Flinchum Brady (Orcid ID: 0000-0003-0395-0450)
Estimating the water holding capacity of the critical zone using near-surface geophysics
Brady A., Flinchum12, W. Steven Holbrook23, Dario Grana2, Andrew D. Parsekian2, Bradley J. Carr2, Jorden L. Hayes4, Jianying Jiao2
1Land and Water, Commonwealth Scientific Industrial Research Organisation (CSIRO), Adelaide, South Australia, Australia
2Department of Geology and Geophysics, University of Wyoming, Laramie, Wyoming, USA
3Department of Geosciences, Virginia Tech, Blacksburg Virginia, USA
4Department of Earth Sciences, Dickinson College, Carlisle, Pennsylvania, USA
Corresponding Author: Brady Flinchum ([email protected])
Abstract
In high-mountain watersheds, the critical zone holds crucial life-sustaining water stores in the form of shallow groundwater aquifers. To better understand role that the critical zone plays in moderating hydrologic response to fluxes at the surface and in the subsurface, the hydrologic properties must be characterized over large scales (i.e., that of the watershed). In this study, we estimate porosity from geophysical measurements across a 58 Ha area to depths of approximately 80 m. Our observations include velocities from seismic refraction, downhole nuclear magnetic resonance logs, downhole sonic logs, and samples acquired by push coring.
We use a petrophysical approach by combining two rock physics models, a porous medium for the saprolite and a differential effective medium for the fractured rock, into a Bayesian inversion. The inverted geophysical porosities show a positive correlation with measured values (R2 = 0.93). We extrapolate the porosity estimates from 30 individual seismic refraction lines to a 3D volume below our study area using ordinary kriging to quantify the water holding capacity of our study area. Our results reveal that the CZ in our study area holds ~2.9 x 106 ± 9.6 x 105 m3 of water, where 34% of this storage is in the saprolite, 55% is in the fractured rock, and the remaining 11% is in the bedrock.
1 Introduction
The critical zone (CZ) is the region near earth’s surface where meteoric water, atmospheric gases, biota, and rock interact; these interactions produce a unique subsurface structure extending from the surface to unaltered bedrock (Brantley et al., 2006, 2007;
Anderson et al.,2007; Chorover et al., 2007). Many studies have proposed physical or chemical process based models to explain CZ evolution (e.g. Riebe et al., 2017 and
references therein; Brantley et al., 2011b). The underlying bedrock is transformed into soil through a wide-range of processes that not only form but depend on porosity in the CZ (Brimhall and Dietrich, 1987; Graham et al., 1997; Navarre-Sitchler et al., 2015). Porosity can be generated by the expansion of biotite (Dong et al., 1998; Murphy et al., 1998; Fletcher et al. 2006; Buss et al. 2008), through chemical dissolution caused by interactions with groundwater (White et al., 1995; Riebe et al., 2004; Lebedeva and Brantley, 2013 ), physical fracturing by frost (Walder and Hallet, 1985; Anderson et al., 2013; Rempel et al., 2016), perturbation by plants (Roering et al., 2010; Pawlik et al., 2016), or by opening existing tectonic fractures by variations in the stress field (Slim et al., 2015; St. Clair et al., 2015).
Thus porosity, especially in eroding landscapes, is linked with the processes that shape and maintain CZ architecture. Furthermore, the resulting porosity creates aquifers that provide water stores for plants (Brooks et al., 2010; McDonnell, 2014), influence solute transport (Holloway et al., 1998; Kuntz et al, 2011; Singha et al., 2011), and control surface and groundwater interactions (Montgomery et al., 1997; Kollet and Maxwell, 2008; Jencso et al., 2010; Katsuyama et al., 2010; Voltz et al., 2013).
To understand the distribution of groundwater within the CZ, hydrological properties must be characterized over large spatial scales (100s of m2). However, parameters that control groundwater flow and storage, such as porosity and hydraulic conductivity, are difficult to measure on this scale. A traditional approach is to estimate changes in
groundwater storage by constructing a water balance (Famiglietti et al., 1992; Flerchinger et al., 2000; Hood et al., 2006; Bales et al., 2011). A successful water balance can quantify changes in groundwater storage, not define the total storage. Furthermore, a water balance
does not provide information about where the changes in storage occurred. Currently, a method to spatially map the water holding capacity of the CZ does not exist.
The water storage capacity of the CZ could be calculated by integrating porosity over the study area—the challenge is obtaining porosity measurements over these large spatial scales. Estimates of porosity in the top 10 to 100 meters of Earth’s subsurface over large spatial domains, particularly in crystalline rock, are uncommon because of the spatially limiting nature of measuring porosity, which often requires a physical sample. Traditionally, porosity is measured by Archimedes method (Taylor et al., 1999), X-ray computer
tomography (CT) (Wellington and Vinegar, 1987; Sarker and Siddiqui, 2009), or laboratory nuclear magnetic resonance (NMR) (Timur, 1969; Hinedi et al., 1993; Merrill, 1993; Keating and Falzone, 2013). Porosity estimates are also made using downhole logging equipment:
NMR (Kenyon, 1997; Freedman, 2006; Walsh et al., 2013), neutron density (Keys, 1990;
Ellis et al., 2003; Ellis et al., 2004), and electrical resistivity (Archie, 1941; Bourlange et al., 2003; Spichak and Zakharova, 2016).
Porosity can be estimated from surface-based geophysical methods with the aid of a rock physics model. The rock physics model defines a relationship between a geophysical property (i.e. seismic velocity, resistivity, or dielectric permittivity) and a hydrologic property of interest (i.e. porosity or hydraulic conductivity). Previous studies have estimated porosity using electrical resistivity (Turesson, 2006; Mota and Monteiro Santos, 2010), ground penetrating radar (Turesson, 2006; Rehman et al., 2016) and seismic refraction (Mota and Monteiro Santos, 2010; Holbrook et al., 2014; Hayes, 2016). The demand of spatially exhaustive geophysical measurements and the procurement of data required to validate the
porosity estimates is one reason why porosity estimates of the CZ over large spatial scales are uncommon. In this study, we map porosity in three-dimensions (3D) over an area of ~58 Ha using rock physics, ordinary kriging, velocities from seismic refraction and sonic logs, downhole NMR measurements, and physical samples.
