A Dynamic Modeling Approach to Simulate Groundwater

KSCE Journal of Civil Engineering (2018) 22(1):341-350

Copyright ⓒ2018 Korean Society of Civil Engineers

DOI 10.1007/s12205-017-0668-9

− 341 −

pISSN 1226-7988, eISSN 1976-3808

www.springer.com/12205

Tunnel Engineering

A Dynamic Modeling Approach to Simulate Groundwater Discharges into

a Tunnel from Typical Heterogenous Geological Media

During Continuing Excavation

Qiang Xia*, Mo Xu**, Han Zhang***, Qiang Zhang****, and Xian-xuan Xiao*****

Received August 27, 2015/Revised 1st: July 11, 2016, 2nd: January 5, 2017/Accepted January 30, 2017/Published Online March 27, 2017

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Abstract

Most of the analytical and numerical models of tunnel groundwater inflow ignore the excavation process, leading to inaccurateprediction of discharge rate. A dynamic modeling approach was introduced by redevelopment of MODFLOW to simulate the changeof groundwater flow step by step in accord with tunnel excavation. The drilling tunnel was conceptualized as a changing boundarycondition, which was modeled by dividing the drilling process into a series of successive steps. The impact of permeabilityheterogeneity on groundwater flow was studied through a comparison between a homogeneous hydraulic conductivity case and asynthetic heterogeneous one. It was found that the discharge rate at drilling front kept stable in the homogeneous case, resulting in alinear increase in the total discharge rate, similar to the analytical solution by Perrochet (2005). In contrast, the front and totaldischarge rate were influenced significantly by the variability of permeability in the heterogeneous case. The time-dependentdischarge rate at a given place was subject to an exponential decay for both cases with the maximum inflow occuring at the beginningof excavation. The relationship between discharge rate and hydraulic properties was further investigated in a high-K zone. It revealedthat maximum discharge rate was proportional to hydraulic conductivity (K) and specific storage (Ss). The decaying rate of dischargewas time-dependent and also proportional to the value of K and Ss. The water budget analysis demonstrated that water released fromstorage of the high-K zone was the major source of tunnel discharge at early times.

Keywords: tunnel discharge, excavation process, numerical simulation, MODFLOW, Hydraulic conductivity heterogeneity, specific

storage

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1. Introduction

Accurate estimate of the groundwater discharge into tunnel isone of the most challenging but essential tasks for geologicalengineering. In the last several decades, many efforts have beenmade on developing more sophisticated analytical solutions toestimate the water inflow in tunnelling (Coli and Pinzani, 2013).However, most of these solutions are developed to predict thedischarge rate per unit length of tunnel, though they areapplicable to the field cases easily (Goodman et al., 1965; Lei,1999; El Tani, 2003; Kolymbas and Wagner, 2007; Park et al.,2008; El Tani, 2010). Based on the analytical solutions, cross-sectional and steady-state models are established by somecommercial codes for numerical simulation of tunnel discharge,

such as ABAQUS (Arjnoi et al., 2009) and COMSOL (Jiang et

al., 2010). In addition, some studies used 3-D numerical modelsto simulate the groundwater flow into tunnel. For example Meiri(1985) developed a model for the solution of non-steadygroundwater flow with a free surface. Yang et al. (2009) usedMODFLOW and FEMWATER to determine the impact oftunneling excavation on the hydrogeological environment, Li et

al. (2009) used FLAC3D to investigate the distribution of the porepressure around tunnels, Chiu and Chia (2012) used MODFLOWto simulate the long-term groundwater discharge into a tunnel.Most of these studies assumed that the whole tunnel isconstructed instantaneously. Only a few work concerned that atunnel excavation was a dynamic process with the constantchange in boundary condition.

