3D hydro-mechanically coupled groundwater flow ...

Computers & Geosciences 67 (2014) 89–99

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Computers & Geosciences journal homepage: www.elsevier.com/locate/cageo

3D hydro-mechanically coupled groundwater flow modelling of Pleistocene glaciation effects Wolfram Rühaak a,n, Victor F. Bense b, Ingo Sass a a b

Technische Universität Darmstadt, Institute of Applied Geosciences, Department of Geothermal Science and Technology, Germany School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, England, UK

art ic l e i nf o

a b s t r a c t

Article history: Received 23 July 2013 Received in revised form 1 March 2014 Accepted 3 March 2014 Available online 13 March 2014

Pleistocene glaciation led to temporal and spatial variations of sub-surface pore fluid pressure. In basins covered by ice sheets, fluid flow and recharge rates are strongly elevated during glaciations as compared to inter-glacial periods. Present-day hydrogeological conditions across formerly glaciated areas are likely to still reflect the impact of glaciations that ended locally more than 10 thousand years before present. 3D hydro-mechanical coupled modelling of glaciation can help to improve the management of groundwater resources in formerly glaciated basins. An open source numerical code for solving linear elasticity, which is based on the finite element method (FEM) in 3D, has been developed. By coupling this code with existing 3D flow codes it is possible to enable hydro-mechanical coupled modelling. Results of two benchmark simulations are compared to existing analytical solutions to demonstrate the performance of the newly developed code. While the result for a fluid–structure coupled case is in reasonable agreement with the analytical model, the result for a classical structure–fluid coupled benchmark showed that the analytical solution only matches the numerical result when the relevant coupling parameter (loading efficiency) is known in advance. This indicates that the applicability of widely applied approaches using an extra term in the groundwater flow equation for vertical stress to simulate hydro-mechanical coupling might have to be re-evaluated. A case study with the commercial groundwater simulator FEFLOW demonstrates the newly developed solution. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Hydro-mechanical coupled modelling Finite element method Porous media Numerical modelling Glaciation

1. Introduction The general concept of thermo-hydro-mechanical (THM) coupled modelling is shown in Fig. 1; for an overview of the current status of thermo-hydro-mechanical-chemical (THMC) coupled modelling see Rutqvist (2011). Besides commercial all-in-one solutions (e.g. COMSOL Multiphysics, ABAQUS, ANSYS) and the research code OpenGeoSys (Kolditz et al., 2012), several couplings of existing groundwater codes with additional mechanical codes exist, e.g. MODFLOW (Leake and Prudic, 1991), TOUGH2 (Rutqvist, 2011; Walsh et al., 2012), HydroGeoSphere (Lemieux et al., 2008a) and SUTRA (Reeves et al., 2000). Here we present a new code that has been developed as a freely available open source code (LGPL licensed), which can be used as a “plug-in” to the commercial software FEFLOW (DHI-WASY, 2012). FEFLOW is one of the most commonly used programs for

n

Corresponding author. Tel.: þ 49 6151 16 3671; fax: þ49 6151 16 6539. E-mail addresses: [email protected], [email protected] (W. Rühaak). http://dx.doi.org/10.1016/j.cageo.2014.03.001 0098-3004/& 2014 Elsevier Ltd. All rights reserved.

groundwater, mass- and heat-transport modelling. FEFLOW was developed to compute thermo-hydro-chemical (THC) coupled processes, however, by adding the newly developed mechanical plug-in it becomes a THMC simulator. To our knowledge no other mechanical coupling extension for FEFLOW exists today. Finally, the coupling of the newly developed code with other groundwater modelling codes is also generally possible.

1.1. Hydro-mechanical coupling The mechanical interaction between the groundwater and the porous geologic media it permeates is a central phenomenon of groundwater flow (Ingebritsen et al., 2006). Calculations which take this interaction into account are called hydro-mechanical (HM) coupled. Various types of mechanical behaviour can be of relevance. In the sub-surface, typically described as porous media, often only elastic dilatation is taken into account. Brittle failure, such as fracturing, is not considered here.

W. Rühaak et al. / Computers & Geosciences 67 (2014) 89–99

Thermo Heat Conduction Fourier‘s Law

Convective Heat Transport Heat Conduction

Hydro Groundwater Flow Darcy‘s Law

Density and Viscosity of the Fluid

y sit n ro tio Po ta la Di th al id m W e r re Th issu F

P Fi erm ss e ur ab E e i lit W y Ch ffec id an tive th ge S ,P s tre or of s os Pr s it y es su re Po te nt ia ls

90

Mechanic Stress - Strain Hooke, Terzaghi, Biot, Biot & Willis Fig. 1. Thermo-hydro-mechanical coupling and the respective constitutive laws.

Although no materials are linearly elastic over a wide range of stresses, elastic constitutive models are often sufficiently accurate to describe rock mechanic deformation (Kolditz et al., 2012). The theory describing the elastic behaviour of porous media is called poroelasticity. Different mathematical approaches are available (e.g. Verruijt, 1969; Smith and Griffiths, 1988; Leake and Hsieh, 1997; Hsieh, 1997; Ingebritsen et al., 2006), to address the time-dependent coupling between rock deformation and fluid flow. A general introduction to the theory of poroelasticity is provided by Wang (2000). The general concept of thermo-hydro-mechanical coupling is depicted in Fig. 1. A description of hydro-mechanical coupling requires simultaneous solution of equations for groundwater flow (Darcy equation) and the mechanical behaviour described, for instance following Biot (1941). For thermo-coupling, the heattransport equation also has to be coupled dynamically. Two basic phenomena underlie poroelastic behaviour (Wang, 2000):

coupled processes are given in Wang et al. (2009), Watanabe et al. (2010) and Kolditz et al. (2012). Thermo-mechanical coupling is generally unidirectional as thermal expansion induces volume changes of the rock (similar to pore pressure changes in case of hydro-mechanical coupling); vice versa the mechanics do not alter the temperature directly but only due to changes of the (convective) flow field.