To estimate porosity over a large spatial domain, we adapt two rock physics models: a porous medium model (Mindlin, 1949; Hashin and Shtrikman, 1963; Dvorkin and Nur, 1996;
Dvorkin et al., 1999) and a differential effective medium model (DEM) (Berryman et al., 2002). The rock physics models drive a Bayesian inversion that incorporates constraints from borehole NMR measurements, saprolite thickness from boreholes, and water level logs from seven boreholes. Porosity estimates are interpolated into 3D using ordinary kriging to
produce a volume of porosities that are used to calculate the water storing capacity of the CZ.
Our results show that saprolite had the highest porosity but is mostly unsaturated. Even though the averaged fractured rock porosity was less than half that of the saproilite, it has the highest water holding capacity because it is ~ 4x thicker.
2 Hydrophysical Setting
The study site is located in the Sherman Batholith in the Laramie Range
approximately 20 km southeast of Laramie, WY (Figure 1). The Sherman granite contains large phenocrysts of potassium feldspar that weathers to a friable coarse-grained material, commonly referred to as grus (Eggler et al., 1969; Evanoff, 1990; Frost et al., 1999). The Sherman granite is comprised of 30-40% microcline, 15-30% quartz, 20% plagioclase, 10- 15% perthite, and 5-10% biotite (Frost et al., 1999; Geist et al., 1999, Edwards and Frost,
2000). The study site receives ~620 mm of precipitation annually, 90% of which occurs in the form of snow (Natural Resources Conservation Service, 2015). A perennial stream in the southern valley (Figure 1) drains a watershed area of ~4.1 km2. During peak snow melt, the northern valley becomes a wetland with a small amount of standing surface water, but is dry by mid to late July.
CZ structure at the study site was interpreted using data from four boreholes and twenty-five seismic refraction profiles which were configured to image the CZ under the small ridge in the study area (Flinchum et al., accepted). The CZ at the study site is divided into three units: saprolite and soil, weathered bedrock, and unfractured bedrock (Flinchum et al., accepted). Here, saprolite is significantly weathered to the point that it is friable but still retains the original fabric of the rock. The weathered bedrock is pervasively fractured near the saprolite/weathered bedrock interface but transitions to approximately 1-3 fractures per meter near the bedrock boundary (Hayes, 2016; Flinchum et al., accepted). The observations from Flinchum et al. (accepted) suggest that the saprolite/weathered bedrock interface is subparallel to the surface topography but the bedrock boundary is inversely related to the surface topography where depth to bedrock is greatest under the ridge and shallow under the valleys.
In total, 30 seismic refraction profiles (~7 line km) exist within the study area (Figure 1). In addition to the seismic profiles, the site contains seven boreholes drilled into bedrock, and seven shallow Geoprobe holes that are limited to the saprolite, and one hand-augered well in the northern drainage (Figure 1). Results from 25 profiles (L1-L28) and data from five of the seven boreholes (BW-1 through BW-5) were presented previously (Hayes, 2016,
Flinchum et al., accepted). Data from five seismic refraction profiles, all of the Geoprobe holes, and two additional boreholes (BW-6 and BW-7) are presented for the first time as part of this study. The Geoprobe boreholes are cased with closed PVC liners (i.e. not screened) and the bedrock boreholes are open below casing. Water levels have been recorded hourly using data logging pressure transducers in five of the seven bedrock boreholes and the hand- augered well in the northern valley (Figure 1) covering water year 2016 (October 1, 2015 to September 30, 2016).
3 Methods
3.1 Seismic Refraction
Seismic refraction generates and records the time required for energy to travel from multiple sources and receivers. The travel-times can be inverted to provide estimates of p- wave velocity underneath a 2D profile. In recent years seismic refraction has emerged as a tool to image shallow CZ structure (Befus et al., 2011; Leopold et al., 2013; Holbrook et al., 2014; St. Clair et al., 2015; Parsekian et al., 2015; Olyphant, et al., 2016). Here, we picked first-arriving energy and inverted the travel-times using travel-time tomography (St. Clair, 2015). Rays were traced through a mesh using a shortest path algorithm (Dijkstra, 1959;
Moser, 1991). The models were updated by solving a regularized, linear inverse problem.
The ray-tracing and inversion steps were iterated until a satisfactory fit to the data was achieved.
The travel-time tomography problem is underdetermined, the solutions are non-
the method described by Flinchum et al. (accepted) and ran 50 inversions with different starting velocities and gradients that were derived from a normal distribution determined by manually selecting values to minimize the initial misfit between observed travel-times and the initial model. Surface velocities ranged from 100 to 250 m/s, and vertical gradients
ranged from 100 to 300 m/s per meter. Using a range of starting models provided an estimate of uncertainty for the velocities in the final tomographic models. P-wave velocity is
influenced by geologic material, fracture density/porosity, density, and water saturation.
Since our study site lies in a granite batholith, we assumed changes in p-wave velocity were due to the weathering of the granite.
Survey geometry, profile locations, velocity models, and model fits for L1-L28 can be found in Flinchum et al. (accepted). An additional five profiles (L29-L33) were collected to improve the ability to interpolate a 3D volume from 2D lines and intersect the two additional boreholes (BW-6 and BW-7) that were drilled (Figure 1). Survey geometry and model fitting criteria for the additional five lines are shown in Table 1. In the text, L29 and L27 are
highlighted because they intersect the majority of the boreholes. L29 was collected in June of 2016 with 1 m geophone spacing and 12 meter shot spacing (Table 1). L27 was collected in September of 2015 with 1 m geophone spacing and 10 m shot spacing (Flinchum et al., 2018). We do not expect different acquisition geometries and collection times to have a substantial effect on the velocities at this scale.
