Using earth-tide induced water pressure changes to measure in-situ permeability: 1

a comparison with long-term pumping tests 2

3

Vincent Allègre (1), Emily E. Brodsky (1), Lian Xue (1), Stephanie M. Nale (1), Beth L. Parker (2), and 4

John A. Cherry (2) 5

(1) Earth and Planetary Sciences Department, University of California, Santa Cruz, United States 6

(2) University of Guelph, G360 Centre for Applied Groundwater Research, Guelph, Canada 7

8

Corresponding author: Vincent Allègre, Earth and Planetary Sciences Department, University of 9

California Santa Cruz, 552 Red Hill Rd., Santa Cruz, CA 95064, USA (vallegre@ucsc.edu) 10

Now at University of Bordeaux, Institute of mechanical engineering, Dept. Civil & Environmental 11

Engineering, CNRS UMR 5295, 33615, Pessac, France. 12

13

14

KEY POINTS 15

● Earth tidal responses yield hydraulic properties consistent with pumping tests 16

● Responses require a vertical flow model of drainage to the water table 17

● Results suggests vertical connectivity of the fracture network 18

19

20

21

22

23

24

ABSTRACT 25

Good constraints on hydrogeological properties are an important first step in any quantitative model of 26

groundwater flow. Field estimation of permeability is difficult as it varies over orders of magnitude in 27

natural systems and is scale-dependent. Here we directly compare permeabilities inferred from tidal 28

responses with conventional large-scale, long-term pumping tests at the same site. Tidally induced 29

water pressure changes recorded in wells are used to infer permeability at ten locations in a densely 30

fractured sandstone unit. Each location is either an open-hole well or a port in a multilevel monitoring 31

well. Tidal response is compared at each location to the results of two conventional, long-term and 32

large scale pumping tests performed at the same site. We obtained consistent values between the 33

methods for a range of site-specific permeabilities varying from ~10-15 m2 to 10-13 m2 for both open 34

wells with large open intervals and multilevel monitoring well. We conclude that the tidal analysis is 35

able to capture passive and accurate estimates of permeability. 36

INTRODUCTION 37

The efficient mapping of subsurface properties is one of the major concerns of field studies in 38

hydrogeology. Applications ranging from groundwater management to hydraulic fracturing rely on 39

accurate, in situ estimates of permeability. The dominant method for determining permeability in situ is 40

the analysis of pumping tests [e.g., Theis, 1935; Jacob, 1940]. Complex geometry, boundary 41

conditions, and parameterizations of heterogeneity have been incorporated into numerous analytical 42

pump test solutions [e.g., Dawson and Istok, 1991]. 43

44

An alternative method to probe permeability is to utilize the tidal response. A well drilled in a confined 45

formation may exhibit water pressure oscillations forced by solid earth tides with dominantly diurnal 46

and semidiurnal periods [Bredehoeft, 1967]. This method is distinct from work that uses the oceanic 47

tide in near-coastal aquifers. Since the solid earth tide exists globally, tidal oscillations are potentially 48

observable in any aquifer on land. A theoretical framework has been proposed to use that signal in 49

order to get permeability by computing the phase and amplitude response of the recorded pressure 50

fluctuations relative to the strains creating them [Hsieh et al., 1987]. The phase shift and amplitude 51

ratio can be converted to hydraulic properties for a given system geometry. 52

53

Although the method has only been sporadically used, tidal response has three major potential 54

advantages for permeability measurements. First of all, tidal responses are sensitive to a scale ranging 55

from one to tens of meters that is difficult to reach with conventional methods [Hsieh et al., 1998]. In 56

natural systems, permeability is highly scale-dependent, especially in fracture-dominated systems such 57

as fault-damage zones, and therefore the fine-scale sensitivity of the tidal responses can potentially 58

separate local effects [e.g., Bense et al., 2012; Caine et al., 1996]. Secondly, tidal responses are a 59

strictly passive method that avoids any possible perturbation of the system through injection or 60

pumping. Finally, continual forcing by the tide opens the door for long-term monitoring of hydraulic 61

conductivity and significant changes have been observed over time in Southern California and Western 62

Sichuan [Elkhoury et al., 2006; Xue et al., 2013]. Conventional pumping tests are unrealistic for long-63

term monitoring of time-dependent permeability. 64

65

Signals driven by the solid earth tide have been used to infer permeability in a handful of studies 66

[Cooper et al., 1965; Bredehoeft et al., 1967; Bower, 1983; Hsieh et al., 1987; Elkhoury et al., 2006; 67

Burbey et al., 2012; Xue et al., 2013; Lai et al., 2014], but to the best of our knowledge analysis of the 68

earth tide response has not yet been compared to conventional methods in a field study. Therefore, in 69

this paper we present such a comparison. After providing details of the field site, we report 70

permeability measurements inferred from the tidal response measured at ten locations, and compare 71

them to permeabilities obtained from the fit of pumping test drawdown curves at the same locations. 72

The permeability values are inferred using a vertical flow model, that is consistent with the vertical 73

connectivity of the fracture network [e.g., Cilona et al., 2016]. We use the range of estimates derived 74

from different data drawdown solutions as a measure of the epistemic uncertainty and then compare 75

this range to the difference between the tidal and pumping test solutions. We find the tidal analysis 76

yields values compatible with the pumping tests. 77

78

2. SITE AND DATA ACQUISITION 79

2.1 Geological and hydrological context 80

The site is located in a section of the Santa Susana Field Laboratory in the Simi Hills in Southern 81

California. This site is an inactive engineering test facility with industrial contamination in portion of 82

the fractured rock system [Sterling et al., 2005]. The geological formation at Santa Susana Field 83

Laboratory is part of the Chatsworth Formation encountered in the Simi Hills west of the San Fernando 84

Valley, and consists of a late Cretaceous to late Pliocene sandstone that was deposited in a turbiditic 85

environment [Link et al., 1984; Cilona et al., 2016]. The average unfractured porosity is around 14% 86