TECHNICAL NOTE

*Assistant Professor, State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu

610059, China (Corresponding Author, E-mail: [email protected])

**Professor, State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059,

China (E-mail: [email protected])

***Associate Professor, Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University, Chengdu 610031, China (E-mail: zhang-

[email protected])

****Associate Professor, State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu

610059, China (E-mail: [email protected])

*****Assistant Professor, State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu

610059, China (E-mail: [email protected])

Qiang Xia, Mo Xu, Han Zhang, Qiang Zhang, and Xian-xuan Xiao

− 342 − KSCE Journal of Civil Engineering

In reality, a tunnel is drilled progressively rather thaninstantaneously. Groundwater flow system, accordingly, shouldevolve with the progressive drilling process. Recently, there hasbeen an increasing interest in estimating the groundwater flowinto the tunnel during the drilling progresses. Analytical solutionsare developed to evaluate the transient, drilling speed-dependentdischarge rates into a tunnel in homogeneous and heterogeneousformation (Perrochet, 2005; Perrochet and Dematteis, 2007;Maréchal et al., 2014). Yang and Yeh (2007) developed amathematical model to describe the groundwater inflow into atunnel in a multi-layer aquifer system. All these analyticalsolutions of groundwater discharge into tunnel are based on anassumption that the drawdown of groundwater table is constant.However, Anagnostou (1995) found that the drawdown ofgroundwater table was considerably affected by the rate ofexcavation advance in some circumstances. Hwang and Lu(2007) proposed a semi-analytical method to analyze the tunnelwater inflow with a constant flow and variable drawdown ofwater level, which is a function of the inflow rate.

In analytical methods, the information of geological structureand rock mass are usually simplified to a few simple input data,which is not used directly to predict the tunnel inflows fromcomplex aquifers. Numerical models can provide more accurateprediction, taking a progressive drilling process into account.Molinero et al. (2002) used TRANMEF-3 to conduct an efficientand accurate simulation of the transient hydrogeological conditionsat and around a tunnel during the excavation process. A series of3D stress-pore water coupled models were developed in ABAQUSto examine the interaction mechanism between tunnelling andgroundwater (Yoo, 2005).

In this paper, a dynamic modeling approach was presented basedon secondly development of MODFLOW and ZONEBUDGET. Apair of simulation tests were carried out to investigate the impactof permeability heterogeneity on groundwater flow into adrilling tunnel. Furthermore, parameter sensitivity were exploredby a series of simulations with different values of hydraulicconductivity and specific storage.

2. Modeling Methodology

MODFLOW-2005 (Harbaugh, 2005) with the Finite DifferentMethod (FDM) was used for modeling in this study. The cellsoccupied by the tunnel need to be specified as an appropriateboundary to simulate the groundwater discharge in a numericalmodel. In this paper, a Dirichlet condition (constant total head)was used for cells lying along the tunnel (Lei, 1999). A moreimportant fact should be taken into account in this simulationwork is that the tunnel boundary condition changes with thedrilling advance. For example, when the tunnel reaches a cell,the cell is assigned as a constant head boundary with the pressurehead of zero and the fixed head equal to the elevation of tunnel.However, the geometry of a boundary condition cannot bemodified within a single MODFLOW model. Instead, a sequence ofmodeling steps were built to represent such boundary changing

process.In a MODFLOW model, the IBOUND variable in the Basic

(BAS) Package handles the administrative task to designate thestate of a cell whether (1) the head varies with time (variable-head cell), (2) the head is constant (constant-head cell), or (3) noflow takes place within the cell (no-flow or inactive cell). Here,we conventionally assigned the IBOUND variable equal to -1 fora constant-head cell, 1 for variable-head cell, and 0 for inactivecell. The value of IBOUND of the preceding step was modifiedfor the next step to connect the two consecutive steps as shownin Fig. 2(c). This modification was repeated until the tunnel iscompletely excavated, meanwhile the entire numerical simulation isfinished.

In addition to the modification of the IBOUND values, theinitial head values in the BAS file are also need to be modified.As the simulation went into the second step, the final calculatedhead in the first step, which is stored in a HDS format file, needto be read and assigned to the next BAS file as the initial head ofthe second step. Namely, the model-generated heads in thepreceding step were assigned as the initial conditions of thefollowing one.

The modifying procedure of BAS file between two steps wascarried on automatically by an outer code written by the authors.Thus, the excavation process of tunnel can be integrated into asingle computer run. Other required packages of MODFLOWkept invariable during the whole simulation, e.g. DIS (DiscretizationPackage), LPF (Layer-Property Flow Package) and RCH (RechargePackage).

