Discrete Feature Modeling of Flow, Mass and Heat Transport Processes

Abstract

The discrete feature approach provides the crucial link between the complex geometries for subsurface and surface, porous and fractured continua as well as to incorporate engineered structures in modeling flow, mass and heat transport processes. In such a holistic approach a 3D geometry of the subsurface domain (aquifer system, rock masses) in describing a porous-medium structure can be combined by interconnected 1D and/or 2D discrete features as shown in Fig. 14.1. In the finite element context the 3D mesh for the porous medium can be enriched by discrete line (channel, borehole, pipe network, tunnel, mine stope) and/or areal (overland, fault, fracture) elements.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction

The discrete feature approach provides the crucial link between the complex geometries for subsurface and surface, porous and fractured continua as well as to incorporate engineered structures in modeling flow, mass and heat transport processes. In such a holistic approach a 3D geometry of the subsurface domain (aquifer system, rock masses) in describing a porous-medium structure can be combined by interconnected 1D and/or 2D discrete features as shown in Fig. 14.1. In the finite element context the 3D mesh for the porous medium can be enriched by discrete line (channel, borehole, pipe network, tunnel, mine stope) and/or areal (overland, fault, fracture) elements.

Fig. 14.1

Schematization of a subsurface modeling system by combining discrete features with volume discretizations of the total study domain: 1D feature elements are used to approximate rivers, channels, wells and specific faults, 2D feature elements are appropriate for modeling runoff processes, fractured surfaces and faulty zones, and 3D elements represent the basic tessellation of the subsurface domain consisting of an aquifer-aquitard system and involving unsaturated and saturated zones

Discrete features are geometric representations of a lower spatial dimension having commonly a significant fluid conductance in comparison to the porous medium. Their conceptual modeling approach and resulting basic equations are thoroughly described in Chap. 4, where a unified basis in form of diffusion-type flow equations, e.g., Darcy, Hagen-Poiseuille or Manning-Strickler laws of fluid motion, as well as mass and heat transport equations is derived. Discrete features are approximated as 1D or 2D finite elements termed as discrete feature elements (DFE’s), which can be mixed with porous-medium elements in two and three dimensions (Fig. 14.2). The 1D and 2D DFE’s share the nodes of the porous-medium elements and can be placed along element edges and faces or can even interconnect arbitrary nodes in a finite element mesh. Both the geometric and physical characteristics of DFE provide a large flexibility in modeling complex situations.

Fig. 14.2

Example of 1D and 2D DFE’s mixed with 3D porous-medium elements

Discrete Feature Master Equations

The governing balance equations for discrete features as listed in Tables 4.5–4.7 for flow, species mass and heat, respectively, can be generalized by the following advection-diffusion-type master equation

$$\displaystyle{ \begin{array}{ll} \mathcal{L}_{F}(\phi ) =&\mathcal{S}_{F} \frac{\partial \phi } {\partial t} +\boldsymbol{ v}_{F} \cdot \nabla \phi -\nabla \cdot (\boldsymbol{\varUpsilon }_{F} \cdot \nabla \phi ) +\varTheta _{F}\phi - Q_{F} + Q_{\phi w_{F}} = 0 \\ &\mbox{ in}\quad \varOmega _{F} \subset {\mathfrak{R}}^{D}\;(D = 1,2),\;t \geq t_{0}\end{array} }$$

(14.1)

which has to be solved for flow (ϕ: = h), species mass (ϕ: = C _k ) and heat (ϕ: = T) in 1D or 2D space Ω _F ⊂Ω of a discrete feature F subject to the Dirichlet, Neumann and Cauchy BC’s as well as well-type SPC as

$$\displaystyle\begin{array}{rcl} \begin{array}{rcll} \phi & =&\phi _{D} &\mbox{ on}\;\varGamma _{D_{F}} \times t[t_{0},\infty ) \\ - (\boldsymbol{\varUpsilon }_{F} \cdot \nabla \phi ) \cdot \boldsymbol{ n}& =&q_{N} &\mbox{ on}\;\varGamma _{N_{F}} \times t[t_{0},\infty ) \\ - (\boldsymbol{\varUpsilon }_{F} \cdot \nabla \phi ) \cdot \boldsymbol{ n}& =& -\varPhi _{F}(\phi _{C}-\phi ) &\mbox{ on}\;\varGamma _{C_{F}} \times t[t_{0},\infty ) \\ Q_{\phi w_{F}} & =& -\sum _{w}{\bigl (\phi _{w} -\phi (\boldsymbol{x}_{w})\bigr )}Q_{w}(t)\delta (\boldsymbol{x} -\boldsymbol{ x}_{w})&\mbox{ on}\;\boldsymbol{x}_{w} \in \varOmega _{F} \times t[t_{0},\infty )\end{array} & &{}\end{array}$$

(14.2)

imposed on the discrete feature boundary $\varGamma _{F} =\varGamma _{D_{F}} \cup \varGamma _{N_{F}} \cup \varGamma _{C_{F}}$ and associated with IC of the form

$$\displaystyle{ \phi (\boldsymbol{x},t_{0}) =\phi _{0}(\boldsymbol{x})\quad \mbox{ in}\quad \bar{\varOmega }_{F} }$$

(14.3)

In the master equation (14.1) ϕ corresponds to a generalized variable, the gradient operator ∇ refers only to 1D or 2D space and $\mathcal{S}_{F}$, $\boldsymbol{v}_{F}$, $\boldsymbol{\varUpsilon }_{F}$, Θ _F , Q _F , $Q_{\phi w_{F}}$ represent specific quantities of storage, advection, dispersion, transfer, supply and well-type sink/source, respectively, which have to be specified from Tables 4.5–4.7 in dependence on the occurring type and dimension of the discrete feature F. The essential parameters required for solving (14.1) with (14.2) and (14.3) are listed in Table I.9 for flow, in Table I.15 for mass transport and in Table I.18 for heat transport of Appendix I. Steady-state situations occur if $\mathcal{S}_{F} = 0$.¹

Finite Element Formulation

The governing ADE (14.1) of flow, mass and heat transport in 1D or 2D discrete features is mathematically similar to the paradigmatic ADE of a scalar variable used in Chap. 8. Based on the principles given there we use now the GFEM to solve (14.1) for the generalized variable $\phi =\phi (\boldsymbol{x},t)$ subject to the corresponding BC’s (14.2) and IC (14.3). Since most of the details are equivalent to the ADE developments given in Chap. 8 we shall focus here only on the specific aspects related to the DFE approach.

Weak Form

In analogy to the statement (8.55) of Sect. 8.5 we find the corresponding weak form for the governing ADE (14.1) in its convective form as

$$\displaystyle\begin{array}{rcl} & & \int _{\varOmega _{F}}w\mathcal{S}_{F} \frac{\partial \phi } {\partial t}d\varOmega +\int _{\varOmega _{F}}w\boldsymbol{v}_{F} \cdot \nabla \phi d\varOmega +\int _{\varOmega _{F}}\nabla w \cdot (\boldsymbol{\varUpsilon }_{F} \cdot \nabla \phi )d\varOmega + \\ & & \qquad \int _{\varOmega _{F}}w[\varTheta _{F}\phi - Q_{F}]d\varOmega +\sum _{w}w(\boldsymbol{x}_{w}){\bigl (\phi _{w} -\phi (\boldsymbol{x}_{w})\bigr )}Q_{w}(t) + \\ & & \int _{\varGamma _{N_{ F}}}wq_{N}d\varGamma -\int _{\varGamma _{C_{F}}}w\varPhi _{F}(\phi _{C}-\phi )d\varGamma = 0,\quad \forall w \in H_{0}^{1}(\varOmega _{ F})\quad {}\end{array}$$

(14.4)

where w is a suitable weighting function and the boundary integrals are suitably separated into their segments $\varGamma _{F} =\varGamma _{D_{F}} \cup \varGamma _{N_{F}} \cup \varGamma _{C_{F}}$ imposed by the Dirichlet, Neumann and Cauchy-type BC’s (14.2) on the discrete feature.

GFEM and Resulting Matrix System

In using the FEM the unknown variable ϕ appearing in the weak statement (14.4) is replaced by a continuous approximation that assumes the separability of space and time (cf. Sect. 8.4). Thus

$$\displaystyle{ \phi (\boldsymbol{x},t) \approx \sum _{j}N_{j}(\boldsymbol{x})\phi _{j}(t),\quad j = 1,\ldots,N_{\mathrm{P}_{F}} }$$

(14.5)

where j designates global nodal indices and $N_{\mathrm{P}_{F}}$ is the number of nodal points related to a discrete feature F. Using the Galerkin method with the weighting function

$$\displaystyle{ w \rightarrow w_{i} = N_{i},\quad i = 1,\ldots,N_{\mathrm{P}_{F}} }$$

(14.6)

and applying the approximate solutions (14.5) in (14.4), the following matrix system of $N_{\mathrm{P}_{F}}$ equations results

$$\displaystyle{ \boldsymbol{O}_{F} \cdot \dot{\boldsymbol{\phi }} +\boldsymbol{K}_{F} \cdot \boldsymbol{\phi }-\boldsymbol{F}_{F} = \mathbf{0} }$$

(14.7)

where

$$\displaystyle\begin{array}{rcl} \boldsymbol{\phi } = \left (\begin{array}{c} \phi _{1}\\ \phi _{ 2}\\ \vdots\\ \phi _{N_{ \mathrm{P}_{F}}} \end{array} \right ),\quad \dot{\boldsymbol{\phi }} = \left (\begin{array}{c} \frac{d\phi _{1}} {\mathit{dt}} \\ \frac{d\phi _{2}} {\mathit{dt}}\\ \vdots \\ \frac{d\phi _{N_{\mathrm{P}_{ F}}}} {\mathit{dt}} \end{array} \right )& &{}\end{array}$$

