2.1 Introduction

Many excellent texts are available on the theory and practice of groundwater flow and transport (e.g., Bear 1972; Freeze and Cheery 1979; de Marsily 1986; Fetter 2000; Ingebritsen et al. 2006). Here we summarize the essentials so that subsequent discussions have some context and theoretical underpinning.

The study of groundwater flow through porous media is important for several related disciplines, including groundwater hydrogeology, contaminant transport, reservoir engineering, chemical engineering and, more recently, earthquake hydrology. It has been an active area of research since Henry Darcy established what we now call Darcy’s law based on column experiments conducted in 1855 and 1856.

An aquifer is a permeable and porous geologic formation or a fracture zone that allows significant fluid flow, thus may serve as an underground source of groundwater. More recently, aquifers have increasingly been used for the storage of wastewaters coproduced from hydrocarbon exploration and nuclear energy production. An aquiclude is an impervious geologic unit that prevents the flow of water, and an aquitard is a semi-confining unit that may allow limited water flow.

Confined aquifers are aquifers that are bounded on both sides by aquitards or aquicludes. Unconfined aquifers (also called phreatic aquifers or water-table aquifers) are aquifers that are bounded on their base by aquitards or aquicludes but by a water table on the top. Leaky aquifers are aquifers that are confined only partially by an aquitard on the top and/or on the base. Most aquifers behave somewhere between the confined and unconfined endmembers. The vertical impedance to flow across the boundary of a confined aquifer is not infinite, and the response of aquifers to applied loads may depend on the time scale of the applied load. In other words, the leakage of an aquifer may be frequency-dependent. For example, a confined aquifer may exchange flow across its boundaries at low frequencies; at the same time, an unconfined aquifer may exhibit some confined behaviors at high frequencies. Furthermore, the permeability of a groundwater system may change when stresses are applied (for example, by earthquakes), and confinement may thus be a dynamic property. For example, a confined aquifer may become leaky after an earthquake, and its permeability may or may not recover with time soon after the earthquake.

A porous geologic unit consists of a solid component and void space. The solid component forms the matrix, and the void space is referred as pores. The matrix of most unconfined aquifers consists of unconsolidated sediments with grain size from fine sand to gravel with a permeability that increases as grain size and sorting increase. Sediments with grain sizes finer than silt usually do not have permeability high enough to allow significant fluid flow. Confined aquifers may consist of either unconsolidated or consolidated sediments (sedimentary rocks). Some igneous rocks, such as fractured granites, pyroclastic deposits and lavas, may have appreciable connected pores and fractures to transmit groundwater and can make good aquifers.

For the study of groundwater as a continuum, a representative elementary volume (REV) is defined, which is sufficiently large relative to the scales of the microscopic heterogeneity (mineral grains, pores) that its averaged hydraulic properties become nearly constant from place to place, but is sufficiently small to be treated as microscopic in the continuum study. Given this definition, we may define porosity as the ratio between the volume of the void space to the bulk volume of a REV of the porous rock or sediments, i.e.,

$$\varphi = \frac{\text{volume of void space in an REV}}{\text{bulk volume of an REV}}.$$
(2.1)

Porosity of crustal materials may be as small as ~0 in some crystalline rocks and as large as >80% in some clay-rich sediments or volcanic deposits. We further differentiate between isolated and connected porosities. Only the connected porosity provides the channels for groundwater flow and is denoted as \(\varphi_{e}\)—the effective porosity. However, the term ‘porosity’ is often used to represent the effective porosity and the subscript of the symbol \(\varphi_{e}\) is often removed.

The top layer beneath the land surface is often unsaturated and the water content generally increases with depth in this layer until the rocks or sediments are fully saturated. Thus, as illustrated in Fig. 2.1, the subsurface may be vertically divided into an unsaturated zone, in which the pore space is partly filled with groundwater and partly with air, and a saturated zone, in which all the pores are filled with groundwater.

Fig. 2.1
figure 1

The unsaturated zone and the saturated zone in the groundwater occurrence. θ is the water content (volume of water divided by bulk volume), θs is the saturated water content and θr is the residual water content

Full size image

The water table is a surface above the groundwater on which the presssure is atmospheric;at equilibrium, its elevation is that of the water level in wells. The vertical position of the water table may change with time in response to precipitation, earthquakes, tides, barometric pressure, and anthropogenic processes such as irrigation.

