approximation and their extended solutions can be sufficiently ‘close’ for integrated quantities over large temperature ranges. However, differences can occur for local quantities. The accurate prediction of the flow field has been shown to be of major concern, and they concluded that the ‘goodness’ of the OB solutions depends on what quantities are of interest in the problem solution.
Furthermore, it is to be noted that under large compression effects when γ l becomes significant the OB solution can considerably violate mass conservativity and the extended OB approximation is to be preferred. However, for a realistically small liquid compressibility γ l, the term \(-\boldsymbol{q}_{l} {\cdot \gamma }^{l}\nabla {p}^{l}\) in (3.269) is usually negligible.
If we replace the mass fraction ω k l by the mass concentration C k l for species k in the governing equations of Table 3.5, we have to neglect density variations in the convective form of the species mass transport equation (3.228) and in the species mass flux vector \(\boldsymbol{j}_{\mathit{lk}}\), (3.187) or (3.183). These assumptions are acceptable within the OB approximation and its extension. This allows to approximate the derivative terms in the convective form as
assuming \((C_{k}^{l}{/\rho }^{l}){\partial \rho }^{l}/\partial t \approx 0\), \((C_{k}^{l}{/\rho }^{l}){\nabla \rho }^{l} \approx \mathbf{0}\), and to write the species mass flux vector \(\boldsymbol{j}_{\mathit{lk}}\) in the form
assuming \((C_{k}^{l}{/\rho }^{l})\boldsymbol{D}_{\mathit{lk}} \cdot {\nabla \rho }^{l} \approx \mathbf{0}\). Note that an evident advantage results in the divergence form (3.227) if \(\omega _{k}^{l}\) is replaced by C k l because assumption (3.271) is not necessary anymore.
Similarly, within the OB approximation and its extension density variations in the terms of the governing heat transport equations, both for the divergence and convective form, are neglected too.
Reformation of Terms
With the replacement of the pressure variable p l (3.108) by the hydraulic head h l (3.260) (or pressure head ψ l (3.259)) and the species mass fraction ω k α (2.123) by the mass concentration C k α (2.117) we have to adjust specific terms in the governing equations of Table 3.5. First, the differential of the liquid density ρ l (3.197) is modified:
to obtain
where h 0 l and C k0 l are reference values of the hydraulic head and mass concentration of species k, respectively, and C ks l represents a given maximum mass concentration of species k, which may be used to estimate the specific solutal expansion coefficient by a linear relation
sometimes called density ratio. A reasonable guess of the liquid density ρ l at maximum concentration C ks l is
where Baxter and Wallace [32] proposed for the factor a = 0. 7 and INTRAVAL project studies [395] used a = 0. 6923. It gives an estimation of the specific solutal expansion coefficient according to
The buoyancy coefficient (3.265) appearing in the Darcy equation (3.262) takes now with (3.274) the form:
where we introduce with
the solutal expansion coefficient of species k. The mass conservation equation of the liquid phase in the formulation of (3.267) can now be written as
to obtain by using (3.274) and (3.260)
where \(S_{o}^{l} =\rho _{ 0}^{l}g{(\varepsilon \gamma }^{l}+\upsilon )\) is the specific storage coefficient, sometimes called specific storativity [38], due to liquid and medium compressibility, and the correction sink term for the EOB approximation (3.268) gives now
To enforce conservativity for any magnitude of the specific storativity S o l ≥ 0, in the EOB approximation (3.282) the correcting divergence term of the liquid compression will be expressed by \(-\boldsymbol{q}{_{l}\gamma }^{l}\rho _{0}^{l}g \cdot \nabla {h}^{l} \approx -\boldsymbol{q}_{l}\frac{S_{o}^{l}} {\varepsilon } \cdot \nabla {h}^{l}\).
Basic Model Equations of Single Liquid Phase Variable-Density Darcy-Type Flow, Mass and Heat Transport in Variably Saturated Porous Media: Second Level Reduction
Applying the above assumptions and derivations of Sects. 3.10.1–3.10.4 to the equations of the first level reduction as listed in Table 3.5, we can now summarize the governing balance laws with their related constitutive relations in Table 3.7 as the basic model equations of second level reduction, which are formulated in the D−dimensional Euclidean space \({\mathfrak{R}}^{D}\) (D = 1, 2, 3). Because we assume that only one dynamic fluid phase, the liquid phase l, is present, we can omit the index l in the symbols for the sake of simplicity. Only the solid phase needs to be further identified by the index s. Typical adsorption relations for the adsorption function \(\varphi _{k}\), the retardation factor ℜ k and the derivative term of retardation \(\acute{\mathfrak{R}}_{k}\) are listed in Table 3.8, which are derived in detail in Chap. 5.