The computer program ZONEBUDGET (Harbaugh, 1990)was applied to calculate the discharge rates into tunnel by usingthe BGT format file, which stores the water budget results ofMODFLOW. ZONEBUDGET uses cell-by-cell flow data savedby MODFLOW to calculate the water budgets.

The flowchart of the proposed methodology is given in Fig. 1.

3. Description of a Numerical Test

3.1 Tunnel and Site Condition

A synthetic numerical test was conducted on the basis of

Fig. 1. Flowchart of the Dynamic Modeling Methodology

A Dynamic Modeling Approach to Simulate Groundwater Discharges into a Tunnel During Continuing Excavation

Vol. 22, No. 1 / January 2018 − 343 −

preceding dynamic approach. We assume a tunnel penetrates asaturated rock mass, which is 500 m long, 305 m wide, and 200m high, as shown in Fig. 2(a). The following assumptionsunderlie our modeling unless specified differently: (i) Thematerial surrounding the tunnel is neither karstic nor highlyfractured, which would fall into the category of ‘hard ground’.The flow in the rock mass obeys Darcy’s law. For this reason, theequivalent continuum model is suitable for the synthetic sitecondition, and adaptable to perform a dynamic simulation. (ii)Water is instantaneously removed from storage. (iii) Excavation-induced mechanical behavior of the rock mass is not taken intoaccount. Hence, permeability change around the tunnel isassumed to be negligible. (iv) External water source includingprecipitation is ignored.

The tunnel is excavated along the +X axis with an origin at x =0 m, y = 152.5 m, namely in the middle of Y direction. Elevationof the whole tunnel is assigned to z = 23 m. The cross sectionperpendicular to the tunnel axis is simplified just as the geometryof a cell within the numerical model, which is a 5 m × 5 m squarewith 25 m2 area. We assume to carry out a full face excavationwith a constant drilling speed of tunnel front, which is specifiedto 5 m/d. The excavation progression is continuous, never stopsdue to a substantial groundwater inflow. The excavated tunnelsection is unlined with no waterproofing engineering, so thegroundwater is drained during and after tunnel excavation.

3.2 Modeling Settings

The modeling domain is discretized into a uniform grid offinite difference cells of 5 m length for each side, thus consistingof 100 columns by 61 rows by 40 layers cells, accordingly 244000 cells are used in total in the model as basic calculation

points. All the four lateral sides and the bottom of the model areassigned as no flow boundary, while neither recharge norevaporation is adopted on the top of the model throughout thewhole simulation period, the boundary conditions is shown as inFig. 2(a) and (b). The water-bearing media is assumed to be fullysaturated before tunnel excavation, so the initial head is assignedas 200 m high for every cell when the most beginning ofsimulation.

Due to the grid discretization, the full long tunnel is represented bythe 100 cells along 31st row and upon 36th layer. Based on theassumption that the advance speed of tunnel front is uniform of 5m/d, it takes 100 days to complete the tunnel excavation giventhe length of tunnel is 500 m. Acoording to our proposeddynamic modeling methodology, the 100-day simulation periodis divided equally into 100 steps. Each modeling step has a daylong simulating time, which is further divided into 10 time stepsto adopt a transient state flow simulation.

4. The Influence of Permeability Variability ofMedia on Drainage Process

Tunnels are commonly drilled in highly heterogeneous andfractured formations in reality. To investigate the effect ofheterogeneity of hydraulic conductivity on this numericalsimulation, two patterns of K distribution were designed in thissection. One is a homogeneous formation case with a uniformvalue of K = 10−3 m/d, while another is a heterogeneous formationcase with a synthetic distribution of K value. In both cases, thepermeability of rock mass was isotropic and the specific storagewas set to be 10−5 1/m.