The unsaturated zone above the saturated zone extends to the surface. . Here the capillary force between pore water and the surfaces of the solid grains acts to pull groundwater upward from the water table against gravity, forming a zone of negative pressure and variable saturation, which is also referred as the “capillary zone”. In response to the capillary force, the saturated zone actually extends a certain distance above the water table into the vadose zone, where pressure is negative. Following previous workers (e.g., Bear 1972; Gillham 1984) we refer to this saturated layer above the water table as the “capillary fringe” (Fig. 2.1).

The capillary zone may significantly affect the hydrogeologic response to earthquakes. Following the 2010 M8.8 Maule earthquake in Chile, for example, Mohr et al. (2015) reported increased stream flow in the Chilean Coast Range and proposed that the increased flow was due to the release of groundwater from the unsaturated zone. Breen et al. (2020) verified the hypothesis with a column experiment and showed that strong vibrations, such as seismic shaking, may disrupt the capillary tension to release water from unsaturated sands. The capillary zone is also known to significantly affect the response of aquifers to ocean tides (e.g., Barry et al. 1996) and solid Earth tides (Wang et al. 2019).

In most studies of groundwater, pore water is treated as fresh water with constant density and viscosity. This may change substantially in some cases in the study of the interactions between water and earthquakes. A notable example is the induced seismicity caused by the injected wastewater coproduced from hydrocarbon production. Here the high density of the injected saline water may drive deep flow and cause persistent induced earthquakes (e.g., Pollyea et al. 2019). On the other hand, in some Enhanced Geothermal Systems (EGS) developed for the recovery of geothermal energy, the injected water may be heated by the high temperature of the crust and expand in volume, and viscosity and density may differ substantially from ambient conditions. The expansion of heated water may cause tensile fractures in rocks and induce non-double-couple earthquakes (e.g., Martínez-Garzón et al. 2017). In such cases, the density and viscosity of water change with pressure and temperature, and thus with space and time.

2.2 Pressure, Hydraulic Head and Darcy’s Law

Groundwater flow is driven by both the gradient of the pressure energy and the gravitational energy (elevation), which are conveniently combined into the hydraulic head h [m]in hydrogeology:

$$h = \frac{P}{{\rho_{f} g}} + z$$
(2.2)

where P [Pa] is fluid pressure, g [m/s2] is gravity, \(\rho_{f}\) [kg/m3] is fluid density and z [m] is elevation. The first term on the right of the equation is the pressure head, the second the elevation head.

Henry Darcy established what we now call Darcy’s law based on column experiments conducted in 1855 and 1856. The permeability unit darcy (~10−12 m2) is named in his honor. According to Darcy’s law, the specific discharge q [m/s] (volume flux per unit area) is given by

$${\bf q} = - \varvec{K} \cdot \,\nabla h$$
(2.3)

where the bold symbols indicate vector or tensor quantities. K [m/s] is the hydraulic conductivity that is a second order tensor and depends on properties of both the fluid (density \(\rho_{f}\) and viscosity µ [Pa s]) and porous material (permeability k [m2])

$$\varvec{K} = \frac{{\rho_{f} g\varvec{k}}}{\mu }.$$
(2.4)

In most groundwater studies, where water temperature is nearly uniform and thus the density and viscosity of water are nearly constant, it is often convenient to use the parameter K instead of k. It is also often convenient to orient the coordinate axes of a Cartesian coordinate system along the principal directions of the K tensor so that the diagonal elements take the principle values Kx, Ky and Kz, and the off-diagonal elements become zero. In this case, Darcy’s law (2.3) takes the following form

$$q_{x} = - K_{x} \frac{\partial h}{\partial x},$$
(2.5a)
$$q_{y} = - K_{y} \frac{\partial h}{\partial y},$$
(2.5b)
$$q_{z} = - K_{z} \frac{\partial h}{\partial z}.$$
(2.5c)

Darcy’s law is found to be a good approximation at Reynolds numbers up to ~10, where the Reynolds number is defined as \(\rho_{f} vd/\mu\), d is the pore dimension and v is the speed of the fluid itself (variably called the interstitial, pore, linear, or seepage velocity), which is different from the specific discharge and can be approximated as

$$\upsilon \sim q/\varphi$$
(2.6)

where φ is porosity.

