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Title The influence of tides, waves, and overtopping on the near-shore water table

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Author Colyar, Maia Astrid

Publication Date 2016

Peer reviewed|Thesis/dissertation

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The inﬂuence of tides, waves, and overtopping on the near-shore water table

THESIS

submitted in partial satisfaction of the requirements for the degree of

MASTER OF SCIENCE

in Civil Engineering

Maia Astrid Colyar

Thesis Committee: Professor Russell L. Detwiler, Chair Professor Brett Sanders Professor Amir AghaKouchak

2016 c 2016 Maia Astrid Colyar DEDICATION

This dissertation is dedicated to my friends and family. Thank you for supporting me in this endeavor and keeping my life balanced.

ii TABLE OF CONTENTS

Page

LIST OF FIGURES v

LIST OF TABLES vii

ACKNOWLEDGMENTS viii

ABSTRACT OF THE THESIS ix

1 Introduction 1 1.1 Motivation ...... 1 1.2 Background ...... 4 1.3 Modeling ...... 7

2 Modeling Methods 10 2.1 The Model ...... 10 2.2 Governing Equations ...... 10 2.3 Model Domain ...... 12 2.4 Grid Selection ...... 14 2.5 Boundary Conditions ...... 15 2.6 Model Parameters and Settings ...... 17 2.7 Processing ...... 18

3 Pﬂotran Model Comparison with Previous Studies 22 3.1 Boussinesq Based Analytical Solution and Simple Field Study ...... 22 3.2 Richard’s Equation Based Analytical Solution ...... 24 3.3 Model Comparison ...... 27 3.3.1 Vertical Beach Results and Analysis ...... 27 3.3.2 Sloped Beach Results and Analysis ...... 29 3.4 Discussion ...... 31

4 Inﬂuence of Additional Forcing and Beach Geometry 33 4.1 Inﬂuence of Additional Forcing ...... 33 4.1.1 Second Tidal Forcing ...... 33 4.1.2 Wave Setup ...... 34 4.1.3 Overtopping ...... 35

iii 4.2 Inﬂuence of Beach Geometry and Geology ...... 39 4.2.1 Depth ...... 39 4.2.2 Beach Slope ...... 40 4.2.3 Porosity ...... 42 4.2.4 Permeability ...... 43 4.3 Discussion ...... 44

5 Cardiﬀ State Beach 47 5.1 Field Data ...... 48 5.2 Cardiﬀ Beach Model ...... 48 5.3 Model Results and Discussion ...... 51

6 Concluding Remarks 54

Bibliography 56

iv LIST OF FIGURES

Page

2.1 Vertical beach domain with tidal forcing...... 13 2.2 Sloped beach domains...... 13 2.3 The steps in creating a voronoi polygon mesh from a Delauney triangle mesh. 14 2.4 Hydraulic head normalized by the boundary values to visualize vertical gradients at low and high tides. The pink line shows the watertable level based on the highest saturated cell, while the black line shows the watertable level based on the head values ...... 21

3.1 Vertical beach domain used for Nielsen and Kong comparison...... 28 3.2 Resulting water table ﬂuctuations for the vertical beach case...... 28 3.3 Sloped beach domain used for Nielsen comparison...... 29 3.4 Close up of measurement observation points...... 29 3.5 Resulting water table ﬂuctuations for the sloped beach case...... 30 3.6 Aerial Image of Barrenjoey Beach...... 32

4.1 Effect of adding a second tidal forcing on the mean water level compared to single tidal forcing. Mean (—) and maximum (- - -) water levels shown. . . . 34 4.2 Effect of adding a time-averaged wave setup on water table levels...... 35 4.3 Time series showing the effect of wave overtopping resulting in various durations of ponding...... 36 4.4 Mean (—) and maximum (- - -) watertable elevations resulting from various durations of ponding...... 38 4.5 Effect of beach depth on water table fluctuations...... 40 4.6 Time series of water levels with various beach slopes...... 41 4.7 Effect of beach slope on mean (—) and maximum (- - -) water levels. . . . . 42 4.8 Time series showing effect of porosity on water table fluctuations...... 43 4.9 Time series showing effect of permeability on water table fluctuations. . . . . 44 4.10 Result from adding the second tidal forcing on the ocean side and wave set up to both sides of the domain compared to the single tidal forcing from the previous chapter and Nielsen field observations...... 46

5.1 Cardiff State Beach field site...... 49 5.2 Model domain used for Cardiff field data comparison...... 50

v 5.3 Resulting water table fluctuations from three different simulations in Pflotran considering no wave effects, wave runup, and wave setup. All three simulations also included tidal and lagoon forcing...... 53

vi LIST OF TABLES

Page

2.1 Soil Property Values ...... 17

3.1 Physical Parameters used for the Analytical Comparisons ...... 27 3.2 Model Settings used for the Analytical Comparisons ...... 27

4.1 Calculation of model ﬁt at Well 7 ...... 45

5.1 Physical Parameters used for the Cardiff Simulations ...... 50 5.2 Model Settings used for the Cardiff Simulations ...... 51 5.3 Model fit at selected sensor locations...... 52

vii ACKNOWLEDGMENTS

This work was supported by funding from California Department of Parks and Recreation, Division of Boating and Waterways.

I also would like to thank Russ Detwiler and Timu Gallien for their time, support, and guidance in this research project.

viii ABSTRACT OF THE THESIS

The inﬂuence of tides, waves, and overtopping on the near-shore water table

Maia Astrid Colyar

Master of Science in Civil Engineering

University of California, Irvine, 2016

Professor Russell L. Detwiler, Chair

Pflotran was used to model the water table response to tides and wave effects in the first couple hundred meters inland from the shoreline. Unlike previous analytical solutions and many numerical models, Pflotran can take into account both the fully and partially saturated zones, irregular boundary shapes, and time-varying boundary conditions such as wave setup and overtopping in addition to the regular tidal harmonics. This research demonstrates the effect of several soil and beach characteristics and the importance of considering all forcings coming together to create boundary conditions. Specifically, the influence of waves and wave overtopping represented by ponding on the beach surface prove to be significant. Field data from Cardiff State Beach quantifies the significant improvement gained by including all of these effects.

ix Chapter 1

Introduction

1.1 Motivation

Coastal aquifers are highly dynamic groundwater systems where the salinity and water level are influenced by the bordering sea. Ocean tides cause the water table of the adjacent aquifer to fluctuate significantly from the mean tide height over the first hundred or so meters inland from their interface (Nielsen, 1990). This region of coastal aquifers that borders the ocean is unique in that the water table cycles through a fairly large range of heights in a daily or semidiurnal pattern. The effects of tidal forcing on the water table have been explored in many studies through field and experimental data collection (Nielsen, 1990; Li et al., 1997a; Raubenheimer et al., 1999; Cartwright et al., 2003; Heiss et al., 2015), analytical modeling (Nielsen, 1990; Kong et al., 2013), and numerical modeling (Li et al., 1997a; Raubenheimer et al., 1999; Robinson et al., 2007; Xin et al., 2010; Pool et al., 2014; Xin et al., 2015). These water table fluctuations are important to understand, because they affect groundwater flooding (Heberger et al., 2009; Rotzoll and Fletcher, 2013), the stability of the beach which in turn affects erosion and accretion tendencies (Grant, 1948), and saltwater intrusion and

1 ﬂow patterns (Rotzoll and Fletcher, 2013; Pool et al., 2014).

Over the past 100 years the sea level has risen about twenty centimeters along the California coast and is predicted to rise another 100 to 140 centimeters by the end of this century (Heberger et al., 2009). It is especially critical for California to thoroughly understand the implications of sea level rise since 87% of the population lives in coastal counties (Hoover et al., 2016). There is also an estimated $100 billion worth of property and infrastructure along the 1800 km California coast that would be vulnerable to flooding if the sea level rose an additional 140 cm from today’s levels (Heberger et al., 2009). This increase in sea level will not only will inundate low lying areas, but will also accelerate erosion rates by exposing new and potentially unstable soils to wave action until a new steady state with respect to the new elevated sea level is achieved, further reducing the area of dry land. In addition to surface flooding by the ocean, areas with shallow aquifers will be more susceptible to groundwater inundation due to the response of the water table to the mean sea level and tidal forcing (Rotzoll and Fletcher, 2013). Even if the groundwater doesn’t fully breach the ground surface, having a shallow water table may interfere with subsurface infrastructure, destabilize soils and affect surface structures, or be a hazard to water quality as it comes into contact with previously hydrologically isolated contaminants (Lerner and Barrett, 1996; Kreibich and Thieken, 2008). Understanding the effect of tidal forcing on the near-shore water table is an important part in better understanding the implications of sea level rise on the near-shore regions of coastal aquifers.