(14.8)

and the matrices and RHS vector

$$\displaystyle\begin{array}{rcl} \mbox{ $\begin{array}{rclcl} \boldsymbol{O}_{F}& = & O_{\mathit{ij}_{F}} & = & \sum _{e}\int _{\varOmega _{F}^{e}}\mathcal{S}_{F}^{e}\,N_{ i}N_{j}{d\varOmega }^{e} \\ \boldsymbol{K}_{F}& = & K_{\mathit{ij}_{F}} & = & \sum _{e}{\Bigl (\int _{\varOmega _{F}^{e}}N_{i}\boldsymbol{v}_{F}^{e} \cdot \nabla N_{ j}{d\varOmega }^{e} +\int _{\varOmega _{ F}^{e}}\nabla N_{i} \cdot (\boldsymbol{\varUpsilon }_{F}^{e} \cdot \nabla N_{j}){d\varOmega }^{e}+\Bigr.} \\ & & & & \;\;{\Bigl.\int _{\varOmega _{F}^{e}}\varTheta _{F}^{e}N_{ i}N_{j}{d\varOmega }^{e} +\int _{\varGamma _{ C_{F}}^{e}}\varPhi _{F}^{e}N_{i}N_{j}{d\varGamma }^{e}\Bigr )} -\delta _{\mathit{ij}}Q_{w}(t)\Big\vert _{i} \\ \boldsymbol{F}_{F}& = & F_{i_{F}} & = & \sum _{e}{\Bigl (\int _{\varOmega _{F}^{e}}N_{i}Q_{F}^{e}{d\varOmega }^{e} +\int _{\varGamma _{ C_{F}}^{e}}N_{i}\varPhi _{F}^{e}\phi _{C}^{e}{d\varGamma }^{e} -\int _{\varGamma _{N_{ F}}^{e}}N_{i}q_{N}^{e}{d\varGamma }^{e}\Bigr )} -\phi _{w}Q_{w}(t)\Big\vert _{ i} \end{array} $}& &{}\end{array}$$

(14.9)

where $(i,j = 1,\ldots,N_{\mathrm{P}_{F}})$ and $(e = 1,\ldots,N_{\mathrm{E}_{F}})$. The integrals appearing in (14.9) are integrated on element level in the local coordinates $\boldsymbol{\eta }$ as described in Sect. 8.12. Analytical evaluations of partial integral terms of (14.9) can be deduced from developments done in Appendix H for selected element types, in particular for the 1D linear line element (Sect. H.1) and the 2D linear triangular element (Sect. H.2). Numerical integration via Gauss-Legendre quadrature is employed for 2D quadrilateral DFE’s.

Assembly of DFE’s into the Global System Matrix

Need for Coordinate Transformation

The matrix system (14.7) written in the form

$$\displaystyle{ \begin{array}{rcl} \boldsymbol{O}_{F} \cdot \dot{\boldsymbol{\phi }} +\boldsymbol{K}_{F}\cdot \boldsymbol{\phi }& =&\boldsymbol{F}_{F} \\ \boldsymbol{O}_{F}& =&\sum _{e}\boldsymbol{O}_{F}^{e} \\ \boldsymbol{K}_{F}& =&\sum _{e}\boldsymbol{K}_{F}^{e} \\ \boldsymbol{F}_{F}& =&\sum _{e}\boldsymbol{F}_{F}^{e}\end{array} }$$

(14.10)

provides elemental matrix and vector contributions for each DFE e of feature F, which must be assembled additionally into the global finite-element matrix systems resulting from the porous-media equations developed in the preceding sections, such as (9.20) for flow in saturated porous media, (10.30) for flow in variably saturated porous media, (11.38) for variable-density flow in porous media, (12.22) for mass transport in porous media and (13.22) for heat transport in porous media.

The integrals (14.9) in $\boldsymbol{O}_{F}^{e}$, $\boldsymbol{K}_{F}^{e}$ and $\boldsymbol{F}_{F}^{e}$ for each DFE e and feature F are performed in the local coordinates $\boldsymbol{\eta }$ for the corresponding Euclidean space ℜ ^D (cf. Sects. 8.8 and 8.11). Usually, 1D finite elements are mapped to the ℜ ¹ space, 2D elements to the ℜ ² space and 3D elements to the ℜ ³ space. In such cases the mapping is strictly one-to-one, that means three global coordinates (x, y, z) are transformed to three local coordinates (ξ, η, ζ) in 3D, two global coordinates (x, y) to two local coordinates (ξ, η) in 2D and one global coordinate (x) to one local coordinate (ξ) in 1D. However, when 1D and 2D DFE’s are generally mapped onto a 3D global space, the number of local coordinates $\boldsymbol{\eta }$ will be less than the number of global coordinates $\boldsymbol{{x}}^{e}$ and the transformation Jacobian $\boldsymbol{{J}}^{e} = \partial \boldsymbol{{x}}^{e}/\partial \boldsymbol{\eta }$ for finite elements (8.115) will not be any more an invertible square matrix (e.g., for the $\xi -\eta -$ system of a 2D DFE mapped onto the global $x - y - z-$ system the third row of $\boldsymbol{{J}}^{e}$ contains zeros, $J_{31}^{e} = J_{32}^{e} = J_{33}^{e} = 0$, because the ζ−coordinate does not exist in 2D elements).

There are at least two ways to overcome this mapping conflict. A more general method has been proposed by Perrochet [414], who uses expressions of gradients in curvilinear coordinates and introduces covariant bases and metric tensors to replace the usual Jacobian. Alternatively, the method of coordinate transformation appears as a cost-effective and simpler method, we shall prefer here. Taking into consideration that all flow and transport processes are invariant with respect to a rotation (orthogonal transformation) of the global coordinates $\boldsymbol{x} =\boldsymbol{ {x}}^{e}$, {∀}e, we can arbitrarily rotate $\boldsymbol{x}$ to the $\boldsymbol{{x}^{\prime}}^{e}-$ coordinates different for each element e by using a suitable rotation matrix of directional cosines $\boldsymbol{{A}}^{e}$ as

$$\displaystyle{ \begin{array}{rcl} \boldsymbol{{x}^{\prime}}^{e}& =&\boldsymbol{{A}}^{e} \cdot \boldsymbol{ {x}}^{e} \\ {\left (\begin{array}{c} x^{\prime}\\ y^{\prime} \\ z^{\prime} \end{array} \right )}^{e}& =&{\left (\begin{array}{ccc} A_{11}&A_{12}&A_{13} \\ A_{21}&A_{22}&A_{23} \\ A_{31}&A_{32}&A_{33} \end{array} \right )}^{e} \cdot {\left (\begin{array}{c} x\\ y \\ z \end{array} \right )}^{e} \end{array} }$$

(14.11)

Taking an appropriate rotation of the global $x - y - z-$ coordinate system in such a way that the resulting local $x^{\prime} - y^{\prime} - z^{\prime}-$ system becomes aligned to the orientation of the 2D or 1D DFE’s in the ℜ ³ space, there will be no more an elemental contribution to the z′−direction for 2D elements and elemental contributions to the y′− and z′−directions for 1D elements (see Fig. 14.3).

Fig. 14.3

Global $x - y - z-$ coordinate system, rotated elemental $x^{\prime} - y^{\prime} - z^{\prime}-$ coordinate system and local $\xi -(\eta )-$ coordinate system for 2D and 1D DFE’s in the ℜ ³ space

The advantages of this coordinate transformation are that the corresponding Jacobian $\boldsymbol{{J}^{\prime}}^{e}$

$$\displaystyle{ \boldsymbol{{J}^{\prime}}^{e} = \frac{\partial \boldsymbol{{x}^{\prime}}^{e}} {\partial \boldsymbol{\eta }} }$$

(14.12)

becomes again an invertible square matrix and the standard metric procedure can be maintained in the assembly process for the global matrix system (14.10). To ease the computations the $x^{\prime} - y^{\prime} - z^{\prime}-$ coordinate system may, in fact, be different for every element e. Actually, the integrals (14.9) in $\boldsymbol{{O}}^{e}$, $\boldsymbol{{K}}^{e}$ and $\boldsymbol{{F}}^{e}$ over Ω _F ^e, $\varGamma _{C_{F}}^{e}$ and $\varGamma _{N_{F}}^{e}$ are performed in the local coordinates $\boldsymbol{\eta }$ which are directly mapped onto the transformed coordinates $\boldsymbol{{x}^{\prime}}^{e}$ (Fig. 14.3):

$$\displaystyle{ \varOmega _{F}^{e} =\varOmega _{ F}^{e}(\boldsymbol{{x}^{\prime}}^{e}(\boldsymbol{\eta })),\quad \varGamma _{ C_{F}}^{e} =\varGamma _{ C_{F}}^{e}(\boldsymbol{{x}^{\prime}}^{e}(\boldsymbol{\eta })),\quad \varGamma _{ N_{F}}^{e} =\varGamma _{ N_{F}}^{e}(\boldsymbol{{x}^{\prime}}^{e}(\boldsymbol{\eta })) }$$

(14.13)

Determination of the Directional Cosines $\boldsymbol{{A}}^{e}$ of DFE e

The directional cosines $\boldsymbol{{A}}^{e}$ are only required for mapping 2D and 1D DFE’s in the ℜ ³ space. Suppose the 3D continuum domain Ω with its boundary Γ is completely filled by 3D finite elements (e.g., hexahedral or pentahedral isoparametric elements), the 1D and 2D DFE’s share the nodal points of the 3D mesh and their geometric extents are aligned to surfaces, edges or diagonals of the 3D porous-medium elements (Fig. 14.4).