[e.g., Sterling et al., 2005]. The predominant coarse sandstone members alternate with finer grain units, 87

which typically consist of inter-bedded sandstone, siltstone, and shale. A detailed stratigraphy and the 88

structural geology study of the area can be found in Cilona et al. [2016]. 89

90

The major structural feature of the site is a 35o stratigraphic dip associated with a series of faults, some 91

of which have previously been shown to generate discontinuities in hydraulic head [Cilona et al., 92

2015]. The studied area is bounded to the west by a major SW-NE fault known locally as the Shear 93

Zone that is characterized by a 60 m hydraulic head difference across it (Fig. 1). A set of sub-parallel 94

faults is oriented ESE-WNW and extends to the Shear Zone fault on their western part. The 95

northernmost of this set is the IEL (Instrument Equipment Lab) fault that was identified by prior 96

hydraulic head mapping as a conductive structure [e.g., MWH, 2009; Meyer et al., 2014]. In the south, 97

the Happy Valley (HV) Fault Zone is delimited by two main fault traces. According to the same 98

hydraulic head dataset there is no evidence that the Happy Valley fault acts either as a barrier or a 99

conduit. 100

101

2.2 Field Experiment 102

We use the water level data monitored before, during, and after two constant-rate pumping tests 103

conducted independently (Fig. 1). The tests were each about one month long, and the recordings lasted 104

approximately 6 months before, during and after the actual test. The pumping wells were open-holes C-105

1 (test I) and RD-10 (test II), which were pumped at an average rate of 40 GPM (2.5x10-3 m3/s) and 106

29.5 GPM (1.9x10-3 m3/s), respectively. The drawdown was recorded at multiple monitoring wells 107

(Fig. 1; Table 1). The baseline records prior to the pumping tests provide an opportunity to measure 108

earth tides. 109

110

The conventional monitoring wells in this study have intervals open to the formation from 146 m to 92 111

m thick (below the top of the casing), while the intervals from the multi-level systems range from 6 m 112

to 1.2 m, and all intersect units from the upper Chatsworth Formation. Both pumping tests were 113

performed in mainly coarse-grained sandstones (Fig. 1). The open wells noted on Table 1 are open to 114

the formation over a large interval and have a free surface. Wells RD-72, RD-31 and RD-103 have 115

hydraulically isolated intervals that are separated by packers using FLUTeTM in RD-72 and Westbay 116

systems (Schlumberger Water Services, SWS) in RD-31 and RD-103. Using both systems, the pressure 117

measurement can be made in each interval by continuously monitoring a port [e.g., Cherry et al., 2007; 118

Keller et al., 2013; Meyer et al., 2014]. The water level data used for both methods (tidal response and 119

aquifer tests) were recorded in open holes using Micro-divers (SWS) during test II, and MiniTroll 120

pressure transducers (In-Situ Inc.) during test I. The ports in Westbay systems were monitored with the 121

buit-in transducers (MOSDAX© pressure probe), and the port in the FLUTeTM system was monitored 122

with a MiniTroll transducer. The dataset is sampled at 10 minute intervals, and sensor accuracy ranges 123

from 1 cm to 3.5 cm of water head in open holes, and is 3.5 cm of water head in multi-level systems. 124

125

Water pressures were monitored at RD-35B and RD-72 during test I, and at RD-01, RD-02, C-2, RD-31 126

and RD-103 during test II (Fig. 1). The well RD-31 had three ports available connected to separate 127

units at different depths. The well RD-103 had two ports opened to the formation. Further specifics of 128

water table depths and length of monitoring interval for each well are reported in Table 1. 129

130

3. TIDAL RESPONSE ANALYSIS 131

3.1 The Tidal Records 132

The pressure heads were monitored for approximately 6 months during both tests. The recordings are 133

very stable until the pumping phase starts (Fig. 2a). For instance, during test II, the pumping phase 134

occurs between March and April, and the pressure head was recorded at the monitoring wells about 135

four months before. The drawdown ranged from 0.5 m to 4 m depending on the location of the 136

monitoring well relative to the pumping well. The entire time series, including the drawdown, is used 137

in the tidal response procedure. 138

In order to proceed with the tidal analysis, the raw measurements are filtered with a zero-phase 2-pass 139

Butterworth filter, designed to keep frequency content ranging from 0.41 to 1.25 cpd (cycles per day), 140

which correspond to 10 hrs and 30 hrs respectively. This procedure allows us to eliminate the high 141

frequency noise, the low frequency trend that may be induced by long-term barometric pressure 142

fluctuations, and the trend of the drawdown occurring during pumping phases. This ability to isolate 143

the tidal response is essential to the success of the method and therefore the method works best in 144

situations with little other noise in the tidal frequency band. The barometric pressure was not directly 145

removed from the raw signals because the barometric forcing contains a strong S2 tidal component and 146

its manipulation would bias the phase response. The filtered signals reveal tidal oscillations of 147

approximately 2-3 cm peak-to-peak amplitude, which is close to the accuracy of the transducers (Fig. 148

2b). The measured amplitude is the same order of magnitude as tidal examples previously published 149

[e.g., Elkhoury et al., 2006, Doan et al., 2006]. The phase response is based on timing and therefore is 150

not sensitive to the amplitude accuracy of the sensors as long as the tidal oscillation is recorded. 151

However, one should interpret specific storage results with caution since it is more likely to be 152

sensitive to the amplitude accuracy of the measurements. 153

154

The filtered data spectra show the earth tide diurnal mode K1 (1 cpd), lunar semi-diurnal mode M2 155

(1.934 cpd), and solar semi-diurnal mode S2 (2 cpd) (Fig. 2g). We focus on the lunar semi-diurnal 156

mode M2 since it is usually the most powerful earth tide mode observed that is distinguished from 157

barometric forcing. Atmospheric tidal modes such as S1 (1 cpd) and S2 (2 cpd) are generally clearly 158

identified in the barometric pressure records [e.g., Hsieh et al., 1987]. The influence of barometric S2 is 159

important to address because it is very close to M2 in the frequency domain (Fig. 2g). The barometric 160

contribution to S2 mode is related to a secondary temperature effect in the high atmosphere [e.g., 161