3.2 Nuclear Magnetic Resonance
Nuclear magnetic resonance (NMR) capitalizes on a nuclear phenomenon that
hydrogen protons in water preferentially align with a magnetic field and oscillate at a specific
frequency called the Larmor frequency (Bloch et al., 1946). An NMR measurement is made by applying a secondary magnetic field to perturb the magnetic moment of the hydrogen protons (Legchenko and Valla, 2002). After removal of the applied field, the protons relax back to equilibrium. The relaxation is primarily a function of the surface to pore volume ratio (Grunewald and Knight, 2011; Keating and Falzone, 2013; Falzone and Keating, 2016). The magnitude of the measured signal is proportional to the number of hydrogen atoms (i.e. water volume). The reader is referred to Behroozmand et al. (2015) for a comprehensive review of environmental NMR principles. To obtain porosity from NMR data, we must assume the pores or fractures are fully saturated. This is a reasonable assumption for the downhole NMR data collected below the water table. In this manuscript, we do not attempt to obtain
porosities in the vadose zone with the NMR data.
In this study, we made downhole NMR measurements in BW-1, BW-4 and BW-5 (Figure 1) using a Javelin JP350 (Vista Clara, Mukilteo, WA) (Walsh et al., 2013). The Javelin measures water content in four cylindrical shells of varying radii (14.0, 15.9, 17.8, and 19.7 cm) and has a vertical zone of sensitivity of 0.46 m (1.5 ft). The data were collected in 0.5 m intervals starting at the bottom of each hole and ending at the surface. The NMR measurement was collected using two different recovery times (Tr): the first enhances data quality for short relaxations (Tr = 800 ms) and the second captures long relaxations (Tr = 4000 ms). The average number of stacks for Tr = 800 ms was 72 and the average number of stacks for Tr = 4000 ms was 20. To improve signal-to-noise, we used a moving average filter over five measurements and combined measurements from all four cylindrical shells. The
vertical moving average increased the signal-to-noise by a factor of three at the cost of reduced vertical resolution.
3.3 Porosity Measurements
In this study, twenty-five samples were obtained using a Geoprobe (Model 7822DT).
The samples spanned depths between 0.8 to 9.1 m. The samples were acquired by push coring until refusal or until the end of the sleeve was reached, augering through the section of push, and pushing again. All samples were acquired within a week of each other and with no major precipitation events occurring between sample acquisitions. Samples were extracted in 1.5 m increments in transparent casing that is 1.25 in (3.2 cm) in diameter. Approximately 2 in (5.1 cm) samples were cut off the bottom of each push core and wrapped in parafin wax to conserve any residual water in the samples. No attempts to prevent compaction were made.
Volumetric water saturation was measured by weighing each sample, drying the samples in an oven at 100 C for 24 hours and re-weighing. The bulk density was calculated by dividing the dry weight of the sample and the volume. Grain density was measured eight times using dried samples from different depths. The average grain density was 2.59 ± 0.04 g cm-3. Porosity was calculated using the volume and the grain density and the relationship shown in Eq. 1:
ϕ = 1 − �ρbulk ρgrain�
(1)
Where ρbulk is the bulk density and ρgrain is the grain density. Compaction was calculated by subtracting the total length pushed from the length of core recovered and then normalized by the length pushed. For the deepest cores (> ~3m depth), compaction was on the order of 0- 10%, whereas the uppermost samples (< ~ 3 m) experienced the highest compaction between 15-50%. Reported porosities are likely underestimated because a correction factor for
compaction has not been applied.
3.4 Sonic Velocities
For this study, full waveform sonic data were collected with a Mount Sopris QL40- FWS (Colorado, USA) and processed with WellCAD (vrs. 5.1 build 504) software using a semblance algorithm that estimates slowness (1/velocity) by searching for similarity across a three-receiver array. In our data, fifteen slowness scans with a 50 µs window width were calculated across the three receivers for each depth. The algorithm sums up the amplitude values within a given time window for one receiver and then stacks this value with
corresponding values from the window of the other two receivers. First arrivals (P -wave, S- wave, Tube wave, etc.) were picked at each depth. The picking was initiated with an
automatic picking algorithm based on time-windowed semblance energy and then checked manually. The logs provided both p-wave and s-wave velocities as a function of depth.
3.5 Bayesian Rock Physics Inversion
Rock physics models provide relationships that link rock and fluid properties to measurable geophysical properties. Our rock physics models predict seismic velocity as a function of mineralogy, porosity, and water saturation of the material. Several rock physics equations have been proposed for fractured rock (Budiansky and O’Connell, 1976; Berryman et al., 2002) and porous media (Mindlin, 1949; Biot, 1941; Gassmann, 1951; Hashin and Shtrikman, 1963; Dvorkin and Nur, 1996; Dvorkin et al., 1999). The saprolite is similar to a porous medium (based on inspection of the samples) but below the saprolite is fractured rock (based on the borehole televiewer logs). Therefore we utilize two rock physics models. For the saprolite, we use a rock physics model based on Hertz-Mindlin theory and Hashin- Shtrikman elastic bounds (Mindlin, 1949; Hashin and Shtrikman, 1963; Mavko et al., 2009).
This model has been used successfully to estimate porosity in saprolite using seismic refraction velocities (Holbrook et al., 2014; Hayes, 2016). For the fractured bedrock, we utilize a differential effective medium (DEM) (Berryman et al., 2002) that estimates seismic velocities by integrating the effect of infinitesimal penny-shaped cracks filled with a material (air, water, or clay).