In the heterogeneous case, we assumed that the tunnel penetratesan inclined formation with a dip of 45° to the +X direction. Thestrike of the formation is exactly along the Y direction. Asynthetic distribution of K(x, z) was generated by usingSequential Gaussian Simulation method (Deutsch and Journel1998; Remy et al., 2009). A spherical model was applied, whichis the most commonly used variogram model. This model has anugget effect C0 = 0.2, a sill Cs = 0.3, as well as the anisotropicrange values ax = 100, ay = 10, and az = 0. One realization waschosen for our simulation test. As a result, the value of log10(K)ranges from -5.56 to -0.49, with an arithmetic mean of -3.04 andthe standard deviation (σ) of 0.78. The log10(K) field shows anormal distribution. The resulting K(x, z) field has a geometric

Fig. 2. (a) A Schematic Plot of the Model, (b) Boundary Condi-

tions, (c) Change of IBOUND Value to Represent the Exca-

vation Process of Tunnel. Constant Head with IBOUND

Value Equal to -1 was Used to Represent the Cells where

Tunnel is Already Excavated, while the Cells Where Tunnel

is Not Excavated Yet Adopt Variable Head Designated by

Number 1

Fig. 3. The Distribution of Hydraulic Conductivity in the Vertical X-

Z Plane for the Heterogeneous Case



mean of 10−3 m/d, showing a logarithmic normal distribution.The spatial distribution of hydraulic conductivity in the longitudinalcross section is illustrated in Fig. 3, representing both randomand systematic variability. As excavated along the +X direction,

the tunnel firstly encounters the high-K belt at 140 m in the Xdirection, which indicates a preferential hydraulic path. Farthertowards +X a less permeable zone occurs between 360 and 380 m.

In the following section, simulation results of groundwater

Fig. 4. Three-dimensional Distributions of Groundwater Heads at Various Time Points, Changing in Consistent with Tunnel Excavation in

a Saturated Homogeneous Media (left side) and a Heterogeneous Media (right side)


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head distribution and the tunnel discharge rate were presented forboth heterogeneous and homogeneous cases. A comparison ofthe two cases was made to assess the effect of permeabilityheterogeneity on groundwater flow.

4.1 Groundwater Heads

The development of groundwater head fields in bothheterogeneous and homogeneous cases as shown in Fig. 4.Three-dimensional hydrographs at t = 10, 40, 70, and 100d inboth cases were chosen for comparison. In the homogeneouscase, calculated groundwater head contours showed a more orless concentric pattern around the tunnel, indicating that thetunnel acts as a groundwater sink as expected to be. The area ofinfluence expanded gradually and evenly as the tunnel advanced.On the other hand, the trend of groundwater flowing towardtunnel in the heterogeneous case was a little similar to that inhomogeneous case. More important is that Fig. 4 further showedthat the spatial variability of permeability had a significant effecton the processes of groundwater flow, comparing with thehomogeneous one. This was mainly caused by the occurrence ofsome preferential paths in complex geometries where groundwaterprefers to flow through.

4.2 Discharge Rate at Front and Along the Excavated

Part of Tunnel

Two discharge rates were considered in the study, thedischarge rate at the drilling front and the total discharge ratealong the already excavated segment of tunnel. The dischargerate at the tunnel face was obtained by water budget analysis inthe most front cell of the excavated part of tunnel. Water budgetcalculation in all cells of the excavated tunnel accounted for totaldischarge rate. Both discharge rates are time-dependent on theadvance of tunnel excavation.

Figure 5 shows the change of the discharge rate at the drillingfront and the total rate along the excavated tunnel. In thehomogeneous case, the total rate of inflow linearly increasedfrom zero to 104.1 m3/d, in response to drilling advance. Thelilinear-increasing pattern is similar to the typical discharge ratecurve by Perrochet (2005). The rate of groundwater inflow wasabout 3.45 m3/d in drilling front at the beginning of excavation. Itdecreased afterward and kept stable around 2.7 m3/d till theexcavation was almost completed, finally underwent a decline inthe last 10 days.

Due to the spatial variability of the K field in the heterogeneouscase, the rate of groundwater inflow at the drilling front variedaccordingly, leading to a considerable fluctuation of the totaldischarge rate. The largest inflow rate was 105 m3/d at the tunnelface, which occured on day 28 at x = 140 m. This can beexplained by the preferential path encountered when the tunnelwas excavated there. Such a large discharge rate at the drillingfront accounted for almost 60% of the total rate (179.9 m3/d) onthat day which can be regarded as a groundwater inrush event.The smallest rate of inflow was 0.115 m3/d at the front on day 74at x = 370 m, indicating that the tunnel intersected a much less

permeable zone there.When the front discharge rate rose to a relative higher value,

the total discharge rate consequently reached to a local peak, andthen followed by a period of recession as shown in Fig. 5(b).Such recession repeated five times right after the occurrences ofa relative larger inflow event at the drilling front in oursimulation. This trend is also observed in some real cases, e.g. atunnel at the Äspö island in Sweden (Molinero et al., 2002), andthe Modane exploratory tunnel (Perrochet and Dematteis, 2007).