Chapman & Lindzen, 1970], and it is not relevant for the current analysis. For short well records, if S2 162

is too large relative to M2, spectral leakage of S2 could occur and the interpretation of M2 would be 163

incorrect. Therefore, wells with spectral energy of M2 less than half S2 were not processed or 164

interpreted further. It is possible that the barometric and earth tide contributions to S2 could potentially 165

be separated through more aggressive signal processing techniques (e.g., component decomposition), 166

but as this study focuses on the most fundamental comparison with conventional techniques, we chose 167

to maintain a more conservative approach and focus only on the wells with robustly resolved M2 168

components. Future work might either use independent barometric measurements or capitalize on the 169

fact that the M2 and S2 earth tide modes must have a similar phase response, but the S2 barometric 170

phase is different. Sparse groundwater temperature measurements were made during the second 171

pumping test, and fortunately did not show any tidal signals. The temperature coupling can be a 172

significant source of noise, as temperature can sometime exhibit semi-diurnal component and can lead 173

to a misinterpretation of the data. 174

175

3.2 Phase and Amplitude Response 176

The first step of the procedure consists in interpreting the observed tidal mode response (phase and 177

amplitude) relative to the theoretical tidal strains. Assuming that ζi and θi are respectively the phase 178

angle of the water level (Fig. 2b) and the imposed dilatation strain (Fig. 2c) at a frequency i 179

corresponding to a single tidal mode, the phase shift between them is defined as: φi = ζi - θi [Hsieh et 180

al., 1987]. Therefore, a negative phase shift means that the water level oscillation lags the imposed 181

tidal strain. The synthetic dilatation strains used in this work (Fig. 2c) are modeled using the software 182

SPOTL [Agnew et al., 2012]. This allows us to include the ocean loading component in addition to the 183

earth tide contribution, which seems relevant for a site located quite close (about 20 km) to the ocean. 184

The earth tide frequencies K1 (1 cpd), O1 (0.93 cpd), M2 (1.93 cpd) and S2 (2 cpd) are chosen to 185

compute the synthetics. We installed a gravimeter on the site for 6 weeks and verified that the synthetic 186

model accurately predicts the local tidal strains. We define the amplitude response here as the ratio of 187

the far-field strains to the measured pressure head with units of m-1. This is slightly different from the 188

approach of Hsieh et al. [1987] that defined the amplitude response as the ratio of the pressure head to 189

the far-field pressure head. 190

191

The phase shift φi and the amplitude response Ai are computed in a least-squares sense for each 192

frequency i included in the synthetics. The values are calculated within a 29.6 days sliding window 193

overlapping by 80% (Fig. 2d-e). This window length is the shortest we can use to correctly resolve 194

both M2 and S2 in the time domain, and therefore avoid spectral leakage between the two modes. 195

Although the phase and amplitude response are computed for each frequency, only M2 is further 196

interpreted in the following. 197

198

The amplitude responses obtained between 10-7 m-1 and 10-3 m-1 similarly for all recordings (e.g., Fig. 199

2e). The phase lags are generally positive with the observed water level leading the dilatational strain. 200

The example given in Fig. 2d exhibits the largest range in the dataset (Table 2). As will be discussed 201

below, these positive phase leads provide a strong constraint on the appropriate physical model. 202

203

3.3 Permeability Time-Series 204

The standard approach described by Hsieh et al. [1987] is to infer the transmissivity T and the 205

storativity S from both the phase and the amplitude assuming a confined, isotropic, and homogeneous 206

porous medium with radial flow to the well. This solution always results in the water level in the well 207

lagging behind the tidal dilatational strain (negative phase lag). The positive phase lags observed here 208

require an alternative solution. Since the oscillations at the lunar tidal period are strong in this selection 209

of wells, and it is clear that the tide must be causing the water level oscillations, rather than the other 210

way around. Some other model of the forcing between the tide and the water level is therefore 211

necessary. There are two physical possibilities: (1) the water level is responding a different component 212

of the tidal strain, e.g., a fault normal stress rather than the volumetric strain or (2) the flow recorded by 213

the well is drainage to the water table, which is driven by the strain rate oscillations at depth rather than 214

the strain oscillations. 215

We investigated the positive phase lags possible for the first possibility by computing the phase 216

lead of alternative horizontal uniaxial strains. The maximum possible phase lead for uniaxial strain 217

occurs for stress oriented at 10° to 15° East of North and the maximum contribution to the phase lead of 218

22°. Table 2 shows that several wells have phase leads exceeding 22o (RD-35B, RD-103 and RD-31) 219

and so the entire dataset cannot be explained by this mechanism. Furthermore, at least one of the 220

extremely high phase well is not on a known fault structure (RD-35B). Therefore, for this study we do 221

not consider this option further. 222

Positive phase lag is predicted vertical flow near a drained surface [Wang, 2000]. The apparent 223

phase leads are due to the constant pressure boundary condition at the water table that makes the 224

driving force effective the tidal strain rate, which is phase shifted from the dilatational strain. 225

The only other modification of the Hsieh et al. [1987] solution was used by Sawyer et al. [2006] 226

which modeled a sealed well with a mathematically identical flow and a correction factor to account 227

for the finite compressibility of the interval, including the instrumental compliance. In practical 228

situations, the correction factor may not be known well and therefore, the Sawyer al. [2006] model is 229

difficult to apply and appears not to be necessary here. It does not address the positive phase issue. 230

In the vertical flow model, the dimensionless amplitude response Ãi and the phase shift φι of the 231

pressure head fluctuations measured at a monitoring well are computed by [Wang, 2000]: 232

= 1 − 2 − − + − / (1) 233

= tan (2) 234

where, Ss is the specific storage, z is the depth below water table of flow into the well, = ; ηr 235

is the hydraulic diffusivity [m2/s] which equals the transmissivity T divided by the storativity S. 236