The inverse problem associated with the estimation of rock and fluid properties from geophysical attributes presents several challenges: the rock physics model is generally non- linear, the measured data are not perfect due to limited signal-to-noise and the resolution of the geophysical tools, and the solution of the inverse problem might not be unique due to the number of rock and fluid configurations with a similar seismic response. To overcome these challenges, we employ a Bayesian inverse method in which the rock physics model is used to
generate a training dataset and the solution is a set of probability distributions of rock and fluid properties. The Bayesian inversion method calculates the posterior distribution of porosity and water saturation given the measured seismic velocities using Bayes’ Rule:
P(m|d) =P(d|m)P(m) P(d)
(2) where 𝑃(𝑚|𝑑) is the posterior distribution of the model given the data, 𝑃(𝑑|𝑚) is the
likelihood function, 𝑃(𝑚) is the prior distribution of the model, and 𝑃(𝑑) is a normalizing constant that guarantees that the posterior distribution is a probability density function (i.e.
the integral of the posterior distribution is 1). In our application, 𝑚 = [𝜙, 𝑠𝑤] is the vector of porosity and water saturation, and 𝑑 = �𝑉𝑝, 𝑉𝑠� is the vector of P-wave velocity and S-wave velocity. The prior distribution 𝑃(𝑚) defines the possible range of the model parameters and is obtained from geological knowledge of the site. The likelihood function 𝑃(𝑑|𝑚) is
estimated from the training dataset built using the rock physics model.
Analytical formulations of the Bayesian inverse problem are available in statistics literature (Tarantola, 2005). However, because the rock physics model is not linear, the solution is not analytically treatable. Iterative methods, such as Markov chain Monte Carlo methods, can be implemented but can be computationally demanding. In our implementation, we adopted a hybrid approach. We generate a training data set comprised of samples from the prior distribution. We compute the velocities for each porosity and water saturation using rock physics models and approximate the joint distribution of model parameters assuming a
multivariate Gaussian distribution. If the joint distribution is Gaussian, then the conditional distribution is also Gaussian and has an analytical solution (Tarantola, 2005). The solution can be uniquely identified by the posterior mean 𝜇𝑚∨,𝑑and posterior covariance matrix 𝛴𝑚,∨𝑑. The two statistical estimators can be then computed as:
𝜇𝑚∨𝑑 = 𝜇𝑚+ 𝛴𝑚𝑑(𝛴𝑑𝑑+ 𝛴𝑒)−1�𝑑̃ − 𝜇𝑑�
(3) and for the conditional variance
𝛴𝑚∨𝑑 = 𝛴𝑚− 𝛴𝑚𝑑(𝛴𝑑𝑑+ 𝛴𝑒)−1𝛴𝑚𝑑𝑇
(4) where 𝜇𝑚 is the prior mean, 𝛴𝑚𝑑is the cross-covariance matrix between the model parameters and the data, 𝛴𝑑𝑑is the covariance matrix of the data, 𝛴𝑒is the covariance matrix of the
measurement errors, 𝜇𝑑is the mean of the data and 𝛴𝑚is the prior covariance matrix.
The calculation of the training data set for the porous medium model can be broken down into a series of steps (details in supplementary information): 1) calculate the elastic properties (bulk modulus, shear modulus, and density) of the solid phase (i.e. rock with zero porosity) using a Voight-Reuss-Hill average (Hill, 1963; Mavko et al., 2009), 2) calculate the elastic properties of the dry rock at the critical porosity (Mindlin, 1949; Helgerud, 2001;
Helgerud et al., 1999), 3) calculate the elastic properties of the dry rock as a function of porosity using the upper and lower Hashin- Shtrikman boundaries (Hashin and Shtrikman, 1963), 4) calculate the elastic properties of the dry rock beyond the critical porosity (Nur et
al., 1998; Dvorkin et al., 1999), 5) calculate the elastic properties of the saturated rock using fluid-mixture elastic averages and Gassmann’s equations (Gassmann, 1951; Brie et al., 1995), and finally 6) calculate the velocities from the elastic properties. We define a range of
possibilities for the stiffness of the rock by using both the lower (soft) and upper (stiff) Hashin-Shtrikman boundaries. The rock physics model shows decreasing velocity as porosity increases (Figure 2a).
For the fractured rock, we generate the training data using the Berryman DEM model (details supplementary information), which can be broken down into a series of steps: 1) calculate the elastic properties of the rock with zero porosity using the Voight-Reuss-Hill average (Hill, 1963; Mavko et al., 2009), 2) calculate the elastic properties of the fluid using Brie’s method (Brie et al., 1995), 3) calculate the elastic properties of the saturated rock with cracks filled with fluid by solving a system of differential equations (Eq. 1 and 2 from
Berryman et al., 2002), and 4) calculate the velocities from the elastic properties. We define a range of possibilities by using two different crack aspect ratios. The area between these two curves defines the range of possible velocities for a given porosity (Figure 2b).
The two rock physics models are used to calculate the likelihood function using Bayes’ Rule (Eq. 2). The selection of the forward model is based on a velocity that is consistent with the casing and augering depths, which represents the boundary between saprolite and fractured rock (see Section 5). If the velocity we are inverting for is above the saprolite/weathered bedrock boundary, we use the Hertz-Mindlin model, otherwise we use the DEM model. The training data set is constructed at every depth by sampling 1000 porosity and water saturation combinations from a prior distribution and using the rock
physics models to calculate corresponding velocities (details in supplementary information).
To estimate the elastic properties of the granite at zero porosity, we follow the mineralogy described by Edwards and Frost (2000) and use the mineral elastic properties found in Mavko et al. (2009) (Table 2). Using a Voight-Reuss-Hill average we expect a p-wave velocity of 5.92 km/s and s-wave velocity of 3.40 km/s at zero porosity for the Sherman granite.
3.6 Porosity Calibration
Rock physics models are site-specific and must be calibrated to match experimental measurements. In our rock physics models, the parameters that need to be determined are the critical porosity, the number of grains per contact (Eq S5), and the aspect ratio of the
fractures (Eq S14). There are a limited number of studies that measure velocity and porosity on weathered granite. Thus, to calibrate the rock physics models we include porosity and velocity measurements from the Oporto granite (Begonha and Braga, 2002), leucocratic granite from the Krudum Massif (Novakova et al., 2012), and granites from Portugal (Sousa et al., 2005). The lowest velocity observed was 1.3 km/s, which had a corresponding porosity of 0.13 m3/m3. The lowest porosity cited was 0.01 m3/m3 which had a velocity of 6.2 km/s.
