Modelling of the hydro-mechanical response of sedimentary rocks of southern

Modelling of the hydro-mechanical response of sedimentary rocks of southern Ontario to past glaciations

O.Nasir a ,M.Fall a ,?,T.S.Nguyen a ,b ,E.Evgin a

a Department of Civil Engineering,University of Ottawa,Canada b

Canadian Nuclear Safety Commission (CNSC),Canada

a b s t r a c t

a r t i c l e i n f o Article history:

Received 17July 2010

Received in revised form 27June 2011Accepted 10July 2011

Available online 12August 2011Keywords:

Past glaciations

Hydro-mechanical coupled processes Deep geological repositories Sedimentary rocks Modeling

Nuclear waste

Glaciation is considered as one of the main natural processes that can have a signi ?cant impact on the long term performance of a deep geological nuclear waste repository (DGR)located in the Northern Hemisphere.The northern part of the American continent has been subjected to a series of strong glaciation and deglaciation events over the past million years.The last glacial cycle in the Northern Hemisphere started approximately 110,000year ago.During that cycle,southern Ontario was buried under a continental ice sheet,with a maximum thickness of up to 3000m at about 20,000years ago.The ice cap retreated approximately 10,000years ago.However,?eld data from deep boreholes in sedimentary rocks of southern Ontario show anomalous pore water pressure including underpressure and overpressure zones.In this paper,a large-scale coupled hydro-mechanical (HM)model is developed to investigate the hydro-mechanical (HM)response of the sedimentary rocks of southern Ontario to past glacial cycles.Particular emphasis has been placed on the evolution of pore water pressures and surface displacements.The HM model is veri ?ed using analytical solutions.The results of the large-scale HM modeling study shows that the past glaciation,particularly the second cycle (22,000apb)had signi ?cant impact on the pore water pressure gradient and effective stress distribution in the sedimentary rocks of southern Ontario.Furthermore,good agreement between the large scale modeling results and anomalous pressures leads us to the conclusion that these anomalies could be glacially induced.The results of this research can provide information that will contribute to a better understanding of the impact of future glaciations on the long term performance of DRGs in sedimentary rocks.

?2011Elsevier B.V.All rights reserved.

1.Introduction

The climate of the Earth is a dynamic system due to its response to external and/or internal forcing mechanisms,such as orbital forcing,tectonic,volcanic,oceanic circulation,atmospheric circulation,and anthropogenic activities (Broecker et al.,1985;Winograd et al.,1988;Maslin and Christensen,2007),which result in dynamic climate outputs of temperature and precipitation.For this study,the main interesting outputs of the climate change are the long term temporal and spatial variation in precipitation and temperature,and in particularly,glaciation –deglaciation.

In the past million years,periodic advance and retreat of major continental ice sheets occurred in many parts of the world,and in particular,the Northern Hemisphere.Geomorphological evidence,including a large number of end moraines,have been used to form the

basis for the reconstruction of the deglacial chronology and the extent of Quaternary glaciers and ice sheets in North America (Sibrava et al.,1986).Models of ice sheets showed that the estimated maximum ice thickness was about 3000m in southern Ontario during the last glacial period (Boulton et al.,1985),with predominant NW –SE direction of ice movement.Glacial cycles are characterized by strong climatic variations with short but intensely cold periods followed by the formation of continental ice sheets,which are responsible for a signi ?cant change in the topography and groundwater regime (Vidstrand et al.,2008).

A key interest for nuclear waste is the long term stability of both the geosphere and engineered components of the geological disposal system (Vidstrand et al.,2008).Climate change is predicted to occur cyclically within the life span of repositories,resulting in the formation of continental ice sheets (Ericsson et al.,1994).The cyclic advance –retreat of these ice sheets is considered as the main natural processes that can signi ?cantly affect deep geological repository (DGR)systems (Nguyen et al.,1993;Chan et al.,2005).Glaciations induce modi ?cations to the thermal,mechanical,hydraulic and chemical conditions at the earth's crust,potentially causing thermo-hydro-mechanical-chemical (THMC)changes at depths where a DGR could be located.The stability of the DGR system can be affected by glaciation –deglaciation cycles.

Engineering Geology 123(2011)271–287

?Corresponding author at:Department of Civil Engineering,University of Ottawa,161Colonel by,Ottawa (Ontario)Canada K1N 6N5.Tel.:+16135625800#6558;fax:+16135625173.

E-mail address:mfall@uottawa.ca (M.

Fall).0013-7952/$–see front matter ?2011Elsevier B.V.All rights reserved.doi:

10.1016/j.enggeo.2011.07.008

Contents lists available at SciVerse ScienceDirect

Engineering Geology

j o u rn a l h o m e p a g e :w ww.e l s evi e r.c o m /l o c a t e /e n g g e o

Advance and retreat of the ice sheets results in a mechanical loading–unloading sequence at the surface that perturb the stress and strain and water pressure regimes in the host rock at depths.Meltwater produced at deglaciation can also potentially in?ltrate the host rock to depths determined by its permeability(e.g.,Hansson et al.,1995).

Modeling the future evolution of a repository site,with emphasis on how this evolution affects the repository safety functions,is a key component of repository performance and safety assessment(SA) (Tsang et al.,2009).The THMC processes are coupled,and can be analyzed by using numerical modeling.

During the last two decades,considerable attention has been given to THMC coupled processes in rocks.These efforts are mainly driven by the concern over the role of such couplings in the performance and safety assessment of heat-releasing nuclear waste repository in the subsurface(Tsang et al.,2009).In addition,THMC coupling has a wider application in geosystems,such as geothermal energy extraction,gas production and others.The?rst studies were performed on binary couplings such as HM.However,for the performance assessment of the repository,it is essential to study the full triple interactive THM coupling,and also THMC coupling.