2.3 Permeability of Layered Media

In the previous  section we showed that permeability in general is a second-order tensor. In this section we show that the average permeability of layered rocks, common in hydrogeological settings, are generally anisotropic, even if the permeability of each individual layer is isotropic. For layered rocks, the average permeabilities parallel to and normal to the bedding of the layered rocks are, respectively,

$$k_{\parallel } = \sum\limits_{i} {k_{i} \left( {\frac{{b_{i} }}{{b_{t} }}} \right)}$$
(2.7a)

and

$$k_{ \bot } = \left( {\frac{{b_{t} }}{{\mathop \sum \nolimits_{i} b_{i} /k_{i} }}} \right),$$
(2.7b)

where \(k_{i}\) is the permeability of the ith layer, \(b_{i} [{\text{m}}]\) is the thickness of the ith layer, and \(b_{t}\) is the total thickness of the layered sediments or rocks. Equations (2.7a, b) show that the average permeability parallel to the bedding of layered rocks or sediments is dominated by the layer with the greatest transmissimivity \((b_{i} k_{i} )\), while the average permeability perpendicular to the bedding is dominated by the layer with the lowest value of \(k_{i} /b_{i}\). Hence the average permeability of layered rocks is generally anisotropic even if the permeability of each individual layer is isotropic.

Permeability may also evolve with time due to ongoing geological and biogeochemical processes such as dissolution, precipitation, and the formation of clay minerals, cracks and fractures. The time scales for calcite dissolution is 104–105 years and for silica precipitation is weeks to years. Permeability may also change suddenly during earthquakes, at time scales of one to tens of seconds. In addition, permeability can be scale dependent and spatially variable.

2.4 Specific Storage and Specific Yield

The specific storage Ss [m−1] is the amount of water released per unit volume of a saturated confined aquifer per unit change of the hydraulic head; i.e.,

$$S_{s} = \frac{1}{{\rho_{f} }}\frac{{\partial \left( {\varphi \rho_{f} } \right)}}{\partial h}.$$
(2.8)

Replacing h by \(P/(\rho_{f} g)\) and applying the chain rule, \(\frac{{\partial \left( {\varphi \rho_{f} } \right)}}{\partial h} = \frac{{\partial \left( {\varphi \rho_{f} } \right)}}{\partial P}\frac{\partial P}{\partial h} ,\) we have

$$S_{s} = \rho_{f} g\left( {\beta_{\varphi } + \varphi \beta_{f} } \right)$$
(2.9)

where \(\beta_{\varphi } \left[ {{\text{Pa}}^{ - 1} } \right] = \frac{\partial \varphi }{\partial P}\) and \(\beta_{f} \left[ {{\text{Pa}}^{ - 1} } \right] = \frac{1}{{\rho_{f} }}\frac{{\partial \rho_{f} }}{\partial P}\) are the compressibilities of porosity and water, respectively. The storativity of a saturated confined aquifer is \(S = bS_{s}\) where b is the thickness of the aquifer. While \(S_{s}\) has a dimension of [m−1], S is dimensionless.

The specific yield \(S_{y}\) of an unconfined aquifer is similarly defined, except that here the amount of water released from a column of sediment or rock is due to a unit change of the water table instead of that of the hydraulic head. The ratio of the volume of water drained to the volume of the aquifer is the specific yield.

2.5 Saturated Flow

2.5.1 Isothermal Flow

Most studies of earthquake-induced hydrological processes consider only the uppermost crust where temperature is nearly constant; as a result, density and viscosity of pore water are also nearly constant, which considerably simplifies the differential equation that controls goundwater flow. This will be studied first.

In the absence of a fluid source, the continuity (conservation of mass) equation for fluid mass in a saturated aquifer is

$$\frac{{\partial \left( {\varphi \rho_{f} } \right)}}{\partial t} = - \nabla \cdot \left( {\rho_{f} \varvec{q}} \right) + \rho_{f}Q,$$
(2.10)

where Q is a fluid source (positive) or a sink (negative) per unit volume.