The accumulative amount of water inflow throughout thewhole drilling period is 5935 m3 and 11787 m3 for homogeneouscase and heterogeneous case, respectively. The latter is twice aslarger as the former, in spite of that the same geometric mean ofhydraulic conductivity of 10−3 m/d in both cases.

4.3 Discharge Rate at Specified Points

Figure 6 illustrated the change of discharge rates at differentgiven locations in response to the construction of the tunnel, (a)is for homogeneous case and (b), (c) for heterogeneous case. Allcurves showed a similar pattern as the maximum water inflowoccurred at the beginning time, the instantaneous moment whenthe tunnel passed through, and then the discharge rate graduallydecreased. The entire reducing process of discharge ratesubjected to an exponential decay.

The largest inflow event occurred at x = 140 m (also, t = 28d)in the heterogeneous case. The curve of inflow rate fitted anexponential model very well with the goodness of fit as high as0.9736. The coefficient of decay is -0.309 as shown in Fig. 6(b).The coefficients were negative for the other decreasing curves

Fig. 5. Trends of Discharge Rate at the Drilling Front and Total

Rate Along the Tunnel, Comparison of: (a) Homogeneous

with, (b) Heterogeneous Media, Total Discharge Stands for

the Total Inflow Amount Along the Excavated Part of Tun-

nel, Front Discharge Means the Inflow Rate at Drilling

Front. In the Homogeneous Case, Various Total Discharge

Rates were Compared Calculated by Both Numerical Simu-

lation and Perrochet Analytical Solution. In the Heteroge-

neous Case, Relatively Larger Groundwater Inrush Incidents

are Highlighted by the Red Strips



and the absolute value of the coefficients for those curves weresmaller than 0.309, indicating a slighter decaying process of thedischarge rate. The maximum discharge rates were very close atthree locations in the homogeneous case, which was in consistentwith the almost constant front discharge as shown in Fig. 5(a).

4.4 Comparison to the Analytical Solution by Perrochet,

2005

Consider an idealized, infinite laterally, homogeneous aquifer offinite length (L) with perfect radial flow toward a tunnel, and theprogressive drilling of the permeable zone at an average drillingspeed v, the specific discharge q(x,t) at any drilled location x can beexpressed by Eq. (1), and total discharge Q(t) by Eq. (2).

(1)

(2)

Where the symbols stand for aquifer hydraulic conductivity(K), specific storage coefficient (Ss), time (t), tunnel radius (ro),specified drawdown at the tunnel (so), and H(L-x) is the Heavisidestep function. Perrochet (2005) developed a convolution integralto solve Eq. (2).

According to Eq. (2) we calculated the total discharge Q(t), theresulting time-dependent curves were shown in Fig. 5(a), allvariables in Eq. (2) were assigned the same value as in numericalmodel, except the drawdown so. In our case, tunnel elevation is23 m, the initial head is 200 m, so the drawdown at tunnel shouldbe so = 200-23 = 177m; however, the analytical solution by thisvalue was larger than the numerical one. This was due to thatPerrochet analytical solution is appropriate for a idealizedconceptual model, which is boundless sidewards, whereas themodel is limited to a definite wide in present numerical test,namely 305 m. This also means that analytical solution couldoverestimate the discharge rate in a real bounded domain. Whenso = 120 m, the analytical solution fits well with numerical results,see Fig. 5(a).

Temporal discharge rate q(t) at x = 50 m with different drawdownso was calculated by using Eq. (1), which also displays anexponential decay just like its numerical analogus as shown inFig. 6(a). Similarily, the solution is most comparative to numercialoutcome when so = 120 m.