The permeability and the specific storage are estimated by fitting eqs. (1) and (2) to the 237

measured phase shift and amplitude response in a least square sense. We apply a large range of 238

permeability and specific storage to eqs. (1) and (2) to fit the observed phase and amplitude response. 239

The inferred permeability and specific storage are the results have the lease misfit. 240

The solution to eqs. (1) and (2) depends on the depth below water table of the open interval. The 241

phase lag decreases with depth, and the amplitude response increases. Therefore, the observed response 242

is most sensitive to water flow into the well from the deeper portions. Since we do not have 243

independent constraints on where the water is entering the well on these historic wells, we infer an 244

upper bound on hydraulic diffusivity by using the deepest possible value of z, i.e., the distance from the 245

water table to the top of the open interval. The results in an upper bound on transmissivity T. Specific 246

storage Ss is less sensitive to this assumption as it appears separately in eq. 1. 247

248

Permeability k and hydraulic conductivity K (m/s) can be inferred from hydraulic diffusivity and 249

specific storage as 250

= = , (4) 251

and, permeability and transmissivity are related through 252 k = , (5) 253

where ρw is the water density, g is the gravitational acceleration, b is the thickness of the saturated open 254

interval of the well, and μ is dynamic viscosity. 255

Notice that in our use of eq. (5), we are using the saturated open interval thickness rather than a 256

thickness of the hydrogeological unit. Discriminate a single unit responsible for the flow within the 257

stratigraphy of this highly fracture reservoir would be extremely difficult and may be inappropriate. 258

Addressing geometry corrections to include the effects of partially penetrating wells might be also an 259

alternate way to proceed, however such corrections are not available for tidal response so far in the 260

literature. In Figure 2f, the phase shift drops during pumping, which would correspond to a 261

permeability increase. This effect could be explained by the drawdown appearing at monitoring well 262

that could create larger vertical flow gradient, within and in the vicinity of the well. However, it is 263

likely an artifact of applying the filter to the non-stationary drawdown in the time-series. Consequently, 264

the part of the phase time-series contaminated by the pumping test period was not used for the 265

comparison. Since month-long windows are used for permeability analysis, we do not utilize the phase 266

inferred from 1 month before the pumping through the end of recordings. We applied this procedure for 267

each studied well to get permeability values and to compare them to pumping test results. 268

Both vertical and horizontal flow may occur at the same time and therefore both contribute to the 269

measured phase response. In this study, the quite large positive phase differences require vertical flow, 270

which does not exclude a coexisting effect of horizontal flow most likely negligible. Combining both 271

effects into a single physical model is not trivial to achieve, since their respective impact on the 272

magnitude of the resulting phase response is extremely difficult to quantify. Further work will 273

investigate a solution that could constrain both vertical and horizontal flow. 274

275

4. COMPARATIVE ANALYSIS 276

4.1 Pumping Test 277

To the best of our knowledge, tidal response has not been compared to conventional hydrogeological 278

techniques such as pumping tests. Prior evaluation of the transmissivity inferred from phase lag has 279

relied on comparisons to either unconventional testing of a unit by earthquake-driven transients [Hsieh 280

et al., 1987] or relative changes in permeability [e.g., Elkhoury et al., 2006]. Here we derive 281

permeability and specific storage from a long-term pumping test for direct comparison with the tidal 282

response. 283

284

Analytic solutions for pumping tests exist for a wide range of boundary conditions and geometries. 285

However, the only analytic solution that exists for the tidal response is for a confined, homogeneous, 286

isotropic, infinite porous medium with a fully penetrating, finite radius well. To make a meaningful 287

comparison, we must therefore select a pumping test solution that makes the identical assumptions as 288

the only extant tidal solution. This approach introduces epistemic uncertainty based on the 289

simplification of the boundary conditions. We will quantify this epistemic uncertainty by comparing 290

the results to an alternative pumping test solution to evaluate how sensitive our results are to the 291

assumptions made. 292

293

Gringarten & Ramey [1973] provide the appropriate pumping test solution for a confined, non-leaking, 294

homogeneous, infinite, isotropic porous medium with a fully penetrating well of finite radius rw. The 295

pressure head drop Δp located at the distance r from the pumping well as a function of time t is 296

τηηη

τ dtr+d

trdI

trQ

Sp

r

2w

r

wt

r

w

s

−

=Δ 4

)(exp2

.2

)(1 2

00

. (6) 297

where rw is the radius of the borehole of the pumping well, Q(τ) is the volume of water extracted per 298

unit of area of source (Fig. 3), ηr is the hydraulic diffusivity, Ss is the specific storage, and I0 is the 299

modified Bessel function of the first kind and order zero. Permeability k and hydraulic conductivity K 300

can be inferred from hydraulic diffusivity and specific storage using eq. (4). 301

We fit the drawdown curves from the 10 locations with significant drawdown to determine the specific 302

storage and permeability separately for each well. The best estimates are obtained in a least square 303

sense using a grid search with 0.05 log unit increments for both Ss and k. The maximum and minimum 304

bounds of the grid search were 10-7 to 10-2 m-1 for Ss (i.e., 10-11 to 10-6 Pa-1) and were 10-14 to 10-9 m2 305

for k (i.e., K=10-7 to 10-2 m/s). 306

307

To quantify the epistemic uncertainty, we also fit the drawdown curve with an alternative geometry. 308

Here we use the Theis [1935] solution as the simplest point of comparison, i.e., 309 ∆p = 4 −−∞ , (7) 310

where 311 u = 24 . (8) 312

The assumptions of Theis [1935] are identical to Gringarten & Ramey [1973] except that the latter 313

considers a finite well. The object of this alternative estimate is to quantify the uncertainty in the fit 314

parameters related to model uncertainty. We further quantify model uncertainty by doing the Theis fits 315

restricted to either short or long-time drawdown data. 316

317

Pump test results are affected by average properties over the entire range between the pumping and 318

observation well. Fitting the individual wells to pump test solutions implicitly assumes a nearly 319

homogeneous system. Local properties perturb the field at each well. Since the values for individual 320

wells inferred here are similar, but not identical, the individual fitting may be a valid approach. 321