Internationally,two main cooperative projects dealt with coupled THMC processes:the DECOVALEX project(abbreviation for the international co-operative project for the development of coupled models and their validation against experiments in nuclear waste isolation)and the BENCHPAR project,sponsored by the European Commission(EC)(Chan et al.,2005;Vidstrand et al.,2008).In the above two projects,glaciation effects have been assessed for Scandinavian and Canadian granitic rock formations.

The safety assessment of DGRs,particularly the transport of radionuclides,requires knowledge of groundwater?ow(hydraulic process)(Rasilainen et al.,1999),which is considered the main agent responsible for radionuclide migration.On the other hand,signi?cant mechanical processes,represented by considerable pressure due to ice loading,play an important role in the hydraulic process.The main objective of the present study is to build a hydro-mechanical(HM)model for the area of southern Ontario to perform coupled HM modeling which can provide the necessary information for the assessment of the in?uence of ice loading on the groundwater regime in sedimentary rocks in southern Ontario.The intended use of the model is to assess the impact of future glaciations on the performance of a DGR in sedimentary rocks in southern Ontario.However,before such an assessment could be performed with con?dence,the model should be used to interpret the effects of past glaciation cycles in light of the hydraulic data obtained from regional studies and existing site-speci?c information.Therefore,the effect of past glaciations is the focus of this paper.

In the?rst section of the paper,we will provide a description of the characteristics of the study area.In the second section,we will show the development of the relevant partial differential equations(PDEs) related to the HM coupled processes.In the third section,the modeling approach for site speci?c conditions is described.The fourth section presents some signi?cant simulation results of the effect of past glaciations on the main processes.Finally,the conclusions and recommendations are presented.

2.Characteristics of the study area

The study area is located in southern Ontario,near the Eastern margin of the Michigan Basin,on the eastern side of the Huron Lake as shown in Fig.1.A two dimensional(2D)model domain encompasses a cross section that is approximately520km in width and1.6k m in depth with the Michigan Basin as the dominant part.A DGR for low and intermediate level nuclear wastes is being proposed at a depth of approximately680m in an argillaceous limestone formation within the study area.A multi-year site investigation program is being conducted at the site of the proposed DGR,consisting of seismic surveys,and a series of deep vertical and inclined boreholes,and hydraulic,petro-graphic,geochemical and mechanical testing performed in-situ and in the laboratory(e.g.,Gartner Lee Limited,2008;Jensen et al.,2009

).

Fig.1.Location of the study area.

272O.Nasir et al./Engineering Geology123(2011)271–287

2.1.Geology and hydrogeology of the study area

A large amount of data related to the geology of southern Ontario is available,including existing published literature,government open ?le reports,etc..Mazurek (2004)and Gartner Lee Limited (2008)compiled signi ?cant geological data for the study area which is situated on the Western margin of the Michigan Basin,consisting of Paleozoic sedimentary formations overlying the Precambrian basement (Figure 2).The geology of the study area (Gartner Lee Limited (2008)seems to be characterized by continuous and predictable stratigraphy as shown in Fig.3.Structurally,the study area is characterized as simple,with no active faults (Gartner Lee Limited,2008).The Michigan Basin started to form more than 450million years ago,during the Paleozoic era.In general,sedimentary rock formations in the Michigan Basin can be typically characterized by:carbonates,shale,limestone,evaporate and sandstone which are located above the pre-Cambrian crystalline basement.The proposed DGR would be located in a Middle Ordovician limestone overlain by approximately 200m of Upper Ordovician shale and an additional 190m of argillaceous dolostones and evaporates of the Upper Silurian Salina Group as shown in Fig.3.The thicknesses and dips of each layer within the Paleozoic formation are variable with increasing thicknesses towards the southwest direction until a maximum total thickness of 4500m is reached at the center of the Michigan Basin and decreasing thicknesses towards the Algonquin

Arch.

Fig.2.Geology of southern Ontario (Source:Sykes et al.,2008

).

Fig.3.Cross section through the Michigan Basin which shows rock formations and location of the proposed hypothetical DGR (vertical is depth).

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Hydrogeologically,the study area includes three main horizons:shallow,intermediate and deep groundwater zones (Sykes et al.,2008).The intermediate and deep zones are mainly characterized by high total dissolved solid of up to 300g/L (Sykes et al.,2008).Moreover,?eld measurements of pore water pressure showed anomalous water pressure distributions with respect to hydrostatic,including under pressure within the Middle and Upper Ordovician formations and over pressure within the Cambrian formation (Jensen et al.,2009).

In this study,and for the purpose of a two dimensional model construction,a geological cross section A –B is constructed as shown in Figs.2and 3.The location and direction of the cross section is selected

based on the NW –SE direction of the ice sheet advance and retreat during the last ice ages which was observed through geomorphologic records (Piotrowski,1987).The cross section is selected as 520km in long and 1653m deep in order to include most of the north east part of the Michigan Basin,and in particular,the formations outcrop up to the Pre-Cambrian basement.2.2.Material properties

Hydraulic and mechanical properties,speci ?cally,hydraulic conductivity,elastic modulus and Poisson's ratio of the

rock

Fig.4.Variation of hydraulic conductivity with

depth.

Fig.5.Variation of elastic modulus with depth used in the model.

274O.Nasir et al./Engineering Geology 123(2011)271–287

formations within the model as shown in Figs.4and 5are collected from the literature and the preliminary site investigations (Lo,1978;Mazurek,2004:Jensen et al.,2009)and used as material property input data for the mathematical model.These numbers represent the average value of the above properties.

Fig.4shows the pro ?le of the horizontal hydraulic conductivity (K h )at the study area.Based on the numerical value of K h ,three main levels can be identi ?ed:the ?rst level (0–600m in depth)which includes the Devonian and Silurian formations with a K h range from 10?11m/s to 10?9m/s,the second level (600–900m in depth)which includes the Upper and Mid-Ordovician formations with a K h range from 10?14m/s to 10?12m/s,and the third level (900–1000m in depth)which includes the Cambrian formation with K h about 10?8m/s.