5. Influences of Hydraulic Properties of a High-KZone on the Discharge Process

In this section, we designed a high-K zone in addition to theforegoing model (see Fig. 7). On the basis of equivalent

2

2( , ) , 0

ln 1

o

s o

Ks xq x t t

vK xt

S r v

π

π

= − >⎛ ⎞⎛ ⎞+ −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

0

2

2 H( )( ) d , 0

ln 1

vto

s o

Ks L x xQ t x t

vK xt

S r v

π

π

−= − >

⎛ ⎞⎛ ⎞+ −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∫

Fig. 6. The Curves of Discharge Rate Versus Time at Some Given

Locations, (a) for Homogeneous Case, (b) and (c) for the

Heterogeneous Case. In plot (a), Analytical Solutions at x =

50 m with Different so Values were Presented Respectively

by Solid Line with Distinct Colors

Fig. 7. (a) Schematic Diagram of Conceptual Model, the Red Belt

Occupying all the Cells in 11th Column to Represent a high-

K Zone Perpendicular to Tunnel, a Pair of Symmetric GHBs

Parallel to Tunnel Serving as Water Sources, (b) Cross-

sectional Plot Illustrating the Principle of a General-head

Boundary, Blue Lines with Arrow Indicating the Groundwa-

ter Flowline


Vol. 22, No. 1 / January 2018 − 347 −

continuum model, a preferential hydraulic path, such as a water-transferring fault even a karst conduit can be conceptualized by azone with relatively greater hydraulic conductivity values. Thiszone is often called a high-K zone.

Since we wish to adopt the dynamic modeling approach topredict transient inflow rates into a tunnel from the high-K zone,not only hydraulic conductivity (K) but also specific storage (Ss)should be taken into account, because both are the crucialhydrogeological parameters to control the hydro-dynamicsbehavior of rock masses (Perrochet and Dematteis, 2007; Mao et

al., 2013; Illman, 2014). Generally speaking, hydraulic conductivitydescribes the ease with which a groundwater can move through,while specific storage evaluates the capacity of an aquifer torelease groundwater.

To test the effect of hydrogeological parameters to transientdischarge process, a series of K and Ss were specified to the high-K zone, ranging from 10−2 to 5 × 10−1 m/d for K, and from 10−5 to10−1 1/m for Ss, respectively. The values indicated for K and Ss

are obtained by referring to some literature (e.g. Perrochet andDematteis, 2007). Except for the high-K zone, the other part ofthe model domain was assigned as low-K bedrock with muchless permeability. The low-K bedrock was assumed to haveconstant K and Ss values of 10−3 m/d and 10−5 1/m, respectively.The permeability of both high-K zone and the low-K bedrock isisotropic.

The modeling domain, grid discretization and tunnel boundaryconditions in this new model kept the same as those in section 3and 4 as shown in Fig. 2, except that two General-headBoundaries (GHB) were added (Fig. 7). The GHB could act asan external source to replenish the volume dewatered by tunnelexcavation. Such lateral boundaries, parallelling to the tunnel,play important roles to the excavation-induced drainage problems inreality. The head of the external source was assigned as 201 m, 1meter higher than the initial groundwater heads (200 m), keepingthe GHB always to be a source term to feed the aquifer.Boundary conductance was set to 0.25 m2/d.

For different K-Ss combinations, daily discharge rates weremeasured at the intersection of tunnel and the high-K zone for allthe tunnel excavation time. The sensitivity of time-dependentdischarge process to hydrogeological parameters was examinedby a comparing analysis. The ZONEBUDGET program wasapplied again to obtain the discharge rate and to carry out a waterbudget analysis.

5.1 Time-dependent Trend of Discharge Rate Versus

Hydraulic Parameters

Daily discharge rates from the high-K zone were recorded byZONEBUDGET when the zone was exposed by excavation.The discharge processes in Fig. 8 displayed the similar pattern tothe exponential decay in Fig. 6. The time-decaying trend ofdischarge rate fitted an exponential equation very well. Themaximum discharge rate in different parameter scenarios occurredon day 11 when the tunnel encountered the high-K zone at x = 55m (also the 11th column in the model).