Fig. 14.4

Exemplified mapping of 2D and 1D DFE’s aligned to surfaces, edges and diagonals, respectively, for a 3D finite porous-medium element. Global and local coordinates

For 2D DFE’s forming surfaces of the 3D porous-medium element it is convenient to derive the directional cosines directly from the shape of the 3D element. We can construct the two directional vectors $\boldsymbol{u}_{1}$ and $\boldsymbol{u}_{2}$ (Fig. 14.4), which are parallel to the local ξ− and η−axes, respectively. They can be found by the following shape-derived relationships

$$\displaystyle{ \boldsymbol{u}_{1} = \left \{\begin{array}{rclll} \left (\begin{array}{c} \frac{\partial x} {\partial \xi } \\ \frac{\partial y} {\partial \xi } \\ \frac{\partial z} {\partial \xi } \end{array} \right )& =&{\left (\begin{array}{c} J_{11} \\ J_{12} \\ J_{13} \end{array} \right )}^{e}&\quad \mbox{ at}&\quad \zeta = \pm 1 \\ \left (\begin{array}{c} \frac{\partial x} {\partial \eta } \\ \frac{\partial y} {\partial \eta } \\ \frac{\partial z} {\partial \eta } \end{array} \right )& =&{\left (\begin{array}{c} J_{21} \\ J_{22} \\ J_{23} \end{array} \right )}^{e}&\quad \mbox{ at}&\quad \xi = \pm 1 \\ \left (\begin{array}{c} \frac{\partial x} {\partial \zeta } \\ \frac{\partial y} {\partial \zeta } \\ \frac{\partial z} {\partial \zeta } \end{array} \right )& =&{\left (\begin{array}{c} J_{31} \\ J_{32} \\ J_{33} \end{array} \right )}^{e}&\quad \mbox{ at}&\quad \eta = \pm 1 \end{array} \right. }$$

(14.14)

$$\displaystyle{ \boldsymbol{u}_{2} = \left \{\begin{array}{rclll} \left (\begin{array}{c} \frac{\partial x} {\partial \eta } \\ \frac{\partial y} {\partial \eta } \\ \frac{\partial z} {\partial \eta } \end{array} \right )& =&{\left (\begin{array}{c} J_{21} \\ J_{22} \\ J_{23} \end{array} \right )}^{e}&\quad \mbox{ at}&\quad \zeta = \pm 1 \\ \left (\begin{array}{c} \frac{\partial x} {\partial \zeta } \\ \frac{\partial y} {\partial \zeta } \\ \frac{\partial z} {\partial \zeta } \end{array} \right )& =&{\left (\begin{array}{c} J_{31} \\ J_{32} \\ J_{33} \end{array} \right )}^{e}&\quad \mbox{ at}&\quad \xi = \pm 1 \\ \left (\begin{array}{c} \frac{\partial x} {\partial \xi } \\ \frac{\partial y} {\partial \xi } \\ \frac{\partial z} {\partial \xi } \end{array} \right )& =&{\left (\begin{array}{c} J_{11} \\ J_{12} \\ J_{13} \end{array} \right )}^{e}&\quad \mbox{ at}&\quad \eta = \pm 1 \end{array} \right. }$$

(14.15)

These directional vectors can be easily used to compute the directional cosines according to

$$\displaystyle{ A_{ij}^{e} =\cos (\boldsymbol{u}_{ i},\boldsymbol{e}_{j}) = \frac{\boldsymbol{u}_{i} \cdot \boldsymbol{ e}_{j}} {\Vert \boldsymbol{u}_{i}\Vert \underbrace{\mathop{ \Vert \boldsymbol{e}_{j}\Vert }}\limits _{=1}}\quad \mbox{ for}\quad \begin{array}{c}\begin{array}{rcl} i& =&1,2\\ j & = &1, 2,3 \end{array} \end{array} }$$

(14.16)

with the base vectors (2.5):

$$\displaystyle{ \boldsymbol{e}_{1} = \left (\begin{array}{c} 1\\ 0 \\ 0 \end{array} \right )\quad \boldsymbol{e}_{2} = \left (\begin{array}{c} 0\\ 1 \\ 0 \end{array} \right )\quad \boldsymbol{e}_{3} = \left (\begin{array}{c} 0\\ 0 \\ 1 \end{array} \right ) }$$

(14.17)

Note that for 2D DFE’s we need only two directional vectors (i = 1, 2), the remaining directional cosines A _3j ^e are meaningless.

Often we can assume that the 2D DFE’s are perfectly plane, i.e., they represent noncurved 2D geometries which occur for arbitrarily oriented linear triangles or for vertical linear quadrilaterals in the 3D space. Instead of using the above shape-derived expressions (14.14) and (14.15), in such cases it is convenient to derive the directional vectors $\boldsymbol{u}_{i}$ in a direct manner as follows (see Fig. 14.5).

Fig. 14.5

Directional vectors $\boldsymbol{u}_{i}\;(i = 1,2)$ for a linear triangular element and a vertically-oriented linear quadrilateral element

We specify the x′−axis along the edge nm of the 2D DFE. The vector $\boldsymbol{u}_{1}$ is accordingly given by

$$\displaystyle{ \boldsymbol{u}_{1} = \left (\begin{array}{c} x_{n} - x_{m} \\ y_{n} - y_{m} \\ z_{n} - z_{m} \end{array} \right ) }$$

(14.18)

The second directional vector $\boldsymbol{u}_{2}$ derived by simple vector algebra yields²

$$\displaystyle{ \boldsymbol{u}_{2} =\boldsymbol{ q} -{\Bigl ( \frac{\boldsymbol{q} \cdot \boldsymbol{ u}_{1}} {\boldsymbol{u}_{1} \cdot \boldsymbol{ u}_{1}}\Bigr )}\boldsymbol{u}_{1} }$$

(14.19)

with the auxiliary vector $\boldsymbol{q}$ formed along the adjacent side lm of the 2D element as

$$\displaystyle{ \boldsymbol{q} = \left (\begin{array}{c} x_{l} - x_{m} \\ y_{l} - y_{m} \\ z_{l} - z_{m} \end{array} \right ) }$$

(14.20)

and the directional cosines $A_{\mathit{ij}}^{e}\;(i = 1,2;j = 1,2,3)$ can be easily computed by using (14.16).

For 1D DFE’s the same procedure can be applied to determine A _ij ^e for $(i = 1;j = 1,2,3)$. Here, only one row A _1j ^e of the rotation matrix is of interest. Taking into consideration that 1D DFE’s can be rather arbitrarily placed at mesh nodes (which are not necessarily connected in one element and oriented along edges) the following direct evaluation procedure can be used to compute A _1j ^e for a 1D linear (noncurved) DFE spanning between the two nodes n and m (cf. Fig. 14.4):

$$\displaystyle{ \boldsymbol{u}_{1} = \left (\begin{array}{c} x_{n} - x_{m} \\ y_{n} - y_{m} \\ z_{n} - z_{m} \end{array} \right ) = \left (\begin{array}{c} \varDelta x\\ \varDelta y \\ \varDelta z \end{array} \right ),\qquad \Vert \boldsymbol{u}_{1}\Vert = \sqrt{\varDelta {x}^{2 } +\varDelta {y}^{2 } +\varDelta {z}^{2}} }$$

(14.21)

$$\displaystyle{ \begin{array}{rcl} A_{11}^{e}& =& \frac{\varDelta x} {\sqrt{\varDelta {x}^{2 } +\varDelta {y}^{2 } +\varDelta {z}^{2}}} \\ A_{12}^{e}& =& \frac{\varDelta y} {\sqrt{\varDelta {x}^{2 } +\varDelta {y}^{2 } +\varDelta {z}^{2}}} \\ A_{13}^{e}& =& \frac{\varDelta z} {\sqrt{\varDelta {x}^{2 } +\varDelta {y}^{2 } +\varDelta {z}^{2}}}\end{array} }$$

(14.22)

Implicit Monolithic Solution of the Coupled Discrete Fracture and Porous-Medium Equations

Discrete features and the porous medium are treated as a monolithic entity, where all components are implicitly integrated in the solution domain

$$\displaystyle{ \varOmega =\varOmega _{P} \cup \sum _{F}\varOmega _{F} }$$

(14.23)

consisting of the joint porous-medium domain Ω _P and a number of nonoverlapping discrete feature domains Ω _F , governed by different balance equations, however, solvable via a common state variable $\phi =\phi (\boldsymbol{x},t)$ (e.g., h for flow, C _k for species mass and T for heat transport). In the finite element context the elementwise continuous approximation for $\phi \approx \hat{\phi }$ allows the assembly of the elemental contributions of porous medium and discrete features in a standard manner such that (cf. Sect. 8.6)

$$\displaystyle{ \begin{array}{rcl} \int _{\varOmega }\{\ldots \}d\varOmega & =&\sum _{e}\int _{{\varOmega }^{e}}\{\ldots \}{d\varOmega }^{e} \\ & =&\sum _{e}{\Bigl (\int _{\varOmega _{P}^{e}}w\mathcal{L}_{P}(\hat{\phi }){d\varOmega }^{e} +\sum _{ F}\int _{\varOmega _{F}^{e}}w\mathcal{L}_{F}(\hat{\phi }){d\varOmega }^{e}\Bigr )} \end{array} }$$

(14.24)

where $\mathcal{L}_{P}(\hat{\phi })$ and $\mathcal{L}_{F}(\hat{\phi })$ represent the governing PDE’s for the porous medium and the discrete features, respectively. Practically, the formation of the global matrix system comprising both the porous medium and the discrete feature entities is simply the assembly of their partial matrix and vector contributions. For example, the matrix system of the porous-medium flow (9.20) extends now to

$$\displaystyle{ \boldsymbol{M} \cdot \dot{\boldsymbol{ h}} +\boldsymbol{ D} \cdot \boldsymbol{ h} -\boldsymbol{ Z} = \mathbf{0} }$$

(14.25)

with

$$\displaystyle{ \begin{array}{rcl} \boldsymbol{M}& =&\boldsymbol{O} +\sum _{F}\boldsymbol{O}_{F} \\ \boldsymbol{D}& =&\boldsymbol{C} +\sum _{F}\boldsymbol{K}_{F} \\ \boldsymbol{Z}& =&\boldsymbol{F} +\sum _{F}\boldsymbol{F}_{F} \end{array} }$$

(14.26)

to solve the hydraulic head $\boldsymbol{h}$, where $\boldsymbol{O}$, $\boldsymbol{C}$ and $\boldsymbol{F}$ are the porous-medium contributions given by (9.22) and $\boldsymbol{O}_{F}$, $\boldsymbol{K}_{F}$ and $\boldsymbol{F}_{F}$ are the discrete feature F contributions given by (14.9). Similarly, for variably saturated flow, variable-density flow, species mass and heat transport the global matrix systems result if we assemble the corresponding porous-medium matrix system (10.30), (11.38), (12.22) and (13.22), respectively, with the matrix system (14.7) for the discrete feature F and associated state variable $\boldsymbol{h}$, $\boldsymbol{C}_{k}$ and $\boldsymbol{T}$ of hydraulic head, species concentration and temperature, respectively.