Fig.5shows the pro ?le of the modulus of elasticity (E)at the study area.Based on the numerical value of E,three main levels can be recognized:the ?rst level (0–650m in depth)which includes the Devonian and Silurian formations with an E ranging from 20to 25GPa,the second level (650–900m in depth)which includes the Upper and Mid-Ordovician formations with an E of about 35GPa,and the third level (900–1000m depth)which includes the Cambrian formation with an E of about 10GPa.An estimated value of 0.2was used for the value of Poisson's ratio based on the available published ?eld data (Lo,1978;Mazurek,2004).

3.Model development 3.1.Introduction

In this work,hydraulic (H)and mechanical (M)processes are presented.Each one of those processes can be the “agent ”that affects the other processes,which are the “objects ”,for example:

Agent ?Object ,H (Agent )?M (Object )will be Hydro-Mechanical (HM)coupled processes,which represent mechanical processes affected by hydraulic processes,while:M (Agent )?H (Object )are MH coupled processes and for that,MH ≠HM.In general,the total number of coupling =n (n ?1),where n=number of processes.

The inclusion of more processes in a model would in theory lead to a closer representation of reality but would substantially increase its complexity as shown in Fig.6.Moreover,this increase in complexity would increase the level of input data requirements and also the reliability of these data.In practice,judgment is required to decide what level of complexity is needed as “it is important to judge whether a given process has relevance to the repository performance and whether increasing the complexity of characterization and modeling is actually required ”(Hudson et al .,2005).In this work,fully coupled Hydro-Mechanical (two-way coupling for both MH and HM)are taken into account.

The hydraulic process represented by the groundwater ?ow is considered as one of the important processes in the safety analysis of a repository system which can be modeled using Darcy's law (Hudson et al.,2005).On the other hand,the mechanical process mainly includes rock deformation and rock stresses.The coupling of H –M processes by determining the transient response of H –M state variables (pore water pressure,effective stress and deformation)has been considered as potentially signi ?cant by most research groups (Hudson et al.,2005).Continuum mechanics that use a macroscopic scale has been commonly adopted and applied to solve partial differential equations (PDEs)derived from conservation principles in porous media (De Marsily,1986).In addition to those PDEs that capture the fundamental physical laws of conservation of mass,momentum,and energy,mathematical equations called constitutive relationships also have to be derived based on the speci ?c experimental behaviour of the porous media under consid-eration (stress –strain relationship,Fourier's law of heat conduction,Darcy's law of pore ?uid ?ow,etc.)(Nguyen,1995;Gatmiri

and

Fig.6.Modeling complexity and acceptable

error.

Fig.7.Ice loading (interpolated from The University of Toronto Glacial Systems Model (GSM)“Peltier's Model ”model nn9930(Peltier,2008)).

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Delage,1997;Yang,2005).Numerical methods such as ?nite difference,boundary element or ?nite element methods (FEMs)are usually employed to solve the partial differential equations with corresponding boundary and initial conditions.In this work,the FEM code COMSOL Multiphysics is used to model HM coupled processes in sedimentary rock in southern Ontario due to past cycles of glaciations.

COMSOL Multiphysics is a well known commercial software for solving https://www.wendangku.net/doc/23771424.html,SOL Multiphysics performs various types of analysis including:stationary and time-dependent,linear and non-linear,eigen-frequency and modal analysis by using the ?nite element method (FEM)with adaptive meshing and error control capabilities and with a variety of numerical solvers (COMSOL,2009).

3.2.Model conceptualization

The host rock could be conceptualized as a porous medium with a solid skeleton and pores,cracks and microcracks ?lled with a ?uid

such as water (the porewater)or a mixture of ?uids (water,gas and/or air).When an ice sheet forms on the surface of the host rock,it imposes a mechanical load that attained maximal values of the order of 30MPa in the last glaciation cycle.According to classical poromechanics theory,that load is shared between the solid skeleton and the pore ?uid in a transient manner.At early stages,most of the load will be taken by the pore ?uid resulting in an increase in pressure and a change in hydraulic gradients.That change in pressure gradients result in pore ?uid redistribution at a rate proportional to the porous medium permeability,according to Darcy's law.Simultaneously with pore ?uid redistribution,the solid skeleton starts to gradually assume parts of the imposed load,and would deform in response to the stress changes.The ice sheet,in addition to the mechanical load due to its weight,would also affect the thermal and hydraulic conditions at the interface between its base and the surface of the host rock.Permafrost conditions might prevail at that interface,at the forefront of the advancing ice sheet.Variable hydraulic conditions can also exist at that interface,that can vary between a “wet ”base case,where a perched water table exist throughout the thickness of the sheet,and a “dry ”base scenario where the interface could be free draining.The model described in this paper is developed based on the above conceptualization,within the framework of poromechanics.In addition,we adopt the following assumptions:

1-Ice loading and surface water pressure due to the last glaciation –deglaciation cycle is generated by the University of Toronto Glacial Systems Model (GSM)“Peltier's model ”(Peltier,2008)as shown in Fig.7.Different loading scenarios are used to apply the top boundary conditions of the model;

2-Permafrost is ignored at the base of the ice sheet and the effect of temperature is not included in this phase of the study,and;

3-The main PDEs are developed to include the HM processes in porous media based on the conservation of mass (solid and ?uid)and momentum for porous media fully saturated with water.This assumption is based on the large scale of the domain which is mainly located under the ground water table.However,it should be mentioned that recent results from the site investigation program has indicated that a separate gas phase might be present in the host rock.The effect of potential gas phase will be implemented in a further study.4)One-dimensional (1-D)and two-dimensional

(2-D).

Fig.8.Ice sheet parabolic

distribution.

500100015002000250030003500

40004500

Distance with respect to ice sheet direction (km)

I c e t h i c k n e s s (m )

Fig.9.Ice sheet parabolic distribution with time (kabp:in kilo year before present)with respect to study area.