Solid-to-fluid coupling occurs when a change in applied stress

where SS is the storage coefficient (m 1), h is the hydraulic head (m), t is the time (s), αb (–) is the Biot coefficient, an empirical constant (Biot, 1941) ranging from 0 to 1, describing to which extent fluid pressure counteracts elastic deformation of the porous rock (Alam et al., 2010), ∂εvol =∂t is the time rate of change of volumetric strain (s 1) and K is the hydraulic conductivity tensor (m s 1), according to

produces a change in fluid pressure or fluid mass.

Fluid-to-solid coupling occurs when a change in fluid pressure or fluid mass produces a change in the volume of the porous material.

1.2. Thermo-hydro-mechanical coupling In recent years the inclusion of thermal effects into the theory of hydro-mechanical (HM) coupling has become increasingly prevalent (e.g. Lee and Ghassemi, 2011). Important applications are for instance the modelling of thermo-mechanical (TM) processes in nuclear waste disposal. In addition, thermo-hydromechanical (THM) induced stress–strain can have an impact on processes related to the usage of geothermal energy. For computing such processes numerically on the respective relevant scales, stress–strain relationships resulting from fluid pressure and temperature have to be computed and dynamically coupled to the regional flow and transport regime. A prototypical application of THM codes is for instance the productivity of a geothermal doublet system with one pumping well and one injection well, which can be influenced by mechanical changes and associated change in hydraulic conductivity (Bundschuh and Suárez Arriaga, 2010). General reviews of THM

2. THM continuity equations For THM coupled modelling, the following continuity equations have to be solved (see Bear, 1972; Diersch and Perrochet, 1999; Alberty et al., 2002; Ingebritsen et al., 2006; Rühaak et al., 2008). Groundwater flow: SS

∂h ∂ε ¼ ∇ U ðK∇hÞ þ αb νol ∂t ∂t

K¼

k ρg

η

ð1Þ

ð2Þ

where k is the permeability tensor (m2), ρ is the density (kg m 3), g is the gravity (m s 2) and η is the dynamic viscosity (kg m 1 s 1). The resulting groundwater flow expressed by the Darcy velocity q (m s 1) is q ¼ KU∇ h

ð3Þ

Coupled conductive and convective heat-transport is expressed by the following equation: ðρcÞg

∂T ¼ ∇ U ðλT ∇T ðρcÞf qTÞ ∂t

ð4Þ

where ðρcÞg is the bulk volumetric heat (J K 1 m 3), T is the temperature (1C), λT is the thermal conductivity tensor (W m 1 K 1), index f denotes fluid properties.


Linear elasticity equation is given by the following equation: ðλ þ μÞgrad div u þ μ∇2 u ¼ f

ð5Þ

where u is the displacement vector (m), and f is the volume force (N), typically resulting from the pore pressure, and λ is Lamé's first parameter (MPa):

λ¼

Eυ ð1 þ υÞð1 2υÞ

ð6Þ

here ν is Poisson's ratio (–); μ is second Lamé parameter (MPa), identical to the shear modulus, often denoted as G

μ¼

E 2ð1 þ νÞ

ð7Þ

where E is Young's modulus (MPa).Eq. (5) is combined with the stress–strain relationship 2 3 2 32 3 εx sx λ þ 2μ λ λ 0 0 0 6s 7 6 76 ε 7 λ þ 2μ λ 0 0 0 76 y 7 6 y 7 6λ 6 7 6 76 7 6 sz 7 6 λ 6 7 λ λ þ 2μ 0 0 0 7 6 7 6 76 ε z 7 ð8Þ 6 7¼6 76 2 γ 7 μ 0 0 76 xz 7 0 0 6 τxz 7 6 0 6 7 6 76 7 6 τyx 7 6 0 6 7 0 0 0 μ 07 4 5 4 54 2γ yx 5

τzy

0

0

0

0

0

μ

2γ zy

where s are stresses (MPa), ε are strains (–), τ and γ denote the shear stress and the shear strain, respectively (MPa). Indexes x, y, z denote the respective Cartesian directions. In Eq. (1) the last term αb ∂ενol =∂t reflects the volume change of the porous medium, which is equivalent to adding or removing fluid. It can therefore be seen as a fluid source/sink term (Neuzil, 2012) and is handled this way in the presented coupling approach, substituting the strain by the volume change (dilation) of the respective elements during the previous time-step (Ahola et al., 1996; Ingebritsen et al., 2006):

αb

∂ενol Vt 1 Vt 1 ffi αb ¼ Qt ∂t V t 1 Δt

ð9Þ

Here Vt 1 is the element volume at time the previous time-step, while Vt is the element volume after actual displacement, Δt is the current time-step size. This source/sink of fluid Qt (s 1) is added to the groundwater flow equation at time t. Within the FEFLOW open programming interface IFM, a call to the IFM API changes the sink/source term of the groundwater flow equation (Eq. (1)). The volume or loading forces f ¼ αb ρf g∇h