Water tables along the coast are rarely influenced by the tide alone. Most beaches are also subject to wave action which elevates the sea surface near shore and may result in overtopping and ponding on the beach. Numerous studies have determined when wave overtopping may occur and pose a flood hazard, but the effect of overtopping on the underlying water table is largely ignored(Dodd, 1998; Hu, 2000; Gallien et al., 2014). Raubenheimer et al. (1999) noticed during a field experiment that the water table was elevated for several days after an

2 overtopping event, but did not attempt to characterize it. Also, some regions are subject to tidal forcing from more than one direction. For instance, narrow spits of land or small islands in the San Fransisco Bay are surrounded by fluctuating water levels even far into the river delta (Elias et al., 2013). Although these locations are also most vulnerable to changes in water levels and flooding, the research is seemingly sparse when it comes to tidal forcing from more than one direction. A key part of this study is aimed to understand how these additional forcings come together to influence the height of the water table in the beach.

Besides regional hydraulic gradients, tides are one of the key drivers for mixing and transport in coastal aquifers. In fact, the nutrient ﬂuxes measured in the subsurface have exceed those at the surface (Moore, 1999). In a study by Boehm et al. (2003), the tide was found to aﬀect pollutant concentrations in the water near shore. During dropping tides the concentration of pollutants increased, indicating that the source of contamination was likely either leaky pipes in the beach or deposited contaminants that were mobilized through groundwater discharge when the tide dropped.

If the groundwater table in the beach is higher than the mean sea level then typically offshore transport is favored and the beach erodes. On the other hand if the groundwater table is lower than the mean sea level, onshore transport is favored and the beach is more likely to grow (Grant, 1948). It is essential to understand the the influence of tidal fluctuations since it affects the water level which in turn affects balance of beach erosion and accretion. This is especially crucial as beachfront property continues to grow in demand and more is invested in developing these areas.

3 1.2 Background

Water table ﬂuctuations driven by tides are often considered in terms of the harmonic constituents of the tidal forcing. The sea level can be described by

X htide(t) = D + Acos(ωjt − φj). (1.1) j

Where htide(t) is the height of the sea [m] at time t [hr], D is the mean sea level in relation to some reference datum, usually an underlying impermeable layer [m], A is the amplitude [m], ω is the angular frequency [rad/hr], and φ is the phase of the harmonic constituent [rad]. This equation allows the tide to be deﬁned by a simple single harmonic or by j number of constituents that produce a more complicated and realistic tidal signal.

A common approach to modeling the water table response to tidal forcing is to consider the Boussinesq approximation for surface waves. The Boussinesq approximation is appropriate for slightly nonlinear waves with a wavelength longer than the water depth. Boussinesq type equations are used to model waves near shore and more recently, porous medium (Kirby, 2002). These account for frequency dispersion while other shallow water equations do not (Schäfferet al., 1993). It is important to consider frequency dispersion when modeling ocean waves and tides since the overall tidal signal is typically composed of several constituents with varying wavelengths. Each of these constituent waves travels at a different phase speed based on its wavelength due to frequency dispersion. With all other conditions the same, a wave with a longer wavelength will travel faster than one with a shorter wavelength.

4 The Boussinesq equation is derived from Darcy’s Law,

∂h q = −K , (1.2) ∂x

And from the continuity equation,

∂h 1 ∂ = − hq . (1.3) ∂t n ∂x

These equations combined with the assumptions of homogenous and isotropic soils with horizontal ﬂow and hydrostatic pressure distributions result in the Boussinesq equation, shown in Equation 1.4, which is the starting point for the Boussinesq based solutions put forward by many researchers (Parlange and Brutsaert, 1987; Nielsen, 1990; Barry et al., 1996; Raubenheimer et al., 1999; Li et al., 2000).

∂h K ∂ ∂h = h (1.4) ∂t n ∂x ∂x

When solving the Boussinesq equation considering tidal forcing, the most simple solution for the describing the water table height in a beach with a vertical boundary is

X −krj x h(x, t) = D + Acos(ωjt − φj − kij x)e . (1.5)

Where h is the height of the water table [m] at time t [hr]and some distance x inland from the ocean-beach interface [m], φ is the relative phase shift between components [rad], and ki and ki are the real and imaginary components of the wavenumber k, respectively. The wavenumber, k, when used in the Boussinesq-type solutions is deﬁned as

r nω k = k + ik = (1 + i) . (1.6) r i 2KD

Where n is the porosity and K is the saturated hydraulic conductivity of the soil [m hr−1].

5 In Boussinesq-based solutions the imaginary, ki, and real, kr, portions of the wave number are assumed to be equal, however, in experimental studies they have been observed to be

diﬀerent with kr > 1 (Nielsen, 1990; Raubenheimer et al., 1999; Cartwright et al., 2003; ki Kong et al., 2013).

There are three characteristics consistent across tidally driven water table fluctuations that were considered when comparing results. The first characteristic of water table fluctuations is that the shape of the oscillations when tracked at a single point typically have a steeper rise and more gradual fall than a perfectly sinusoidal wave would have. This is due to the water filling into the beach more quickly during a rising tide than it can drain out of the beach during a falling tide (Nielsen, 1990).

Another characteristic of tidally driven ﬂuctuations is the way they are transmitted inland. As the ﬂuctuations are transmitted inland, the amplitude decreases the further from the

ocean boundary it gets. This attenuation is approximately proportional to e−krx. This means that each tidal constituent with a unique period gets attenuated at a diﬀerent rate compared to the distance traveled, with the longer period waves attenuating more slowly. The beach essentially acts like a low pass ﬁlter quickly damping out high frequency oscillations driven by waves and transmitting longer period tidal oscillations far from the beach interface (Parlange et al., 1984). Similarly, the phase of the wave can be followed at a phase velocity described

ω kix by vp = . This means that travel time of a single oscillation can be found by t = . ki ω Although the ﬁt from the ﬁeld data of Nielsen (1990) showed kr > 1, he used the average ki of the two and assumed that the imaginary and real components of k were equal in his

modeling. Raubenheimer et al. (1999) also found that kr > 1 when fit to field observations ki of oscillation amplitude and phase lag. They concluded that the nonuniform aquifer depth to the underlying low permeability formation is likely the cause for the difference between the imaginary and real parts of the wavenumber. However, in this Pflotran modeling study

kr was found to be larger than ki even when the depth was uniform. Meanwhile, Kong et al.

6 (2013) attributed the difference between the imaginary and real parts of the wavenumber to capillary effects and vertical flow in the beach.

The final characteristic discussed in this study is the effect tidal forcing has on the mean water table elevation. It has been observed in field and modeling studies that the mean water table position increases going inland, and this difference between the mean sea level and mean water table has been coined the ”overheight” (Parlange et al., 1984; Nielsen, 1990; Ian L. Turner, 1997; Ataie-Ashtiani et al., 2001; Cartwright et al., 2003; Kong et al., 2013). The Richard’s Equation based solution from Parlange et al. (1984) determined that

2 the overheight was approximately equal to Hover = D[1 + 0.25(A0/D) ]. Similarly, Nielsen (1990) also concluded from his Boussinesq-based solution that the overheight is related to A2/D2. Kong et al. (2013) arrived at a similar equation for the inland overheight, but with the addition of a coeﬃcient Nover that accounts for the distance between the water table and beach surface. This results in an overheight slightly smaller than originally predicted

2 by Parlange Hover = D[1 + 0.25Nover(A0/D) ], where Nover < 1 for most physical systems.

1.3 Modeling

Analytical models are nice because they are easy to use, however, they are limited in several ways. One is that the exit point at the beach face and the sea level are always coupled. Realistically, the tide often recedes faster than the exit point causing a decoupling during the dropping tide (Nielsen, 1990; Bruce J. Hegge, 1991). The stretch of beach surface between the water table exit point and the sea level is called the seepage face. This means the water table will fall more slowly and average higher levels than predicted by the analytical models which keep the water table coupled to the sea level (Nielsen, 1990).

Also, it is nearly impossible to consider the full depth of the variably saturated zone above

7 the water table in an analytical model. This unsaturated zone plays a critical role in coastal aquifer dynamics (Ataie-Ashtiani et al., 2001). Many groups have tried to incorporate the zone above the water table by including the capillary fringe in their modeling (Gillham, 1984; Barry et al., 1996; Li et al., 1997b; Kong et al., 2013). It was also observed that there is fluid and solute flow in the capillary zone which confirms that it plays a role in the transmission of pressure oscillations from tides (Silliman et al., 2002; Horn, 2006).