Basic Model Equations of Variable-Density Darcy-Type Flow, Mass and Heat Transport in Groundwater: Third Level Reduction
The equations of Table 3.7 can be simplified for flow, mass and heat transport in groundwater, the fully saturated porous medium. In this case
-
The saturation is set to s = 1
and the following system of equations results which is summarized in Table 3.9.
Basic Model Equations of Vertically Averaged Flow, Mass and Heat Transport in Unconfined and Confined Aquifers: Specific Case of Third Level Reduction
Flow, mass and heat transport, which are essentially horizontal in an aquifer, can be vertically averaged as described in Sect. 3.5. 2D depth-integrated balance equations result as described in Sect. 3.7.8 for which constitutive relations have to be added similar to those as developed for the full 3D problems above. In doing this, the following simplifications for the 2D, vertically averaged, essentially horizontal flow and transport processes in aquifers hold:
-
The aquifer forms a layer of a saturated porous medium of thickness B = B(x 1, x 2, t). While the bottom of the layer is considered stationary, on top the saturated zone is bounded by a possibly moving phreatic surface, so that as shown in Fig. 3.13:
$$\displaystyle{ \begin{array}{lcll} B(x_{1},x_{2},t)& =&h(x_{1},x_{2},t) - {f}^{B}(x_{1},x_{2})\quad &\mbox{ for unconfined condition} \\ B(x_{1},x_{2}) & =&{f}^{T}(x_{1},x_{2}) - {f}^{B}(x_{1},x_{2})\quad &\mbox{ for confined condition}\\ \end{array} }$$(3.283)where h = h(x 1, x 2, t) is the hydraulic head (3.260), \({f}^{T}(x_{1},x_{2})\) and \({f}^{B}(x_{1},x_{2})\) are the top and bottom bounding surfaces, respectively.
-
The coordinate direction of integration x j (commonly x j = x 3 = z) coincides with direction of gravity, i.e., \(\nabla x_{j} =\boldsymbol{ e}\). Accordingly, gravitational effects on liquid density disappear. (Extensions will be treated in Sect. 11.9).
Unconfined and confined conditions in an aquifer (vertical cross section x j = x 3)
The boundaries of the aquifer on top and bottom, respectively, can be expressed by their surface functions, cf. (2.112)
For unconfined conditions the top boundary moves with the velocity \(\boldsymbol{w}\) and in accordance with (2.113)–(2.115) it is
with the outward-pointing unit normal vector to the surface F T = 0
and the normal component of the moving surface with \({F}^{T} = x_{3} - h = 0\)
where we have assumed that \(\Vert \nabla {F}^{T}\Vert =\Vert \nabla (x_{3} - h)\Vert \approx \Vert \nabla x_{3}\Vert = 1\), i.e., the water table is approximately horizontal. For the stationary boundaries F T in the case of confined aquifer and F B we have
Now, let us consider the vertically averaged mass balance equation (3.76) written in the form:
We can replace the external mass supply \(Q_{\mathrm{ex}}\) by the interface relation (3.75) of the upper phreatic surface specified for mass
where \(\boldsymbol{w}\) designates the macroscopic surface velocity and \(\boldsymbol{n}\) is the outward unit normal vector to the moving surface F T. The phreatic surface separates the fully saturated zone from the unsaturated zone, where we assume that the interface is sharp. It forms the water table with h = x 3. For the unsaturated zone we assume that the liquid in the void space is at the residual (irreducible) saturation s r . On the upper side of the phreatic surface we take into account the possibility of accretion \(\boldsymbol{P}\), e.g., from precipitation, as depicted in Fig. 3.14.