276O.Nasir et al./Engineering Geology 123(2011)271–287

Fig.7shows the time history of the normal stress on the ground surface due to the ice during the past 70,000years for sites A and B,both located on the selected geological cross section A –B for the study area.Along the line A –B,the pro ?le of the ice sheet load is a parabolic shape with increasing height towards the northwest direction as shown in Fig.7for the time of 19,500abp (annum before the present).

The direction of the line A –B will be assigned an assumed x-coordinate parallel to the direction of ice sheet movement,as shown in Fig.8.In order to take into account the actual parabolic shape of the ice load distribution with respect to time and location in southern Ontario,a steady-state ice sheet on a horizontal bed is assumed with a parabolic ice sheet distribution (as shown in Fig.7,modi ?ed after

Paterson (Paterson,1994))which can be represented by the following equation:h =3:4L ?x eT

1=2

e1T

This equation can be used to estimate the length of the covered area (L,in the direction of the ice sheet movement)beyond the highest thickness h point (at x =0):

L =

h 211:

56

Fig.10.Two dimensional model

mesh.

Fig.11.Initial pore water pressure at time 70,000year before present.

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The history of the maximum thickness of the ice used to estimate L is adopted from the Peltier's model (Peltier,2008).Fig.9shows the pro ?le of the ice thickness for six selected times (63,000abp to 14,000abp).In this ?gure distance is measured along line A –B,with distance=0and 375km at A and B,respectively.

It should be emphasized that the aforementioned GSM or Peltier's model (1988)incorporates a number of interacting components including:a 3D thermo-mechanically coupled ice-sheet model that

includes a model of sub-surface thermal evolution,a representation of fast ?ow due to subglacial till-deformation,a model of visco-elastic bedrock response,a surface mass-balance module,an ice calving module,and ?nally a fast dynamical melt water surface routing and storage solver (Tarasov and Peltier,2004).The most important element is the deterministic model of continental ice sheet evolution (Tarasov and Peltier,1999,2002,2004,2006).Peltier (2008)executed the GSM over the past 120ka in North America with a 1.0°longitude by 0.5°latitude grid resolution in order to produce a data set for surface elevation,ice sheet thickness,relative sea level and subglacial melt rate.The GSM model has a large number of parameters,many of which are well known.Many parameters are “ensemble parameters ”that lie within a given range.The ensemble parameters have been undergoing calibration for North American deglaciation against a large set of observational constraints,including,a large set of high-quality relative sea-level histories,a space geodetic observation of the present-day rate of vertical motion of the crust from Yellowknife and a traverse of absolute gravity measurements from the west coast of Hudson Bay southward into Iowa (Tarasov and Peltier,2004).In the study area,the Laurentide Ice Sheet covered the region during the Last Glacial Maximum (around 18,000years BP).By 14,000years BP,the ice sheet had rapidly https://www.wendangku.net/doc/23771424.html,rge proglacial lakes formed in front of the retreating ice sheet.During the advance and retreat of the ice sheet,ice-induced hydraulic loading at the ice/bed interface is in ?uenced by many factors,including temperature and melting pressure,surface transmissivity,surface geometry,glacial lake,and tunnel channel system utilization (e.g.,Brennand et al.,2006;McIntosh and Walter,2006).

3.3.Initial conditions

For the 2D model shown in Fig.10,the initial hydraulic conditions are set as linear hydrostatic as for the initial condition,and at the same time,the self-weight of the rock formations is assumed to be based on a rock density of 2500kg/m3for the mechanical initial stresses at 10million years before the present.The above time is selected after several trials for achieving self weight mechanical and hydraulic equilibrium.At the time of equilibrium,both hydraulic pressure and effective stresses were used as initial conditions for the analysis before the application of ice loading.Fig.11shows the pore water pressure

Table 1

Material properties for the validation model.Parameters

Value Hydraulic conductivity (m/s)2E ?8Initial water pressure (Pa)0Modulus of elasticity (Pa)4E7Poisson ratio

0.3

Fig.12.Boundary conditions for the veri ?cation 1D

model.

https://www.wendangku.net/doc/23771424.html,parison between modelling results (COMSOL)and Terzaghi's analytical solution.

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distribution used as the initial pore water pressure at the start of the last glaciation cycle.3.4.Boundary conditions

The advance and retreat of ice sheet caused transient mechanical and hydraulic boundary conditions.Boulton et al.(2009)suggested that there is a coupling between subglacial channels and groundwater which play an important role in the subglacial hydraulic pressure regime,and thereby the glacier dynamic regime.Boulton et al.(2007a,b)developed a channel-groundwater ?ow theory.In this theory,it is suggested that water pressure along tunnels is low and predictable,which draws groundwater to ?ow toward tunnels.In addition to that,surface melted water drainage and pressure represented by the frequencies of eskers are affected by surface rocks,in which there are higher frequencies for the shield area as compared to younger sedimentary rocks (Clark and Walder,1994;Boulton et al.,2009).In contrast,evidence from Quaternary glacial environments supports the cold base glacial theory.Lloyd Davies et al.(2009)work addressed the behaviour of cold based glaciers and ice sheets.In this study,three scenarios of surface water pressure are considered as explained below.

In this work,the hydraulic boundary conditions are set as no ?ux at both sides and the bottom.On the other hand,to cover the diversity in theories,spatial and temporal variation of subglacial pore water pressure (e.g.,Boulton et al.,2007a,b;Jansson and N?slund,2009),

Formation Pressure (kPa)

D e p t h (m B G S )

0100200300400500600700800900

010*******

400500600700800900

02000

4000

6000

8000

10000

D e p t h (m B G S )

Formation Pressure (kPa)

DGR3COMSOL (1D)Hydrostatic (0 TDS)

a

b

https://www.wendangku.net/doc/23771424.html,parison of experimental ?eld surface).a.)Free draining conditions.b.)Comparison of three hydraulic boundary at the surface (1/3p,);and (iii)hydraulic head equal to 80%of the ice thickness at the surface (80%p,).Kv:vertical hydraulic conductivity;Kh:horizontal permeability.