ð10Þ

are evaluated as a vector in the centre of gravity. They represent the pore-pressure (Ingebritsen et al., 2006)

sef f ¼ stotal ppore

ð11Þ

The head gradient ∇h inside each tetrahedron is computed using a code from Abriola and Pinder (1982). In a THM problem the expansion of the rock matrix due to temperature changes has to be taken into account; concomitant changes of the fluid properties are not discussed here because they are handled by FEFLOW directly (for details see Diersch and Kolditz, 2002). The volume of a solid increases or decreases with temperature changes, and homogenous bodies expand evenly in all directions when temperature increases. If deformation is not possible, the internal stresses increase or decrease with temperature changes (Kolditz et al., 2012). This unidirectional stress f' is added to the volume force following Ahola et al. (1996): f' ¼ DβΔT

ð12Þ

91

where β is the volumetric thermal expansion coefficient (K 1), and D is the bulk modulus (Ingebritsen et al., 2006): D ¼ λþ

2μ E ¼ 3ð1 2υÞ 3

ð13Þ

The hydraulic and mechanical equations are coupled because pore pressure appears in the linear elasticity equation, and because volumetric strain appears in the fluid-flow equation. Due to the mechanical deformation a non-linear system results, which requires additional treatment in order to obtain convergence. The strain modifies the fluid-pressure, while the fluidpressure is a force which is biasing the strain. Compared to the other TH coupled processes, mechanical compaction can be considered to be an instantaneous process (Kolditz et al., 2012). As stress and strain have a negligible impact on the temperature (Kolditz et al., 2012), a backward coupling with the temperature is not performed. Due to the strain, a change of the porosity and permeability is likely. This can be taken into account by reducing the porosity linearly with the change of volume and simultaneously changing the permeability in accordance with the porosity. 2.1. Uncoupled approach for purely vertical strain A commonly applied approach to simplify hydro-mechanical effects in simulations of basin-scale fluid flow is to add a source/ sink term to the fluid flow equation to account for the pore pressure effects of mechanical loading (e.g. Bense and Person, 2008; Lemieux et al., 2008a): dh dL ∇q ¼ Ss ð14Þ dt dt in which dL=dt is the loading rate (m s 1), and Ss is the onedimensional specific storage (m 1) given by (e.g. Domenico and Schwartz, 1998) the following equation: Ss ¼ ρf gðβ v þnβ f Þ

ð15Þ

in which βv and βf are the compressibility (Pa ) of the rock matrix and fluid respectively, and n is the porosity of the medium (–) and ζ is the one-dimensional loading efficiency (–) given by (Lemieux et al., 2008a,b) 1

ξ¼

βv β v þ nβ f

ð16Þ

Neuzil (2003) gives a different equation for the loading efficiency:

ξ¼

Bð1 þ υÞ 3ð1 υÞ 2αb Bð1 2υÞ

ð17Þ

where B denotes the Skempton coefficient or three-dimensional loading efficiency (–). For highly compressible media, B approaches unity while in very stiff media it can be close to zero (Neuzil, 2003). By rewriting Eq. (17)

αb ¼

1þυ 3ð1 υÞ 2ð1 2υÞ 2Bð1 2υÞ

ð18Þ

it can be seen that the Biot coefficient αb can be expressed as a difference between terms containing one- and three-dimensional loading efficiency. Although the one-dimensional loading efficiency can be calculated, the paucity of representative data introduces considerable uncertainty (Lemieux et al., 2008b). One of the most important restrictions of the use of Eq. (14) to describe hydro-mechanical effects is that purely vertical strain needs to be assumed (e.g. Ingebritsen et al., 2006). Wang (2000) and Neuzil (2003) suggest that the assumption of purely vertical strain is a reasonable one for both two- and three-dimensional

92


D C B

Confining Unit

C B

Aquifer

E

Drainage Boundary

A A

120 m

B

A

Aquifer

440 m A

A

4000 m 5200 m Fig. 2. Benchmark example following Leake and Hsieh (1997). Aquifer C and the confining unit are both 20 m thick (view is 5-times exaggerated).

flow systems, which only results in small errors where overburden changes are laterally extensive. However, a thorough evaluation of the error introduced by the application of this assumption does not seem to have been carried out in the literature (Ingebritsen et al., 2006). The use of Eq. (14) is often called an “uncoupled” approach to consider hydro-mechanical effects. 2.2. 3D finite element implementation for linear elasticity Flow and heat transport are computed with FEFLOW as usual. Corresponding stress and strain are computed with the new IFM plug-in. Initially, the stiffness matrix of the linear system, Eq. (5), is computed once with the callback PreSimulation. As this computation is the most expensive one it is advantageous to perform this calculation only once. However, this is only possible because the changes of the grid geometry due to the mechanical displacement are not taken into account. FEFLOW allows a moving mesh in vertical but not horizontal direction. Due to this limitation the plug-in does not modify the problem's geometry according to the actual displacement. This simplification is mainly justified by the relatively small amounts of displacement of the order of typically less than 1%. However, the output in VTK and TECPLOT format does allow visualisation of the 3D deformed mesh. The mechanical calculations are called before the calculation of the hydraulic head with the FEFLOW IFM callback PreFlowSimulation. Convergence can be improved by performing additional inner iterations (for details see Rutqvist, 2011) to minimise the variance of the steady-state strain. This can be achieved in the subsequent callback PostFlowSimulation. However, currently only the socalled loose or weak coupling is possible, where values are exchanged after each time-step. The associated error due to this procedure depends on the type of coupling but is always small as long as the time steps are not too big. Options to improve the coupling are discussed later in the conclusions. The routines for solving the Navier–Lamé equation (Eqs. (5) and (8)) for linear elasticity are part of the ffp library (free finite element program), while the plug-in itself is called mcf (mechanical coupling feflow). All routines are open source (LGPL1) and available from http://sourceforge.net/projects/ffp/. Boundary conditions (Neumann – scalar weight and Dirchlet – vectorial displacement) and the elemental parameters (Lamé's 1