We use porosities from Holbrook and others (2014) and our measured porosities (Table 3) and assume that the velocity from the nearest seismic refraction profile is equivalent to the velocity where the sample was taken (Figure 2c and 2d). To our knowledge, these are the only data that provide a relationship between porosity and p-wave velocity of saprolite.
To calibrate the Hertz-Mindlin model, we tried different combinations of the critical porosity and grains per contact (Eq S5) to encompass as many calibration points as possible.
The final model used a critical porosity of 0.4 and a grain contact value of 4 (Figure 2c).
Most calibration data occurred near the lower Hashin-Shtrikman boundary (Figure 2). We used a skewed Gaussian distribution with a mean near the lower boundary to incorporate this observation into the training data set (Figure 2c). The Hertz-Mindlin model failed to fit low porosity samples, no matter which parameters were used (Figure 2c). If we used only the Hertz-Mindlin model, the fractured bedrock porosities would be underestimated by 0.01-0.05 m3/m3 (Figure 2c).
To calibrate the DEM model we followed a similar approach. We selected a range of crack aspect ratio values that encompassed our calibration points. To provide a range we selected the upper and lower values of the alpha parameter (Figure 2d). The lower boundary corresponds to a crack aspect ratio of 0.005 and the upper boundary has a crack ratio of 0.025 (Equation S13 and S14) (Figure 2d). The calibration data were in the center of these
boundaries, thus we used a Gaussian distribution between the two boundaries (Figure 2d). No matter what crack aspect ratio is chosen, the DEM model cannot fit the weathered rock data and the saprolite simultaneously, emphasizing the need for two different rock physics models.
3.7 Steady State Water Table
Due to the saturation influence on velocity (Nur and Simmons, 1969; Murphy, 1984;
Bachrach and Nur, 1998; Holbrook et al., 2014) we need to constrain water saturation at every seismic transect. Using water level measurements from six bedrock boreholes, a hand augered hole in the northern drainage (Figure 1), and assuming that the stream in the southern drainage is connected to the groundwater, we modeled the potentiometric water surface across our study area using data from September 22, 2015. The water table was inverted
following the methods in Jiao and Zhang (2014). Unlike the objective-function-based inversion techniques, this inverse method does not require forward simulations to assess measurement-to-model misfits; thus the knowledge of aquifer boundary conditions is not required. Instead, the method employs a set of local approximate solutions of flow to impose continuity of hydraulic head and Darcy fluxes. Given observed hydraulic heads and
conductivity, the water table can be obtained. We assumed a homogenous hydraulic conductivity of 8.5 x 10-3 m/day following Freeze and Cherry (1979). The results of the steady state inversion indicate approximately 30 m gradient over 1 km, consistently sloping towards the west southwest under the ridge with the water table coming near the surface in the northern valley and at the surface in the southern valley (Figure 3a and 3b).
3.8 Drilling
Because we use two rock physics models, one for the saprolite and one for the fractured rock, we must define a boundary that separates them. We rely on casing depths of seven deep boreholes and the depths of refusal of the Geoprobe boreholes in conjunction with seismic refraction tomograms to approximate the velocity of the boundary separating
saprolite and weathered bedrock. We make two assumptions using this analysis: 1) drillers seek a solid surface to set casing and 2) the Geoprobe auger will not penetrate solid rock. We calculated the seismic velocity at the casing depth or refusal depth of each hole from the nearest seismic transect (Figure 3c). From the 13 estimates, we find that the casing depths or refusal depths occur in conjunction with an average velocity of 1.1 +/- 0.18 km/s at depths ranging from 4.1 to 17.9 m (Figure 3c).
4 Results
4.1 Laboratory Porosity Measurements
The 25 geoprobe samples, which only sampled saprolite, had a minimum porosity of 0.32 m3/m3 and a maximum of 0.43 m3/m3 (Figure 4; Table 3). The average porosity was 0.36
± 0.03 m3/m3. The samples were orange in color and contain large k-spar crystals intermixed with fine grains (Figure 4b to 4e). Residual granitic structure could still be observed through the plastic casing. Upon removal from the casing, samples would crumble to a loose gravel with visible fine-grained material. There was no trend of porosity with depth (Figure 4a).
Uncertainty was calculated by substituting one standard deviation of the measured grain density (Eq. 1).
Residual water content for all 25 samples was at or below 0.10 m3/m3 (Table 3). We assumed that all water in the sample was preserved from collection and that the water content is equally distributed throughout the 5.1 cm sample. The highest water content occurred in a clay rich sample that was visually different from other samples (Figure 4b). The remaining 24 samples had volumetric water contents less than 0.04 m3/m3 (Table 3). Despite the relatively high measured porosities (~0.32 m3/m3), the results suggest that the saprolite is unsaturated to at least 9 m depth. No samples for porosity analysis were taken from below the water table.
4.4 Surface Geophysical Results
Seismic refraction lines showed slow velocities (Vp < 2 km/s) dominating under the ridge and high velocities (Vp > 4 km/s) coming close to the surface in the valleys (Figure S1).
Line geometries and final model RMS fits are shown in Table 1. We focus on the results of two profiles, L29 and L27, because they intersect most of the boreholes and Geoprobe holes (Figure 5) but the velocity profiles for all 30 profiles can be found in the supplementary information (Figure S1).
L29 intersects BW-1 and BW-4 and is oriented along the crest of the ridge (Figure 5a) and L27 is located directly on top of the ridge and intersects BW-2 and BW-5 (Figure 5b).
The seismic results show a large section of slow velocities (< 2 km/s) under the ridge,
extending to greater than 40 -50 meters depth (Figure 5). In L29 the vertical thickness of low velocities thins eastward as the ridge transitions into the valley, where the 4 km/s velocity contour is often within 5 meters of the surface under both valleys (Figure 5a). We expect the bottom of saprolite to occur within the region defined by 1.1 ± 0.18 km/s (Figure 5), where this velocity region is based on the average velocity at the bottom of casing and refusal depth of the Geoprobe holes (Figure 3c). There was no clear refractor in the seismic data and without the drilling results we would not have interpreted this boundary. The region defined by the 1.1 ± 0.18 km/s velocities contours appears to be correlated with the water table (Figure 3c; Figure S1).
