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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 ([email protected]) 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
([email protected]). 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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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)