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two cases of the surface hydraulic boundary conditions are assumed.The ?rst case is a zero pressure hydraulic boundary conditions,and the second case is a transient condition as a function of the ice loading history.However,in this paper we mainly focus on the ?rst case as shown in Fig.10.The stress at the ground surface due to ice is directly adopted from the main outputs of the (GSM)(Peltier,2008)and the geometry of water pressure at the ice/bed interface is assumed to be linearly related to the surface stress and taken as a fraction of the stress at surface with different value of fraction (0,1/3and 80%)to cover a wide range of potential ice/bed interface boundary conditions results presented in Fig.14for 1D model.

For the mechanical conditions,roller boundary conditions are assumed for the two sides and bottom and free deformation at the surface.Transient normal stress which represents the ice load is applied at the surface as shown in Fig.10.

3.5.Finite element discretization

A ?nite element mesh is generated by dividing the rectangular domain of 520km×1653m into 3539elements using Lagrange-quadratic triangular elements.These types of elements are chosen for

their irregular subdomain shapes.The mesh for the 2D model is shown in Fig.10.

3.6.Mathematical formulations

The governing PDEs are derived from the consideration of conservation of mass and momentum.In the following,Eqs.2and 3express the conservation of mass for both ?uid and solid,respectively,which can be written as (De Marsily,1986):?·ρf U f +?ρf n +ρf q =0

e2T

?·ρs U s eT+

?

?t ρs

1?n eTeT+ρq =0e3T

where:ρis density,U are ?ctitious velocities,t is time,n is porosity,q is mass source,s is the solid and f is the ?uid.In the above equations,the mean velocities for ?uid and solid can be de ?ned as:u =U n ,and u s =U s

Fig.15.Surface displacement history at the location of the study

area.

Fig.16.Pore water pressure history at the Proposed DGR level.

280O.Nasir et al./Engineering Geology 123(2011)271–287

Darcy's law can be expressed in terms of the mean velocities as:u ?u s eT=?

κη

?p +ρf g ?D

e4T

where:κis permeability,ηis dynamic viscosity,p is pressure,D is the direction of gravitational acceleration (g).

Combining Eqs.2and 3,and using Darcy's law we obtain (De Marsily,1986):

?·ρf κη?p +ρf g ?D

!=n ?ρf +ρf ?n ?ρn ρs d ρS e5T

Based on the compressibility of ?uid,solid and skeleton of the rock

components,the term ?n

?t

(time variation in porosity)can be

represented with:dn

dt

=α?n eTde ff dt

+

α?n eTK s dp dt

+?α?n eTβ+1?n eTβ?βS eTeTdT

dt

e6T

The ?nal equation of pore ?uid and solid mass conservation is:?·ρf κη?p +ρf g ?D !=n γeT?C ?t

+ρf α0de ff

+ρf eα0s ?n S +n K f T

dp dt

+ρf en βS ?α0

β+β?βS eT?n βf T

dT

e7Twhere α0=α?n eT1?n eT,α=1?K D

K S ,K D ,K S and K f are the bulk moduli of the solid matrix,solid grains and water ?uid,respectively,β,βs and βf are

thermal expansion coef ?cients for the solid matrix,solid grains and

water ?uid,respectively.

Eq.7includes the concentration (C)of dissolved solids in the pore ?uid and the local average temperature (T)of the porous medium.The density of the pore ?uid is assumed to vary with dissolved solid concentration according to the following equation:ρf =ρfo +γC

e8T

where ρfo is the initial ?uid density,and γis a concentration –density

coef ?cient.The numerical values of both ρfo and γare taken to be 1000kg/m 3and 2/3for a range of C from 0to 300kg/m 3(Sykes et al.,2008).

The total dissolved solid in the pore ?uid is assumed to be able to migrate by advection,dispersion and molecular diffusion mechanisms.With the consideration of mass conservation for the total dissolved solids and the above transport mechanisms,the governing equation of transport can be derived as follows (e.g.Fetter,1999):θs

?c

?t

+?·?θs D L ?c +u c ? =S c e9T

where:θs is porosity;D L is the hydrodynamic dispersion tensor;u is vector of pore ?uid velocities;and S c is the solute source.

Assuming isotropic linearly elastic rocks and Biot's effective stress principle,the equation of momentum conservation,becomes (see e.g.Nguyen,1995):

G ?2u i j j +G +λeT?2

u j i j ?α?p i ?βK D ?T

i

+F i =0e10

T

Fig.17.Pore water pressure history at different depths.

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where:u is the displacement,G is the shear modulus,λis Lamé's?rst parameter,αis Biot coef?cient,and T is the temperature.

Mechanical deformation is an important process because it can affect the porosity and the intrinsic permeability,hence the hydraulic conductivity of sedimentary rocks.In this work,the change in intrinsic permeability due to change in porosity is modeled using the Carman–Kozeny relationship(Kozeny,1927;Carman, 1937):

k t=

n3

1?n2

àá

"#

t1?n2

n3

2

4

3

5

initial

k initial

where k t[L2]represents permeability at time t,and k initial [L2]represents the initial permeability.In this work,initial poro-sity and permeability are assumed to be equal to the current?eld values.

3.7.Model testing for con?dence building

Before a numerical model could be used for predictive purposes, con?dence must be gained in its adequacy to accurately solve the equations it is supposed to solve,and also to adequately capture the main physical processes that occur in reality.The?rst activity,called veri?cation,is usually performed by comparing the numerical results with the results of analytical solutions when they exist.The second activity is sometimes called validation,however there is a tendency to avoid that term in the geoscienti?c community,since no model can exactly replicate the real world.We will show here the results of the above two activities.

Time=65,000 year before present MPa Time=63,000 year before present MPa

Time=59,000 year before present MPa Time=55,000 year before present MPa

Horizontal distance (m)

Time=55,500 year before present MPa Time=40,000 year before present MPa

Fig.18.Pore water pressure history and distribution of65,000to40,000years before the present for part of the Michigan Basin.