http://www.gnu.org/copyleft/lesser.html.

Table 1 Parameters used for the Leake and Hsieh (1997) example. Value Ss K λ ¼μ ν αb

Aquifer 5

1 10 2.89 10 4 320 0.25 1

Confining 4

1 10 1.16 10 7 32

Unit m1 m s1 MPa

first and second parameters) can be conveniently assigned as “user data” using the graphical user interface of FEFLOW. For the 3D solution of the IFM plug-in a tetrahedron based FEM approach is used; other element types are currently not supported. This FEM implementation is based on the approach presented by Alberty et al. (2002), with the code written object orientated in standard ANSI/ISO C þ þ. The most common geometrical element type used in FEFLOW is a pentahedron (a wedge or a triangular prism). Tetrahedrons are not supported. The plug-in therefore generates three tetrahedrons for each FEFLOW pentahedron (for details see Dompierre et al., 1999; Cordes and Putti, 2001). For the meshing TetGen (Si, 2010) is used. The main motivation for using TetGen and tetrahedrons is to easily enable future subgrid approaches. For solving the linear systems the direct PARDISO solver (Schenk and Gärtner, 2004) is used. A conjugate gradient solver utilising the GPU (using NVIDIA’s CUDA functions) is currently under development.

3. Verification An important part of developing numerical programs is to verify the code in order to ensure agreement with known solutions. 3.1. Fluid-to-solid The first verification example compares the numerical solution with a solution according to Terzaghi (1943), following Leake and Hsieh (1997). Increased effective stress due to pumping can result in compaction of an aquifer. The reduction of pore space brings about more fluid movement and the solid frame compacts further. Accordingly, the stress and fluid flow affect each other. This verification example is therefore only a suitable verification for the fluid-to-solid coupling part of the solution.


Three sedimentary layers overlay impermeable bedrock in a basin where faulting creates a bedrock step near the mountain front (Fig. 2). The sediment stack is 440 m thick at the deepest point of the basin (x ¼0 m) but thins to 120 m above the step (x 44000 m). The total width is 5200 m. The top two layers of the sequence are each 20 m thick. The first and third layers are aquifers; the middle layer is relatively impermeable to flow. The materials given in Table 1 are homogeneous and isotropic within a layer. Hydraulic and mechanical boundary conditions are given in Table 2. The flow field is initially at steady state, but pumping from the lower aquifer reduces hydraulic head by 6 m per year at the basin centre. Edge E (Fig. 2) represents a drainage boundary, where the hydraulic head lowers at a rate of 6 m per year. The total simulation time is 10 years. As presented in the paper by Leake and Hsieh (1997), the result here is compared with an approach based on the Terzaghi theory (Terzaghi, 1943). One-dimensional vertical compaction, Δb, is computed using the relation

Δb ¼ Ssk bΔh;

ð19Þ

93

where Ssk is the skeletal component of specific storage, b is the thickness of compacting sediments, and Δh is the change in head computed by FEFLOW. Compaction is computed for each cell, and vertical displacement is the compaction for a cell plus the sum of compaction values for the cells below. Vertical displacements calculated by both models are compared in Fig. 3 and are generally in good agreement. Differences occur at the bedrock on the right-hand side and on the lefthand side. The numerical solution is full 3D using FEFLOW; so at the left-hand side there is considerably more displacement due to the additional deformation in both the x- and ydirections (see Fig. 4). The so-called roller boundary condition, where only displacement in the vertical direction is possible, would be necessary to replicate the Terzaghi approach. The model boundary can also move horizontally and – because the model is in fact 3D – also in the direction perpendicular to the view. The discrepancies between the numerical and analytical solution near the bedrock step are similar to those observed by Leake and Hsieh (1997), who also identified them as a result of the onedimensional limitation of the Terzaghi solution.

Table 2 Hydraulic and mechanical boundary conditions.

3.2. Solid-to-fluid

Mechanical

Hydraulic

A B C D E

Fixed Free Free Free Free

Flux ¼0 m s 1 (Neumann) Flux ¼0 m s 1 (Neumann) h ¼ h0 h ¼ h0 h ¼ h(t)

0

Boundary conditions

-100 6

-0.06m

-0.06

-200 -300

0.

-0.1

-0.04m

-0.04

-0.02m

0

1000

2000

3000

z (m)

-0 . 08 -0.04 -0.0

-0.08m

-0.12

Table 3 Model properties for the solid-to-fluid verification case.

-400

8

-0.14

m -0.14 -0.12m

2

2 -0 . .0 -0

m -0.16

-0.04

.