Applying the chain rule \(\frac{{\partial \left( {\varphi \rho_{f} } \right)}}{\partial t} = \frac{{\partial \left( {\varphi \rho_{f} } \right)}}{\partial h}\frac{\partial h}{\partial t}\), Darcy's law (2.3) and the definition of the specific storage (2.8) we obtain the differential equation for saturated flow

$$\rho_{f} S_{s} \frac{\partial h}{\partial t} = \nabla \cdot \left( {\rho_{f} \varvec{K}\nabla h} \right) + \rho_{f}Q.$$
(2.11)

In most studies where the fluid density \(\rho_{f}\) is constant we have the more familiar form of the equation for saturated flow

$$S_{s} \frac{\partial h}{\partial t} = \nabla \cdot \left( {\varvec{K}\nabla h} \right) + Q.$$
(2.12a)

We adopt a Cartesian coordinate system along the principle directions of K. If Kx, Ky and Kz are spatially constant, Eq. (2.12a) may be expressed as

$$S_{s} \frac{\partial h}{\partial t} = K_{x} \frac{{\partial^{2} h}}{{\partial x^{2} }} + K_{y} \frac{{\partial^{2} h}}{{\partial y^{2} }} + K_{z} \frac{{\partial^{2} h}}{{\partial z^{2} }} + Q.$$
(2.12b)

In the study of groundwater flow to wells, polar coordinates are used. Here the aquifer is usually treated as a horizontal and laterally isotropic layer, and the principal components of K are thus Kr and Kz. Equation (2.12a) may then be expressed as

$$S_{s} \frac{\partial h}{\partial t} = K_{r} \left( {\frac{{\partial^{2} h}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial h}{\partial r}} \right) + K_{z} \frac{{\partial^{2} h}}{{\partial z^{2} }} + Q.$$
(2.12c)

2.5.2 Flow Through Variable Temperatures

In the cases where groundwater flows across considerable depth or occurs in geothermal areas where temperature changes along the flow paths, the assumptions made for isothermal flow break down. Here we need to ‘couple’ the groundwater flow equation to the heat transport equation such that the temperature dependence of the physical parameters may be calculated as a function of time and, at the same time, the effect of groundwater flow on the groundwater temperature may be calculated at the same time. Here we only consider the effect of groundwater temperature on the flow equation. Considering only the effects of temperature on fluid density and fluid viscosity, we may write the two-dimensional flow equation as follows:

$$S_{s} \frac{\partial h}{\partial t} = \frac{\partial }{\partial x}\left( {\frac{{\rho_{w\left( T \right)}gk_{H} }}{\mu \left( T \right)}\frac{\partial h}{\partial x}} \right) + \frac{\partial }{\partial z}\left( {\frac{\rho_{w\left( T \right)}gk_{V} }{\mu \left( T \right)}\frac{\partial h}{\partial z}} \right),$$
(2.13)

where kH and kV are, respectively, the horizontal and vertical permeability, \(\rho_{w}\) and μ are, respectively, density and viscosity of water; Ss is the specific storage defined as

$$S_{s} = \frac{1}{{\rho_{w} \left( T \right)}}\frac{{\partial \left[ {\varphi \rho_{w} \left( T \right)} \right]}}{\partial h}.$$
(2.14)

2.6 Unsaturated Flow

Only isothermal flow is considered here. Richards (1931) showed that the basic proportionality between flow and the driving force in Darcy’s law, as shown previously for saturated flow (2.3), remains true for unsaturated flow. The essential difference between the saturated and unsaturated flows is that the hydraulic conductivity K for the latter is a function of the water content \(\theta\) of the porous medium, defined as

$$\theta = \frac{\text{volume of water in an REV}}{\text{bulk volume of REV}}.$$
(2.15)

Thus, Darcy’s equation for unsaturated flow has the following form (Richards 1931)

$$\varvec{q} = - \varvec{K}\left( \theta \right) \cdot \nabla h,$$
(2.16)

where h is the hydraulic head, identical to (2.2), but has a negative value owing to the capillary force and is often called suction or the ‘matric potential’. \(\varvec{K}\left( \theta \right)\) is the unsaturated hydraulic conductivity tensor and is often expressed as \(\varvec{K}\left( \theta \right) = k_{r} \left( \theta \right)\varvec{K}_{s}\) where \(k_{r} \left( \theta \right)\) is the relative conductivity that varies between 0 and 1, and \(\varvec{K}_{s}\) is the saturated hydraulic conductivity tensor.