282O.Nasir et al./Engineering Geology123(2011)271–287

3.7.1.Veri ?cation with analytical solution

For that veri ?cation purpose,the results obtained from the model are compared with the analytical solution for one and two dimensional consolidation equations by Terzaghi (1943).

u 0z ;t eT=Δσv ∑m =02sin M z exp ?M 2T v :::8eTM =π2m +1eT=2;and T v =c v t H

where:u ’is the pore water pressure,Δσv is the change in vertical stress,z is depth,H is drainage path length and c v is the coef ?cient of consolidation.

Table 1shows the material properties and initial conditions for the veri ?cation model,while Fig.12shows the hydraulic and mechanical boundary conditions.As explained in Section 3.3,the assumed boundary conditions are:zero ?ux for the bottom and sides,zero

pressure for the surface,roller (mechanical)for the bottom and sides,and free deformation for the surface.The results obtained by using the developed HM model (COMSOL)show a very good agreement with the analytical solution proposed by Terzaghi (1943)as indicated in Fig.13.

3.7.2.Simulation of present day pore water pressure pro ?les

A 1-D HM model,using the input data pro ?le as shown in Figs.4and 5is used to simulate the present pore pressure pro ?le in response to the past glaciation cycle.The calculated pressure pro ?le is compared with ?eld pore water pressure pro ?le (Jensen et al.,2009)as shown in Fig.14a.Despite some differences in modeling

Time=30,000 year before present MPa Time=24,000 year before present MPa

Time=19,500 year before present MPa Time=14,000 year before present MPa

Time=11,500 year before present MPa Time=0 year before present (Now) MPa

Horizontal distance (m)

V e r t i c a l d i s t a n c e (m )

Fig.19.Pore water pressure history and distribution 30,000years before the present to now for part of the Michigan Basin.

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(COMSOL)results and ?eld data at a greater depth,good agreement is achieved for the depth range of 0to 700m which includes the location of a potential DGR.These results are consistent with the

?eld measurements.The differences in results may be due to some assumptions adopted in this work,such as a linear elastic model,homogenous and isotropic materials in addition to uncertainties in geological and glacial data.

The measured ?eld pore water pressure is also compared with the modeling results for the three potential cases of surface hydraulic boundary conditions as discussed in Section 3.2:(i)free draining conditions (0p,Figure 14b);(ii)hydraulic head equal to 30%of the ice thickness at the surface (1/3p,Figure 14b);and (iii)hydraulic head equal to 80%of the ice thickness at the surface (80%p,Figure 14b).The comparison results are shown in Fig.14b.From this ?gure,it can be observed that the 0p shows an underpressure in the Ordovician that is closest to the ?eld data,however,no overpressure is predicted for the Cambrian.For the 1/3p case,the underpressure in the Ordovician formation is predicted,however it is much lower than the measured one.The 1/3p case calculates overpressure in the Cambrian which is consistent with the ?eld measurements.The 80%p case overestimates the pressure,especially in the Ordovician.

4.Simulation of the hydro-mechanical response of the study area The HM model is used to simulate the impact of the past glacial cycle on the hydraulic and mechanical responses of sedimentary rocks by using one and two dimensional

models.

Fig.20.Pore water pressure pro ?le history (65,000years before the present to now with the uppermost line at 19,500years before the present).

Time=65,000 year before present MPa

Time=63,000 year before present MPa

Time=59,000 year before present MPa

Time=55,000 year before present MPa

Horizontal distance (m)

Time=55,500 year before present MPa Time=40,000 year before present MPa

V e r t i c a l d i s t a n c e (m )

Fig.21.Effective stress history and its distribution over 65,000–40,000years before the present for part of the Michigan Basin.

284O.Nasir et al./Engineering Geology 123(2011)271–287

Some selected results are presented,particularly the time evolution of surface displacement and water pressure pro ?le.Fig.15shows the surface displacement under the impact of the past glaciations cycle.Two main episodes of loading –unloading,with peaks at approximately 60and 20kybp (Figure 7),can be detected;each one is mainly characterized by the shape of surface loading with a maximum surface displacement of about 1.2m.This displacement is only due to the mechanical deformation and consolidation of the rock formations relative to the base of the model.It does not include the ?ow of the mantle underneath the earth crust,which contributes to the majority of the absolute displacement of the earth surface.It can be seen that the surface displacement is consistent with the ice loading history (Figure 7).This is a result of fast consolidation as compared to the time of loading.

The variation of water pressure with time at a depth of 680m (the same depth as the proposed DGR)is presented in Fig.16.Two peaks are noticed which are related to each glaciation episode with a slight drop in water pressure after the second ice unloading (around 11,000years before the present).That drop is induced by an elastic rebound.It takes a signi ?cant amount of time for the pressure drop induced by unloading to recover.The model predicts that the pressure at 680m in depth is slightly lower than the hydrostatic pressure.In general,the shape of the time history of pore water pressure at a depth of 680m is still similar to the ice loading history (Figure 7)due to the fast consolidation as compared to the loading rate.The HM response of the host rock is mainly affected by two main factors,?rst the location within the host rock with respect to its surface boundary,and the second factor is the hydro-mechanical properties.Fig.17shows the pore water pressure history at different depths.At a depth of 650m where the hydraulic conductivity is very low;the hydraulic response is characterized by a signi ?cant drop in the pore water pressure following the unloading stage.This prediction is consistent with ?eld measurements from boreholes at the site which shows an under-pressure zone at the same level.However,the value of the predicted underpressure at present time is smaller than the measured ones.Differences between the measured and predicted values could be related to the assumed boundary conditions,assumed homogeneity in material properties and more sources of pore pressure change (such as chemical reaction,and gas pressure).

Time and space transient mechanical conditions,represented by ice sheet loading (shown in Figure 9)will mainly in ?uence the groundwater ?ow.This change can be presented in the form of water pressure changes.Figs.18and 19show the predicted changes in water pressure for selected times during the last 65,000years.