Since the DFE’s share the same nodal points with the porous medium, a natural result of the assembly process is in the parallel behavior of exchanging (advective and conductive/diffusive) fluxes between porous medium and discrete features. Suppose K _P and K _F represent characteristic conductivities of porous medium and discrete feature, respectively, at the same node, an exchanging flux between porous medium and discrete feature is affected by its effective conductivity $K = K_{P} + K_{F}$. If K _F ≫ K _P the flux becomes dominated by the discrete feature property, however if K _F → 0 the effect from DFE vanishes and the exchanging flux is determined by the porous-medium property alone. A disadvantageous consequence of the latter is that there is no possibility to model clogging or sealing effects by simply using DFE’s with sharing nodes because the exchanging flux can never be smaller than that of the porous medium since K ≥ K _P , except for K _P → 0.

Time Integration

The global matrix systems for flow such as (14.25), and similar to mass and heat transport comprising both the porous medium and the discrete feature entities have to be solved in time t with the associated IC’s via suitable single-step semi-implicit or fully implicit time marching recurrence schemes as described in Sect. 8.13. The GLS predictor-corrector time stepping method combined with an automatic error-controlled time step selection strategy is usually preferred. Its solution steps applied to the global matrix systems are fully equivalent to the procedures as thoroughly described above in Sect. 8.13.5 (summarized in Table 8.7) for a general ADE, in Sect. 10.7.5 for unsaturated flow, in Sect. 11.6.4 for density-variable flow, in Sect. 12.3.3 for reactive mass transport and in Sect. 13.3.3 for heat transport.

Computation of Velocity Fields and Budget Analysis

The flow vectors for the porous medium $\boldsymbol{q}_{P}$ and for the discrete features $\boldsymbol{v}_{F}$ are at first separately evaluated by using smoothing techniques as thoroughly described in Sect. 8.19.1. For the porous medium continuous Darcy velocities $\boldsymbol{q}_{P}$ at the nodal points are derived such as given in (10.120) for variably saturated media and in (11.69) for variable-density flow. The same procedures are applied to compute the discrete velocities at the nodes of a discrete feature F. For example, for the Hagen-Poiseuille flow velocity (4.51) and for the overland and channel flow velocity (4.63) the following discrete evaluation is performed

$$\displaystyle{ \boldsymbol{v}_{F}(\boldsymbol{x},t_{n+1}) = -\sum _{j}\boldsymbol{K}_{F}f_{\mu } \cdot {\bigl [\nabla N_{j}(\boldsymbol{x})\,h_{j}(t_{n+1}) +\chi \boldsymbol{ e}\bigr ]} }$$

(14.27)

by using the known hydraulic head values $h_{j}(t_{n+1}) =\boldsymbol{ h}_{n+1}$ at nodes j of discrete feature F at time plane n + 1, where $\boldsymbol{K}_{F}$ corresponds to a generalized hydraulic conductivity of discrete feature F specifying the different flow laws according to (4.51) or (4.63). Note that the velocity $\boldsymbol{v}_{F}$ is only smoothed separately for the contributions of the discrete feature nodes j, however, no smoothing is performed with the velocity contributions of the porous medium. Finally, the total velocity $\boldsymbol{q}$ is a result of superimposing the velocity of porous medium and discrete feature at given location $\boldsymbol{x}$ and time stage t _n+1, viz.,

$$\displaystyle{ \boldsymbol{q}(\boldsymbol{x},t_{n+1}) =\boldsymbol{ q}_{P}(\boldsymbol{x},t_{n+1}) +\boldsymbol{ v}_{F}(\boldsymbol{x},t_{n+1}) }$$

(14.28)

Note, however, the evaluations of the advective terms in the corresponding mass and heat transport equations are always based on their separate (nonsuperimposed) flow fields, i.e., porous-medium equations take the porous-medium Darcy velocities $\boldsymbol{q}_{P}$ and fracture equations use the fracture velocities $\boldsymbol{v}_{F}$.

The precise budget analysis for flow, mass and heat transport problems which are mixed with discrete features is fully analogous to the technique as described for porous-media processes in Sects. 9.7, 10.11, 11.8, 12.4 and 13.4 based on CBFM introduced and thoroughly described in Sect. 8.19.2. Similarly, we use the basic weak statement (14.4) of the discrete feature transport equation to express the corresponding boundary flux on the discrete feature boundary Γ _F as

$$\displaystyle\begin{array}{rcl} & & \qquad \int _{\varGamma _{F}}N_{i}\,q_{n_{F}}\,d\varGamma = -\int _{\varOmega _{F}}N_{i}\mathcal{S}_{F} \frac{\partial \phi } {\partial t}d\varOmega -\int _{\varOmega _{F}}N_{i}\boldsymbol{v}_{F} \cdot \nabla \phi d\varOmega - \\ & &\int _{\varOmega _{F}}\nabla N_{i} \cdot (\boldsymbol{\varUpsilon }_{F} \cdot \nabla \phi )d\varOmega -\int _{\varOmega _{F}}N_{i}(\varTheta _{F}\phi - Q_{F})d\varOmega - (\phi _{w}-\phi )Q_{w}(t)\vert _{i}{}\end{array}$$

(14.29)

which leads to the matrix system

$$\displaystyle{ \boldsymbol{M}_{F} \cdot \boldsymbol{ q}_{n_{F}} = -\boldsymbol{O}_{F} \cdot \dot{\boldsymbol{\phi }}-\boldsymbol{K}_{F}^{\dag }\cdot \boldsymbol{\phi } +\boldsymbol{F}_{ F}^{\dag } }$$

(14.30)

with

$$\displaystyle{ \begin{array}{rclcl} \boldsymbol{M}_{F}& =&M_{\mathit{ij}_{F}} & =&\int _{\varGamma }N_{i}N_{j}d\varGamma \\ \boldsymbol{O}_{F}& =&O_{\mathit{ij}_{F}} & =&\int _{\varOmega _{F}}\mathcal{S}_{F}N_{i}N_{j}d\varOmega \\ \boldsymbol{K}_{F}^{\dag }& =&K_{\mathit{ij}_{F}}^{\dag }& =&\int _{\varOmega _{F}}N_{i}\boldsymbol{v}_{F} \cdot \nabla N_{j}d\varOmega +\int _{\varOmega _{F}}\nabla N_{i} \cdot (\boldsymbol{\varUpsilon }_{F} \cdot \nabla N_{j})d\varOmega + \\ & & & &\int _{\varOmega _{F}}\varTheta _{F}N_{i}N_{j}d\varOmega -\delta _{\mathit{ij}}Q_{w}(t)\Big\vert _{i} \\ \boldsymbol{F}_{F}^{\dag }& =&F_{i_{F}}^{\dag } & =&\int _{\varOmega _{F}}N_{i}Q_{F}d\varOmega -\phi _{w}Q_{w}(t)\Big\vert _{i} \end{array} }$$

(14.31)

for solving the continuous boundary flux vector $\boldsymbol{q}_{n_{F}}$ of discrete feature F. To compute the boundary flux of the total system comprising both the porous medium and the discrete feature entities we can simply assembly their partial matrix and vector contributions. For example, the matrix system of porous-medium flow budget (9.65) extends now to

$$\displaystyle{ \boldsymbol{{M}}^{\ddag }\cdot \boldsymbol{ q}_{ n}^{\ddag } = -\boldsymbol{{O}}^{\ddag }\cdot \dot{\boldsymbol{ h}} -\boldsymbol{ {C}}^{\ddag }\cdot \boldsymbol{ h} +\boldsymbol{ {F}}^{\ddag } }$$

(14.32)

with

$$\displaystyle{ \begin{array}{rcl} \boldsymbol{q}_{n}^{\ddag }& =&\boldsymbol{q}_{n} +\sum _{F}\boldsymbol{q}_{n_{F}} \\ \boldsymbol{{M}}^{\ddag }& =&\boldsymbol{M} +\sum _{F}\boldsymbol{M}_{F} \\ \boldsymbol{{O}}^{\ddag }& =&\boldsymbol{{O}}^{\dag } +\sum _{F}\boldsymbol{O}_{F} \\ \boldsymbol{{C}}^{\ddag }& =&\boldsymbol{{C}}^{\dag } +\sum _{F}\boldsymbol{K}_{F}^{\dag } \\ \boldsymbol{{F}}^{\ddag }& =&\boldsymbol{{F}}^{\dag } +\sum _{F}\boldsymbol{F}_{F}^{\dag } \end{array} }$$

(14.33)

to solve the total boundary flux vector $\boldsymbol{q}_{n}^{\ddag }$, where $\boldsymbol{M}$, $\boldsymbol{{O}}^{\dag }$, $\boldsymbol{{C}}^{\dag }$ and $\boldsymbol{{F}}^{\dag }$ are the porous-medium contributions given by (9.66) and $\boldsymbol{M}_{F}$, $\boldsymbol{O}_{F}$, $\boldsymbol{K}_{F}^{\dag }$ and $\boldsymbol{F}_{F}^{\dag }$ are the discrete feature F contributions given by (14.31). Note that in the budget analysis the total integral flux Q _n ^‡ is directly evaluated at each boundary node by

$$\displaystyle{ \begin{array}{rcl} Q_{n}^{\ddag }& =& -\boldsymbol{ {M}}^{\ddag }\cdot \boldsymbol{ q}_{n}^{\ddag } \\ & =&\boldsymbol{{O}}^{\ddag }\cdot \dot{\boldsymbol{ h}} +\boldsymbol{ {C}}^{\ddag }\cdot \boldsymbol{ h} -\boldsymbol{ {F}}^{\ddag } \end{array} }$$

(14.34)

Similarly, for variably saturated flow, variable-density flow, species mass and heat transport the global matrix systems for budget analysis result if assembly the corresponding porous-medium matrix system (10.123), (11.73), (12.47) and (13.30), respectively, with the matrix system (14.30) for the discrete feature F and associated state variable $\boldsymbol{h}$, $\boldsymbol{C}_{k}$ and $\boldsymbol{T}$ of hydraulic head, species concentration and temperature, respectively.