-0.16

-0.2

-0.22m -0.18

-0.0

-0.2m2 -0.2 m -0.18

For verifying the solid-to-fluid coupling the numerical solution is compared with an analytical solution for the fluid pressure change due to an increasing load on top of a 10 km long 1D column, following Lemieux et al. (2008a) and Walsh et al. (2012). The example is intended for testing the fluid flow effects of mechanical loading by a 3 km thick ice-sheet during a 10,000 year period. The verification example settings are given in Table 3. The applied stress on top of the column is continually increased as a linear function of time. The top of the column is drained (hydraulic head is held constant at zero) and the base of the column is at an infinite distance. The analytical solution to this problem is

4000

5000

x (m) Fig. 3. Result of the benchmark simulation. The isolines depict the displacement (m) in vertical direction; either computed with ffp/mfc (blue lines and labels) or with the classical Terzaghi approach (black lines and labels) following Leake and Hsieh (1997).

K¼ 1 10 3 (m a 1) Ss ¼ 1 10 6 (m 1) λ ¼μ ¼5886 (MPa) E ¼14,715 (MPa) ρf ¼ 1000 kg m 3 g ¼ 9.81 m s 2

stress Z

X

Y

5000 m 4000 m 3000 m 00 m 2000 m 00 m 1000 m 500 m 0m -500m

0m

Fig. 4. The deformed mesh (displacement is 500 times exaggerated).

1 MPa 0.9 MPa 0.8 MPa 0.7 MPa 0.6 MPa 0.5 MPa 0.4 MPa 0.3 MPa 0.2 MPa 0.1 MPa

94


according to Lemieux et al. (2008a) and Walsh et al. (2012): " rffiffiffiffiffiffiffi # ζ dszz z z t z2 pffiffiffiffiffiffi þ z erf c hðz; tÞ ¼ t tþ exp 2D ρg dt πD 4Dt 2 Dt

1000

ð20Þ

βv ¼

3ð1 2νÞ E

ð21Þ

Using Eq. (13) with ν ¼ 0.25, the respective Lamé parameters can be calculated. The relation between stress and loading due to an increasing ice-sheet height H (m) is

s ¼ H U ρf U g

ð22Þ

Using the values given above, smax ¼29.43 (MPa). The density difference between ice and fluid water is neglected in general in this work. The stress rate is therefore Δs MPa ð23Þ ¼ 0:002943 a Δt A strong misfit between the analytical and the numerical solution is observed. For instance after 10,000 years at a depth of 500 m the head is 613 m for the numerical solution while it is 990.2 m for the analytical solution. To investigate the difference between approaches a stand-alone 1D solution was developed using MATLAB.2 The flow equation is calculated using a conventional 1D implicit finite differences approach while the compaction is calculated using a slightly modified 1D FEM MATLAB code from Bower (2009). This MATLAB code uses Eqs. (1), (4) and (5) and gives an almost identical solution as FEFLOW/mcf (Fig. 5). As no sufficiently accurate analytical solution exists, it can be used as a benchmark. Neither the body forces (fluid–structure coupling) nor the movement of the mesh (20 m after 10,000 years of a 10 km long row) have a notable impact on the resulting head. Following Hook's law, the given stress leads to a deformation of approximately 2 m per thousand years with respect to the 10 km long row. The deformation is slightly lower due to the fluid pressure which is acting against the compaction, but the difference is too small to be of relevance. Based on the equation Q¼

umax ; Lt

ð24Þ

with umax ¼20 m, L ¼10,000 m and t ¼10,000 365 days, a fluid source of Q¼ 5.48 10 10 d 1 is obtained. Using this value in any groundwater flow model for such a 10 km long row gives almost identical results after 10,000 years. 2

Available from www.wolfram-ruehaak.net/matlab/mcf-test.zip.

800

600 Head (m)

where dszz/dt is the stress application rate (Pa a 1), z is the depth (m) and D¼ K/Ss is the hydraulic diffusivity (m2 s 1). Analogous to the original work a loading efficiency ξ ¼ 1 is applied. A similar system was modelled using FEFLOW/mcf. For this verification the full 3D solution was replaced by a 1D FEM solution from Bower (2009). The fully 3D ffp solution is not suitable for such a 1D example, as deformation in the other directions has a notable impact on the solution. To convert the specific storativity into a respective elastic modulus, Eq. (15) is used. Since the compressibility of water is approximately 10 8 (m 1) (Talebian et al., 2013), and this value is significantly smaller than the bulk compressibility of 1 10 6 (m 1), only the matrix compressibility is taken into account. This is related to Young's modulus according to Ingebritsen et al. (2006):

Matlab Numerical - 1 ka Matlab Numerical - 5 ka Matlab Numerical - 10 ka FEFLOW/mcf - 1 ka FEFLOW/mcf - 5 ka FEFLOW/mcf - 10 ka Analytical LE=1 - 1 ka Analytical LE=1 - 5 ka Analytical LE=1 - 10 ka Analytical LE=0.62 - 1 ka Analytical LE=0.62 - 5 ka Analytical LE=0.62 - 10 ka

400

200

0 0

200

400

600

800

1000

Depth (m) Fig. 5. Comparison of the numerical FEFLOW/mcf and Matlabs solution with the analytical solution assuming loading efficiency (LE) ¼1.0, and analytical solution assuming loading efficiency ¼0.62.

If a loading efficiency of 0.62 is used, the fit of the uncoupled analytical solution is close to the numerical solution.