Another aspect of the unsaturated zone is the Wieringermeer effect. It was first noted in the early 1900s that the water table would drop more than expected for the amount of water released. Since then, lab experiments have confirmed a similar effect when the capillary fringe extends near the surface (Gillham, 1984; Ibrahimi Mohamed et al., 2011). The opposite is also true; the water table will rise disproportionately to the amount of water added and this phenomenon has been named the reverse Wieringermeer effect. Essentially, a small volume of water will fill the unsaturated and capillary zone and quickly increase the height of the water table more than the amount of water added would suggest. In one lab experiment 0.3 cm of water applied to the surface resulted in a 30 cm rise of the water table over the course of 15 seconds (Gillham, 1984). This is seen in the field at the beach face when wave run-up results in a near instantaneous increase in the local water table (Bruce J. Hegge, 1991; Ian L. Turner, 1997). Ignoring these effects would likely result in underestimating the distance fluctuations travel inland and the level of the watertable in the beach thereby potentially missing flood hazards and incorrectly predicting flow patterns.

Despite the large amount of previous research, significant questions remain unanswered. For instance, the role of the unsaturated zone in the propagation of tidally forced water table fluctuations in sloped beaches needs to be better understood and modeled. Also largely yet unexplored is the effect that multiple forcing have on the water table. This includes tidal forcings from more than one direction - ie under a peninsula or small island - or waves, such as run-up effects and wave overtopping and subsequent ponding and infiltration.

8 This paper aims to fill in these gaps left by previous studies by selecting a groundwater flow model capable of simulating variably saturated flow in complex geometries with multiple forcing boundary conditions. This model will be evaluated by first comparing against accepted analytical models and data from simple field sites. Once proven to be reasonable, the effects of various beach conditions and additional forcings will be examined. Finally, the model will be used to investigate the driving forces acting on the water table at a complex field site at Cardiff State Beach with an irregular geometry and forcing from tides, waves, and a lagoon. This paper aims to fill in these gaps left by previous studies by selecting a groundwater flow model capable of simulating variably saturated flow in complex geometries with multiple forcing boundary conditions. This model will be evaluated by first comparing against accepted analytical models and data from simple field sites. Once proven to be reasonable, the effects of various beach conditions and additional forcings will be examined. Finally, the model will be used to investigate the driving forces acting on the water table at a complex field site at Cardiff State Beach with an irregular geometry and forcing from tides, waves, and a lagoon.

9 Chapter 2

Modeling Methods

2.1 The Model

Pflotran was developed by a team of researchers at various US National Labs to model reactive flow and transport. It is highly customizable, easy to run in parallel, and it is built on physically meaningful systems of equations (Lichtner et al., 2013). The ability to include flow in both fully and partially saturated soils was an important deciding factor in selecting Pflotran. Pflotran solves the system of nonlinear partial differential equations using a fully implicit Newton-Raphson algorithm.

2.2 Governing Equations

The model was set to use the Richard’s mode for subsurface simulation, which allows for the variably saturated processes to be captured. Richard’s, van Genuchten, and Mualem equations were used to describe the flow, saturation, and permeability, functions respectively. Similar to Darcy’s flow equation, Richard’s equation relates the amount of flow to the of the

10 local hydraulic gradient. However, Richard’s equation accounts for the fact that the hydraulic conductivity decreases with decreasing saturation (Richards, 1950). This allows the variably saturated region above the water table to be included in the calculations. Pﬂotran uses the following pressure based equations. The governing mass conservation equation is

∂ nθρ + ∇ · (ρq) = Q , (2.1) ∂t w where q is the Darcy ﬂux deﬁned by

kk (θ) q = − r ∇(P − W ρgz), (2.2) µ w

and where Qw is the source/sink term deﬁned by

qm Qw = − ∇(r − rss). (2.3) Ww

Here, n is porosity [-], θ is saturation [m3m−3], ρ is water density [kmol m−3], q is the Darcy

−1 2 ﬂux [m s ], k is intrinsic permeability [m ], kr is relative permeability [-], µ is viscosity

3 [Pa s], P is pressure [Pa], z is the vertical position [m], qm is mass rate [kg/m /s], r is the

position vector, rss is the position vector of the location of the source/sink, and Ww is the molar weight of water [kg/kmol].

The van Genuchten equation relates the capillary pressure and saturation of the soil with the equation,

n−m pc θ − θr se = 1 + 0 = . (2.4) pc θs − θr

3 −3 Here, se is the eﬀective saturation [m m ], pc is the capillary pressure [Pa], θr is the residual

3 −3 3 −3 water content [m m ], θs is the maximum or saturated water content [m m ]. m, n, and

0 pc are constants that are found by ﬁtting to experimental data or looking up approximate

11 values for certain soils. Rearranging the above equation gives

0 −1/m 1/n pc = pc (se − 1) . (2.5)

The Mualem version of the van Genuchten function relates n and m by the equation

1 m = 1 − . (2.6) n

This results in the Mualem function for relative permeability, which scales the hydraulic conductivity according to the degree of saturation,

√ 1/m m 2 kr = se[1 − (1 − se ) ] . (2.7)

2.3 Model Domain

The domain was oriented so that the positive x direction corresponds with increasing distance inland from the ocean and the positive z direction points upward and corresponds with increasing elevation. The y direction was taken to be longshore and a unit length, which allowed the model to essentially run in 2D as the y direction could be ignored.

Three general domain shapes were used during this study. These include a vertical beach face, sloped beach face, and a field based domain with a more complicated profile. The vertical beach domain allows direct comparisons with the vertical Nielsen (1990) and Kong et al. (2013) analytical solutions. The single sloped beach face allows a more realistic simulation and can also be compared against the field data and sloped analytical solutions from Nielsen. The final and most complicated domain was used to model water table fluctuations to be compared against field data which was subject to multiple forcings.

12 Figure 2.1: Vertical beach domain with tidal forcing.

(a) With single tidal forcing.

(b) With double tidal forcing.

Figure 2.2: Sloped beach domains.

13 2.4 Grid Selection

Pflotran can either generate a structured grid internally or read in a user-defined unstructured mesh to run a simulation. The ability to use an unstructured mesh was an important component in modeling the sloped face of a beach. After testing structured, triangle, and voronoi polygon meshes, the voronoi polygons proved to be the best choice. The Pflotran model conserved mass when using the voronoi polygons and was able to model a sloped geometry which better simulates a natural beach surface. This mesh was made by first generating a Delauney triangle mesh using Triangle by Shewchuk (1996) and then using the vertices of the triangle mesh as the centroids of the new voronoi mesh generated by LaGrit developed by the Los Alamos National Lab (LANL, 2010). These steps of the mesh- generating process is shown in Figure 2.3.

Figure 2.3: The steps in creating a voronoi polygon mesh from a Delauney triangle mesh.

14 2.5 Boundary Conditions

Pflotran uses pressure as the forcing for a flow simulation. The pressure is defined at a datum and is applied across a region either as hydrostatic or seepage pressure to create the boundary condition. In the hydrostatic case the pressure decreases with increasing elevation, and the moisture above the water table has a pressure less than atmospheric, whereas in the seepage case the pressure above the water table remains equal to atmospheric. The seepage face condition allows the model to mimic the physical seepage face that forms when the tide drops faster than water can exit the beach, which causes the water table in the beach to be disconnected from and higher than the sea level.

The top and western domain boundaries were set to be a seepage face which allows water to exit the domain if the pressure exceeds the atmospheric pressure. When wave overtopping was simulated as a ponding event the first 2 m of the top boundary inland from the beach crest was designated as the ponding region and had alternating seepage face or hydrostatic conditions applied to mimic ponding. In most simulations the eastern face was set to be a no flow boundary unless it had a time varying water level as a second tidal forcing, in which case it was assigned the same seepage face flow condition as the western boundary. In the cases where the eastern inland face was set to be a no flow boundary, a sufficient distance was needed between the ocean and inland faces to minimize any artificial effects caused by the fluctuations reflecting off of the eastern end within the timescale of the analysis.

The ocean boundary condition was composed of either just a tidal signal or wave-set up superimposed on the tidal signal. The tidal signal was deﬁned by a simple single harmonic or by looking up the historical water levels at the nearest station from the National Oceanic and Atmospheric Administration. The wave-set up was calculated based on the period and amplitude following Stockdon et al (Stockdon et al., 2006). First, the wavelength was found

15 by

gT 2 L = . (2.8) 0 2π

−2 Where L0 is the wavelength [m], g is the gravitational acceleration [ms ], and T is the wave period [s]. This then is used to ﬁnd the time averaged mean set-up driven by waves, < η > [m],

p < η >= 0.35βf H0L0. (2.9)

Where βf is the slope of the beach foreshore [m/m] and H0 and is the deep water wave height [m].

The runup height was found by using the equation from Hunt (1959),

R = H0ξ, (2.10) with

p ξ = βf (H0/L0). (2.11)

Where R is the runup height [m], and ξ is the Irribaren Number, or surf similarity parameter from Battjes (1974).