4.5 Borehole Geophysical Results
We collected downhole NMR, Vp and Vs sonic velocities in BW-1, BW-4, and BW-5 (Figure 6). Sonic velocities were limited to fully saturated regions in the borehole because
water is required to couple the source and receiver to the surrounding rock. Unlike the sonic logs, the NMR logs are not inhibited by PVC casing. Unfortunately, BW-1 was cased in steel, which prevents NMR logging along the cased interval. Instead, we logged the nearby
Geoprobe well BWG-6 (11 m west) in the unsaturated zone to assemble a more complete depth profile (Figure 6c). We are still missing 5 m of NMR data due to the instrument length, our vertical averaging scheme, and the fact that BWG-6 was approximately 2 m short of the casing depth in BW-1.
In general, sonic velocities are slower in significantly fractured zones that can be observed in the optical televiewer logs (Figures S6-S8) and also show significant variability (Figure 6). The sonic velocities (both P-wave and S-wave) in BW-4 are slower near the top of the hole (~2 km/s) but steadily increase (~5 km/s) toward the bottom of the hole. In BW-5 the velocities at the base of casing are already high (~3.8 - 4 km/s) and do not show much
variation. BW-1 is similar to BW-5, in that at the base of casing, the p-wave velocities are high (~3.8 – 4 km/s), but has more variation that BW-5.
The NMR measurement allows the quantification of the volumetric water content in any material as long as the material lies within operating range of the instrument. If we assume full saturation (i.e. below the water table) the NMR-derived water content is equal to the porosity. Water contents in BW-4 are consistent around 0.1 m3/m3 until they drop to 0.03 m3/m3 below 50 m. In BW-5 there is a thick region of high water contents located above the casing. Given that it is above the casing, this large zone of higher water content (~0.1 m3/m3) is probably in the saprolite. Finally, BW-1 shows low water contents throughout the borehole, consistent with the high velocities and lack of fractures in the optical televiewer images
(Figure S6-S8). Although the water contents are small, the signal was acceptable after
stacking, and the small water contents are not a result of an inversion artifact (Figure 6d). All of the NMR soundings near the surface show water contents less than 0.1 m3/m3, which are consistent with the water contents we measured in the porosity samples (Table 3).
5 Prior Construction and Sensitivity
The Bayesian inversion allows the incorporation of prior knowledge in the form of mathematical functions. For example, based on the results from the lab measurements, we expect porosity of the saprolite at our site to range from 0.2 to 0.5 m3/m3 (Figure 4). Based on the NMR data (Figure 6), the fractured rock porosity should be no higher than 0.2 m3/m3. We use the steady state water table model (Figure 3a and 3b) to constrain saturation. We use the velocities from the seismic refraction surveys to constrain the boundary between saprolite and fractured bedrock (1.1 +/- 0.18 km/s). We combine this information to construct prior
functions that vary across the landscape.
The prior functions are used to minimize the uncertainty associated with the
inversion. Based on lab measured porosities (Figure 4) and borehole NMR data (Figure 6), we know that porosity must decrease across the saprolite/fractured bedrock boundary. Based on the average velocity of casing, we interpreted the saprolite/fractured bedrock boundary to occur near the 1.1 km/s velocity contour (Figure 3c), although a strong refractor is not present in the seismic refraction data. One possible interpretation for this lack of boundary is that the boundary dividing saprolite and fractured rock is transitional. Thus, we define a transition zone where the porosity range decreases from 0.2 – 0.5 m3/m3 (saprolite) to 0 - 0.2 m3/m3
(fractured bedrock) (Figure 7) over four meters. The prior function varies along each seismic transect where we define the weathered bedrock boundary by a velocity of 1.1 km/s. The saturated boundary is constrained by assuming it occurs at the same boundary (Figure 3). Due to uncertainty in the modeling effort we use a similar transition zone (Figure 7) where the transition zone increases the water content from 0 – 0.1 m3/m3 to a near full saturated condition (0.9 – 1.0 m3/m3) (Figure 7).
We ran a series of sensitivity tests to determine how much the inverted porosity sections depend on the prior functions (supplementary information). These tests show that it is important to constrain the water saturation but regardless of the prior function, the porosity at depths less than about 10 m (velocities < 1.1 km/s) and greater than about 40 meters (velocities > 4.0 km/s) are not strongly dependent on the prior function. They also show that when the water table and weathered bedrock boundary are disconnected, the inversion produces porosity artifacts (Figure S2). The area in the model most influenced by the prior function, and thus the highest uncertainty, is the fractured bedrock (1.1 < Vp < ~4 km/s). We feel justified to constrain these porosities between 0-0.2 m3/m3 based on the measured water contents in the three borehole NMR soundings (Figure 6) and literature values (Begonha and Braga, 2002; Novakova et al., 2012; Sousa et al., 2005; Goodfellow et al., 2016).
6 Porosity Inversion Results
We investigate the inversion’s performance in the fractured rock by comparing inverted porosities from the sonic velocities to the downhole NMR porosities. All of the data presented were inverted using a prior of 0.2 – 0.5 m3/m3 that transitions to a range of 0-0.2
m3/m3 when the velocities are greater than 1.1 km/s (Figure 7). The water saturation prior ranges between 0-0.1 m3/m3 and transitions to near full saturation for velocities greater than 1.1 km/s. We assume that the water saturation and weathered bedrock boundary occur at the same location given the correlation between the casing velocities (Figure 3d) and the velocity at the water table (Figure 3c). This assumption slightly alters the hydrologic condition of the study area, but will reduce porosity artifacts in the final inversions (Figure S2). We use the Hertz-Mindlin rock physics model for velocities less than 1.1 km/s and the DEM model for velocities greater than 1.1 km/s.