Figure 8 also demonstrated that the discharge process was verysensitive to the hydraulic parameters. The maximum dischargerate increased with the value of K as well as Ss. As plot (a2)shown, the maximum discharge rate was in a linear directproportion to the value of K. Similarly, the peak discharge ratewas in an exponential direct proportion to the value of Ss in plot(b2). Each time-decaying discharge process fits an exponentialequation with a decay exponent. For example, the decayexponent is -0.143 for the case K = 0.5 m/d in plot (a1). Therelations between exponents and hydraulic parameters werepresented in plot (a3) and (b3). Decay exponent linearly decreasedwith K value in plot (a3), while the linear relation between the Ss

values and decay exponents was not very solid in terms of the R2

number of 0.6895 in plot (b3).The time-dependent curve of discharge rate fits an exponential

equation as , where q is the discharge rate; t is timeq α tβ–

⋅=

Fig. 8. The Time-dependent Change of Discharge Rate Under Dif-

ferent Parameter Conditions, (a1) Different K, the Same Ss

= 10−3 1/m, (b1) Different Ss, the Same K = 0.1 m/d, all the

Discharge Rates in Spite of Different Parameter Scenario

were Measured at the Same Location where the Tunnel

Intersected the high-K zone, Black Circles are the Records

of Discharge Rate from MODFLOW, Blue Solid Line is the

Fitted Curve Labeled with its Fit Exponential Equation, Red

Asterisk is the Maximum Discharge Rate; (a2) and (b2)

Illustrating the Relation between the Maximum Discharge

Rates and Parameter Values; (a3) and (b3) Indicating the

Relation between the Decay Exponents and Parameter

Values



since the high-K zone exposed; α is a coefficient which reflectsthe magnitude of initial discharge rate; β is the decay exponent.The decaying rate of discharge rate therefore could be obtainedby taking the derivative of q with respect to t, and the equation

goes to . The derivative equation illustrates

that not only decay exponent β but also the α coefficientdetermines the decaying rate. Besides, the decaying rate changeswith time as well.

Figure 9 showed that the decaying rate decreased sharply atearly times, due to the exponential time-decaying pattern of the

discharge rate. When we zoomed in to a narrow time span from10d to 20d, the relation between decaying rate and parameterswere shown clearly. Taking t = 15d for example, the decayingrate was in direct proportion to K value as well as Ss, whichmeans that the greater hydraulic parameter is, the quicker time-dependent discharge rate decreases.

5.2 Water Budget Analysis on High-K Zone

A water budget analysis was conducted in the cells of high-Kzone. According to the conceptual model shown in Fig. 7, thesource terms included three parts: (1) the water released from the

dq

dt------ α β t

β– 1–( )⋅ ⋅–=

Fig. 9. Decaying Rate of Groundwater Discharge Versus Time: (a) Different K Values the Same Ss, (b) Different Ss Values the Same K

Fig. 10. Different Compositions of Discharge Rate Under Five Parameter Scenarios, Noting Scenario (a), (b), (c) have the Same Ss = 10−3,

while Scenario (d), (b), (e) have the Same K = 0.1 m/d. The Percentage of Every Water Source was Calculated at the Start Time (t

= 11d) and in the End (t = 100d) Respectively


Vol. 22, No. 1 / January 2018 − 349 −

storage of high-K zone in the light of a transient simulation, (2)water from the General Head Boundaries (GHB), and (3) waterfrom the neighboring low-K bedrock. The sink term aredischarge into tunnel along with some water flow from the high-K zone to the low-K bedrock. The GHB could not be a sink termin this study as its external head was assigned as 201 m, slighthigher than the initial head 200 m.

Figure 10 shows the various compositions of discharge rateunder five parameter scenarios. Here we took plot (a) for a detailexplanation. In plot (a), the discharge rate at the beginning fromthe storage of high-K zone, GHB and low-K bedrock were63.5%, 23.3% and 13.1%, respectively. At the end of this dischargeprocess, the percentage of water from storage fell to 25.8%,while the percentage of water from GHB and low-K bedrockincreased to 52.7% and 21.4%, respectively.

Figure 10 demonstrated that tunnel discharge was the primarysink term in the high-K zone budget analysis. The sum of theflow rates of the three source terms almost equated to the tunneldischarge rate. At early times of case (a) and the full time of case(e), the total source amount was a little larger than discharge rate,which indicated some water flowed from high-K zone to thelow-K bedrock, but the tunnel discharge rate was still the majorsink term.