It can be seen that the maximum water pressure is accrued at the time around 19,500years before the present (Figure 19)at the Cambrian formation,with an excess water pressure of about 15MPa,causing a pressure of up to 25MPa at a depth of 1000m.However,the water pressure at the upper formation (Devonian)has a higher

Time=30,000 year before present MPa Time=24,000 year before present MPa

Time=19,500 year before present MPa

Time=14,000 year before present MPa

Time=11,500 year before present MPa Time=0 year before present (Now) MPa

Horizontal distance (m)

V e r t i c a l d i s t a n c e (m )

Fig.22.Effective stress history and its distribution over 30,000years before present to now for part of the Michigan basin.

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dissipation rate due to its location near the free draining surface,as well as a relatively high hydraulic conductivity of 10E ?8m/s.Starting from 14,000ybp,the pore water pressure distribution starts to return to the hydrostatic conditions,with some underpressure and over-pressure that are still evident as shown in Fig.19.

Fig.20shows the pro ?le of water pressure with depth for different times (65,000years before the present to now).In this ?gure,pressure dissipation is relatively fast in the upper 600m.In addition to that,under pressure due to unloading accrued at a depth of 600m to 700m,particularly at the Upper Ordovician formation with a maximum value of 2.5MPa,as compared to a hydrostatic pressure of 6.4MPa (about a reduction of 4MPa).At the same time,the deeper parts of the Cambrian formation have an excess water pressure of about 10MPA (causing a pressure increase from 17.5MPa to 27.5MPa).The underpressure and overpressure is consistent with ?eld observations.The persistence of these anomalous pressures for more than ten thousand years could only be possible with very low permeability of the Ordovician layers that results in low rates of hydraulic dissipation.

In addition to its effect on the hydraulic regime,large normal stress due to ice loading also contributes to signi ?cant changes of effective stress.Figs.21and 22show the normal effective stress distribution and its development with time (65,000years before present to now).In general,the trend of normal effective stress follows the trend of loading and unloading,except at the early stages of loading,where the effective stress decreases as a result of increase in excess water pressure.However,some formations have a different response than others due to different poroelastic response;for example,the Devonian formation will have a quick response as compared to the Ordovician formation due to the differences in both hydraulic conductivity and modulus of elasticity.

Two distinctive layers can be identi ?ed from Fig.22:the ?rst is the Mid-Ordovician with lower effective stress due to higher excess pore pressure,and the lower Silurian with higher effective stress due to fast consolidation.

Fig.23shows the change in effective stresses at the level of the proposed DGR.It can be seen that the general trend of change is similar to the trend of loading (Figure 7).However,at the DGR level,a small amount of effective stress reduction is observed during the early time of loading (65,000and 23,000years before the present)due to high excess of pore water pressure at the early loading stages,and then,the evolution of the effective stress starts to follow that of the ice load as the time of loading is much longer than the time of excess pore

water pressure dissipation.On the other hand,at a depth of 100m,the change in effective stress is about three times the change in effective stress at a depth of 680m.This result could explain the low RQD values recorded at shallow depths (?rst 200m)in the study area and high RQD values (almost 100%)at higher depths,thereby providing an additional argument for long term safety under the impact of future glaciations.5.Conclusions

In this paper,a numerical study for the HM processes associated with past glaciation cycles in sedimentary rocks in southern Ontario using a 2D model has been conducted.The main HM coupled equations are derived from the conservation of mass and momentum,coupled with Darcy's law for pore water ?ow,Terzaghi's effective stress principle,and Hooke's law of linear elasticity for the solid skeleton.The initial hydraulic conditions for 700,000years before the present is assumed as hydrostatic,and ice loading on the surface is generated based on the University of Toronto Glacial Systems Model (GSM).Based on the results obtained from this study,the following conclusions can be drawn.First,past glaciation,particularly the second cycle (22,000abp)had a great impact on the pore water pressure gradient and effective stress distribution.The results are consistent with the ?eld observations of persistent pressure to the present time.However,the predicted values of anomalous water pressure is less than the observed values at the site,which could be attributed to additional sources,such as gas or somatic pressure.Moreover,additional factors such as thermal,chemical and 3D effects should be included into the developed model.Data uncertainties also need to be included using suitable statistical methods.Furthermore,in this study we focused only on one loading history with three scenarios of water pressure at surface.Thus,varying loading histories and (thermal,hydraulic,mechanical)boundary conditions should be incorporated in the future model to improve the analysis ability of the developed model.

The results of this research can provide valuable information that will contribute to a better understanding of the impacts of future glaciations on the long term performance of DGRs in sedimentary rocks.

Acknowledgement and disclaimer

The authors would like to thank the Canadian Nuclear Safety Commission (CNSC)and the University of Ottawa (UO)for their ?nancial support.The opinions expressed in this paper are the authors ’and do not necessarily re ?ect the UO's or CNSC's.References

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尊重的素材

尊重的素材（为人处世） 思路 人与人之间只有互相尊重才能友好相处 要让别人尊重自己，首先自己得尊重自己 尊重能减少人与人之间的摩擦 尊重需要理解和宽容 尊重也应坚持原则 尊重能促进社会成员之间的沟通 尊重别人的劳动成果 尊重能巩固友谊 尊重会使合作更愉快 和谐的社会需要彼此间的尊重 名言 施与人，但不要使对方有受施的感觉。帮助人，但给予对方最高的尊重。这是助人的艺术，也是仁爱的情操。—刘墉 卑己而尊人是不好的，尊己而卑人也是不好的。———徐特立 知道他自己尊严的人，他就完全不能尊重别人的尊严。———席勒 真正伟大的人是不压制人也不受人压制的。———纪伯伦 草木是靠着上天的雨露滋长的，但是它们也敢仰望穹苍。———莎士比亚 尊重别人，才能让人尊敬。———笛卡尔 谁自尊，谁就会得到尊重。———巴尔扎克 人应尊敬他自己，并应自视能配得上最高尚的东西。———黑格尔 对人不尊敬，首先就是对自己的不尊敬。———惠特曼