For all boundary conditions, the desired water level in meters was converted to absolute pressure in Pascals using

PBC = Patm + Hobsρg. (2.12)

Where PBC is the generated boundary condition pressure, Patm is the atmospheric pressure

16 Table 2.1: Soil Property Values

m3 Porosity 0.355 m3 Tortuosity 0.5 - Permeability 5.374x10−11 m2 m3 Residual Saturation 0.087 m3 Mualem-van Genuchten m 0.444 - Air entry value 5.43x10−4 Pa−1 Gardner α value 0.16 m−1

taken to be a constant 101325 Pa, Hobs is the observed or calculated water level [m], ρ is the

kg m density of seawater taken to be 1025 m3 , g is gravitational acceleration 9.806 s2 .

2.6 Model Parameters and Settings

Parameter values for a generic sand and loam soil were estimated from the mean van Genuchten parameter values found by Hodnett and Tomasella (2002). When comparing the Pflotran model to existing studies or field data I used the same parameters in both the analytical and Pflotran modeling. In the case of the Kong, et al. model the alpha parameter for Gardner’s equation had to be chosen. The value was found by comparing the hydraulic conductivity curves resulting from van Genuchten and Gardner. Pareto front optimization was used with L1 and L2 norms as the two objective functions to be minimized.

The Newton solver was used with an absolute tolerance of 1x10−15 and a maximum of 50 iterations was allowed. Initially 10 iterations were suﬃcient, but for the more complicated scenarios such as the sloped beach face and water ponding up to 50 iterations were needed in order to ensure convergence.

17 2.7 Processing

Pflotran outputs a file with the hydraulic pressure head in Pascals at each specified location at each time. To convert the Pflotran result back into length units to find the surface of the water table the following equation, which assumes a hydrostatic pressure distribution, was used. As seen in Figure 2.4, the fully saturated region follows the hydrostatic assumption.

Htotal = (P − Patm)/(gρ) + Hz (2.13)

Where Htotal is the total hydraulic head [m], P is the pressure [Pa] output from Pﬂotran at a certain position x [m] horizontally, and Hz [m] is the height of the observation point measured from the no-ﬂow boundary at the bottom of the domain. Here Patm, g, and ρ are the atmospheric pressure, gravitational acceleration, and water density constants with the values described previously.

In order to compare results from diﬀerent beach geometries, the coordinates were normalized after the water table elevations were determined. Vertical coordinates were referenced to mean sea level so that the mean sea level was at an elevation of 0 m. Similarly, the horizontal coordinates were referenced to the point where the mean sea level intersects the beach face so that x=0 where z=0 at the beach surface.

One way to succinctly compare results is to consider the rate of amplitude attenuation and the time lag for a single wave to travel into the beach. By passing the timeseries of water table elevations at a single point through a Fourier transform the amplitude and phase shift can be extracted. This was repeated along points traveling inland from the shoreline to get the oscillation amplitude and phase shift as a function of distance inland. The real and imaginary parts of the wavenumber, k, are a convenient way of representing the rate of damping and phase shifting. The real part, kr, corresponds to the amplitude damping and

18 was found by ﬁtting an equation of the form,

A = e−krx. (2.14) A0

Where A0 is the amplitude of the tide at the boundary with the beach [m] and A is the amplitude of the water table oscillation some distance x from the boundary [m]. The imaginary part of the wavenumber can be found by ﬁtting a line through the phase lag, φ, as a function of x so that

φ = kix. (2.15)

In the cases where there is observational data, the model can be evaluated using four different measures of fit. The first is the correlation coefficient, which shows the linear dependence between the model and observations. Ideally the correlation coefficient is equal to 1.

Pn h (t) µ =< h >= t=1 x (2.16) x x n

v u n u 1 X 2 σx = t (hx(t) − µx) (2.17) n t=1

n 1 X < hmodel(t) − µmodel > hobs(t) − µobs c = (2.18) 1 − n σ σ t=1 model obs

The second measure is the bias which demonstrates the difference between the observed and modeled value. Ideally the bias is equal to 0. The bias should be considered in tandem with at least one other measure of fit since positive and negative differences will cancel each other

19 out.

n X B = (hmodel(t) − hobs(t)) (2.19) t=1

The Nash-Sutcliffe model efficiency coefficient measures the predictive power of a model (Nash and Sutcliffe, 1970). It is usually used to asses hydrological models and the discharge values, but can be used for other models as well. Possible values of the coefficient range from − inf to 1, with 1 being the ideal value. A value of 0 means that the model doesn’t provide any more information than the average of all the observations.

Pn (h (t) − h (t))2 E = 1 − t=1 obs model (2.20) Pn 2 t=1(hobs(t)− < hobs(t) >)

The ﬁnal measure calculated was the root mean square error (RMSE). Ideally the RMSE is equal to 0.

r n P (h (t) − h (t))2 RMSE = t=1 model obs (2.21) n

20 (a) Low tide

(b) High tide

Figure 2.4: Hydraulic head normalized by the boundary values to visualize vertical gradients at low and high tides. The pink line shows the watertable level based on the highest saturated cell, while the black line shows the watertable level based on the head values

21 Chapter 3

Pﬂotran Model Comparison with Previous Studies

3.1 Boussinesq Based Analytical Solution and Simple

Field Study

In order to test the Pflotran model a simple field study with observational data was desired. The paper, Tidal Dynamics of the Water Table in Beaches, by Nielsen (1990) was chosen because it presented field data from a beach with negligible wave action, a simple sinusoidal tide, and fairly constant beach slope. By choosing a beach with no wave action the effects from the tide could be singled out and ensure that the model accurately captured the water table fluctuations due to tides.

The data was collected over a 25 hour period at Barrenjoey Beach, a sheltered beach north of Sydney, Australia, in April 1989. Measurements were taken every 30 minutes at eleven stilling wells spaced 2.5 meters apart. Nielsen (1990) determined the tide throughout the

22 duration of the experiment was made up of two harmonic constituents. The ﬁrst had a period of 12.25 hours and amplitude of 0.516 meters, while the second had a period of 6.125 hours and an amplitude of only 0.014 meters. Since the amplitude of the second constituent was less than three percent of the amplitude of the primary constituent, it was ignored in the modeling.

Nielsen derived three Boussinesq-based analytical models in this paper. All three assume the oscillations become completely damped out far from the beach so that

∂h → 0 as x → ∞. (3.1) ∂t

At the other end of the modeled domain, it was assumed the water table matched the sea level tidal height at the beach face. This means that no seepage face could form and the ocean and water table were always coupled so that

h((htide − D)cotβ, t) = htide. (3.2)

Where h is the height of the water table at the beach face [m] at time t, htide is the tidal height of the ocean [m], D is the depth to lower impermeable layer [m], and β is the slope of the beach. The ﬁrst solution additionally assumed a nearly vertical beach face and that the tidal amplitude was much smaller than the depth from the underlying impermeable layer to the mean sea surface. The solution to this most simple case is

h(x, t) = D + Acos(ωt − φ − kx)e−kx. (3.3)

Where h(x, t) is the height of the water table [m] x meters inland from the mean sea level boundary at time t [hr], D is the mean sea level in relation to some reference datum [m], A is the amplitude [m], ω is the angular frequency [rad/hr], and φ is the phase of the harmonic constituent [rad], and k is the wavenumber [m−1]. The second solution also assumed a nearly

23 vertical beach face, but that the tidal amplitude was not negligible compared to the depth from the underlying impermeable layer to the mean sea surface. The solution to this case is

A2 √ √ h(x, t) = D+Acos(ωt−φ−kx)e−kx + 1−2cos2(ωt−kx)e−2kx +cos(2ωt− 2kx)e− 2kx . 4D (3.4)

The ﬁnal solution took into account the slope of the beach surface and like the ﬁrst case, assumed the tidal amplitude was negligible compared to the beach depth. The Boussinesq type equation was solved using the perturbation parameter

= kAcot(β). (3.5)

The solution to this case, correct to the ﬁrst order of , is

√ 1 2 π √ √ h(x, t) = D + Acos(ωt − φ − kx)e−kx + A + cos 2ωt + − 2kx e− 2kx . (3.6) 2 2 4

The solution, to the second order of , is

√ 1 2 π √ √ h(x, t) =D + Acos(ωt − φ − kx)e−kx + A + cos 2ωt + − 2kx e− 2kx + 2 2 4 √ (3.7) 1 2 √ √ 2A − sin(ωt − kx)e−kx + sin(3ωt − 3kx)e− 3kx . 4 2

3.2 Richard’s Equation Based Analytical Solution

Kong et al. (2013) created a fundamentally different solution by starting with the 2-D Richard’s Equation which allowed both vertical and horizontal flows to be considered. This paper did not have a field data component for comparison, so this model was compared to the field data and vertical beach model from the Nielsen study. The 2-D Richard’s Equation