Examples

Solute Diffusion into Porous Matrix from a Single Fracture

The single solute transport through fractured media was studied by Grisak and Pickens [215] for the case of a thin single fracture situated in a saturated porous rock as illustrated in Fig. 14.6. Advective transport is dominant in the fracture, while diffusive solute transport is usually dominant in the adjacent porous matrix. The diffusion into the porous matrix reduces the solute advancement in the fracture and thereby delays the migration of the solute, which acts as a diffusive loss for the fracture. Grisak and Pickens [215] used the standard FEM to model the fracture-matrix system, where the single fracture is discretized by thin areal 2D elements (i.e., no DFE).

Fig. 14.6

Schematic sketch of the fracture-matrix system

An analytical solution for the fracture-matrix system of Fig. 14.6 has been developed by Tang et al. [506] by using Laplace transforms, which includes (1) advective transport along the fracture, (2) longitudinal dispersivity in the fracture, (3) molecular diffusion within the fracture, in the direction of the fracture axis x, (4) molecular diffusion from the fracture into the matrix, in the y−direction perpendicular to the fracture axis, (5) linear adsorption onto the face of the matrix, (6) linear adsorption within the matrix and (7) linear radioactive decay. It solves the coupled system of single solute mass balance equations governing in the fracture domain (0 ≤ x ≤∞, 0 ≤ y ≤ a) as

$$\displaystyle{ \mathfrak{R}\frac{\partial C} {\partial t} + v\frac{\partial C} {\partial x} - D_{\mathit{xx}}\frac{{\partial }^{2}C} {\partial {x}^{2}} + \mathfrak{R}\vartheta C -\left.\frac{\varepsilon D^{\prime}_{\mathit{yy}}} {a} \frac{\partial C^{\prime}} {\partial y} \right \vert _{y=a} = 0 }$$

(14.35)

associated with the IC and BC’s

$$\displaystyle{ C(x,0) = 0,\quad C(0,t) = C_{D},\quad C(\infty,t) = 0 }$$

(14.36)

and governing in the porous matrix domain (a ≤ y ≤∞) as

$$\displaystyle{ \mathfrak{R}^{\prime}\frac{\partial C^{\prime}} {\partial t} - D^{\prime}_{\mathit{yy}}\frac{{\partial }^{2}C^{\prime}} {\partial {y}^{2}} + \mathfrak{R}^{\prime}\vartheta C^{\prime} = 0 }$$

(14.37)

associated with the IC and BC’s

$$\displaystyle{ C^{\prime}(x,y,0) = 0,\quad C^{\prime}(x,a,t) = C(x,t),\quad C^{\prime}(x,\infty,t) = 0 }$$

(14.38)

with the retardation factors $\mathfrak{R} = 1 + \frac{{{K}^{d}}^{\prime}} {a}$ and $\mathfrak{R}^{\prime} = 1 + \frac{\rho _{s}{K}^{d}} {\varepsilon }$ as well as the dispersion coefficient $D_{\mathit{xx}} = D +\beta _{L}v = \mathcal{D}$ in the fracture and diffusion coefficient $D^{\prime}_{\mathit{yy}} = \mathcal{D}^{\prime}$ in the porous matrix, where all symbols quoted with^′ refer to the porous matrix, unquoted symbols are related to the fracture or being indifferent, C and C′ are the single solute concentrations in the fracture and in the porous matrix, respectively, a is the half of fracture width (see Fig. 14.6), K ^d′ and K ^d are the distribution coefficients for the porous matrix and fracture, respectively (cf. Table 3.8), ρ _s is the bulk density of the porous matrix, v is the groundwater velocity in the fracture (positive in x−direction), $\varepsilon$ is the porosity of the porous matrix, $\vartheta$ is the decay rate and β _L is the longitudinal dispersivity.

Tang et al.’s general solution [506] of (14.35)–(14.38) takes the form of an integral which must be evaluated by numerical quadrature for each point in space and time (actually, Gaussian quadrature is used). On the other hand, a closed analytical transient solution can be derived for the simpler case which assumes negligible dispersion within the fracture, i.e., $\mathcal{D}\equiv 0$. It yields [506] the solute distribution within the fracture as

$$\displaystyle\begin{array}{rcl} & & \mbox{ $ \frac{C} {C_{D}}\,=\,\frac{1} {2}\exp {\Bigl (\! -\frac{\vartheta \mathfrak{R}x} {v} \Bigr )}{\biggl [\exp {\biggl (\! -\frac{\varepsilon \sqrt{\vartheta \mathfrak{R}^{\prime}\mathcal{D}^{\prime}}} {\mathit{av}} x\biggr )}\,\mathrm{erfc}{\biggl ( \frac{\varepsilon \sqrt{\mathfrak{R}^{\prime}\mathcal{D}^{\prime}}} {2\mathit{av}\mathfrak{R}\sqrt{t - x\mathfrak{R}/v}}x\,-\,\sqrt{\vartheta }\sqrt{t - x\mathfrak{R}/v}\biggr )}+\biggr.}$} \\ & & \qquad \qquad \mbox{ ${\biggl.\exp {\biggl (\frac{\varepsilon \sqrt{\vartheta \mathfrak{R}^{\prime}\mathcal{D}^{\prime}}} {\mathit{av}} x\biggr )}\,\mathrm{erfc}{\biggl ( \frac{\varepsilon \sqrt{\mathfrak{R}^{\prime}\mathcal{D}^{\prime}}} {2\mathit{av}\mathfrak{R}\sqrt{t - x\mathfrak{R}/v}}x + \sqrt{\vartheta }\sqrt{t - x\mathfrak{R}/v}\biggr )}\biggr ]} $} \\ & & \qquad \mbox{ $\mbox{ if}\;\;(t - x\mathfrak{R}/v) > 0 $} \\ & & \qquad \mbox{ $\mbox{ otherwise}\;\; \frac{C} {C_{D}} = 0 \mbox{ if} (t - x\mathfrak{R}/v) \leq 0 \;\;\;$}{}\end{array}$$

(14.39)

and the solute distribution within the porous matrix as

$$\displaystyle\begin{array}{rcl} & & \qquad \mbox{ $ \frac{C^{\prime}} {C_{D}} = \frac{1} {2}\exp {\Bigl ( -\frac{\vartheta \mathfrak{R}x} {v} \Bigr )}\;\times $}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \;\; \\ & &\mbox{ ${\biggl [\exp {\biggl (\!-\frac{\varepsilon \sqrt{\vartheta \mathfrak{R}^{\prime}\mathcal{D}^{\prime}}} {\mathit{av}} x-\sqrt{\vartheta }A(y)\!\biggr )}\mathrm{erfc}{\biggl (\! \frac{\varepsilon \sqrt{\mathfrak{R}^{\prime}\mathcal{D}^{\prime}}} {2\mathit{av}\sqrt{t- x\mathfrak{R}/v}}x+ \frac{A(y)} {2\sqrt{t- x\mathfrak{R}/v}}-\sqrt{\vartheta }\sqrt{t- x\mathfrak{R}/v}\!\biggr )}+\biggr.}$} \\ & & \mbox{ ${\biggl.\exp {\biggl (\frac{\varepsilon \sqrt{\vartheta \mathfrak{R}^{\prime}\mathcal{D}^{\prime}}} {\mathit{av}} x\,+\,\sqrt{\vartheta }A(y)\biggr )}\mathrm{erfc}{\biggl ( \frac{\varepsilon \sqrt{\mathfrak{R}^{\prime}\mathcal{D}^{\prime}}} {2\mathit{av}\sqrt{t- x\mathfrak{R}/v}}x\,+\, \frac{A(y)} {2\sqrt{t- x\mathfrak{R}/v}}+\sqrt{\vartheta }\sqrt{t - x\mathfrak{R}/v}\biggr )}\!\biggr ]}$} \\ & & \qquad \mbox{ $\mbox{ if}\;\;(t - x\mathfrak{R}/v) > 0 \;\;$} \\ & & \qquad \mbox{ $\mbox{ otherwise}\;\; \frac{C^{\prime}} {C_{D}} = 0 \mbox{ if} (t - x\mathfrak{R}/v) \leq 0 \;\;$}{}\end{array}$$

(14.40)

where

$$\displaystyle{ \mbox{ $A(y) = \sqrt{ \frac{\mathfrak{R}^{\prime}} {D^{\prime}}}\,(y - a)$} }$$

(14.41)

Note that for evaluating the analytical exp(. )erfc(. ) expressions appearing in (14.39) and (14.40) the more suitable exf(. , . ) function is used as already introduced in Sect. 12.5.1.

To predict the ultimate penetration distances steady-state solutions can be useful. Closed analytical steady-state solutions can be found [506] without the need for neglecting dispersion within the fracture so as necessary for the transient solutions (14.39) and (14.40). The steady-state solute distribution within the fracture results as

$$\displaystyle{ \frac{C} {C_{D}} =\exp \;{\Biggl [{\Biggl ( \frac{v} {2\mathcal{D}}-\sqrt{ \frac{{v}^{2 } } {4{\mathcal{D}}^{2}} + \frac{\vartheta +\varepsilon \frac{\sqrt{\mathcal{D}^{\prime}\vartheta }} {a} } {\mathcal{D}}} \;\;\Biggr )}x\;\Biggr ]} }$$

(14.42)

and the steady-state solute distribution within the porous matrix results as

$$\displaystyle{ \frac{C^{\prime}} {C_{D}} =\exp \;{\Biggl [{\Biggl ( \frac{v} {2\mathcal{D}}-\sqrt{ \frac{{v}^{2 } } {4{\mathcal{D}}^{2}} + \frac{\vartheta +\varepsilon \frac{\sqrt{\mathcal{D}^{\prime}\vartheta }} {a} } {\mathcal{D}}} \;\;\Biggr )}x\;\Biggr ]}\exp {\biggl [ -\sqrt{ \frac{\vartheta } {\mathcal{D}^{\prime}}}\,(y - a)\biggr ]}\quad }$$

(14.43)

Equation (14.42) can be used to estimate the penetration depth d _δ into the fracture at steady state for a given concentration of $\delta = C/C_{D}$. It gives

$$\displaystyle{ d_{\delta } = \frac{\ln \delta } { \frac{v} {2\mathcal{D}} -\sqrt{ \frac{{v}^{2 } } {4{\mathcal{D}}^{2}} + \frac{\vartheta +\varepsilon \frac{\sqrt{\mathcal{D}^{\prime}\vartheta }} {a} } {\mathcal{D}}} } }$$

(14.44)

We compare the analytical solutions given by (14.39) for the solute behavior in the fracture and by (14.40) for the solute behavior in the porous matrix with FEFLOW’s finite-element simulations based on the spatial discretization shown in Fig. 14.7. The symmetric half of the fracture-matrix domain is discretized by only 50 × 25 quadrilaterals in variable thicknesses in y−direction. The fracture is modeled by using 50 1D DFE’s sharing the corresponding quadrilateral element edges of the porous matrix at y = a, 0 ≤ x ≤ L (Fig. 14.7). Note that the discretized 2D domain measures L × (D − a) in x− and y−direction (Fig. 14.6) while the thickness (aperture) of the fracture is integrated in the 1D parameters of the used DFE’s.