6.1 Surface
The inversion results based on surface seismic velocities show that porosities between 0.1 and 0.4 m3/m3 dominate the 2D sections (Figure 8). The inverted water saturation profiles are shown to illustrate the influence of the prior constraint (Figure 8). Combined, these two profiles provide porosity estimates that cover horizontal distances of ~430 meters and depths of approximately 80 meters. The boundary between the saprolite and fractured rock that can be observed in the sections are a result of the prior functions (Figure 8). The inverted
porosities are in good agreement with the measured porosities obtained from the Geoprobe campaign (Figure 8c – 8f). A quantitative comparison between the lab measured data and seismically estimated porosities can be found in the supplementary material (Figure S3a).
6.2 Borehole
The borehole porosities are in agreement with the water contents obtained via NMR (Figure 9). For all three holes, the sonic velocity porosity estimate and NMR porosity estimate are positively correlated (R2= 0.68) and never off by more than 0.1 m3/m3 (Figure S3b). The decreasing trend from about 0.18 m3/m3 to 0 m3/m3 in BW-4 is picked up by the inversion. In BW-5, there is a low velocity spike in the sonic velocities, which results in a spike of porosity to about 0.12 m3/m3. This spike was below the resolution of the NMR, or it is a fracture that is unsaturated. In almost every case, the sonically estimated porosity is higher than the NMR porosity (Figure 9). One possible explanation for this difference is that the NMR data will only measure fractures filled with water, whereas the seismic velocities will be sensitive to all fractures—similar to the difference in total and effective porosity.
Furthermore, if fractures along the borehole are sporadic, the sonic wave field, which sampled a slightly larger volume, would have a higher probability of encountering more fractures, producing a lower velocity and a higher porosity estimate. A quantitative
comparison between the downhole NMR porosities and sonically estimated porosities can be found in the supplementary material (Figure S3b).
6.3 Extrapolation to 3D
One goal of the study is to calculate the total water storage capacity of the CZ—
specifically understanding the storage capacity over watershed scales. Here, we interpolate the porosity values along all measured seismic profiles (Figure 10a) to a 3D volume using
extrapolation into 3D follows three steps: 1) remove topography, 2) calculate and fit experimental variograms (Figure S4), 3) apply the kriging algorithm, 4) reintroduce topography, and 5) remove values that had a kriging variance greater than 0.01 (m2/m2)2. Details pertaining to the variogram analysis and the resulting kriging interpolation can be found in the supplementary information.
7 Discussion
7.1 Defining the water storage capacity of the CZ
Using the volume of porosities (Figure 10c) we can characterize porosity and
calculate the water storage capacity of the CZ. In general, porosity decreases from about 0.4 m3/m3 near the surface to just above 0 m3/m3 in the deepest regions of the model area (~80 m). Because of the high spatial density of the data set we can observe significant spatial patterns (Figure 11a-d). At shallow depths, the porosity under the ridge is higher than that of the valleys (Figure 11a). At greater depths, the large porosities under the ridge are
emphasized. At 25 m below the surface, there is a section of relatively high porosities located directly under the ridge (Figure 11c). Remnants of this high porosity region can be seen all the way to 35 m below the surface (Figure 11d). This spatial pattern of porosity is consistent with the interpretation that the fresh bedrock is deepest under the ridge (Flinchum et al., accepted).
Here, we calculate the water holding capacity, or the total integrated porosity (e.g.
Holbrook et al., 2014; Klos et al., 2018) of the CZ. This value represents the amount of water that can be held in the subsurface assuming that all of the porosity is filled with water. In map
view, we normalize by the area of each pixel (100 m2) so that the water holding capacity is expressed as depth (m). The water holding capacity map (Figure 11e) illustrates how many meters of water could be stored in each pixel. The water holding capacity emphasizes the patterns observed in the depth sections (Figure 11e). Without other hydrological information, it is not possible to determine if this region of high porosities is connected to the stream, riparian zones but based on spatial patterns, the hillslope/ridge in the study area could have a large influence on the local hydrology.
We can break down the spatial patterns of porosity in an even more detailed way using the corresponding 3D seismic velocity volume. The seismic velocity volume can be used to map the three main units discussed by Flinchum et al. (accepted): saprolite and soil, fractured bedrock, and fresh bedrock. We recalculated the volume presented in Flinchum et al. (accepted) to incorporate the additional five seismic refraction lines that were collected (supplementary information). To define the bottom of saprolite we extract the 1.1 km/s velocity contour from the volume (Figure 12a). This velocity is based on the casing and final depths of tge Geoprobe boreholes, and represents the boundary separating the saprolite and weathered bedrock (Figure 12c). To define the bottom of the fractured rock layer we extract the 4.0 km/s velocity contour from the volume (Figure 12b). This velocity was selected based on a seismic refraction survey conducted on an unfractured outcrop in the area (Flinchum et al., accepted). The average thickness of the saprolite and fractured bedrock is 4.9 ± 2.8 m and 20.8 ± 11.0 m, respectively. The fractured bedrock layer is approximately four times thicker than the saprolite, and is thickest underneath the ridge. The fresh bedrock layer extends
infinitely into the subsurface, but in the model area is defined by the depth of the deepest ray path and limited to no more than 80 m.
We can use the porosity volume and the boundaries from the velocity volume to calculate the water holding capacity for each unit (Figure 12d-12f). The saprolite layer has the greatest storage capacity under the ridge top and minimal storage under the valleys. The saprolite has the highest average porosity of all three units (0.34 ± 0.06 m3/m3) but is thin compared to the fractured bedrock. Most of the saprolite is unsaturated and the water table occurs near this boundary. The majority of the water storing capacity for the fractured
bedrock layer occurs under the ridge. Although the average porosity of this unit is half that of the saprolite (0.15 ± 0.05 m3/m3), it can hold more water since it is ~4 times thicker than the saprolite. Finally, the unit we call fresh bedrock is just the remainder of the porosity volume, which had the lowest average porosity (0.03 ± 0.02 m3/m3). The saprolite makes up about
~34% of the total water storage, the fractured bedrock makes up ~55%, and the fresh bedrock (down to ~80 m) accounts for the remaining ~11% of the water storage (Figure 12c).