However, it is worth comparing the results to values inverted from a simultaneous well in order to 322

arrive at an alternative quantification of the epistemic uncertainty. 323

324

We inverted a single permeability for all the wells involved in test II (RD-01, RD-02, C-2, RD-31 and 325

RD-103) using Gringarten & Ramey [1973] model. The inversion was done the same way as the 326

calculation already presented. We performed a grid search the parameter space [T, S] and found the 327

best fit for the drawdown curves combined, with only a single curve for multi-level RD-31 and RD-103 328

that depicts the most significant drawdown. We are able to fit all the curves correctly using a 329

permeability of k = 7x10-14 m² and a specific storage varying from Ss = 3.3x10-10 Pa-1 to Ss = 5.4x10-9 330

Pa-1. A single [T, S] couple is difficult to reach with the same fit quality (Figure 3d). However, we 331

consider that obtaining a specific storage varying within an order of magnitude is already satisfying and 332

the results are consistent with our earlier analysis. We computed the corresponding RMS deviations 333

from the residuals between each best model and the drawdown curve. We found that those deviations 334

are 0.065 m, 0.07 m, 0.04 m, 0.053 m and 0.057 m, for RD-01, RD-02, C-2, RD-31 and RD-103 335

respectively. Fitting a single couple [T, S] provides an average storage of 5.9x10-10 Pa-1. Comparing this 336

value to those reported just above leads to a difference ranging from 10% to an order of magnitude. 337

338

4.2 Estimates of Permeability and Storage 339

Figure 4a reports the permeability inferred using the best estimate of storage and hydraulic diffusivity 340

from eq. (6) and using the entire observed drawdown curves. For each well, we compare the result 341

with the permeabilities inferred from tidal response averaged for times out of the pumping phase for 342

each well. The resulting values range between 10-15 and 3x10-13 m2 (K=10-8 to 10-6 m/s, fig. 4a). We 343

also perform a fit for comparison with only the short time data with the same Gringarten & Ramey 344

[1973] solution. For the purpose of this study, the first 6 days were used as the short time data for test 345

II and the first day for test I. These time intervals were selected because other studies on the same site 346

had previously identified these time intervals as unaffected by leakage based on a conceptual model 347

[MWH, 2014]. 348

349

On each well, we also explore the alternative model solution of Theis [1935] to get permeability from 350

the entire curve. Short-term fits lead to slightly larger and more scattered permeability values going 351

from 10-13 m2 to 2x10-12 m2 (i.e., from K ~ 10-6 to 2x10-5 m/s), while the whole-curve fits lead to almost 352

the same values as inferred by Gringarten & Ramey [1973]. 353

For both open-holes and multi-level wells, the tidal and pumping test results using eq. (6) (red 354

symbols in Figure 4a) are closer to each other than the range of possible pumping test solutions. This 355

consistency shows that the uncertainty introduced by using the alternative method of tidal response 356

analysis is less than the uncertainty introduced by routinely required decisions on model selection. 357

Therefore, we conclude that the tidal responses are providing as useful a quantification of the 358

permeability as the pumping tests. A significantly smaller permeability is obtained from the tidal 359

response at RD-72 that shows k=10-15 m². The small open interval of RD-72 (6 m) which is located at 360

shallow depth (around 50 m) can explain that the hydraulic diffusivity estimate is smaller. The 361

permeability obtained for wells with the smallest open intervals seem to be slightly smaller. However, 362

we do not observe a systematic correlation between hydraulic properties and thickness of the open 363

interval. The consistency of a vertical flow model with tidal response results suggests a significant 364

connectivity of the fracture network. Possible influence of vertical flow is consistent with the 365

knowledge of the complex fault architecture on that site [e.g., Cilona et al., 2016]. 366

367

The specific storage estimated by the Gringarten & Ramey [1973] solution ranges from 3x10-6 m-1 to 368

10-4 m-1 (equivalent to compressibilities of 10-10 Pa-1 to 10-6 Pa-1), and tidal analysis leads to lower 369

values by an order of magnitude on average (Fig. 4b). Tidal response storage estimates are very close 370

to the fit obtained with eq. (7) for a single inversion of all drawdown curves. Storage values are 371

reasonably consistent from one well to another, regardless of the location or the thickness of the open 372

intervals. The corresponding storativity S can be estimated using open interval thickness provided in 373

Table 1, and varies from 10-6 to 10-2. Large storage values are likely related to the porosity and larger 374

compressibility due to the dense fracture network (e.g., van der Kamp and Gale, 1983). Specific 375

storage also depends on the scale of measurements, and values from single hole pump tests are 376

generally smaller than ones from large-scale tests [e.g., Quinn et al., 2015]. Generally smaller tidally 377

induced specific storage is consistent with a smaller domain investigated by earth tides. 378

379

380

381

382

4.3 Reliability of Estimates 383

4.3.1 Scale of investigation 384

Alternative estimates of the hydraulic conductivity and permeability exist from prior work at the site. 385

Estimates of hydraulic conductivity K calibrated within a global groundwater and contaminants flow 386

model range from 10-7 m/s to 10-6 m/s for the coarse grained unit, which corresponds to permeability 387

ranging from 10-14 m2 to 10-13 m2, and drop to K = 10-9 m/s for fine grained unit such as shale beds 388

[Cherry et al., 2009]. These values are consistent with both pumping test and tidal response results in 389

Table 3. On the other hand, core sample permeability measurements performed in the laboratory on 390

unfractured rock core lead to permeability of 10-16 m2 on average with the permeability dropping to 10-391

18 m2 in finer grain units [e.g., MWH, 2014]. The core values are at least two orders of magnitude lower 392

than the field estimates. 393

394

The discrepancy could arise from fractures at scales larger than the cores dominating the field 395

measurements of permeability. The approximate sampling scale of the early time pumping test data is 396

the distance between the pumping well and the monitoring well. In our case this distance goes from 397