每当人们不尊重我们时，我们总被深深激怒。然而在内心深处，没有一个人十分尊重自己。———马克·吐温 忍辱偷生的人，绝不会受人尊重。———高乃依 敬人者，人恒敬之。———《孟子》 人必自敬，然后人敬之；人必自侮，然后人侮之。———扬雄 不知自爱反是自害。———郑善夫 仁者必敬人。———《荀子》 君子贵人而贱己，先人而后己。———《礼记》 尊严是人类灵魂中不可糟蹋的东西。———古斯曼 对一个人的尊重要达到他所希望的程度，那是困难的。———沃夫格纳 经典素材 1元和200元 （尊重劳动成果） 香港大富豪李嘉诚在下车时不慎将一元钱掉入车下，随即屈身去拾，旁边一服务生看到了，上前帮他拾起了一元钱。李嘉诚收起一元钱后，给了服务生200元酬金。 这里面其实包含了钱以外的价值观念。李嘉诚虽然巨富，但生活俭朴，从不挥霍浪费。他深知亿万资产，都是一元一元挣来的。钱币在他眼中已抽象为一种劳动，而劳动已成为他最重要的生存方式，他的所有财富，都是靠每天20小时以上的劳动堆积起来的。200元酬金，实际上是对劳动的尊重和报答，是不能用金钱衡量的。 富兰克林借书解怨 （尊重别人赢得朋友）

那一刻我感受到了幸福_初中作文

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感受幸福作文650字 第1篇感受幸福作文650字 什么是幸福？幸福是各种各样的，不同的人对幸福的感受也各不相同。假如我们不会说话，那就不能表达自己内心的想法，不能向朋友倾诉自己的喜悦与烦恼，那是一种不幸，反之就是一种幸福。假如我们听不见，就不能感受到一切美妙的声音，如叮叮咚咚的琴声，哗哗的流水声，以及父母、老师、同学的说话声，那也是一种不幸，反之也是一种幸福。假如我们看不见，就不能欣赏到这个大千世界形形色色的样子，看不见斑斓的色彩，更看不见那美丽的太阳和月亮……这也是一种不幸，反之，，也是一种幸福。父母爱着我，就是一种幸福，别人帮助我，是一种幸福，能吃到美味的食物，也是一种幸福，快乐地生活着，更是一种幸福。被人信任是一种莫大的幸福。我平时有个坏习惯，不喜欢洗脸，一般都要妈妈督促着才勉强去洗。可有一次，妈妈回家比较晚。她到家时，我们都已经在吃晚饭了，正吃得津津有味时，妈妈回来了，她的第一句话就是“你今天有没有洗过脸？”，我赶紧回答：“我洗过了”。妈妈半信半疑：“你到底洗过没有？”，并盯着我，脸上的表情十分严肃。她这样做，其实是测试我有没有说谎。于是我更加坚定并大声地说：“我真的洗过了！”这次，她不再问了，并且笑了，我知道她相信了我的话，这时，我真的感受到了一种从未有过的幸福！其实，幸福还有很多很多，如爷爷奶奶的宠爱，父母亲的关怀，亲戚朋友们的赞美，老师们的期盼，同学们的友爱…。.都值得我们去好好珍惜，让我们常怀一颗感恩的心，时刻感受生活中的幸福吧！ 第2篇感受幸福作文650字 欧文曾说过,“人类的一切努力的目的在于获得幸福。幸福是什么？幸福是一种感觉，感动就是一种幸福。幸福就像作家毕淑敏说的：世上有预报台风的，有预报蝗虫的，有预报瘟疫的，有预报地震的，但没有人预报幸福，其实幸福和世界万物一样，有它的征兆。清晨，旭日东升，温柔的阳光洒向人间，抚摸着我的脸，心，感觉暖暖的。这，就是幸福的征兆。当我们在写作业时，一边写，一边贪婪地记下知识的结晶。这时，笔尖沙沙的响声，就是幸福的征兆。走在小河边，两岸的杨柳随风摆动，鱼儿和鹅卵石正在编制着充满活力的动态画面。这时，小河潺潺的流水声就是幸福的征兆。幸福的感觉是美好的，幸福的味道是甜美的。只不过它比玫瑰更芬芳，比棒棒糖更甜蜜，比巧克力更浓郁，比咖啡更香醇......幸福是什么？“幸福是贫困中相濡以沫的一块糕饼,患难中心心相印的一个眼神,父亲一次粗糙的抚摸,女友一个温馨的字条......这像一粒粒缀在旧绸子上的红宝石,在凄凉中愈发熠熠生辉。”幸福在哪里？当我们与家人一起散步时，幸福就在那温馨的眼神里；当我们和朋友们一起玩耍时，幸福就在那灿烂的笑容里；当我们与同学在一起探讨难题时，幸福就在那激烈的争辩里；当我们想象着自己的未来时，幸福就在那美好的憧憬里。幸福有时不一定是甜的，快乐有时也不一定就是幸福。快乐只是表面的开心，而幸福是发自内心的快乐和欣慰。因此，只要你是幸福的，那你一定是快乐的。让我们一起追求幸福吧！在生活中快乐成长，在成长中感受幸福。 第3篇感受幸福作文650字 幸福是什么？幸福对于每个人，都是不一样的！其实幸福可以很小，很简单，它时刻陪伴在我们的身边，也许是因为人们平时没有去在意，身边的那些本可以让人暖心的普通点滴，只要我们好好的去感受幸福！ 幸福就好比如这样：天悄悄拉下夜幕，同学们渐渐散了，不知什么时候，课室里只剩下我和她，她的画，也终于画好了。而我还是差一些才能搞定，她便开始收拾起来，时间更加紧迫，因此，我的心更加紧张，一滴晶莹的汗滴在了画纸上，我觉得实在画不下去了：小姨妈（外号），你觉得我的画还有有救吗！她凑过脑袋：来，我看看，当然有啦，快画吧！我看她望了望校门口，她一定是在找寻来接她回家的爸爸，我看看空空的课室：小姨妈，很晚