Phreatic surface with accretion
The mass balance at the phreatic surface requires that the mass flux through the lower side of the interface at the saturated zone is equal to the mass flux through the upper side of the interface at the unsaturated zone, viz.,
The accretion is \(\boldsymbol{P} =\varepsilon \boldsymbol{ v}\bigr |_{\mathrm{unsat}}\). For a vertically downward-oriented accretion we use
where P corresponds to the rate of infiltration or groundwater recharge. Using (3.287) with \(\varepsilon \vert _{\mathrm{unsat}} =\varepsilon s_{r}\) and \(\boldsymbol{w}\bigr |_{\mathrm{sat}} =\boldsymbol{ w}\bigr |_{\mathrm{unsat}}\) we find
Then, with (3.291), (3.292), and (3.293) the interface BC reads
and the flux BC of a phreatic (free) surface results
where
is referred to as the specific yield (also called storativity or drainable and fillable porosity) of a phreatic aquifer and \(\boldsymbol{q} \cdot \boldsymbol{ n}\) is the positive outward normal flux of liquid leaving the saturated zone through the phreatic surface.
Using (3.294) the mass balance equation (3.289) for the unconfined aquifer can be written as
Since B = h − f B (3.283) the vertically averaged mass balance equation (3.297) for an unconfined aquifer finally takes the form:
where the derivations of ∂ ρ∕∂ t and \(\partial \varepsilon /\partial t\) have been developed in the same manner as described in Sect. 3.10.4. To simplify the notation we shall designate depth-integrated quantities by an overline and define
so that (3.298) can be written as
The remaining vertically averaged balance equations (3.77)–(3.82) for species mass, momentum and energy can now be similarly developed, where the same principles for the constitutive relations are applied as described in Sect. 3.10.6 for the fully saturated porous medium (groundwater). The resulting model equations of vertically averaged flow, mass and heat transport in an unconfined aquifer are summarized in Table 3.10.
Under confined aquifer conditions the boundary surfaces F T and F B are assumed stationary, so that ∂ B∕∂ t = 0 and \(B\varepsilon Q_{\mathrm{ex}} = P\) and the mass balance equation (3.300) reduces to the simple form
Usually, for confined aquifers the product of hydraulic conductivity \(\boldsymbol{K}\) (3.263) and aquifer thickness B = f T − f B (3.283) is combined in the tensor of transmissivity \(\boldsymbol{T}\) defined as
which represents an aquifer property measured as the flow rate per unit width through the entire aquifer thickness. The concept of transmissivity is only applicable to vertically averaged, essentially horizontal flow in confined aquifers. In Table 3.11 we summarize the governing model equations of vertically averaged flow, mass and heat transport in confined aquifers.
Standard Model Equations for Solving Flow, Mass and Heat Transport in Porous Media
The equation systems derived in Sect. 3.10 provide the general physical modeling basis for solving 3D and 2D (including axisymmetric and vertically averaged) variable-density flow, multispecies (chemically reactive) mass and heat transport processes in variably saturated porous media. They are summarized in Tables 3.7, 3.9, 3.10, and 3.11 for the variably saturated porous medium, for groundwater, for 2D unconfined and confined aquifers, respectively. In general, the equations are nonlinearly coupled due to density effects, dependencies by variable saturation (or presence of phreatic surface), chemical reactions, non-Fickian mass flux and viscosity effects. Four problem classes are distinguished:
-
1.
Flow: Solving the flow equations in a separate manner, there are no density and viscosity effects.
-
2.
Flow + mass: Solving flow and mass transport, which can be coupled by density, viscosity, chemical reaction and non-Fickian mass flux.
-
3.
Flow + heat: Solving flow and heat transport, which can be coupled by density and viscosity.
-
4.
Flow + mass + heat: This represents the most complex model for simultaneous solution of flow, mass and heat transport, which can be coupled by density, viscosity, chemical reaction and non-Fickian mass flux. If the non-isothermal mass transport is related to salinity, the processes are often termed as thermohaline. Since, in general, mass and heat have different diffusivities new phenomena can result for this problem class termed as double-diffusive convection (DDC).
With respect to the temporal dependency of the governing flow and transport equations we can choose three time classes:
-
(i)
Transient flow/transient transport: Both flow and (mass/heat) transport are simulated in their fully temporal dependency as formulated in the basic equations of Tables 3.7, 3.9, 3.10, and 3.11.
-
(ii)
Steady-state flow/transient transport: The flow process is considered stationary, i.e., ∂ h∕∂ t = 0, while (mass/heat) transport remains fully transient. This exceptional case is useful if the temporal, often short-term variations of flow are negligible in comparison to the temporal, often long-term variations of mass and/or heat transport.