24 is

∂θ ∂ ∂Φ ∂ ∂Φ = K(Ψ) + K(Ψ) . (3.8) ∂t ∂x ∂x ∂z ∂z

Here, θ is the soil water content, Ψ is the pressure head, z is the elevation, Φ = Ψ + z is the hydraulic head, and K(Ψ) is the hydraulic conductivity. Since the goal was to include capillary eﬀects on the water table ﬂuctuations, the zone above the water table needed to be considered. Gardner’s equation was used to describe the water retention and hydraulic conductivity as follows:

αΨ θ = (θs − θr)e + θr for Ψ < 0 (3.9)

θ = θs for Ψ ≥ 0

αΨ K(Ψ) = Kse for Ψ < 0 (3.10)

K(Ψ) = Ks for Ψ ≥ 0

−1 Where α is the inverse of the length of capillary rise [m ], θs is the saturated water content m3 m3 [ m3 ], θr is the residual water content [ m3 ], and Ks is the saturated hydraulic conductivity m [ hr ]. Similar to the Boussinesq Equation based solutions, they first assumed pressure to be hydrostatic and the bottom and top boundary of the beach to be no flow boundaries. Kong et al. (2013) only produced a solution for the vertical beach face, so the boundary condition is defined at the vertical face. They eventually added in a correction to allow nonhydrostatic pressure by including dynamic pressure head, P , into the hydraulic head

∂P w equation, Φ = Ψ+z+P , where ∂z = − K , and w is the vertical Darcy ﬂow in the unsaturated

25 zone. Through perturbation parameters they were able to solve for the analytical solution,

−xkUS F1 h = D + Ae cos(ωt − xkUSF2)

where, r R1ω kUS = 2R2 R2 NUS = R3ω s N N (3.11) F = US + US 1 p 2 1 + N 2 1 + NUS US s N N F = US − US 2 p 2 1 + N 2 1 + NUS US

α(D−Z0) R1 = n(1 − e ) 1 R = K D + K (1 − eα(D−Z0)) 2 s s α n2 D2α2 R = (2eα(D−Z0) − 2 − α(D − Z )eα(D−Z0) + α(D − Z ) + ). 3 α 0 0 3

In this solution, Z0 is the depth from the lower impermeable layer to the upper beach surface.

If this solution is made to ﬁt the form given by the most simple solution given above in

−krx equation 3.3, h(x, t) = D + Acos(ωt − φ − kix)e , we get that

kr = kUSF1 (3.12) and

ki = kUSF2. (3.13)

26 Table 3.1: Physical Parameters used for the Analytical Comparisons

m3 Porosity 0.355 m3 Tortuosity 0.5 - Permeability 5.374x10−11 m2 m3 Residual Saturation 0.087 m3 Mualem-van Genuchten m 0.444 - Air entry value 5.43x10−4 P a−1 Gardner α value 0.16 m−1 Atmospheric Pressure 101325 P a kg Water Density 1025 m3 Depth to MSL (D) 5 m Depth to Surface (Z0) 8 m

Table 3.2: Model Settings used for the Analytical Comparisons

Newton-Solver Iterations 100 Absolute Tolerance 10−15 Time step 0.25 hr

3.3 Model Comparison

An analogous model was run in Pﬂotran using the measured distances given in the paper by Nielsen (1990). Values for hydraulic conductivity, porosity, or depth to underlying impermeable layer were unknown so estimates were made for a typically sandy beach.

3.3.1 Vertical Beach Results and Analysis

In the vertical case only the observations landward of the mean tide line were considered, which corresponds to wells 5 through 11.

Pﬂotran simulation tracks the timing and peak height values of the observations better than both analytical models. The Kong et al. (2013) solution agrees with the peak values and timing more than the vertical Nielsen (1990) model which dampens much more quickly and the peak time is shifted over an hour later in Well 11. This supports the theory that the

27 Pflotran Model Domain 4

3 Model Domain 2 Mean Sea Level Observation Points 1

−1

Elevation [m] −2

−3

−4

5 6 7 8 9 1011 −5

−6 0 20 40 60 80 100 120 140 160 180 200 Distance Inland from MSL [m]

Figure 3.1: Vertical beach domain used for Nielsen and Kong comparison.

Well 7 Well 9 Well 11

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

−0.2 −0.2 −0.2 Height [m above MSL] Height [m above MSL] Height [m above MSL]

−0.4 −0.4 −0.4

0 10 20 0 10 20 0 10 20 Time [h] Time [h] Time [h] (a) Well 7 (b) Well 9 (c) Well 11

Figure 3.2: Resulting water table fluctuations for the vertical beach case. unsaturated zone facilitates the transmission of pressure oscillations that drive the water table fluctuations, because Nielsen (1990) model only considers the saturated zone and horizontal flow while the Kong et al. (2013) and Pflotran model consider the the region above the water table and vertical flow. The Pflotran result appears to dampen the fluctuations more quickly than the Kong et al. (2013) model, but less so than the Nielsen (1990) model. All the models, however, underestimate the observed and remain centered near the mean sea level height, failing to capture the inland overheight.

28 3.3.2 Sloped Beach Results and Analysis

Pflotran Model Domain 4

−1

Elevation [m] −2

−3 Model Domain Mean Sea Level −4 Observation Points 1 2 3 4 5 6 7 8 9 1011 −5

−6 0 50 100 150 200 Distance Inland from MSL [m]

Figure 3.3: Sloped beach domain used for Nielsen comparison.

3 Model Domain Mean Sea Level 2 Observation Points

−1

Elevation [m] −2

−3

−4

1 2 3 4 5 6 7 8 9 10 11 −5

−10 −5 0 5 10 15 20 25 Distance Inland from MSL [m]

Figure 3.4: Close up of measurement observation points.

The characteristic shape of the tidally forced fluctuations, where the rising limb is steeper than the falling, is seen in the sloped solutions from Nielsen and in the Pflotran model result more readily than in the vertical models. As seen in Figure 3.5, the Pflotran simulation tracks the general shape of the oscillation better. The sloped Nielsen solutions draw out the falling limb more than was observed, however, the mean height more closely matches the

29 observed. The peak values and timing are similar between the two analytical solutions and Pﬂotran.

Well 7 Well 9 Well 11

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

−0.2 −0.2 −0.2 Height [m above MSL] Height [m above MSL] Height [m above MSL] −0.4 −0.4 −0.4

0 10 20 0 10 20 0 10 20 Time [h] Time [h] Time [h] (a) Well 7 (b) Well 9 (c) Well 11

Figure 3.5: Resulting water table ﬂuctuations for the sloped beach case.

The inland overheight is higher than in the vertical beach case. The mean water level asymp- totes to about 3.7 cm in the sloped Pflotran model while the observed levels were recorded to be about 5 cm. According to Nielsen, the overheight is due to some combination of the beach slope, non-linearity in the system, and the seepage face which forms when the tide falls faster than the water table at the beach face. The analytical solutions, however, cannot account for the seepage face formation and this is reflected in the results in that the analytical solutions underestimate the inland overheight. On the other hand, the Pflotran model allows the water table near the beach surface boundary to lag behind the tidal movement, but it’s inland overheight is still below the observed. The observed overheight at the most landward well was 0.29 m above mean sea level. In comparison, the mean water levels of the analytical models are 0.13 m and the mean water level of the Pflotran model is 0.08 m. The observed water levels are higher than all three sloped model solutions so it seems there may be more at play than simple tidal forcing.

From the time series at each location the wavenumber can be extracted using the fast Fourier

30 transform in Matlab as described in the methods section. The wavenumber found from the amplitude field data was 0.093 m−1, while the Pflotran simulation yielded a damping rate of 0.10 m−1, and the sloped Nielsen solution 0.0967 m−1. The wavenumber extracted from the phase lag observations was equal to 0.056 m−1, while the Nielsen model gave wavenumber of 0.0955 and, and the Pflotran simulation resulted in a wave number of 0.072.

3.4 Discussion

The discrepancies between modeled and observed mean height of the water table, amplitude damping rate, and the elevated trough of the oscillations are of interest. While Nielsen proposes some ideas for the cause of this, they are all based on the Boussinesq approximation which oversimplifies some physical processes. The higher observed mean water level and elevated trough may be due to a westward flux of water or wave setup, both of which were ignored by Nielsen. Looking at the field site, shown in Figure 3.6, suggests that both of these may be important factors. It is not mentioned in the paper that the Barrenjoey Beach field site sits on the eastern side of a thin peninsula whose western side, Palm Beach, is exposed to ocean tides and significant wave action later recorded by Nielsen and Hanslow (1991).