150 to 800 meters. We can use the drawdown expression reported by Hsieh et al. [1987] to estimate the 398

volume investigated by tides following the method outlined in Xue et al. [2013]. We calculate the 399

distance from the wells at which the tidally induced drawdown reaches a negligible percentage (5%) of 400

its value at the wells and use this distance as an effective radius of influence. Based on the values in 401

Table 3, the resulting radii investigated around the wells ranges from ~30-100 m. Both the pumping 402

test and tidal response scales are significantly larger than laboratory measurements for both methods as 403

expected. 404

Intriguingly, the pumping test and the tidal response yield similar permeability values, despite the 405

smaller scale of the region sampled by the tidal response. One interpretation is that within the volume 406

investigated by tides, the wells are efficiently interconnected to the fracture network. This indication of 407

pervasive fracturing at a relatively fine scale requires further investigation. Future work may focus on a 408

systematic comparison at different scales by using smaller scale tests such as packer tests [e.g., Quinn 409

et al., 2015]. 410

411

4.3.2 Uncertainty on estimates 412

For the formal inversion error from the pumping tests, we perform a posteriori calculation. The 413

measure of goodness of fit here is the objective function fobj = log10 Σ (Δpo – Δpf)2 where Δpo and Δpf 414

are the observed and fit pressure drawdown, respectively, and the sum is over all sample points. 415

Examining fobj over parameter space shows that both parameters are quite well resolved around their 416

best estimate (Fig. 3c). Based on the slope of this objective function at the global minimum, we 417

calculate a covariance matrix and find that the formal standard deviation on the parameter estimates is 418

<0.1% of their best-fit values. We also observe an indication of covariance between ηr and Ss for 419

hydraulic testing in the dome-shape of the objective function, which results in a similar formal error 420

(Fig. 3c). 421

422

For the tidal responses, the inversion error inherent in the method was examined in detail by Xue et al. 423

[2013]. The phase shifts as measured have sub-sample resolution due to the long time windows. The 424

previous work found formal errors of 0.3°, which corresponds to 0.7 minute delay at the frequency of 425

M2 [Xue et al., 2013]. The data in that study was sampled at 2 minute resolution and the subsample 426

accuracy in phase timing is due to the long time window fit. We use a similar time window here with 427

10 minute data sampling. Therefore, we expect at most to have 5 minute error in the delay, which 428

corresponds to a phase error of at most 2.4°. 429

430

All of these formal errors are dwarfed by the uncertainty introduced by model assumptions, i.e., 431

epistemic uncertainty. Since formations are intrinsically complex, a useful interpretation of 432

hydrogeological behavior often relies on the knowledge of the conceptual model of the site. The 433

conceptual models drive decisions on which part of the drawdown curve to use. We provide pumping 434

test solutions from both the whole drawdown measurements and early stages. These different decisions 435

give different permeability results as expected. The permeability values obtained from early time are 436

higher than the long-term drawdown estimates by almost an order of magnitude (Fig. 4). That 437

difference between results is characteristic of the epistemic uncertainty due to the model assumptions 438

relative to the 3-D nature of the field. We conclude from this discussion that the difference between 439

model choices provides the best characterization of the uncertainty inherent in pumping test and tidal 440

response interpretation. The formal errors are secondary. 441

442

4.3.4 Tidal model limitations 443

The range of transmissivity that is accessible by tidal responses is limited by the detectability threshold 444

for both amplitude and phase response. Both sensor resolution and local hydraulic properties 445

(permeability and storage) contribute to detectability. For this dataset, the strongest constraint on 446

structure is given simply by the fact that the observed phase lags are strongly positive. That observation 447

in itself requires a vertical flow in order to be realizable. If the phase lags had been negative, the results 448

would be most sensitive to transmissivity ranging from 10-6 m2/s to 10-4 m2/s (Fig. 5). 449

450

Since the inversion errors of the vertical flow model is very small, the uncertainties of phase and 451

amplitude response determine the uncertainties of the inverted permeability and specific storage. As 452

mentioned before, the uncertainty of the phase response is 2.4°. The resolution of the pressure sensor 453

determines the uncertainty of the amplitude response, so we use 10-9 m/strain as the uncertainty of 454

amplitude response. The sets of permeability and specific storage whose fitting residues are within the 455

uncertainties of phase and amplitude give the range of the inverted permeability and specific storage. 456

Since the phase and amplitude response are not a linear function, the uncertainty of different phase and 457

amplitude response are different. Phase response is more sensitive to the diffusivity which is 458

proportional to the ratio of permeability to specific storage. The amplitude response is most strongly 459

controlled by the specific storage. When the phase is close to zero, the inversion has a larger 460

uncertainty because of the uniform undrained response beyond that depth. When the phase lag is larger 461

than 43°, the amplitude is less than 10% of the far-field pressure head value which is most likely not 462

resolvable. For our study, the phase range is 1°-32°, and the amplitude is 104-105 m/strain. The 463

corresponding uncertainties of the permeability and specific storage are 2-13% and 11-26% 464

respectively. Some of the wells do not show any earth tide signal (Table 1). For most of them, it seems 465

clear that it can be explained by too low signal to noise ratio, which could potentially be related to 466

locally low transmissivity. If a site has too large of a range of transmissivities, the tidal response should 467

not be used. 468

469

CONCLUSIONS 470

To the best of our knowledge, this study is the first comparison of the tidal response method to 471

conventional pumping tests. We obtained consistent permeability results for all the wells, and require 472

vertical flow to the water table for the tidal response. The site appears homogeneous at the scale of 473

measurement for both methods, which is on the order of hundreds of meters. The result is slightly 474

surprising given the structural complexity of the region. The permeability values inferred from in situ 475

measurements are larger than core-sample measurements by at least two orders of magnitude as is 476

expected for the fracture-dominated system. The use of a vertical flow model for the tidal response 477

suggests the connectivity of the fracture network vertically. Future work will have take into account 478

both horizontal and vertical flow to properly account for anisotropy. The effect of the length of the 479

connected interval of the monitoring wells will also have to be further investigated and compared to 480

small scale hydraulic tests such as packer tests. 481

482

483

484

Acknowledgments 485

Data supporting this work are open and are available upon request to the corresponding author 486