Fig. 14.7

Used nonuniform finite element mesh (vertical exaggeration 300:1) of the half-space fracture matrix domain consisting of 50 × 25 2D quadrilateral porous matrix elements combined with 50 1D DFE’s located at y = a

Table 14.1 Parameters and conditions used for the fractured media diffusion problem

The parameters and conditions used in the numerical simulations are summarized in Table 14.1. Unspecified BC’s for flow and solute transport represent boundaries at which natural BC’s are imposed, i.e., $-(\boldsymbol{K} \cdot \nabla h) \cdot \boldsymbol{ n} = 0$ and $-(\boldsymbol{D} \cdot \nabla C^{\prime}) \cdot \boldsymbol{ n} = 0$, respectively. For the fracture a Hagen-Poiseuille law of flow motion is assumed (cf. Sect. 4.3.2.2 and Table 4.5). Since the analytical solution (14.39) is only valid for negligible dispersion within the fracture $(\mathcal{D}\equiv 0)$, we also set the dispersion to zero for the DFE’s in the numerical approach. To stabilize the numerical simulations we prefer the GLS 1st-order accurate FE/BE predictor-corrector time stepping method, however, no resort to upwinding is necessary.

FEFLOW’s finite-element results are compared in Figs. 14.8–14.10 with the analytical findings. The agreement is rather well, although the used mesh is relatively coarse. Figure 14.8 shows the solute breakthrough curves in the fracture for values of different matrix diffusion $\mathcal{D}^{\prime}$. Differences to the analytical solutions are only revealed for very small diffusion, i.e., for cases where the advective solute transport is dominant in the fracture. This is also seen in the computed solute profiles into the porous matrix as depicted in Fig. 14.10, where the used discretization in y−direction is obviously insufficient for a small matrix diffusion of $\mathcal{D}^{\prime} = 1{0}^{-14}$ m² s⁻¹. The results for this case can be improved by using more refined meshes.

Fig. 14.8

Simulated versus analytical solute breakthrough curves in the fracture (y = a) at distance of x = 0. 76 m from the source point for values of matrix diffusion $\mathcal{D}^{\prime}$ in the range of 10⁻¹⁰–10⁻¹⁴ m² s⁻¹

Fig. 14.9

Simulated versus analytical solute profiles at t = 4 days along the porous matrix in x−direction at $y = 1{0}^{-4}$ m for values of matrix diffusion $\mathcal{D}^{\prime}$ in the range of 10⁻¹⁰–10⁻¹⁴ m²s⁻¹

For the solute profiles in the porous matrix in longitudinal x−direction shown in Fig. 14.9 we can observe that the accuracy of the numerical results expectedly decreases with increasing matrix diffusion, such as revealed in particular for $\mathcal{D}^{\prime} = 1{0}^{-10}$ m² s⁻¹ in Fig. 14.9. A more refined mesh could also improve the accuracy for those cases of dominant matrix diffusion. The numerical simulations required numbers of adaptive time steps ranging between 113 and 266 for simulating a time period of 4 days in dependence on the used matrix diffusions $\mathcal{D}^{\prime}$.

Fig. 14.10

Simulated versus analytical solute profiles at t = 4 days into the porous matrix in y−direction at x = 0. 76 m for values of matrix diffusion $\mathcal{D}^{\prime}$ in the range of 10⁻¹⁰–10⁻¹⁴ m² s⁻¹

Density-Dependent Solute Transport in a 45^∘−Inclined Single Fracture Embedded in a Low-Permeable Porous Matrix

Graf and Therrien [198] have studied density-dependent solute transport in single fractures of arbitrary inclination embedded in a low-permeable porous matrix. We shall benchmark their results for the 45^∘−inclined fracture problem against FEFLOW and the research code Ground Water (GW) developed by F. Cornaton [102]. This single fracture problem is shown in Fig. 14.11. The fracture inclined by 45^∘ is discretized by using correspondingly inclined 1D DFE’s. The left and right boundaries of the L × H enclosing porous matrix domain are assumed to be impermeable. The top and bottom boundaries are modeled as open boundaries with a constant hydraulic head h (set to zero). A contaminant source of constant solute concentration C = C _s overlies groundwater of initial concentration C = C ₀, where $C_{0} = 0 < C_{s} = 1$. The simulations cover a time of 20 years. The model parameters and conditions are summarized in Table 14.2. It is assumed that the porous matrix is isotropic and homogenous and that the entire domain is completely saturated. BC’s unreported in Table 14.2 for flow and solute transport represent boundaries at which natural BC’s are imposed, i.e., $-(\boldsymbol{K} \cdot \nabla h) \cdot \boldsymbol{ n} = 0$ and $-(\boldsymbol{D} \cdot \nabla C) \cdot \boldsymbol{ n} = 0$, respectively.

Fig. 14.11

Single 45^∘−inclined fracture in a porous matrix; 2D geometry, BC’s and IC’s (Modified from [198])

Table 14.2 Parameters and conditions used for the inclined single fracture problem

Graf and Therrien [198] tested different fracture slopes θ and mesh refinement levels. The present study focuses on the 45^∘−inclined fracture problem at their highest grid refinement level, consisting of 12,221 nodes and 24,000 triangles as shown in Fig. 14.12. We use two time stepping strategies: (1) in agreement to Graf and Therrien [198] a fully implicit time step marching scheme (combined with a Picard iteration) with a constant time step length Δ t of 0.2 years, and (2) alternatively, the adaptive GLS 2nd-order accurate predictor-corrector AB/TR time stepping using a RMS tolerance error of 10⁻⁴. No upwinding is employed in all simulations. The computation of the consistent velocity fields is performed by using FKA. The inclined fracture is modeled by 100 1D DFE’s fitted to the edges of the corresponding triangular elements (Fig. 14.12). For the flow in the fracture the Hagen-Poiseuille law is applied. Fluid viscosity is considered independent of the concentration $\mu =\mu _{0} =\mathrm{ const}$. Graf and Therrien’s variable-density computations employed standard Oberbeck-Boussinesq (OB) approximation (cf. Sect. 3.10.3).

Fig. 14.12

2D triangular finite element mesh with 1D DFE’s used for FEFLOW and GW simulations

Fig. 14.13

Computed concentration distributions and velocity/pathline field after 2, 4 and 10 years simulation. Comparison of FEFLOW results obtained by using AB/TR time stepping (right) to findings by Graf and Therrien [198] modeled by a fully implicit constant time stepping (left). OB approximation is used

Fig. 14.14

Breakthrough curves at observation point $x = y = 6$ m (Fig. 14.11). Comparison of Graf and Therrien’s results [198] to GW [102] (with adaptive time stepping) and FEFLOW (with constant and adaptive time stepping) in using OB approximation

For the 45^∘−inclined fracture problem the results obtained by Graf and Therrien [198] and by FEFLOW in form of computed concentration distributions as well as velocity fields and pathline patterns at 2, 4 and 10 years simulation time are shown in Fig. 14.13. It reveals how the solutes migrate from the fracture into the adjoining porous matrix mainly governed by hydrodynamic dispersion and to a small degree by convection. As a typical feature of the problem two convection cells form above and below the fracture with increasing extent in time. Both cells move downward in time. Note that the cell above the fracture moves faster downward than the lower cell. Both convection cells remain separated by the high-conductive fracture, therefore, acts as a barrier to convection.

At a first glance, FEFLOW and Graf and Therrien’s results agree very well. However, as already seen in Fig. 14.13 the advance of solute transport in the fracture seems slightly faster at early times in Graf and Therrien’s predictions compared to the FEFLOW results. Indeed, this can be confirmed if as shown in Fig. 14.14. While the FEFLOW curves for adaptive time stepping (taking 236 steps) and for constant time steps (100 implicit steps with each of 0.2 years length) provide reasonably close solutions, Graf and Therrien’s breakthrough curve is apparently advanced at early times. Due to the high velocity contrasts between matrix and fracture, the influence of early times on the spreading of solute in the depth is crucial and requires further model comparisons.

Fig. 14.15

Computed solute concentration contours at t = 15 years: FEFLOW versus GW results in using OB approximation

Fig. 14.16

Breakthrough curves at the observation point $x = y = 6$ m (Fig. 14.11). Comparison between OB approximation and EOB approximation. Adaptive time stepping is used for FEFLOW’s solutions

The problem was also simulated by using the GW finite-element simulator [102]. The GW results provide a nearly perfect agreement with the FEFLOW predictions (cf. Figs. 14.14–14.16). As evidenced in Fig. 14.14 FEFLOW’s and GW’s breakthrough curves are very close. This could be confirmed by using both adaptive and constant time stepping strategies. Note further that the type of solving the resulting sparse equation systems did not influence the outcome. Direct and iterative equation solvers were tested in FEFLOW. Additionally, the extended Boussinesq approximation (EOB), cf. Sect. 3.10.3, is also performed. As indicated in Fig.14.16 the breakthrough curve for the EOB is slightly shifted in advance compared to FEFLOW’s OB solution, however, remains further behind Graf and Therrien’s OB solution. It can be concluded that the discrepancies between Graf and Therrien’s findings and the results simulated by FEFLOW or GW are not attributed to different time stepping strategies, Boussinesq approximations and different sparse matrix solvers. Furthermore, more spatially refined meshes did not change notably anymore the solutions because the mesh convergence is practically achieved at the analyzed mesh refinement level.