4. Application example: hydro-mechanical modelling of the impact of the Pleistocene glaciation Mechanical loading and unloading by an ice sheet leads to small changes in pore water volume that occur when a porous medium is mechanically compressed (loaded) or expands (unloaded), with resulting changes in pore water pressure (e.g. Domenico and Schwartz, 1998). In highly permeable rock these transient changes in fluid pressure will quickly dissipate. However, in low-permeability units like shales, clay, or crystalline rocks, the effects of loading and unloading can induce anomalous fluid pressures that require thousands of years to return to equilibrium conditions (e.g., Neuzil, 1993). Bense and Person (2008) have shown that, in basins which were formerly covered by an ice sheet, fluid flow and recharge rates are strongly elevated during glaciations as compared to inter-glacial periods. Steady state hydrodynamic conditions are probably never reached during a 32.5 thousand year cycle of advance and retreat of a wet-based ice sheet. Present-day hydrogeological conditions across formerly glaciated areas are likely to still reflect the impact of glaciation that ended locally more than 10 thousand years before present. Modelling can help to improve the management of groundwater resources in formerly glaciated basins and to evaluate the viability of nuclear waste repositories at high latitudes on geological timescales. To investigate the impact of slowly increasing and decreasing glaciation, a two-dimensional situation with two non-continuous clay layers in a 3 km thick sedimentary basin is computed using the simplified FlexPDE model (Fig. 6). Results are compared with the new FEFLOW/mcf code. The model starts with a glacier thickness of 0 m (Fig. 7). The thickness is increased at a rate of 0.2 m a 1 until it reaches a maximum thickness of 2 km after ten thousand years. This thickness persists for ten thousand years. Finally the glacier shrinks to a thickness of zero during the last ten thousand years.


2000

750 m

Clay Lens Sand

B

Clay Lens

1600 1200 800 400 0

3 (km)

500 m

Glacier thickness (m)

250 m 250 m

Transient changing weight of the glacier

A

95

0

10000

20000

30000

Time (a)

1250 m

Fig. 7. Time varying loading and unloading due to the glacier thickness.

Bedrock Sand

3 (km) Fig. 6. Schematic diagram of the glaciation example. (A) and (B) depict positions where head-time changes are logged for later comparison.

The maximum thickness of 2 km corresponds to a loading weight of 19.62 MPa (Eq. (22)).

Table 4 Parameters used for the glaciation example with FlexPDE. Parameter

Sand

Clay

Unit

βv (Eq. (21)) ρ Ss (Eq. (15)) βf Ε kx ky η ξ

7.5 10 10 1000 7.35 10 6 0 2000 10 13 10 14 10 3 0.75

7.5 10 9 1000 7.35 10 5 0 200 10 16 10 17 10 3 0.75

Pa 1 kg m 3 m1 Pa 1 MPa m2 m2 kg m 1 s 1 –

4.1. FlexPDE In the FlexPDE simulations, fluid compressibility (βf) again is considered to be zero. The matrix compressibility relates to Biot parameters following Eq. (21). As discussed in Section 3.2, the uncoupled approach leads to overestimation of the head. For correction a loading efficiency of 0.75 is used. All parameters used in the FlexPDE simulations are given in Table 4. 4.2. FEFLOW The parameters used for the mechanical coupled FEFLOW simulation are given in Table 5 and are similar to the FlexPDE values. The time varying stress on top of the model is given by a time varying Neumann stress boundary condition. The model base is fixed using Dirichlet boundary conditions of zero in the x-, y- and z-direction. The flow boundary condition is a Dirichlet free outflow head boundary condition of zero at the top of the model. All other boundaries are impermeable. As FEFLOW/mcf is a pure 3D solution, the model in fact has a 2 m extension in the third direction. However, the deformation in this direction is very small compared to that in the horizontal and vertical directions. For the calculation FEFLOW’s automatic time stepping procedure is used (forward Adams–Bashford; backward trapezoid). 4.3. Result The FEFLOW result is shown in Fig. 8. The maximum head of 68 m occurs in the lower clay layer at the end of the loading phase, when the glacier has reached its maximum thickness of 2 km after ten thousand years. From then on, the head decreases rapidly over the following 2000–3000 years, while the load is constant and the matrix is in equilibrium. During the unloading phase the head becomes negative due to expansion of the formation and reaches a minimum of 65 m after 24,000 years.

Table 5 The parameters used for the glaciation example with FEFLOW. Parameter

Sand

Clay

Unit

Ss kx ky η E ν λ ¼μ αb

7.35 10 6 10 13 10 14 10 3 2000 0.25 800 1

7.35 10 5 10 16 10 17 10 3 200 0.25 80 1

m1 m2 m2 kg m 1 s 1 MPa – MPa –

At the end of the unloading phase after 30 thousand years the head value is still relatively low with a minimum of 58 m. The FEFLOW result is compared with the FlexPDE result in Fig. 9; slight differences exist. These differences are partly related to the three dimensionality of the FEFLOW/mcf solution. As can be seen in Fig. 8, in addition to the major displacement in the vertical direction, significant deformation occurs in the horizontal direction.