At its thinnest point the peninsula is about 190m wide and may be narrow enough for the ocean side to influence the observations at the calm eastern side. This could explain why the observed overheight is higher than any of the models predicted. According to years of swell data at the ocean side, the average wave setup for the time of year of the field study is 0.57m (Nielsen and Hanslow, 1991; MHL, 2013a,b). This is not a negligible amount and could set up a sloped water table with westward flux under the beach. The influence of tidal forcing from the ocean side and wave setup is explored in the next chapter.

The rate of amplitude attenuation is inﬂuenced by soil properties and the depth to the

31 Nielsen Field Site

¯ 0 1,250 2,500 5,000 Meters

Source: Esri, DigitalGlobe, GeoEye, Earthstar Geographics, CNES/Airbus DS, USDA, USGS, AEX, Getmapping, Aerogrid, IGN, IGP, swisstopo, and the GIS User Community Figure 3.6: Aerial Image of Barrenjoey Beach. underlying impermeable layer. As stated before, there were no field measurements of soil properties and Nielsen inferred these by fitting the field data to his solutions to the Boussineq equations which ignore several physical processes at work in the system. His conclusions are also inconsistent with assumptions made in getting the solutions. For instance, in two of the three solutions the tidal amplitude is assumed to be very small compared to the depth in the derivation of the solution, however, Nielsen claims that the aquifer depth is be about 0.51 m thick which is approximately equal to the tidal amplitude of 0.516 m.

In order to better understand the role of boundary conditions and soil properties the next chapter explores the eﬀects of soil porosity, hydraulic conductivity, depth, wave setup, and additional tidal forcings on the resulting water table ﬂuctuations in the beach.

32 Chapter 4

Inﬂuence of Additional Forcing and Beach Geometry

4.1 Inﬂuence of Additional Forcing

4.1.1 Second Tidal Forcing

Adding the same tidal forcing to the opposite end of the domain mimics what occurs under areas of land- such as islands, peninsulas, and ithsmus’- that are surrounded by water whose height goes through a cyclic pattern such as the ocean, a bay, or a dam-controlled river. The same beach slope was used at both ends of the domain and the tidal forcing was applied in phase at either end.

The peak value of the mean overheight remained about the same changing only from 10.5 cm to 11 cm. The most notable diﬀerence is that the mean water level is approximately equal through the entire beach at about 10 cm above mean sea level, whereas, in the case with only a single tidal forcing, the mean water level drops oﬀ asymptotically after 11 m

33 inland to a small overheight of 3.4 cm.

Looking at the midpoint between the two tidal boundaries we can see a change in the maximum heights the water table reaches. In the case with tidal forcing from one side, the water table is about 7 cm above mean sea level, while tides at both ends caused the water table to be 11 cm above mean sea level. The diﬀerence only becomes more pronounced closer to the east boundary.

Mean and Maximum Water Level [m] 0.6

0.5

0.4

0.3 Height [m above MSL]

0.2

0.1

0 20 40 60 80 100 120 140 160 180 200 Distance Inland from Mean Tide Line [m]

Figure 4.1: Eﬀect of adding a second tidal forcing on the mean water level compared to single tidal forcing. Mean (—) and maximum (- - -) water levels shown.

4.1.2 Wave Setup

Rather than resolving the eﬀect of individual waves occurring with a period of less than a minute the average wave setup was calculated using the method described in Stockdon et al. (2006). Adding a ﬁxed wave setup to the tidal boundary condition essentially increases the water table at all locations by that same amount. However, if the wave setup were

34 0m Inland 10m Inland 1.5 1.5

1 1

0.5 0.5

0 0

−0.5 −0.5 Water Table Height [m above MSL] 0 5 10 15 20 25 Water Table Height [m above MSL] 0 5 10 15 20 25 Time [h] Time [h]

50m Inland 100m Inland 1.5 1.5

1 1

0.5 0.5

0 0

−0.5 −0.5 Water Table Height [m above MSL] 0 5 10 15 20 25 Water Table Height [m above MSL] 0 5 10 15 20 25 Time [h] Time [h]

Figure 4.2: Effect of adding a time-averaged wave setup on water table levels. different over time to reflect current wave conditions it would change the tidal boundary condition non-monotonically, which would be translated inland as spatially varied water table elevations compared to the tide only case.

4.1.3 Overtopping

Overtopping occurs when a wave runs up past the crest of the beach and ponds on the upper beach surface. The ponded water then seeps into the sand aﬀecting the local water table. This phenomenon was modeled by applying a hydrostatic pressure to the crest section of the

35 0m Inland 10m Inland 2 2

1.5 1.5

1 1

0.5 0.5

0 0

−0.5 −0.5 0 100 200 300 400 500 0 100 200 300 400 500 Water Table Height [m above MSL] Water Table Height [m above MSL] Time [h] Time [h] 50m Inland 100m Inland 2 2

1.5 1.5

1 1

0.5 0.5

0 0

−0.5 −0.5 0 100 200 300 400 500 0 100 200 300 400 500 Water Table Height [m above MSL] Water Table Height [m above MSL] Time [h] Time [h]

Figure 4.3: Time series showing the eﬀect of wave overtopping resulting in various durations of ponding. beach surface, at 30 to 32 m inland, to mimic about 2 cm of ponded water. There was about a 10 minute delay from the time ponding started until it percolated to the water table. The duration of ponding has a large eﬀect on how high and for how long the water table becomes elevated. Directly under the ponding region, 15 minutes of ponding resulted in maximum height of 0.24 m above MSL and the water table returned within 1cm of its normal level after 86 hours. Meanwhile two hours of ponding resulted in 0.88 m increase and it took 230 hours to recover and 12 hours of ponding resulted in a 1.9 m rise that took more than 400 hours to recover. After 300 hours the water table directly under the ponding location was

36 still about 20 cm higher than the simulation without ponding. The water table elevation eﬀects are diminished but the recovery time takes signiﬁcantly longer farther from shore. At 100 m inland from the shoreline, a short 15 minute event results in a slightly elevated water table that remains 11 cm over it’s steady state for more than 300 hours, or 12 days. Twelve hours of ponding results in a water table height that peaks at 42 cm above MSL and remains 30 cm higher for about 100 hours and stays 20 cm higher for more than 12 days.

As see in Figure 4.4, once the water reached the water table, the water table rose both inland and seaward of the ponded region. Rather than soley draining out to sea as might be expected, a significant portion of the infiltrated water moved inland and caused the water table to rise even far from the ponding location. This means that if there were multiple wave overtopping events or storms within a couple days of each other, this effect would add up and could result in groundwater inundation even far from the shoreline. This behaviour agrees with the hysteretic effects observed during increased wave action by Cartwright (2014); Xin et al. (2015). This stresses the importance of including wave overtopping and subsequent ponding in the modeling otherwise the water table elevations will be significantly underestimated.

37 Mean and Maximum Water Level [m] 3

2.5

1.5

1 Height [m above MSL]

0.5

0 0 20 40 60 80 100 120 140 160 180 Distance Inland from Mean Tide Line [m]

Figure 4.4: Mean (—) and maximum (- - -) watertable elevations resulting from various durations of ponding.

38 4.2 Inﬂuence of Beach Geometry and Geology

4.2.1 Depth

Three model domains, ranging from 2.5 to 10 m between mean sea level and the no flow bottom, were created to test the effect of depth on water table fluctuations. Changing the distance between the mean sea level and the lower bound of the domain had a marked effect on the water table fluctuations. Decreasing the depth caused the time averaged water levels to increase near the tidal boundary condition and even far inland remained slightly above simulations with a larger depth. When looking at the water table fluctuations at a single point, as in the 10m case of Figure 4.5, the depth influences how the oscillation strays from the normal sine wave of the tide. A shallower depth causes the falling water level to be more drawn out and the rise to remain steeper, while a deeper depth had less distortion from the normal sine wave shape. Decreasing the depth caused the amplitude of the fluctuations to attenuate faster and decreased the velocity of the oscillation traveling through the region.

39 0m Inland 10m Inland 0.6 0.6

0.4 0.4

0.2 0.2

0 0

−0.2 −0.2

−0.4 −0.4

Water Table Height [m above MSL] 0 5 10 15 20 25 Water Table Height [m above MSL] 0 5 10 15 20 25 Time [h] Time [h]

50m Inland 100m Inland 0.6 0.6

0.4 0.4

0.2 0.2

0 0

−0.2 −0.2

−0.4 −0.4

Water Table Height [m above MSL] 0 5 10 15 20 25 Water Table Height [m above MSL] 0 5 10 15 20 25 Time [h] Time [h]

Figure 4.5: Eﬀect of beach depth on water table ﬂuctuations.