-
(iii)
Steady-state flow/stationary transport: For both flow and (mass/heat) transport only steady-state solutions are searched, i.e., ∂ h∕∂ t = 0, ∂ C k ∕∂ t = 0 and ∂ T∕∂ t = 0. We note, however, under nonlinear conditions in general, a unique steady-state solution must not exist.
It is obvious that the Darcy law of momentum conservation, written in the general form
is very well suited for substituting the Darcy velocity \(\boldsymbol{q}\) in the mass conservation
to obtain the flow equation
which represents a generalized form of the so-called Richards’ equation named after L.A. Richards [440], who firstly derived and published such a type of flow equation for unsaturated porous media in 1931. The advantage is that only one primary variable h (or ψ) remains to be solved for the flow problem, while the Darcy velocity \(\boldsymbol{q}\) appears now as a secondary variable, which can be easily solved from (3.303) with known h. Accordingly, the solution is reduced to only 1 flow equation, N ∗ mass transport equations and 1 heat transport equation of scalar primary variables h, C k and T, respectively, as summarized in Table 3.12.
Notes
- 1.
It denotes a balance statement in its basic conservation formulation.
- 2.
It denotes a balance statement in which mass conservation is substituted.
- 3.
The equivalence of the area- and volume-averaged fluxes is shown for the interface term A α(ρ ψ) of (3.71), cf. [229]. The volume-averaged flux describes the REV average in the form:
$$\displaystyle{[\langle \rho \rangle _{\alpha }{\overline{\psi }}^{\alpha }(\boldsymbol{{v}}^{\alpha } -\boldsymbol{ W}) -\boldsymbol{ {j}}^{\alpha }] \cdot \boldsymbol{ {n}}^{\mathrm{TB}} = \frac{1} {\mathit{dV}}\Bigl (\int {_{{\mathit{dV}}^{\mathrm{TB}}}\gamma }^{\alpha }[\rho \psi (\boldsymbol{v} -\boldsymbol{ w}) -\boldsymbol{ j}]dv\Bigr ) \cdot \boldsymbol{ {n}}^{\mathrm{TB}}}$$Let us assume that the interface has a thickness D, the volume integral may be written
$$\displaystyle{ \frac{1} {\mathit{dV}}\int _{-D/2}^{D/2}\Bigl (\int {_{{ \mathit{dS}}^{\mathrm{TB}}}\gamma }^{\alpha }[\rho \psi (\boldsymbol{v} -\boldsymbol{ w}) -\boldsymbol{ j}] \cdot \boldsymbol{ {n}}^{\mathrm{TB}}\mathit{da}\Bigr )dl \approx \frac{D} {\mathit{dV}}\int {_{{\mathit{dS}}^{\mathrm{TB}}}\gamma }^{\alpha }[\rho \psi (\boldsymbol{v} -\boldsymbol{ w}) -\boldsymbol{ j}] \cdot \boldsymbol{ {n}}^{\mathrm{TB}}\mathit{da}}$$where mean values are used to replace the line integral. With dS = dV∕D we find finally
$$\displaystyle{[\langle \rho \rangle _{\alpha }{\overline{\psi }}^{\alpha }(\boldsymbol{{v}}^{\alpha } -\boldsymbol{ W}) -\boldsymbol{ {j}}^{\alpha }] \cdot \boldsymbol{ {n}}^{\mathrm{TB}} = \frac{1} {\mathit{dS}}\int {_{d{S}^{\mathrm{TB}}}\gamma }^{\alpha }[\rho \psi (\boldsymbol{v} -\boldsymbol{ w}) -\boldsymbol{ j}] \cdot \boldsymbol{ {n}}^{\mathrm{TB}}\mathit{da}}$$which corresponds to a α(ρ ψ) in (3.71).
- 4.
They represent constitutive relations originally found from the observation that fluxes of extensive quantities (e.g., mass, heat, momentum) are produced by the nonuniform distribution of their state variables (e.g., concentration gradient, temperature gradient, velocity difference). Frequently, a simple proportionality between fluxes and gradients of state variables is postulated using a parameter taken to be a property of the material (e.g., diffusivity, conductivity, friction).
- 5.