Fig. 14.17

Schematic representation of the fractured sandstone block

Wendland and Himmelsbach’s Experiment: Solute Transport in a 3D Fracture-Matrix System

Wendland and Himmelsbach [560] conducted laboratory experiments and numerical computations of solute transport in a fractured sandstone block. The sandstone block has a length of 24 cm, a width of 21 cm and a height of 24 cm (Fig. 14.17). The fracture with a mean aperture of >350 μm divides the block into two parts. The geometric details of the fracture plane are shown in Fig. 14.18. Water is pumped from below flowing upwards at a constant rate of Q = 4. 57 ml h⁻¹ through the fracture plane. A multi-tracer experiment with pyranine and cadmium as solutes was performed. The tracer is injected in the fracture at the bottom and the tracer breakthrough is observed at the outlet on the top of the fracture (Figs. 14.17 and 14.18).

Fig. 14.18

Geometry of the fracture plane at $z = L/2 = 120$ mm

Table 14.3 Parameters and conditions used for Wendland and Himmelsbach’s 3D fracture-matrix problem

In the tracer experiment the injection of the solutes is considered as a pulse directly into the fracture. The injection of the total tracer mass of 32. 2 μg lasted less than 1 min. For modeling purposes, Wendland and Himmelsbach [560] smoothed the pulse injection over a time interval of 6 min, which is still small relative to the duration of the tracer experiment. Geometric relations, IC, BC’s and material parameters used for the present simulation are summarized in Table 14.3. BC’s unreported in Table 14.3 for flow and solute transport represent boundaries at which natural BC’s are imposed, i.e., $-(\boldsymbol{K} \cdot \nabla h) \cdot \boldsymbol{ n} = 0$ and $-(\boldsymbol{D} \cdot \nabla C) \cdot \boldsymbol{ n} = 0$, respectively. The flow is modeled steady-state while the solute transport is transient.

In Wendland and Himmelsbach’s simulation [560], the sandstone block was discretized into 8,668 3D elements for the porous matrix and 435 2D elements for the plane fracture. They used a symmetric streamline stabilization technique (cf. Sect. 8.14.5) and an implicit time stepping with 2,000 constant time steps. In the present FEFLOW simulation the spatial discretization is largely similar to Wendland and Himmelsbach’s mesh in that, considering the expected concentration profile in the porous matrix, logarithmic grid spacing is employed. The first nodal row is located at a distance of 2 ⋅ 10⁻⁵ m from and parallel to the fracture interface. The subsequent nodes are at distances of 5 ⋅ 10⁻⁵, 1. 4 ⋅ 10⁻⁴ and 4 ⋅ 10⁻³ m. All further nodes parallel to the vertical fracture are located at a constant horizontal distance of 1 cm, except the last two slices having distances of 4 cm. The resulting finite element mesh of the entire sandstone block is shown in Fig. 14.19. Note that in the FEFLOW simulations only the symmetric half of the domain is considered. This leads to a half-mesh consisting of 20,160 3D linear brick elements for the porous matrix and 1,904 2D linear quadrilateral fracture elements. In order to account for the sealed areas in the fracture plane (Fig. 14.18), the corresponding 2D elements of the fracture were deleted from the mesh.

Fig. 14.19

FEFLOW’s finite element mesh of the sandstone block with a vertical fracture: view of the entire block and magnified mesh at the fracture

Fig. 14.20

Computed stationary pathlines in the fracture and head distributions in the contacted sandstone (half-space view)

In the FEFLOW simulation the adaptive GLS FE/BE predictor-corrector scheme is applied with an initial time step of 10⁻⁵ d and a RMS error criterion of 10⁻⁴. The simulations are performed for a period of 660 min, which required 173 variable time steps. Similar to Wendland and Himmelsbach [560] a PGLS upwind technique (cf. Sect. 8.14.5) is used to stabilize the numerical solution.

Fig. 14.21

FEFLOW results of solute distribution within the fracture $(z = L/2)$ at final simulation time t = 660 min

Figure 14.20 illustrates the resulting flow field and head distribution in the fracture-matrix system. The computed distribution of solutes within the fracture at the final time of 660 min is shown in Fig. 14.21. Wendland and Himmelsbach’s [560] results are displayed in Fig. 14.22. A comparison of the FEFLOW results with solution given by Wendland and Himmelsbach [560] reveals differences in the solute concentration at the sealed areas and near the outlet of the fracture. We note, however, that the magnitudes of solute concentrations are in good agreement (the same concentration levels are used both in Figs. 14.21 and 14.22). Perhaps more significant are the results of the breakthrough behavior at the outlet shown in Figs. 14.23 and 14.24. The agreement with Wendland and Himmelsbach’s [560] measurements is quite well. Wendland and Himmelsbach obtained a higher peak concentration in their simulations compared to the measurements (Fig. 14.24) and the FEFLOW simulation (Fig. 14.23). Obviously, the solute diffusion into the matrix and its accurate numerical representation in the 3D fracture-matrix system is of high importance. The better agreement of the FEFLOW results can result from the more refined spatial resolution.

Fig. 14.22

Wendland and Himmelsbach’s [560] simulation results of solute distribution within the fracture at final simulation time t = 660 min

Fig. 14.23

Breakthrough curve at the outlet: FEFLOW results compared to the measurements [560]

Fig. 14.24

Measured and simulated breakthrough curves at the outlet obtained by Wendland and Himmelsbach [560]

Flow and Solute Transport in a Fracture Network of Rock Mass

The simulation of flow and transport processes in a collection of individual fractures (fracture network) is a challenging task due to the inherent geometric complexity and its required numerical resolution.³ While a small set of individual fractures can still often be described in a deterministic way, a fracture network, where a whole set of crossing and intersecting fractures is typical, necessitates more advanced modeling approaches [5]. Fracture networks are usually described either via stochastic or fractal approaches [80] or by using mechanical parameters in combination with statistical rules for the underlain rock masses [293, 294].

We consider a 2D example of a sedimentary rock mass measuring B × L = 250 × 500 m (Fig. 14.25). A fracture network is generated by using the algorithm developed by Josnin et al. [293] based on stochastic and mechanical parameters given for a tabular stratified rock. A discontinuity network results which is composed of two orthogonal joint sets normal to bedding in the tabular sedimentary rock mass, controlled by two shape parameters: the half-wide E and parameter F for adjusting joint overlap. The resulting orthogonal fracture network shown in Fig. 14.25 was generated by choosing $E = F = 0.5$ m.

Fig. 14.25

Study domain and fracture network generated by Josnin et al.’s algorithm [293] using shape parameters $E = F = 0.5$ m. At central LHS boundary (95 m ≤ x ≤ 155 m, y = L) a solute source is imposed

Table 14.4 Parameters and conditions used for the fracture network model problem

The fracture network geometry (Fig. 14.25) is mapped onto a regular finite-element mesh consisting of 305 × 984 linear quadrilateral elements. The individual fractures are assigned to the edges of corresponding quadrilaterals. In doing so, 68,488 1D DFE’s finally result to model the fracture network in the spatially discretized domain. For the fracture network the Hagen-Poiseuille law of flow with uniform apertures of 100 μm is assumed. A steady-state flow is modeled by prescribing a hydraulic gradient of 1 % between the LHS boundary at $y = L = 500$ m and the RHS boundary at y = 0, ($0 \leq x \leq B = L/2 = 250$ m). At the central LHS boundary a single-species solute intrudes into the domain with a constant concentration C _D , migrates through the fracture network and penetrates the porous matrix over time. The simulations cover a time of 1,000 years. The model parameters and conditions are summarized in Table 14.4. It is assumed that the porous matrix is isotropic and homogenous and that the entire domain is completely saturated. BC’s unreported in Table 14.4 for flow and solute transport represent boundaries at which natural BC’s are imposed, i.e., $-(\boldsymbol{K} \cdot \nabla h) \cdot \boldsymbol{ n} = 0$ and $-(\boldsymbol{D} \cdot \nabla C) \cdot \boldsymbol{ n} = 0$, respectively.

Fig. 14.26

Steady-state hydraulic head distribution $h(\boldsymbol{x})$ in the fracture network domain: FEFLOW vs. GW simulation results

In the FEFLOW simulations the GFEM (without any upwind) and the adaptive GLS 2nd-order accurate predictor-corrector AB/TR time integrator with a RMS tolerance error of 10⁻⁴ are used. To evaluate the computational results, comparisons to the finite-element research code Ground Water (GW) [102] are performed. GW is independently developed and uses differently implemented solution techniques. FEFLOW and GW can run on the same mesh and fracture network data.

The steady-state hydraulic head distribution $h(\boldsymbol{x})$ in the fracture network domain is compared in Fig. 14.26 between FEFLOW and GW revealing a nearly perfect agreement. This can also be evidenced in more detail for h−profiles such as exemplified in Fig. 14.27 at y = 400 m, 0 ≤ x ≤ B. For the transient solute transport through the fracture network domain we also recognize very good agreements between FEFLOW’s and GW’s computational results. This is evidenced in Fig. 14.28 comparing the solute distributions at three selected time stages, in Fig. 14.29 showing the FEFLOW vs. GW solute breakthrough curves at four points selected in the fracture network domain and in Fig. 14.30 comparing C−profiles for the cross section at y = 400 m, 0 ≤ x ≤ B. FEFLOW took 226 variable AB/TR time steps for the simulation period of 1,000 years. A different, but likewise variable time stepping of 2nd-order accuracy was used in the GW simulations.

Fig. 14.27

Hydraulic head profiles at y = 400 m, 0 ≤ x ≤ B, in the fracture network domain simulated by FEFLOW and GW

Fig. 14.28

Comparison between FEFLOW’s and GW’s solute distributions simulated in the fracture network domain at different times t (years)

Fig. 14.29

Comparison between FEFLOW’s and GW’s solute breakthrough curves at points P1(x, y) = (125 m, 400 m), P2(x, y) = (125 m, 300 m), P3(x, y) = (125 m, 200 m) and P4(x, y) = (125 m, 100 m) in the fracture network domain

Fig. 14.30

Concentration profiles at y = 400 m, 0 ≤ x ≤ B, in the fracture network domain for different times t (years) simulated by FEFLOW and GW

Fig. 14.31

Schematic representation of the aquifer-aquitard-aquifer system with the abandoned borehole in the center of the aquitard

Thermohaline Variable-Density Convection in an Aquifer-Aquitard-Aquifer System with Abandoned Borehole

In this hypothetical example we study the effect of a single abandoned borehole causing a short-circuit flow situation in a deep stratified aquifer-aquitard-aquifer system driven by heavy saltwater and buoyant thermal gradients (Fig. 14.31).⁴ The abandoned borehole is to be modeled via the discrete feature approach. The borehole bridges very locally the upper and lower aquifer so that saltwater and heat can be efficiently exchanged over this preferential flow channel. The study domain measures L × H × B = 100 × 100 × 100 m for a 3D schematization and L × H = 100 × 100m for a 2D cross-sectional schematization as shown in Fig. 14.31. The upper and lower aquifers have thicknesses of each 20 m, the aquitard in between is 60 m thick. In the center of the domain the abandoned borehole is located, which interconnects the upper and the lower aquifer in a vertical distance of 60 m. Traces of the abandoned borehole in the aquifers are neglected.