5. A 3D model of glaciation for an intercratonic sedimentary basin To demonstrate the potential of the newly developed code, a 2D study of Bense and Person (2008) describing the hydrodynamic consequences of glaciation/deglaciation within an intercratonic sedimentary basin is recalculated in 3D. However, in contrast to the original study, heat- and mass-transport are not covered; only mechanical coupling is studied. The geometry of the idealised sedimentary basin considered here (Fig. 10) is inspired by the model shown in Bense and Person (2008) which is meant to represent an intercratonic sag basin. The basin consists of four aquifers (A1–A4) separated by three


0m

96

-80 m

-55 m

m

-32 m

-30

-500 m

m -42 -13 m

-12

-1000 m

m

-8 m

-8 m

m -2 2

m -32

-3 m

20 45 -22 m

-1500 m

-3 m

20 -55 m 45

20

-12 m

45

-12mm -3

20

-30 m

45

-30 m

Head

20

-2000 m

20

80 m

-2500 m

70 m 60 m

-5 m -2 m

50 m

-5 m

-2 m

-3000 m

40 m 30 m 20 m

0m

10 m 0m

-500 m

-34 m

-0.85 m

.9 -0

-5

m

-5

-0.85 m

-10 m -20 m

-14

m

-5

0.1 m

-5

-44 m

-24 m

-5

-30 m -0.6 m

-0.4 m

-5

-5

-55 4m -2

-30 -0.9 m

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-55 -14 m -30 -5

-2000 m

-5

-60 m

-0.35 m

-55 45mm -0. -0.3 -30 -5

-30

-50 m -5

-70 m

0.1 m

-1500 m

-5 -30

-40 m

-0.6 m

-34

m

-1000 m

-5

- 30

-80 m

-5

-4 m

-4 m

-0.1 m

-0.1 m

-3000 m

-2500 m

-4 m

0m

500 m

1000 m

1500 m

2000 m

2500 m

3000 m

0m

500 m

1000 m

1500 m

2000 m

2500 m

3000 m

Fig. 8. Head (contours), displacement in vertical (solid line) and horizontal (dashed line) direction after 5000 years (A), 10,000 years (B), 25,000 years (C) and 30,000 years (D).

semiconfining units (S1–S3) representing shale beds. The sedimentary units thicken toward the basin centre. The values of compressibility, permeability and porosity for all units are shown in Table 6. Relative to Bense and Person (2008), the sediment compressibility is reduced for all units except the basement to a tenth of the original value, because the original values resulted in an unrealistic vertical deformation of more than 400 m. The respective Lamé parameters are calculated by applying Eq. (13). The 3D model which has been set-up for this study has a slightly modified geometry. The model has an extension of 800 km in north–south and 500 km in east–west direction, which is some 200 km smaller than the model of Bense and Person (2008). The actual basin is positioned in the middle and is elliptic with an extension of 500 km in north–south and 250 km in east–west direction.

The ice sheet moves linearly for 10,000 years from the western border continuously into the model till a maximum of length L (t) ¼575 km is reached. The ice sheet height at the western boundary increases linearly during this time to a maximum of H (t) ¼3200 m. The ice sheet geometry is approximated as a function of distance along the x-direction following van der Veen (1999) and Bense and Person (2008) with " #1:5 x 2:5 HðxÞ ¼ HðtÞ 1 ð25Þ LðtÞ The effective stress is calculated according to Eq. (19). The model starts with a glacier thickness of 0 m. The thickness is increased until it reaches a maximum of 3.2 km after ten thousand years. This thickness persists for ten thousand years. Finally the glacier


shrinks again to a thickness of zero during the last ten thousand years. The resulting glacier positions are depicted for the first 10,000 years in Fig. 10. 5.1. Results Fig. 11 shows the spatial displacement within the basin due to glacier weight at the beginning of maximum glaciation after 10,000 years. Fig. 12 shows the temporal change of the head inside the four aquifer units in the middle of the western half of the basin. After a continuous increase of head due to the increase in glacier thickness, the head decreases once the glacier thickness is for 10,000 years continuous. The decrease of the glacier evokes a continuous decrease in head. 6. Conclusions and outlook An open source plug-in for mechanical coupling is available which allows the popular commercial groundwater and heat- and mass-transport code FEFLOW to be used as a THMC simulator. Alternatively, the code could be used in other groundwater codes with relatively small effort. The code is used to model anomalous pressures due to glaciation. The result is in good agreement with previous work where

97

the impact of the glacier was modelled with a simpler approach using the storativity term of the groundwater equation. However, in addition to a better representation of the physics, the new plugin allows such processes to be readily modelled in 3D. Loading efficiency is not necessary in the full numerical solution, but is required for the uncoupled approach. Several authors have modelled the impact of glaciation on groundwater head using this simplified approach (e.g. Bense and Person, 2008; Lemieux et al., 2008a; Nasir et al., 2011). The present study indicates that the use of an appropriate loading efficiency between 0.6 and 0.8 should be considered for such simplified models to avoid overestimating resulting heads. Bethke and Corbet (1988) noted that simplified models seem to overestimate the pressure which is the case here, too. A general revaluation of using loading efficiencies together with an uncoupled approach seems necessary. mcf/ffp is a purely 3D solution which has only limited capability for 1D or 2D solutions. For a 3D model the common approach of placing the borders sufficiently far from the region of interest should be followed. Another possible option is to use stiff surrounding material. Thermo coupling, although already implemented, is not discussed in this paper. Mechanically induced stresses, for instance due to hydraulic stimulation or reservoir operation, can be modelled and studied (Rühaak and Sass, 2013). A potential application of the plug-in is to model stress–strain behaviour close to a borehole heat exchanger (BHE), where the efficiency of such a BHE may be influenced by thermal alteration of the surrounding grouting (Laloui et al., 2006). However, before such applications are considered, validation will be necessary, using either a thermotriax device or analytical solutions. Another interesting application of the plug-in would be landslide modelling. Existing FEFLOW models could be easily extended to allow the stress to be calculated, although it is not possible to calculate the explicit failure criteria.