4.2.2 Beach Slope

The influence of beach slope was tested by creating several model domains that had the same x dimensions, but varied in the z direction to achieve beach slopes ranging from 0.02 to 0.2. The most notable effect was that decreasing the beach slope increased the time averaged water level. This is probably because the shallower slope slows down water draining out of the beach and is why many solutions that derived using a simplified vertical beach face underestimate the location of the water level. Luckily, measuring the beach profile and deriving the slope is one of the easier parameters to collect in fieldwork. As seen in Figure 4.7, a beach slope of 0.02 resulted in an overheight of 21 cm, while a beach slope of 0.2 only produced an overheight of about 4 cm. However, the results from beach slopes of 0.1, 0.15 and 0.2 are nearly identical so it seems that the water table fluctuations are more sensitive to changes in shallow slopes than they are over some threshold near the 0.1 slope. This agrees

40 with the claim by Nielsen and Hanslow (1991) that a slope of 0.1 is the cutoff between shallow and steep beaches. Similarly, the location of the maximum of the time averaged water table height shifts further inland from the mean tide location with decreasing beach slope. When considering the fluctuation of the water level at a single location, as shown in Figure 4.6, the beach slope greatly affects the rate of amplitude attenuation and also the timing of the peaks.

0m Inland 10m Inland 0.6 0.6

0.4 0.4

0.2 0.2

0 0

−0.2 −0.2

−0.4 −0.4

Water Table Height [m above MSL] 0 5 10 15 20 25 Water Table Height [m above MSL] 0 5 10 15 20 25 Time [h] Time [h]

50m Inland 100m Inland 0.6 0.6

0.4 0.4

0.2 0.2

0 0

−0.2 −0.2

−0.4 −0.4

Water Table Height [m above MSL] 0 5 10 15 20 25 Water Table Height [m above MSL] 0 5 10 15 20 25 Time [h] Time [h]

Figure 4.6: Time series of water levels with various beach slopes.

41 Mean and Maximum Water Level [m] 1

0.9

0.8

0.7

0.6

0.5

0.4 Height [m above MSL]

0.3

0.2

0.1

0 0 20 40 60 80 100 120 140 160 180 Distance Inland from Mean Tide Line [m]

Figure 4.7: Eﬀect of beach slope on mean (—) and maximum (- - -) water levels.

4.2.3 Porosity

The soil porosity was varied from 0.2 to 0.5, which represents the full range of fine clays to coarse sand and gravel. Decreasing porosity drives higher water table near shore, but lower far from shore. However, there was no significant variation in water levels compared to wide range of porosity values tested. Increasing the porosity raises the low point of the water table oscillations while maintaining similar peak values and times. This is reflected in the plot of wavenumbers where kr, the rate of amplitude attenuation, changes much more rapidly than ki, the rate of change of the phase lag, which correlates to the time lag.

42 0m Inland 10m Inland 0.6 0.6

0.4 0.4

0.2 0.2

0 0

−0.2 −0.2

−0.4 −0.4

Water Table Height [m above MSL] 0 5 10 15 20 25 Water Table Height [m above MSL] 0 5 10 15 20 25 Time [h] Time [h]

50m Inland 100m Inland 0.6 0.6

0.4 0.4

0.2 0.2

0 0

−0.2 −0.2

−0.4 −0.4

Water Table Height [m above MSL] 0 5 10 15 20 25 Water Table Height [m above MSL] 0 5 10 15 20 25 Time [h] Time [h]

Figure 4.8: Time series showing eﬀect of porosity on water table ﬂuctuations.

4.2.4 Permeability

The eﬀect of permeability on water table ﬂuctuations was tested by varying the permeability between 9.95x10−10 and 9.95x10−14m2 which corresponds with hydraulic conductivity rang-

− − m ing from 10 2 to 10 6 s . Lower soil permeability was associated with higher time averaged water level near the tidal boundary. Decreasing the hydraulic conductivity caused both the high and low points of the oscillations to increase. The timing of the peaks remained the nearly the same with the lower conductivity having a slower decline which caused the peak height to occur slightly earlier and the low slightly later.

Unfortunately, hydraulic conductivity is one of the hardest parameters to accurately measure.

43 Depending on where the water levels are needed to be known, however, it may be suﬃcient to approximate with the correct order of magnitude. After 20m inland on a 0.1 slope beach the peak water table levels are within 20 cm of each other and after 100m inland within 10 cm of each other.

0m Inland 10m Inland 0.6 0.6

0.4 0.4

0.2 0.2

0 0

−0.2 −0.2

−0.4 −0.4

0 5 10 15 20 25 0 5 10 15 20 25

50m Inland 100m Inland 0.6 0.6

0.4 0.4

0.2 0.2

0 0

−0.2 −0.2

−0.4 −0.4

0 5 10 15 20 25 0 5 10 15 20 25

Figure 4.9: Time series showing eﬀect of permeability on water table ﬂuctuations.

4.3 Discussion

Values for soil parameters and beach geometry of possible beach material ranging from ﬁne silty sand to gravel were considered. Since ﬂooding is one of the main concerns when

44 considering water table fluctuations this means that accurately capturing the peak water levels is most important. Based on the model results, it appears that beach slope, wave setup, and hydraulic conductivity affected the peak water table heights the most. Also, including wave overtopping events is crucial to capturing accurate water levels in the simulations. The shape of the water table fluctuations, considering the minimum water table level and rate of dropping, was most influenced by beach depth, beach slope, and hydraulic conductivity.

Considering what was discovered in testing the effects of various parameters, I tried to improve the Pflotran model of the Nielsen (1990) field data. In order to better represent the physical system present in the Nielsen study, wave set up was added to both the study side and the ocean side of the beach domain, and the depth was decreased to 2.5 m. The wave setup was set at 10 cm on the sheltered side of the domain, where the study was conducted, and 57 cm on the ocean side of the domain. These values of wave set up were determined using average historical offshore swell observations (1.6 m swell at 10.3 second interval (Nielsen and Hanslow, 1991; MHL, 2013a,b)) and the author’s estimations of observed wind waves on the protected side (5 cm waves with 10 second interval (Nielsen, 1990)). As seen in Figure 4.10, the adjusted pflotran model better matches the observed water levels. By taking into account waves, even with a simple setup approximation, the model results much more closely fit what was observed in the field.

Table 4.1: Calculation of model ﬁt at Well 7

Measure of Fit Single Tidal Forcing Double Tidal Forcing and Wave Setup Correlation 0.96 0.97 Bias -7.05 m -1.84 m Nash-Sutcliﬀe -0.03 0.73 RMSE 0.16 m 0.08 m

45 Well 11 Well 7 Well 9 0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

−0.2 −0.2 −0.2 Height [m above MSL] Height [m above MSL] Height [m above MSL] −0.4 −0.4 −0.4

0 10 20 0 10 20 0 10 20 Time [h] Time [h] Time [h] (a) Well 7 (b) Well 9 (c) Well 11

Figure 4.10: Result from adding the second tidal forcing on the ocean side and wave set up to both sides of the domain compared to the single tidal forcing from the previous chapter and Nielsen ﬁeld observations.

46 Chapter 5

Cardiﬀ State Beach

Water level data was collected at Cardiff State Beach in Southern California by our collab- orators at the Scripps Institute of Oceanography (SIO). The study area sits on a sand spit 190 m wide between the Pacific Ocean and San Elijo Lagoon. The water in the lagoon is hydraulically linked to the ocean through a direct surface connection at its northwestern end so its level changes with the tides. This means the groundwater under the sand spit is receiving tidal forcing from two sides. Six pressure transducers were buried in the beach going inland perpendicular to the shoreline and measured pressures at a 2 HZ frequency. Two sensors were placed at different depths at the same coordinates 75 meters inland from the mean tideline, just past the crest of the beach surface. One was placed below the water table to measure the water table fluctuations and the other was placed above the highest tide level to measure only wave overtopping events. The furthest inland transducer was placed in a channel of the lagoon on the east side of the highway in order to record the water level in the lagoon.

The soil in the study area is primarily poorly graded sand mixed with some silt and gravel in a massive structure. A nearby borehole from July 1990 provided soil type and grain size

47 data for the top 30.78 meters. The upper samples were approximately 89-96% sand and 5-10% silt until 21 meters below the surface where it became 85% gravel, 9% sand, and 6% silt.

The San Elijo Lagoon Conservancy publishes weekly water quality measurements of the lagoon taken at several locations. Their measurements indicate salinity levels of 30-35 ppt in the area of the pressure transducers which recorded water levels in this study (Nussbaum, 2016). This means that variable density ﬂow doesn’t need to be considered as the bodies of water on either side of the model domain have the same characteristics.

5.1 Field Data

The data was collected at 2 HZ and then averaged into the desired frequency by SIO. The water table oscillations were averaged into 6 minute intervals and normalized relative to measured atmospheric pressures and then converted into a length equivalent assuming a hydrostatic pressure distribution. The P3 sensor was smoothed into 6 second intervals to preserve the detail of overtopping events. The following results and analysis correspond to the time period from January 1, 2015 12:00 am to January 24, 2015 11:54 pm.