Neglecting a fluid source, the continuity equation for fluid in unsaturated rocks is

$$\frac{{\partial \left( {\theta \rho_{f} } \right)}}{\partial t} = - \nabla \cdot \left( {\rho_{f} \varvec{q}} \right).$$
(2.17)

If we assume the water in unsaturated flow is incompressible, (2.17) becomes

$$\frac{\partial \theta }{\partial t} = - \nabla \cdot \varvec{q}.$$
(2.18)

Combining (2.16) and (2.18) we obtain the equation that governs unsaturated flow

$$C\left( \theta \right)\frac{\partial h}{\partial t} = \nabla \cdot \left[ {\varvec{K}\left( \theta \right)\nabla h} \right].$$
(2.19)

where \(C\left( \theta \right)\) [m−1] is the specific water capacity defined as (Bear 1972, p 496)

$$C\left( \theta \right) = \frac{\partial \theta }{\partial h}.$$
(2.20)

For horizontally layered porous media, with water content \(\theta\) changing only in the vertical direction, we choose a Cartesian coordinate system with the z-axis in the vertical direction and the x- and y-axes in the horizontal plane and along the principle directions. \(K_{z} \left( \theta \right)\) thus depends on z, and \(K_{x} \left( \theta \right)\) and \(K_{y} \left( \theta \right)\) are spatially constant. Equation (2.19) reduces to

$$C\left( \theta \right)\frac{{\partial h}}{{\partial t}} = {K_x}\left( \theta \right)\frac{{{\partial ^2}h}}{{\partial {x^2}}} + {K_y}\left( \theta \right)\frac{{{\partial ^2}h}}{{\partial {y^2}}} + \frac{\partial }{{\partial z}}\left( {{K_z}\left( \theta \right)\frac{{\partial h}}{{\partial z}}} \right).$$
(2.21)

Similarly, for the study of groundwater flow to wells, we have,

$$C\left( \theta \right)\frac{\partial h}{\partial t} = K_{r} \left( \theta \right)\left( {\frac{{\partial^{2} h}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial h}{\partial r}} \right) + \frac{\partial }{\partial z}\left( {K_{z} \left( \theta \right)\frac{\partial h}{\partial z}} \right).$$
(2.22)

Comparing Eqs. (2.21) and (2.22) with the equations for saturated flow (2.12b ) and (2.12c ) shows that the differences between the saturated and unsaturated flows are that the conductivity for unsaturated flow is a function of water content and that the specific storage \(S_{s}\) is replaced by the specific water capacity \(C\left( \theta \right)\).

2.7 Heat Transport

Heat transport in groundwater systems occurs through both conduction and advection by fluid flow. The conductive transport is governed by Fourier’s law

$$\varvec{q}_{h} = - \varvec{K}_{h} \cdot \nabla T$$
(2.23)

where \(\varvec{q}_{h}\) [W/m2] is the heat flux by conduction, Kh [W/(m-K)] is the thermal conductivity tensor, and T [oK] is temperature. While the hydraulic conductivity of rock varies by 16 orders of magnitude, the average thermal conductivity \(K_{h}\) varies by less than a factor of five. Clay, one of the least conductive materials, has \(K_{h} = 1\;{\text{W}}/({\text{m K}})\), while granite, a relatively good thermal conductor, has \(K_{h} = 3\;{\text{W}}/({\text{m K}})\). For saturated porous media, the average thermal conductivity may be estimated with

$$K_{h} = K_{f}^{\varphi } K_{r}^{1 - \varphi }$$
(2.24)

where Kf and Kr are, respectively, the thermal conductivity of the pore fluid and the solid rock. At 25 °C, Kf is about 0.6 W/(m K) so that the thermal conductivity of saturated porous rocks is dominated by the mineralogy.