7.2 Uncertainty Estimates
There are two major uncertainties associated with the porosity estimates. The first is associated with the porosity inversion. This uncertainty is characterized by a standard
deviation outputed by the Bayesian inversion. The second is associated with the extrapolation from 2D to 3D which can be characterized using the kriging variance. We look at the
uncertainty in terms of the maximum volume of water that can be held in our study region (m3). To quantify the uncertainty associated with the inversion we input the 30 inverted
porosity transects minus one (-1σ) and plus one standard deviation (+1σ) from the Bayesian Inversion into the kriging algorithm. We use the same variograms and kriging parameters to calculate a minimum and maximum volume. The volume is calculated by summing the porosities vertically and multiplying by the area of each pixel. We show the total water holding capacity and then express the fractions of saprolite, fractured rock and bedrock as percentages for each of these calculations (Table 4). The water holding capacity of the volume is 2.8e6 m3 (for Figure 11e). If the interpolation/extrapolation had no uncertainty, we would expect the water holding capacity of the study area to be ~2.8e6 +/- 0.5e6 m3.
To quantify the uncertainty associated with interpolation we use the kriging variance (Figure 10d). The total water storing capacity of the volume plus one standard deviation from the kriging variance is 4.2e6 m3 and the volume subtracting one standard deviation is 1.7e6 m3 (Table 4). This value is much larger than the uncertainty of the inversion. This value is so much larger because we are interpolating ~7 line kilometers of data to an area 58 ha.
There is one more uncertainty that must be discussed even though it is difficult to quantify—namely the issue of scale. By definition, the porosity is the percentage of void space in the material. If the porosity or fracture density is not uniform, then the porosity will be dependent on the volume sampled. In this paper the downhole NMR sampled a volume of
~0.12 m3, the sonic velocity sampled a volume somewhere between 0.37 and 0.39 m3 (velocity dependent), and the volume sampled by seismic refraction is difficult to quantify but is much larger (likely on the scale of 100’s m3). The larger sample volume means that the velocities derived from seismic refraction velocity likely integrate more fracture zones, which would reduce the overall predicted velocities. This difference in velocity, associated with the
scale of the measurement, would produce higher porosity estimates, resulting in an overestimate in the total volume that is extremely difficult to quantify.
7.3 Implications for Critical Zone Evolution
Understanding the processes responsible for generating porosity in the deep critical zone is vital to improve understanding of near-surface processes, especially over landscape scales. These processes convert bedrock into a material that can sustain life by storing water and providing nutrients (Graham et al., 2010). If the weathered material has not been
expanded or perturbed (i.e. strain = 0) then porosity can be used to map chemical weathering (Hayes, 2016). If we could determine strain, the porosity volume (Figure 10c) could be used to map chemical weathering.
Models that provide mechanisms to understand critical zone structure over large spatial scales often rely on proxies like topographic attributes (i.e. slope, aspect, or elevation) (Chorover et al., 2011; Riebe et al., 2015), lithology (Bazilevskaya et al., 2015; Brantley et al., 2011a), temperature and water availability (Anderson et al., 2013), regional stresses (Slim et al., 2015; St. Clair et al., 2015), or water level measurements (Rempe and Dietrich, 2014;
Goodfellow et al., 2011; Braun et al., 2016). This study provides porosity, a possible proxy to chemical and physical weathering. Estimating and mapping spatial patterns of porosity on these large spatial scales opens the door for improving and understanding the processes that shape the deep critical zone. The ability to map and quantify porosity in weathered crystalline environment 3D could reveal clues about the processes that dominate the evolution of deep critical zone structure.
7.4 Highlighting Key Challenges
The analysis presented in this manuscript highlights some of the difficulties in estimating porosity over large spatial scales using near-surface geophysics. A key challenge is the calibration of the rock physics models. But in weathered materials, specifically weathered granite, hydrological and geophysical parameters are rarely quantified
simultaneously—making calibration difficult. There is a need to define the relationships between geophysical and hydrologic parameters in weathered granites so that this type of analysis can be expanded elsewhere. The inversion framework presented here does not need to be limited to a single geophysical parameter (i.e. seismic velocities); it just requires a working and calibrated rock physics model and an independent method to validate the inversion throughout the study area. The caveat is that in weathered crystalline rock environments, it might be important to combine different rock physics models because the physics that govern porous media are different from those that govern fractured media.
8 Conclusions
In this paper we present a novel method to estimate porosity over large spatial scales (~ 58 Ha). The Bayesian porosity inversion utilized two rock physics models: a porous medium rock physics model for the saprolite (velocities <= 1.1 km/s), and a differential effective medium model for the fractured rock portion of the subsurface. To improve porosity estimates we used a modeled map of water table depths based on 7 observation wells. Our porosity volume revealed a large region of high porosity under the ridge. Because of the spatial coverage of the data, we were also able to quantify the water holding capacity of the
saprolite, fractured bedrock, and remaining fresh bedrock. Overall, the saprolite had the highest average porosity but only made up about 34% of the total water storage. The
fractured bedrock had a relatively low average porosity, but because it was ~4 times thicker than the saprolite comprises about 55% of the water storage. The remaining 11% is found in the fresh bedrock unit which had a low average porosity of 0.03 ± 0.02 m3/m3.
9 Acknowledgments
This work was supported by NSF EPS-1208909, NSF 1531313, and the Society of
Exploration Geophysicists (SEG) Near-Surface Geophysics award (2016). All data used in the publication is hosted on the Wyoming Center for Hydrology and Environmental Geophysics data discovery portal at http://wycehg.wygisc.org. We would like to thank the numerous undergraduates who helped with geophysical acquisition over the course of two summers, under the leadership of Mathew Provart. I am also extremely grateful to many others for thoughtful discussions and constructive criticism to improve the manuscript. We are grateful for the three reviewers who provided excellent suggestions that helped improve this manuscript.
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