In index notation we derive (dropping phase indices for the sake of simplicity)
$$\displaystyle{\begin{array}{rcl} \frac{\partial } {\partial x_{j}}\bigl [\varepsilon \tfrac{1} {2}( \frac{\partial v_{i}} {\partial x_{j}} + \frac{\partial v_{j}} {\partial x_{i}})\bigr ] & = & \tfrac{1} {2} \frac{\partial } {\partial x_{j}}\bigl [\frac{\partial (\varepsilon v_{i})} {\partial x_{j}} + \frac{\partial (\varepsilon v_{j})} {\partial x_{i}} - v_{i} \frac{\partial \varepsilon } {\partial x_{j}} - v_{j} \frac{\partial \varepsilon } {\partial x_{i}}\bigr ] \\ & = & \tfrac{1} {2} \frac{{\partial }^{2}(\varepsilon v_{i})} {\partial x_{j}\partial x_{j}} + \tfrac{1} {2} \frac{{\partial }^{2}(\varepsilon v_{j})} {\partial x_{i}\partial x_{j}} -\tfrac{1} {2} \frac{\partial } {\partial x_{j}}\bigl (v_{i} \frac{\partial \varepsilon } {\partial x_{j}}\bigr ) -\tfrac{1} {2} \frac{\partial } {\partial x_{j}}\bigl (v_{j} \frac{\partial \varepsilon } {\partial x_{i}}\bigr ) \\ & = & \tfrac{1} {2} \frac{{\partial }^{2}(\varepsilon v_{i})} {\partial x_{j}\partial x_{j}} + \tfrac{1} {2}\varepsilon \frac{{\partial }^{2}v_{j}} {\partial x_{i}\partial x_{j}} -\tfrac{1} {2}v_{i} \frac{{\partial }^{2}\varepsilon } {\partial x_{j}\partial x_{j}} -\tfrac{1} {2}\bigl ( \frac{\partial v_{i}} {\partial v_{j}} -\frac{\partial v_{j}} {\partial v_{i}}\bigr ) \frac{\partial \varepsilon } {\partial x_{j}} \end{array} }$$ - 6.
In 3D Cartesian coordinates, with v 1, v 2 and v 3 denoting the velocity components in the x 1, x 2 and x 3 directions, respectively, and \(v =\Vert \boldsymbol{ {v}}^{fs}\Vert\), we obtain from (3.182), dropping phase indices for convenience
$$\displaystyle{\begin{array}{rcl} D_{\mathrm{mech},11} & = & \beta _{T}v + (\beta _{L} -\beta _{T})\frac{v_{1}^{2}} {v} = \frac{1} {v}{\bigl (\beta _{L}v_{1}^{2} +\beta _{T}v_{2}^{2} +\beta _{T}v_{3}^{2}\bigr )} \\ D_{\mathrm{mech},22} & = & \beta _{T}v + (\beta _{L} -\beta _{T})\frac{v_{2}^{2}} {v} = \frac{1} {v}{\bigl (\beta _{T}v_{1}^{2} +\beta _{L}v_{2}^{2} +\beta _{T}v_{3}^{2}\bigr )} \\ D_{\mathrm{mech},33} & = & \beta _{T}v + (\beta _{L} -\beta _{T})\frac{v_{3}^{2}} {v} = \frac{1} {v}{\bigl (\beta _{T}v_{1}^{2} +\beta _{T}v_{2}^{2} +\beta _{L}v_{3}^{2}\bigr )} \\ D_{\mathrm{mech},12} & = & (\beta _{L} -\beta _{T})\frac{v_{1}v_{2}} {v} = D_{\mathrm{mech},21} \\ D_{\mathrm{mech},13} & = & (\beta _{L} -\beta _{T})\frac{v_{1}v_{3}} {v} = D_{\mathrm{mech},31} \\ D_{\mathrm{mech},23} & = & (\beta _{L} -\beta _{T})\frac{v_{2}v_{3}} {v} = D_{\mathrm{mech},32} \end{array} }$$ - 7.