Combining Fourier’s law (2.23) with the conservation law for thermal energy, we obtain the differential equation for the thermal transport of heat by conduction

$$\rho c\frac{\partial T}{\partial t} = \nabla \cdot \left( {\varvec{K}_{h} \nabla T} \right) + Q_h,$$
(2.25)

where \(\rho\) [kg/m3] and \(c\) [J/(kg-K)] are, respectively, the bulk density and specific heat of the aquifer Qh is a heat source (positive) or heat sink (negative) per unit volume. If  Qh = 0 and the aquifers is uniform and isotropic (i.e., constant \(K_{h}\)), Eq. (2.25) takes the simplified form:

$$c\rho \frac{\partial T}{\partial t} = K_{h} \nabla^{2} T$$
(2.26)

The product \(c\rho\) for a porous rock with porosity \(\varphi\) may be estimated from the arithmetic mean of the solid and fluid components of the aquifer, i.e., \(c\rho = \varphi \rho_{f} c_{f} + \left( {1 - \varphi } \right)\rho_{r} c_{r}\), where \(c_{f}\) is the specific heat of the pore fluid and \(c_{r}\) that of the rock matrix.

Fluid flow can be effective at transporting heat. The amount of advective transport is proportional to the gradient of the thermal energy and the specific discharge. Hence heat transport in groundwater consists of both a conductive process and an advective process, and the governing equation becomes

$$\left[ {\varphi \rho_{f} c_{f} + \left( {1 - \varphi } \right)\rho_{r} c_{r} } \right]\frac{\partial T}{\partial t} = K_{h} \nabla^{2} T - \rho_{f} c_{f} \varvec{q} \cdot \nabla T,$$
(2.27)

The specific discharge q in the equation couples groundwater flow to heat transport. In most studies, the effect of temperature on flow is small and the ‘coupling’ between flow and heat transport is treated ‘one-way’, meaning that groundwater discharge is included in the transport equation, but the flow equation assumes an isothermal condition (2.12a).

The isothermal assumption is acceptable when the temperature of the groundwater is nearly constant. However, in situations where groundwater flows across steep geothermal gradients such as in geothermal systems, temperature may affect the density and viscosity of water, which in turn affect the velocity and direction of groundwater flow. The effect of pressure is relatively small and usually neglected. In such cases, the equation for groundwater transport becomes nonlinear and may be expressed as

$$\frac{{\partial \left[ {\rho \left( T \right)c\left( T \right)T} \right]}}{\partial t} = \nabla \cdot \left[ {K_{h} \left( T \right)\nabla T} \right] - \varvec{q} \cdot \nabla \left[ {\rho_{f} \left( T \right)c_{f} \left( T \right)T} \right],$$
(2.28)

where c and \(\rho\) are, respectively, the specific heat and density of the bulk sediments or rocks, \(K_{h}\) the thermal conductivity, and the subscript f refers to the properties of water. The bulk properties of the saturated sediments may be approximated as the linear mixture of solid grains and pore water, e.g., \(K_{h} = \varphi K_{f} + ( 1- \varphi )K_{s}\), etc., where the subscript s indicates solid grains. Here the coupling between groundwater flow and transport is ‘two-way’, meaning that the temperature affects the flow (2.13, 2.14) and the latter affects the groundwater temperature. Solution of equation (2.28) requires the simultaneous solution of groundwater flow (2.13). Numerical procedures are often required for solving the coupled nonlinear equations.

The relative significance of the advective versus the conductive heat transport may be accessed by using the following procedure. Since the magnitude of the conductive transport across a region with temperature difference ΔT is of the order of Kh ΔT/L2 and the magnitude of the advective transport is of the order of \(\rho_{f} c_{f} q\Delta T/L\), where L is the linear dimension of the studied region, the relative significance of the advective versus the conductive transport is given by the ratio

$${\text{Pe}} = \frac{{\rho_{f} c_{f} q\Delta T/L}}{{K_{h}\Delta T/L^{2} }} = \frac{{\rho_{f} c_{f} qL}}{{K_{h} }}$$
(2.29)

which is known as the Peclet number. If Pe is greater than 1, the advective heat transport is more important than the conductive heat transport, and vice versa.

Some other dimensionless number related to heat transport in porous flow include the Nusselt number Nu which is the ratio between the total heat transport and the conductive transport:

$${\text{N}}_{\rm u} = \frac{{c_{f} \rho_{f} qT + \frac{{K_{h} \Delta T}}{L}}}{{\frac{{K_{h} \Delta T}}{L}}}$$
(2.30)

and the Rayleigh number Ra which indicates the tendency of the pore fluid towards free convection, i.e., flow driven purely by density differences.

$${\text{Ra}} = \frac{{\left[ {\frac{{k\alpha_{f} \rho_{f} g \Delta T}}{\mu }} \right]\rho_{f} c_{f} L}}{{K_{h} }}$$
(2.31)

where \(\alpha_{f}\) is the thermal expansivity of the pore fluid. Fluid will start to convect when Ra exceeds some critical value depending on the boundary conditions.