Table 6 Physical properties of the stratigraphic units, modified after to Bense and Person (2008).

Fig. 9. Head variation during the simulation at the left boundary in the middle of the clay layers (A is in the upper clay layer, B in the lower clay layer).

Layer Unit

Max. depth (m) K (m s 1)

1 2 3 4 5 6 7 8

1000 1500 2000 2500 3250 4000 5000 7500

Aquifer 1 Aquitard 1 Aquifer 2 Aquitard 2 Aquifer 3 Aquitard 3 Aquifer 4 Basement

1 1 1 1 1 1 1 1

North

10 5 10 9 10 6 10 10 10 7 10 11 10 7 10 12

S (m 1) 1 5 1 1 5 5 5 1

10 5 10 4 10 5 10 4 10 6 10 5 10 6 10 7

n

λ ¼ μ (MPa)

0.4 0.6 0.3 0.5 0.2 0.4 0.2 0.1

6000 120 6000 600 12,000 1200 12,000 60,000

South

Elevation (km)

2 2000 a

0

4000 a

6000 a

8000 a

10000 a

A1 S1 A2 S2 A3 S3

-2 -4

A4

Basement

-6 0

100

200

300

400 Distance (km)

500

600

700

800

Fig. 10. Cross section through the basin and position of the observation points (black dots at x¼ 300 km). The position of the progressing glacier over time is also shown (view is 20 times exaggerated).

98


0m 000 m 000 m 000 m 0m 0m 100000 m 200000 m

Z

200000 m 400000 m

X

-4 m -8 m -12 m -16 m -20 m -24 m -28 m -32 m -36 m -40 m

300000 m 600000 m

Y

400000 m 800000 m

500000 m

Fig. 11. Vertical displacement after ten thousand years (view is 20 times exaggerated).

120 A B C D

Head (m)

80

The developed code is available as open source from http:// sourceforge.net/projects/ffp/. The code will be continuously improved in the future, and interested developers are invited to cooperate. In addition to the solution for the Navier–Lamé equation an additional solution for Laplace like steady-state equations, e.g. heat-diffusion is already included in the code.

40

7. Technical details 0

In the graphical user interface of FEFLOW the user has to load the plug-in, afterwards the following reference data have to be created:

-40

Nodal – “Neumann”: The Neumann boundary conditions (unit -80 0

10000

20000

30000

Time (a) Fig. 12. Hydraulic head at the observation points inside the four aquifer units (A is the uppermost) in the middle of the western half of the basin (see Fig. 10).

A fully coupled THM analysis generally includes a number of explicit and implicit assumptions and approximations, as the relevant mechanical and hydraulic parameters and boundary conditions are only poorly known (Neuzil, 2012). Thus comparison with experimental results will be performed in future. Discrete coupling is challenging due to its non-linearity. Mechanical computations are costly in terms of memory and CPU time consumption. Future developments will focus on mechanical subgrids to avoid calculations in regions of little interest. Several previous studies have shown (Minkoff et al., 2003) that a loose time-step coupling is generally feasible, and that the associated bias is typically small and therefore tolerable. Nevertheless, enabling a more rigorous and tighter coupling is desirable (Rutqvist, 2011). To do so, access to the Jacobi matrices of FEFLOW would be necessary (compare Forsyth et al. 1995), which is not the case. Future work will be done to test an alternative procedure shown by Matthies and Steindorf (2001). The development of a stand-alone open source solution including a separate FEM flow computation is planned for the future and would allow an iterative coupling using the Newton or Picard method.

is MPa); have to be assigned to an area of horizontal three or vertical four nodes. Nodal – “DirichletX”: The boundary condition for the displacement in the x-direction (unit is m). Nodal – “DirichletY”: The boundary condition for the displacement in the y-direction (unit is m). Nodal – “DirichletZ”: Boundary condition for the displacement in z-direction (unit is m). Elemental “Parameter1”: Lamés first parameter λ (unit is MPa). Elemental “Parameter2”: Lamés second parameter μ (unit is MPa).

For generating VTK outputfiles which can be used to display the deformed 3D mesh a time series with the number 9999 has to be created and has to contain the times for which an output is wanted. For more complex Neumann boundary conditions and additional settings an input file (named settings.dat, stored in the FEFLOW Folder structure import þexport) can be defined. FEFLOW and also the presented plug-in can be run under Microsoft Windows and Linux (32 bit and 64 bit). However, only a Windows version is currently available. Developers wanting to build a Linux version should contact the corresponding author in advance for advice. The current version of ffp and mcf is developed based on the Intel c þ þ compiler and the Intel MKL library. However, a change to gþ þ and/or the GSL library is possible without too much effort.


The usage of other free libraries for linear algebra, like for instance Lis (Nishida, 2010) is planned for future releases.

Acknowledgements The first author was partly funded for this project by the German Science Foundation – DFG (RU 1837/1-1). We thank three anonymous reviewers for valuable comments which helped to improve this paper.

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