Using calculus manipulations the material derivative of E f with respect to the density ρ f can be alternatively developed for the \(\tfrac{{D{}^{f}\rho }^{f}} {\mathit{Dt}}\) term:
$$\displaystyle{{\frac{\varepsilon _{f}} {\rho }^{f}}{\Bigl ({p}^{f} - {T}^{f} \frac{\partial {p}^{f}} {\partial {T}^{f}}\Bigr )}\frac{{D{}^{f}\rho }^{f}} {\mathit{Dt}} ={ \frac{\varepsilon _{f}{p}^{f}} {\rho }^{f}} \frac{{D{}^{f}\rho }^{f}} {\mathit{Dt}} +\varepsilon _{f}{T{}^{f}\beta }^{f}\frac{{D}^{f}{p}^{f}} {\mathit{Dt}} }$$where the thermal expansion coefficient (3.197), \({\beta }^{f} = -(1{/\rho }^{f})({\partial \rho }^{f}/\partial {T}^{f})\), is inserted.
- 8.
It can be alternatively expressed by introducing the relationships (3.95) and (3.100) of the solid displacement \(\boldsymbol{{u}}^{s}\):
$$\displaystyle{ \frac{\partial } {\partial t}(\varepsilon {_{s}\rho }^{s}) +\varepsilon { _{s}\rho }^{s}\biggl (\boldsymbol{{m}}^{T} \cdot {\Bigl (\boldsymbol{ L} \cdot \frac{\partial \boldsymbol{{u}}^{s}} {\partial t} \Bigr )}\biggr )} + \nabla (\varepsilon {_{s}\rho }^{s}) \cdot \frac{\partial \boldsymbol{{u}}^{s}} {\partial t} ={\rho }^{s}Q_{s}$$ - 9.
Sometimes, the volumetric flux density is simply represented by the so-called Dupuit-Forchheimer relationship [389], which is a bulk flux in the form \(\boldsymbol{v}_{f} =\varepsilon _{f}\boldsymbol{{v}}^{f}\). This quantity has been given various names by different authors (e.g., seepage or filtration velocity). We shall prefer the term Darcy velocity \(\boldsymbol{q}_{f}\) emphasizing the correct relationship (3.240) for the flux.
- 10.
While a parallel behavior occurs in most of the natural porous media, there could be a porous-medium structure and orientation, where the heat conduction takes place in series. In this case, the heat flux can pass serially though the solid and the fluid, such that the overall thermal conductivity is a harmonic mean \(\boldsymbol{\varLambda }_{0}^{-1} =\varepsilon {s}^{f}{{(\varLambda }^{f}\boldsymbol{\delta })}^{-1} + (1-\varepsilon ){{(\varLambda }^{s}\boldsymbol{\delta })}^{-1}\). The arithmetic mean and harmonic mean represent upper and lower bounds, respectively, for the overall thermal conductivity \(\boldsymbol{\varLambda }_{0}\). Other, more empirical arrangements for \(\boldsymbol{\varLambda }_{0}\) can be made up for certain porous media as discussed in [305].
- 11.
From (3.260) it is \({p}^{l} =\rho _{ 0}^{l}g({h}^{l} - x_{j})\) and with \(\boldsymbol{e} = \nabla x_{j}\) we find \(\nabla {p}^{l} =\rho _{ 0}^{l}g(\nabla {h}^{l} -\boldsymbol{ e})\). Now expanding
$$\displaystyle\begin{array}{rcl}{ \frac{\boldsymbol{k}} {\mu }^{l}} =\underbrace{\mathop{ \frac{\boldsymbol{k}\rho _{0}^{l}g} {\mu _{0}^{l}} }}\limits _{\boldsymbol{{K}}^{l}}\,\underbrace{\mathop{{ \frac{\mu _{0}^{l}} {\mu }^{l}} }}\limits _{f_{\mu }^{l}}\, \frac{1} {\rho _{0}^{l}g} =\boldsymbol{ {K}}^{l}f_{\mu }^{l} \frac{1} {\rho _{0}^{l}g}& & {}\\ \end{array}$$and inserting into (3.258) with (3.261), we obtain
$$\displaystyle{\boldsymbol{q}_{l} = -k_{r}^{l}\boldsymbol{{K}}^{l}f_{\mu }^{l} \cdot \bigl (\nabla {h}^{l} + \tfrac{{\rho }^{l}-\rho _{ 0}^{l}} {\rho _{0}^{l}} \boldsymbol{e}\bigr )}$$
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Diersch, HJ.G. (2014). Porous Medium. In: FEFLOW. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38739-5_3
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