5.2 Cardiﬀ Beach Model

The model domain was based on UAV and LIDAR surface profile measurements of the field site provided by SIO. Approximations for the soil characteristics used accepted parameter values for the soil types described in the nearby soil profile. The average values of the top ten meters were used to make the model a homogenous approximation of the field site.

kg The density of water was taken to be 1025 m3 based on local salinity and temperature

48 Cardiff Field Site ¯ ¯ 0 500 1,000 2,000 Meters 0 500 1,000 2,000 Kilometers

Source: Esri, DigitalGlobe, GeoEye, Earthstar Geographics, CNES/Airbus DS, USDA, USGS, AEX, Getmapping, Aerogrid, IGN, IGP, swisstopo, and the GIS User Community, Esri, HERE, DeLorme, Esri, HERE, DeLorme, MapmyIndia, © OpenStreetMap contributors, and the GIS user community MapmyIndia, © OpenStreetMap contributors, and the GIS user community

P4P2 P6 P3 P7

Borehole ¯ 0 25 50 100 Meters

Source: Esri, DigitalGlobe, GeoEye, Earthstar Geographics, CNES/Airbus DS, USDA, USGS, AEX, Getmapping, Aerogrid, IGN, IGP, swisstopo, and the GIS User Community, Esri, HERE, DeLorme, MapmyIndia, © OpenStreetMap contributors, and the GIS user community Figure 5.1: Cardiﬀ State Beach ﬁeld site.

49 Pflotran Model Domain

P2 P6

0 P4 P1 P7

−5 Elevation NAVD88 [m]

Model Domain Mean Sea Level Observation Points −10

−50 0 50 100 150 200 Distance Inland from MSL [m]

Figure 5.2: Model domain used for Cardiﬀ ﬁeld data comparison.

Table 5.1: Physical Parameters used for the Cardiﬀ Simulations

m3 Porosity 0.355 m3 Tortuosity 0.5 - Permeability 5.5x10−11 m2 m3 Residual Saturation 0.087 m3 Mualem-van Genuchten m 0.444 - Air entry value 5.43x10−4 Pa−1 Atmospheric Pressure 101325 Pa kg Water Density 1025 m3 Depth to MSL (D) 10 m measurements.

The ocean water levels were created using historical tidal data from the La Jolla NOAA station and combining it with wave data from the Torrey Pines CDIP buoy (NOAA, 2016; OREG, 2016). The Torrey Pines Outer buoy (Station 100) provided historical wave height and period about 13.6 km south west of the pressure sensors. Data were available in 30 minute increments which were interpolated to match the 10 minute interval of the tidal data using the 1-D piecewise cubic interpolation method in Matlab. The time averaged wave setup was calculated following Stockdon et al. (2006). The measurements recorded at the

50 Table 5.2: Model Settings used for the Cardiﬀ Simulations

Newton-Solver Iterations 100 Absolute Tolerance 10−15 Time step 0.1 hr pressure transducer on the lagoon side were used as the inland lagoon boundary condition of the model. The third source of forcing was occasional wave overtopping. This was modeled by assigning a region of the beach to have some ponding when overtopping occurred. The duration of ponding was determined using data from the P3 sensor buried near the surface that showed a response to wave overtopping events. The remaining pressure sensors were used to assess the Pﬂotran model result.

The P3 sensor succeeded in recording overtopping events and also showed that there is a vertical gradient in the beach. The average observed gradient between the P3 and P4 sensors over the 25 day period in January was 0.1 m/m. This means that the previous approximations that assume only horizontal ﬂow in analytical solutions may not be appropriate in regions close to the ocean boundary or above the water table.

5.3 Model Results and Discussion

The model was first run considering forcing from the tide and lagoon water levels only. Shown in blue in Figure 5.3, only considering the tide and lagoon severely underestimates the water level under the beach at sensor P4. Next, wave effects were added as wave-set and wave runup height in addition to the tide and lagoon forcing. These improved the overall result, but still underestimated the times when ponding occurred. Adding ponding greatly improved the model fit at the P4 sensor around January 21st, shown in red and magenta in Figure 5.3. This shows the importance of considering wave overtopping when looking at water levels near shore. However, there were still some times where the model

51 under-predicted the observed water level around January 11th and from January 22nd-25th.

Even considering possible run-up effects there is a mismatch between the model and observations around 1/22/2015. It is possible that a portion of the beach near the transect was overtopped and the resulting ponded water infiltrated and flowed toward the P4 sensor, but was low enough to miss the P3 sensor. This theory matches the relatively smaller increase in water level observed than when overtopping occurred directly above the sensors.

Table 5.3 shows that the model ﬁt measurements indicate using runup instead of set up improves the simulated water table heights at the P4 sensor while it over estimates the heights at P7. At the P1 sensor, the model agreement with the observations is predictably near perfect since the recorded lagoon water levels were used as the boundary condition.

Overall, the Pﬂotran model was able to simulate water table levels more accurately than any previous model. This was enabled by being able to consider not only the tide, but also various wave eﬀects including setup, run-up, and overtopping.

Table 5.3: Model ﬁt at selected sensor locations.

Measure of Fit Tide + Lagoon Tide + Lagoon + Tide + Lagoon + Setup + Ponding Runup + Ponding P7 Correlation 0.965 0.967 0.961 Bias 354 m 824 m 1730 m Nash-Sutcliffe 0906 0.820 0.416 RMSE 0.130 m 0.180 m 0.324 m P4 Correlation 0.792 0.905 0.902 Bias -1540 m -906 m -276 m Nash-Sutcliffe -0.636 0.449 0.780 RMSE 0.331 m 0.192 m 0.121 m P1 Correlation 0.999 0.999 0.999 Bias -1.29 m -0.830 m -0.385 m Nash-Sutcliffe 0.998 0.998 0.998 RMSE 0.008 m 0.008 m 0.008 m

2.4

2.2

1.8

1.6

1.4 Water Table Height [m above MSL] 1.2

1 01−Jan−2015 00:00:00 07−Jan−2015 06:00:00 13−Jan−2015 12:00:00 19−Jan−2015 18:00:00 26−Jan−2015 00:00:00 Date

(a) P1

3.5

2.5

1.5 Water Table Height [m above MSL] 1

0.5 01−Jan−2015 00:00:00 07−Jan−2015 06:00:00 13−Jan−2015 12:00:00 19−Jan−2015 18:00:00 26−Jan−2015 00:00:00 Date

(b) P4

2.5

1.5

Water Table Height [m above MSL] 0.5

0 01−Jan−2015 00:00:00 07−Jan−2015 06:00:00 13−Jan−2015 12:00:00 19−Jan−2015 18:00:00 26−Jan−2015 00:00:00 Date

Figure 5.3: Resulting water table fluctuations from three different simulations in Pflotran considering no wave effects, wave runup, and wave setup. All three simulations also included tidal and lagoon forcing.

53 Chapter 6

Concluding Remarks

The previous chapters have demonstrated the success with which Pflotran can model water table levels. Unlike previous analytical models, Pflotran can take into account irregular boundary shapes and time-varying boundary conditions such as wave setup and overtopping in addition to the regular tidal harmonics. The comparison with the Nielsen (1990) and Cardiff field data show the advantage of this flexibility.

It is not enough to only consider the tide from one side and simply assume the oscillations die out at some distance from the ocean-beach interface when looking at water levels in a region with multiple forcings; the forcings from all sides need to be considered. As demonstrated with the Nielsen (1990) data, once the tides and wave setup from both the Barrenjoey and Palm Beach sides of the sandspit were considered, the Pflotran simulation results closely matched the observed water levels. Similarly, modeling the Cardiff field data required con- sideration of the wave setup, run-up and overtopping events in order to have any meaningful results.

This study demonstrates the importance of considering wave eﬀects in addition to tidal forcing when modeling water table levels near the coast. It also demonstrates the need for better

54 handling of wave run-up. This study showed considerable improvement in modeling water table fluctuations when the wave run-up was considered, even with only a simple approximation for run-up height. A better way to determine an appropriate run-up height value, similar to the time-averaged wave setup, is needed to include the role of wave run-up. A controlled study at various temporal discretizations ranging from resolving individual wave run-up events up to calculating some effective aggregate run-up value on the timescale of interest for water table fluctuations is ultimately desired. This would provide a better understanding of how wave run-up affects water table fluctuations in the near-shore environment. Essentially, improving our understanding of wave run-up would improve water table modeling results. Given the current estimates of sea level change, and the danger of groundwater flooding in areas with shallow aquifers where a fraction of a meter could mean a big difference, a better understanding of wave effects is essential to providing more accurate water table level predictions in the near shore zone of coastal aquifers.

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