2.8 Solute Transport

The concentration of a solute in groundwater can be expressed either as a mass fraction or mass per unit volume. Here we use the latter, i.e., mass per unit volume, as the definition. Let \(C_{c}\) [kg/m3] be the concentration of a chemical component in the fluid. If there is a gradient in \(C_{c}\), diffusion will occur, with flux \(\varvec{q}_{d} \left[ {{\text{kg}}/\left( {{\text{m}}^{2} {\text{s}}} \right)} \right]\) given by Fick’s law

$$\varvec{q}_{d} = - D_{w} \nabla C_{c} ,$$
(2.32)

where \(D_{w}\) [m2/s] is the coefficient of molecular diffusion in water.

In isotropic porous media, the diffusion of chemical components is impeded by the tortuous paths, in addition to the limited pore space, and the diffusion coefficient for the porous media \(D_{m}\) may be related to \(D_{w}\) by

$$D_{m} = \frac{{\varphi_{e} }}{\tau }D_{w} ,$$
(2.33)

where \(\varphi_{e}\) is the effective porosity (i.e., pore space in which chemical components can move with flow) and \(\tau\) is tortuosity (i.e., the ratio between the actual path length of the solute molecule through the porous medium from one point to another and the straight-line distance between the two points). Typical values for the diffusion coefficients for geologic media range from 10−13 to 10−11 m2/s. Here Fick’s law for diffusion is modified as

$$\varvec{q}_{d} = - D_{m} \nabla C_{c} .$$
(2.34)

The continuity equation for solute transport is

$$\frac{{\partial \left( {\varphi_{e} C_{c} } \right)}}{\partial t} = - \nabla \cdot \varvec{q}_{d} .$$
(2.35)

Combining (2.34) and (2.35) and assuming that \(\varphi_{e}\) is constant, we have

$$\varphi_{e} \frac{{\partial C_{c} }}{\partial t} = D_{m} \nabla^{2} C_{c} .$$
(2.36)

Porous flow is another important mechanism for solute transport, and the amount of solute transport by this mechanism is proportional to the velocity of groundwater flow. Combining the diffusive solute transport (Fick’s law) and the advective solute transport, we have the governing equation for solute transport with groundwater flow in a porous medium

$$\frac{{\partial C_{c} }}{\partial t} = \nabla \cdot \left( {\varvec{D } \cdot \nabla C_{c} } \right) -\varvec{\nu}\cdot \nabla C_{c} + Q_{c} ,$$
(2.37)

where \(\varvec{D} = \varvec{D}_{m} /\varphi_{e}\), \(\varvec{\nu}= \varvec{q}/\varphi_{e}\) and \(Q_{c}\) is the rate at which mass of the solute is produced (positive) or removed (negative) per unit volume by chemical reactions.

Microscopic heterogeneities and velocity variations across pores in the porous medium cause mechanical dispersion that results in dispersion anisotropy even if the porous medium is macroscopically isotropic. The combined effect of the mechanical dispersion and the molecular diffusion along and normal to the direction of groundwater flow may be represented in terms of the longitudinal and transverse dispersivities, \(\alpha_{L}\) and \(\alpha_{T}\),

$$D_{L} = \alpha_{L} \left| \nu \right| + D$$
(2.38a)
$$D_{T} = \alpha_{T} \left| \nu \right| + D$$
(2.38b)

where DL and DT, respectively, are the longitudinal and transverse hydrodynamic dispersion coefficients.

The relative significance of the advective versus the diffusive solute transport may be accessed by a procedure similar to that discussed in the last section on heat transport. Since the magnitude of the diffusive transport is of the order of \(D\Delta C/L^{2}\) and the magnitude of the advective transport is of the order of \(v\Delta C/L\), the relative significance of the advective versus the diffusive transport is given by the ratio

$${\rm Pe} = \frac{{v\Delta C/L}}{{D\Delta C/L^{2} }} = \frac{vL}{D},$$
(2.39)

which is the Peclet number for solute transport. If Pe is greater than 1, the advective solute transport is more important than the diffusive solute transport, and vice versa.