(vallegre@ucsc.edu). This work was supported by the site owner, the Boeing Company, and benefited 487

from the help of their project manager, Mike Bower. The authors would like to thank Nicholas Johnson 488

from MWH Americas Inc. for fruitful discussion on previous version of the manuscript. The support 489

provided by UoGuelph research staff Amanda Pierce, Ryan Kroeker, and Dan Elliott with field 490

deployments and discussions with Dr. Jessica Meyer regarding data interpretations from multilevel 491

system designs is greatly appreciated. 492

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494

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599

TABLE CAPTIONS 600

601

Table 1. Wells specifications. The notation “Well/#” indicates the port number for RD-31 and RD-602

103. Water depth is recorded at the beginning of the pumping test and therefore does not include 603

drawdown. The terms ET and PT stand for Earth Tide and Pumping Test, and the symbol ● indicates 604

the existence of the corresponding solution. The parameters k and S stand for permeability and 605

specific storage respectively. For multi-level systems, the water table depth is identical for all ports. 606

Well Types: (1) Pumping Well, (2) Monitoring Well/Open-hole, (3) Monitoring Well/Multi-level. 607

608

Table 2. Observed Amplitude and Phase Responses. For each well, the phase and amplitude 609

response to the dilatational tide are measured over the period prior to pumping. 610

611

Table 3. Inferred Hydrogeologic Properties. Permeability and specific storage estimates from the 612

tidal response (kET; SET) and pumping tests analysis, from fits of the entire test duration to Gringarten & 613

Ramey's solution and the Theis solution (kPT,2; SPT,2) as well as the Gringarten & Ramey solution 614

solution for the early stages (kPT,3 ; SPT,3). Permeability reported in units of 10-14 m2 and specific storage 615

reported in units of 10-10 Pa-1. Well Types: (1) Pumping Well, (2) Monitoring Well/Open-hole, (3) 616

Monitoring Well/Multi-level. 617

Table 1. Wells specifications. The notation “Well/#” indicates the port number for RD-31 and RD-618

103. Water depth is recorded at the beginning of the pumping test and therefore does not include 619

drawdown. The terms ET and PT stand for Earth Tide and Pumping Test, and the symbol ● indicates 620

the existence of the corresponding solution. For multi-level systems, the water table depth is identical 621

for all ports. Well Types: (1) Pumping Well, (2) Monitoring Well/Open-hole, (3) Monitoring 622

Well/Multi-level. 623

624

Water Table Depth

Borehole radius (rw)

Open Interval Depth [Open Interval Thickness]

Well (Test #) Type (m) (cm) (m) ET PT

C-1 (I) PW(1)

31 10.6 91-183 [92] / / RD-10 (II) 56 5.1 9-121 [112] / /

RD-35B (I)

MW/O(2)

27 12.5 92.3-98.7 [6.4] ● ● RD-01 (II) 79 10.95 8-154 [146] ● ● RD-02 (II) 55 10.95 8-122 [114] ● ●

C-2 (II) 59 21.9 18-121 [103] ● ●

RD-72/1 (I)

MW/ML(3)

28 6.35 46-52 [6] / / RD-72/2 (I) 28 / 52-58 [6] ● ● RD-31/1 (II) 53 4.85 135.3-138.9 [3.6] ● /

RD-31/2 53 / 139.9-141.1 [1.2] ● ● RD-31/3 53 / 152.4-154.5 [2.1] ● ● RD-31/4 53 / 159.7-163.7 [4] / ●

RD-103/1 (II) 69 5.1 93.6-94.8 [1.2] ● ● RD-103/2 69 / 97.9-100 [2.1] ● ●

625 626

627

628

629

630

631

Table 2. Observed phase and amplitude response. For each well, the phase and amplitude response 632

to the dilatational tide are measured over the period prior to pumping. 633

Well Phase (o) 2σ (o)Amplitude

(x105 m/strain) 2σ

(x104 m/strain)

RD-35B 25.65 9.28 1.35 1.76 RD-72/1 3.11 1.61 3.18 1.5

C-2 11.52 6.96 2.09 1.92 RD-01 4.72 10.48 1.28 1.36 RD-02 15.83 24.03 9.53 2.24

RD-103/1 58.36 9.23 6.47 1.59 RD-103/2 32.58 7.33 2.57 3.11 RD-31/1 32.98 10.82 2.23 5.07 RD-31/2 4.8 4.71 4.62 3.41 RD-31/3 13.88 11.78 1.44 2.75 RD-31/4 0.94 3.25 4.75 1.8

634

Table 3. Inferred Hydrogeologic Properties. Permeability and specific storage estimates from the 635

tidal response (kET; SET) and pumping tests analysis, from fits of the entire test duration to Gringarten 636

& Ramey's solution (kPT,1; SPT,1) and the Theis solution (kPT,2,; SPT,2) as well as the Gringarten & 637

Ramey solution for the early stages (kPT,3 ; SPT,3). Permeability reported in units of 10-14 m2 and 638

specific storage reported in units of 10-10 Pa-1. Well Types: (1) Pumping Well, (2) Monitoring 639

Well/Open-hole, (3) Monitoring Well/Multi-level. 640

641

Well (Test #) Type kET kPT,1 kPT,2 kPT,3 SET SPT,1 SPT,2 SPT,3

C-1 (I)

PW(1) / / / / / / / /

RD-10 (II) / / / / / / / /

RD-35B (I)

MW/O(2)

26 9 4 / 5.1 100 9 50 RD-01 (II) 9 2.8 8.3 18.3 8.5 80 470 170 RD-02 (II) 33 4.7 4.7 36.9 9.1 4 0.54 22

C-2 (II) 5.6 4.3 3.0 0.4 5.0 39 3.1 6

RD-72/2 (I)