The five plots in Fig. 10 indicated that storage of the high-Kzone contributed most to the maximum discharge rate at thebeginning with the percentage ranging from 57.2% to 97.3%.Given the same K value of 0.1 m/d in case (d), (b), and (e), theproportion of water from storage increased in response to the Ss

value from 10−4 to 10−1.

6. Discussions

Although the proposed dynamic modeling methodology cansimulate groundwater flow toward a tunnel during excavation,this approach has some limitations, e.g. (1) the tunnel wasrepresented as constant head Dirichlet-type, not taking tunnellining into consideration in the process of construction. If thelining condition is required in a modeling, nodes lying along thetunnel was suggested to be designed as a Cauchy-type boundarycondition (Molinero et al., 2002), or a General Head Boundary(GHB), or a time-varied and head-dependent boundary (Chiuand Chia, 2012). However, all of these boundary conditions willcomplicate the determination of relevant parameters. (2) Thismodel is applicable to fractured rock, while not able to simulatethe flow in the medium with developed conduits by strongkarstification. The latter could be modelled by coupling theMODFLOW-CFP (Conduit Flow Process) module into themodel. (3) The numerical test in this study were conducted in thesaturated rock, which did not take water table decline intoaccount. The decreasing water pressures due to tunnel drainagemay result in the drying up of springs, and ground settlement.Moreover, the effect of effective stresses on hydrogeologicparameters is neglected, which implies an overestimation of theflow rate, especially for deep tunnels (Preisig et al., 2014). (4)

Technologies such as hydro-geophysical prospecting andhydraulic tomography could be useful tools for improving thedistribution of hydraulic properties, and in turn the accuracy ofprediction of discharge into tunnels in subsurface (Jiang et al.,2010).

7. Conclusions

A dynamic modeling approach has been developed to simulatethe evolution of groundwater flow during a tunnel excavationprocess of. To represent the continuous change of boundarycondition of a tunnel, the tunnelling process was discretized intoa series of successive simulation time steps. The Basic PackageFile was modified by secondly development of MODFLOW torepresent the change of tunnel boundary condition and initialgroundwater head for each step. Accordingly, it is possible tosimulate the excavation process of a tunnel in a single computerrun. The simulation results demonstrated that the proposeddynamic modelling approach can predict the groundwater flowtowards tunnel appropriately.

To assess the effects of spatial variability of permeability onthe pattern of groundwater flow into tunnel, a comparison wasconducted between a homogeneous and a heterogeneous hydraulicconductivity (K) fields. Results showed that the hydraulicconductivity heterogeneity has a significant impact on thegroundwater flow field. The rate of water inflow at drilling frontis relatively stable for the homogeneous K-field, leading to agradual increase in the total rate of inflow. In contrast, the rate ofwater inflow at drilling front varies dramatically in theheterogeneous case due to the different permeability of thestrata. Groundwater gushing events could occur when thepreferential hydraulic path is exposed, followed by a recession inthe total discharge rate. The time-dependent curve of dischargerate at a given location is subject to an exponential time-decayingpattern for both cases.

To investigate the effect of hydrogeologic parameter to thedischarge process, a variety of hydraulic conductivities (K) andspecific storages (Ss) was assigned to a high-K zone, a preferentialflow path. The various time-dependent discharge rates weremeasured under different parameter conditions. It is revealed thatmaximum discharge rate occurred when the high-K zone wasexposed, and the rate was in direct proportion to both K and Ss.The decaying rate of discharge was also in direct proportion to Kand Ss, namely, larger parameter value result in faster decayingprocess. Water budget analyses in the high-K zone indicated thatwater released from storage was the major sources of tunneldischarge at early times, and the proportion of storage enlargedin responese to the increase of Ss.

Acknowledgements

This research is financially supported by the PostdoctoralResearch Fund established by the Department of Human Resourcesand Social Security of Sichuan Province, China and partially



supported by the National Natural Science Foundation of China(Grant No. 41502237 and 41472275). The authors wish to thankthe anonymous reviewers for their constructive comments.

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A Dynamic Modeling Approach to Simulate Groundwater