Table 14.5 Parameters and conditions used for the abandoned borehole problem

At initial time, the aquifer system is in a stable hydrostatic equilibrium: the model domain contains freshwater and is subjected to a thermal gradient increasing linearly with depth from 10 to 60 ^∘C. On the top and bottom surface corresponding conditions for hydraulic head h, salinity C and temperature T are held constant. In the simulation a heavy saltwater starts to enter on the top surface. It initializes cellular convective currents in the upper aquifer layer, where the saltwater sinks down, enters the abandoned borehole and salinates the lower aquifer layer. At the same time cooler water reaches the lower aquifer layer via the abandoned borehole. This thermohaline convection process is purely driven by the saltwater density and affected by thermal buoyancy.

The used model parameters and conditions are summarized in Table 14.5. We assume isotropic and homogeneous material conditions for each layer of the aquifer system. The flow in the abandoned borehole is described by Darcy law. The simulations covering a time of 1 year are fully transient both for flow, saltwater and heat transport. BC’s unreported in Table 14.5 for flow, saltwater and heat transport represent boundaries at which natural BC’s are imposed, i.e., $-(\boldsymbol{K} \cdot \nabla h) \cdot \boldsymbol{ n} = 0$, $-(\boldsymbol{D} \cdot \nabla C) \cdot \boldsymbol{ n} = 0$ and $-(\boldsymbol{\varLambda }\cdot \nabla T) \cdot \boldsymbol{ n} = 0$, respectively.

Fig. 14.32

Salinity patterns for 2D meshes of refinement levels ℓ = 0, 1, 2 $(N_{\mathrm{E}} = 1{0}^{4},4 \cdot 1{0}^{4},1.6 \cdot 1{0}^{5})$ and different times t = 5, 10, 20, 100 (d) simulated by FEFLOW. Color sequence blue-lightblue-green-yellow-orange-red depicts normalized salinity C∕C _s from 0 to 1 using 20 intervals

We simulate the thermohaline convection process by using both 2D and 3D models with different spatial resolutions. Regular meshes of linear quadrilateral elements in 2D and linear brick elements in 3D are chosen. With increasing mesh refinement level of ℓ = 0, 1, 2, …, the resulting number of elements N _E and nodes N _P are

$$\displaystyle{ \begin{array}{rcl} N_{\mathrm{E}} & =&{(1{0}^{2} \cdot {2}^{\ell})}^{D} \\ N_{\mathrm{P}} & =&{(1{0}^{2} \cdot {2}^{\ell} + 1)}^{D} \end{array} }$$

(14.45)

where D = 2, 3 represents the dimension. The abandoned well is embodied in the meshes by using 60 ⋅ 2^ℓ 1D DFE’s both in 2D and 3D schematizations. For all FEFLOW simulations we use GFEM (without any upwinding), adaptive GLS 2nd-order accurate predictor-corrector AB/TR time integrator, FKA consistent velocity and OB approximation. Comparisons will be given to the computational results obtained by the finite-element research code Ground Water (GW) [102] using same mesh and DFE data.

Fig. 14.33

Temperature patterns for 2D meshes of refinement levels ℓ = 0, 1, 2 $(N_{\mathrm{E}} = 1{0}^{4},4 \cdot 1{0}^{4},1.6 \cdot 1{0}^{5})$ and different times t = 5, 10, 20, 100 (d) simulated by FEFLOW. Color sequence blue-lightblue-green-yellow-orange-red depicts temperature T from 10 to 60^∘C using 20 intervals

Fig. 14.34

Salinity breakthrough curves at entry point point P1(x, y) = (50 m, −20 m) and exit point P2(x, y) = (50 m, −80 m) of the abandoned borehole simulated by FEFLOW for 2D meshes of refinement levels ℓ = 0, 1, 2

Fig. 14.35

Temperature breakthrough curves at entry point point P1(x, y) = (50 m, −20 m) and exit point P2(x, y) = (50 m, −80 m) of the abandoned borehole simulated by FEFLOW for 2D meshes of refinement levels ℓ = 0, 1, 2

Due to the layer structure and the presence of hydrodynamic dispersion it is obvious that the quantification of the convective regime via solute and thermal Rayleigh numbers, Ra _c (11.25), Ra _t (11.26), is not possible. In particular, the dispersivities β _L , β _T introduce additional nonlinear dependences of saltwater mixing and thermal conduction on the convective velocity. If we disregard dispersivity effects (acceptable at initial phase) and consider only the top aquifer layer we can make a rough estimate from a HRL problem equivalence (cf. Sect. 11.5) and assess a solutal Rayleigh number of $\mathrm{Ra}_{c} = -4 \cdot 1{0}^{6}$ and a thermal Rayleigh number of Ra _t = 33, which clearly indicate a monotonic convection representing a fingering regime in the CSA quadrant of the DDC stability diagram of Fig. 11.8. A Turner number (11.29) of Tu = 100 indicates the gravitational dominance of the saltwater. As a consequence, we must expect a strong primarily solute-driven free convection behavior which is sensitive to inherent perturbation and discretization effects. We note, however, that dispersion effects can significantly reduce $\vert \mathrm{Ra}_{c}\vert$ and $\vert \mathrm{Ra}_{t}\vert$ because the effective ‘diffusion’ increases with $\varepsilon D +\beta _{L}\vert q_{c}\vert$, where the density-dependent Darcy velocity $\vert q_{c}\vert$ could be in the range $0 \leq \vert q_{c}\vert \lesssim K$.

Fig. 14.36

Comparison between FEFLOW’s and GW’s salinity patterns for 2D mesh of refinement level ℓ = 1 (N _E = 4 ⋅ 10⁴) and different times t = 5, 10, 20, 100 (d). Color sequence blue-lightblue-green-yellow-orange-red depicts normalized salinity C∕C _s from 0 to 1 using 20 intervals

Fig. 14.37

Comparison between FEFLOW’s and GW’s salinity breakthrough curves at entry point point P1(x, y) = (50 m, −20 m) of the abandoned borehole for 2D meshes of refinement levels ℓ = 0, 1

Fig. 14.38

Comparison of salinity breakthrough curves at entry point point P1(x, y, z) = (50 m, −20 m, 0 m) and exit point P2(x, y, z) = (50 m, −80 m, 0 m) of the abandoned borehole simulated by FEFLOW for 3D and 2D meshes of refinement level ℓ = 0

Fig. 14.39

Fifty percentage salinity isosurface and temperature field for 3D mesh of refinement level ℓ = 0 (N _E = 10⁶) at different times t = 5, 10, 20, 100 (d) simulated by FEFLOW. Color sequence blue-lightblue-green-yellow-orange-red depicts temperature T from 10 to 60^∘C using 20 intervals

The evolution of salinity and temperature for 2D meshes of three consecutive refinement levels ℓ = 0, 1, 2 simulated by FEFLOW is shown in Figs. 14.32 and 14.33, respectively. It clearly reveals the dependence of the spatial resolution on the convection process. While a coarser mesh with ℓ = 0 produces symmetric patterns, more refined meshes lead always to unsymmetric patterns in the salinity and, correspondingly, in the temperature field. It is obvious, a higher resolution implies more inherent perturbing noise, which triggers the convective instability in the upper boundary layer of salinity at certain locations in a random manner. Notice, for the present simulations we do not induce extra perturbations on the top boundary. As illustrated in Fig. 14.32 the salinity reaches the bottom of the upper aquifer layer after about 10 days and leads to a breakthrough of salinity in the abandoned borehole. Once saltwater enters the borehole a fast descent into the lower aquifer occurs, where saltwater spreads conically over time. On the other hand, the temperature field features a negative image to the salinity pattern (Fig. 14.33). With the sinking of heavy saltwater the aquifer layers and the borehole are cooled down. It is remarkable that fingering convection only occurs at beginning in the upper aquifer layer. At later times this effect vanishes and the solution approaches to a steady state equivalent for all mesh resolutions. As a consequence, the saltwater and temperature breakthrough in the borehole at beginning is determined by the history of convection in the upper aquifer layer, which implies mesh dependency as evidenced in Figs. 14.34 and 14.35. In dependence on the actual history of free convection developing in the upper aquifer the simulated breakthrough curves can be nonmonotonic and lagged.

In the FEFLOW simulations the number of AB/TR adaptive time steps took about 1,200 for refinement level ℓ = 0 (both in 2D and 3D), about 1,800 for for refinement level ℓ = 1 and about 3,700 for for refinement level ℓ = 2. In Fig. 14.36 FEFLOW’s salinity results for the 2D mesh with refinement level ℓ = 1 are compared to the findings obtained by the finite element simulator GW [102]. It indicates that both codes simulate quite different convection patterns at beginning. It is obvious that in FEFLOW’s computations the finger evolution is faster and the resulting saltwater breakthrough in the borehole is more advanced. This is also shown in the breakthrough curves of Fig. 14.37 for the two refinement levels ℓ = 0, 1.

FEFLOW simulations are also performed for the equivalent 3D problem by using a mesh of refinement level ℓ = 0 (N _E = 10⁶). The 3D breakthrough histories in comparison to 2D are given in Fig. 14.38 for ℓ = 0. It reveals that the breakthrough in 3D is clearly faster than in 2D. The developments of salinity and temperature for the 3D model are shown in Fig. 14.39. It illustrates how the heavier and cooler saltwater intrudes very locally via the tubular borehole.