MW/ML(3)

0.1 8 3 90 3.4 50 55 80 RD-31/1 (II) / / / / / / / /

RD-31/2 3 9 11 55 2.3 40 5.2 79 RD-31/3 26 7 9 32 6.9 13 1.6 18 RD-31/4 3 5 7 44 2.3 16 2.1 35

RD-103/1 (II) / 7 9 28 / 63 8.0 63 RD-103/2 6 20 22 28 2.1 79 10 63

642

643

644

645

646

647

648

649

650

FIGURE CAPTIONS 651

652

Figure 1. (a) Location map with California shown in red (modified from Cilona et al., 2015). (b) 653

Location of the study area in southern California. (c) Map of the Santa Susana site. The pumping test I 654

is performed at C-1, while the water levels were monitored at RD-35B and RD-72. The test II is 655

performed at RD-10, and the water levels are monitored at RD-01, RD-02, C-2, RD-31 and RD-103. 656

The grey lines indicate a contour map of the interpolated water table depth in meters inferred from 657

water level before test II (orange dots). The dashed line indicates the location of the cross-section 658

(Bottom panel). The well RD-10 is located inside the Happy Valley (HV) fault-zone. RD-31 intersects 659

both coarse and fine grain unit. The fault indicated near RD-02 (vertical dashed line) is poorly exposed 660

and the extent is speculative. 661

662

Figure 2. a) Relative change in water level at monitoring well RD-01 (Test II). The gray area indicates 663

the pumping phase at RD-10. b) Filtered water level data. c) Theoretical volumetric strain computed 664

with SPOTL. d) Phase response and e) amplitude response computed using SlugTides. Note that 665

positive phase differences cannot be solved using radial flow, and the permeability results are therefore 666

not physical (see text for details). f) Amplitude spectrum of the filtered water level (panel b). The 667

contributions of barometric pressure are identified by S1 (1 cpd), S2 (2 cpd) and S3 (3 cpd). 668

669

Figure 3. Pumping test solution. a) Example of the best fit of eq. (7) to the entire drawdown data 670

(green line) and for early times only (red line). This leads to the best estimate of ηr and Ss. Thin lines 671

indicate intermediate solutions for other combinations ηr and Ss. b) (inset) Geometry of the solution 672

from Gringarten et al. [1973]. c) Example of an a posteriori objective function calculation for a range 673

of hydraulic diffusivity and storage. The white circle locates the best estimate in the [kr, Ss] space, 674

corresponding to the green line above (3a). The objective function is the logarithm of the sum of the 675

least square differences between measurements and model computed at each sample. (d) Simultaneous 676

fit of drawdown curves (C-2, RD-01, RD-02, RD-31/3 and RD-103/1) with the Gringarten & Ramey 677

[1973] solution. 678

679

Figure 4. Permeability (a) and specific storage (b) computed from tidal response (orange circles), from 680

aquifer test using eq. (7) for both entire drawdown (orange squares) and early time (t < 6 days or t < 1 681

day, green diamonds), and from Theis solution for entire drawdown (blue diamonds) for all monitoring 682

wells. Values from tidal response are average of the permeability time-series, inferred from T using eq. 683

(6), before the pumping phase only. Permeability values for aquifer test are computed with eq. (4) 684

using the best estimate of [ηr, Ss]. The horizontal dashed lines indicate the permeability (a) and storage 685

(b) estimated using all the drawdown curves in a single inversion (see text). c) Permeability results as a 686

function of the thickness of the well open or screened interval. 687

688

Figure 5. Transmissivity computed from tidal response using the vertical flow model for both open 689

wells and isolated intervals (orange circles), from aquifer test using eq. (6) (red squares), from aquifer 690

test using Theis solution on the entire drawdown (blue diamonds). 691

692

693

HVfault zone

RD-72

RD-35B

RD-02

N

shear

zone

IEL fault

580

430

A RD-02 RD-31RD-10 B C580

430

Elev

atio

n (m

)

A

B

C

0 430meters

scale

finer grainunits

faults damagedzonepumpingwell

HVfault zone

IEL fault

c.

a. b.

25.5 31.7

37.9

37.9

44.2

44.2

50.4

50.4

50.4

56.7

56.7

56.6

62.9

62.969.175.4

81.687.9

94.1

RD-31

RD-10

RD-01C-2 RD-103

C-1

20

-1.50

1.5

-40040

∆h (m

)∆ε

∆h*(

cm)

x10-9

Nov Dec Jan Feb Mar Apr May

a.

b.

c.

date

0 0.5 1 1.5 2 2.5 3 3.5 4frequency (cpd)

10-7

10-5

10-3 M2 S2 S3 f.

050

100

-50-100

1.5

2

1

0.5

x10-5

d.

e.

K1/S1

ampl

itude

resp

onse

(1/m

)ph

ase

shift

(deg

rees

)

RD−35B

RD−01 RD−02

C−2

RD−72 10-11

10-10

10-9

10-8

10-7

Spec

i�c

stor

age

(1/P

a)

RD−31/2

RD−31/3

RD−31/4

RD−103/2

b.

RD−103/1

10-1

100

101

10210

-15

10-14

10-13

10-12

Open/Screened interval thickness(m)

c.

From earth tides analysisFrom aquifer test (Gringarten & Ramey)From aquifer test (Theis)From aquifer test (Gringarten & Ramey, early times)

10-15

10-14

10-13

10-12

a.

From earth tides analysisFrom aquifer test (Gringarten & Ramey)From aquifer test (Theis)From aquifer test (Gringarten & Ramey, early times)

Perm

eabi

lity

(m2)

Perm

eabi

lity

(m2)

10-1

100

101

102

10-8

10-7

10-6

10-5

10-4

10-3

Tran

smis

sivi

ty (m

2/s)

Open/Screened interval thickness (m)

From earth tides analysisFrom aquifer test (Gringarten & Ramey)From aquifer test (Theis)From aquifer test (Gringarten & Ramey, early times)