Comparison of NMR‐Derived Hydraulic Conductivity with Various Hydraulic Testing Methods

Abstract

Borehole nuclear magnetic resonance (NMR) can be used to estimate the hydraulic conductivity (K) of unconsolidated materials. Various petrophysical models have been developed to predict K based on NMR response, with considerable efforts on optimizing site‐specific constants. In this study, we assessed the utility of NMR logs to estimate K within highly heterogeneous glaciofluvial deposits by comparing them with K measurements from three types of co‐located hydraulic testing methods, including permeameter, multi‐level slug, and direct‐push hydraulic profiling tool (HPT) logging tests. Four NMR models, including Schlumberger‐Doll Research (SDR), Seevers, Sum‐of‐Echoes (SOE), and Kozeny‐Godefroy (KGM), were applied to construct K profiles at four locations with model constants optimized using permeameter‐based K. Model constants suitable for glaciofluvial deposits were provided. Results showed that NMR logging can provide reliable K estimates for interbedded layers of sand/gravel, silt, and clay. Through cross‐hole comparison of NMR‐derived K profiles, the trends and magnitudes of K for aquifers/aquitards were readily mapped. Quantitatively, the NMR‐derived K coincided with hydraulic‐testing K, with optimal model fits within one order of magnitude. We noticed that (1) Seevers performed similarly but no better than SDR in predicting permeameter and slug testing measurements; (2) SOE yielded slightly better predictions than SDR; (3) the removal of porosity in SDR did not deteriorate its prediction, and the optimized SDR constant resembled the literature‐based values for glacial deposits; and (4) KGM yielded the optimal fits with slug‐based K, demonstrating its reliable performance. Lastly, we made recommendations on selecting suitable petrophysical models.

Introduction

Borehole nuclear magnetic resonance (NMR) tools have been used in the petroleum industry for over 50 years to determine formation properties such as permeability and porosity. With the development of slim‐hole NMR logging tools (e.g., Sucre et al. ²⁰¹¹; Perlo et al. ²⁰¹³; Walsh et al. ²⁰¹³; Zhu et al. ²⁰²¹), applications have extended to the characterization of unconsolidated sediments within the environmental and engineering sectors. Borehole NMR for unconsolidated sediments provides high‐resolution site characterization of hydraulic properties (e.g., hydraulic conductivity [K] and water content) and features a fast in‐situ application deployed either in wireline logging or through a direct‐push technique (Morozov et al. 2024). Unlike conventional neutron source tools, NMR does not have an active nuclear source, making it easier to deploy in the environmental sector.

Various NMR models have been developed to estimate K from NMR signals. Most models were empirically derived with model constants acting as fitting parameters (e.g., Schlumberger‐Doll Research [SDR], Seevers, and Sum‐of‐Echoes [SOE]). The selection of NMR signals (e.g., porosity and relaxation time) in NMR models also affects the quantitation of model constants. For instance, Maurer and Knight et al. (2016) found that NMR‐derived porosity does not improve the K prediction in the SDR model. In addition, attempts were made to replace empirical with physical parameters, e.g., the Kozeny‐Godefroy model (KGM, see Dlugosch et al. ²⁰¹³).

Several studies have explored the predictive performance of these models across different geological conditions and have found that model constants in unconsolidated materials often differ significantly from consolidated materials due to varying sedimentation and tortuosity. Therefore, the site‐specific optimization of NMR models through calibration with hydraulically derived K measurements is usually performed for more reliable K estimation. Dlubac et al. (2013) were the first study to optimize model constants for unconsolidated materials, with numerous subsequent studies presenting unique optimizations based on site conditions (Parsekian et al. 2015; Knight et al. ²⁰¹⁶; Maurer and Knight ²⁰¹⁶; Kendrick et al. ^{2021, 2023}; Crow et al. ²⁰²²). The comparison of NMR‐derived K with hydraulic testing methods in these previous studies has generally shown good agreement. For example, Knight et al. (2016) optimized the SDR constants with K calibration datasets from three sites within alluvial deposits (fluvial, riparian, agricultural sediments) and found only minor differences in the SDR constants. This finding indicates the potential of NMR logging to provide K estimates with reasonable uncertainty for unconsolidated materials when site‐specific calibration is not available. However, on occasion, model constants have been shown to be inconsistent between sites, indicating potential limitations in model utility, particularly in lower‐K materials. For instance, Kendrick et al. (2023) have compiled model constants for the SDR and SOE models from the literature (Dlubac et al. 2013; Walsh et al. ²⁰¹³; Knight et al. ²⁰¹⁶; Kendrick et al. ²⁰²³), which were calibrated to different types of sediments. Results show that these model constants can vary over an order of magnitude across different geological materials.

Another potential limitation of existing model constants is the nature of the hydraulic testing method used to calibrate NMR models. Various hydraulic testing methods have been used to calibrate these empirical models, including direct‐push permeameter (DPP) tests, wellbore flowmeter (WBF) logging tests, and multi‐level slug tests. Varying scales and components of K are measured and related to NMR response, potentially leading to a study‐specific calibration bias. For example, DPP tests (Butler et al. 2007) have been applied in several NMR comparison studies (e.g., Maurer and Knight ²⁰¹⁶; Kendrick et al. ^{2021, 2023}). They are conducted by injecting a stream of water horizontally and measuring the pressure difference at two transducers with a spacing of ~0.4 m vertically for K estimation by Darcy's law, assuming pressure equality at the same distance spherically. Furthermore, NMR data have been compared with WBF logging tests by Dlubac et al. (2013), which measure the effective horizontal K component at meter scale, assuming that the aquifer is uniformly layered and K is proportional to the inflow rate at each layer. Multi‐level slug tests were utilized to fit with the SDR model by Walsh et al. (2013) and Crow et al. (2022). K measurements from slug tests are biased toward the horizontal component, assuming radial flow toward the screen. Various hydraulic testing methods can cause considerable differences in the optimization of NMR‐derived K estimates, with implications for predictions of groundwater flow and contaminant transport. Sun et al. (2024) evaluated K estimates from various hydraulic testing methods, including grain‐size analysis, multi‐level slug tests, permeameter tests, direct‐push hydraulic profiling tool (HPT) logging, and hydraulic tomography (HT), by comparing observed drawdown from pumping tests with numerically modeled drawdown. Results show that different hydraulic testing methods have led to varying drawdown prediction performances.

Unlike conventional hydraulic testing methods, a NMR probe stimulates the porewater within a circular distance around the borehole and measures the mobility of the hydrogen within the pore space to infer K without assuming any potential directional component of K. However, given the heterogeneous and anisotropic characteristics of geologic material, hydraulic testing methods often yield different representative estimates of K. Therefore, it is valuable to investigate how NMR‐derived K models fit with co‐located K measurements from different hydraulic testing methods, in which varying scales and components of K are sampled.

In this work, we examined the capability of NMR logging to characterize subsurface heterogeneity and estimate K in highly heterogeneous glaciofluvial deposits ranging from gravel, sand, silt, to clay, which exhibit more than a five‐order range in K (m/s). Permeameter‐based K measurements sampled at a similar interval to NMR logging were used as the K calibration dataset. Then, we assessed the K predictive performance of NMR logging with permeameter tests, multi‐level slug tests, and direct‐push HPT tests, by comparing their co‐located K profiles. Four NMR petrophysical models, including SDR, Seevers, SOE, and KGM, were used to quantitatively evaluate model fits with physical K measurements. The calibrated model constants were listed and compared to the literature‐based constants from various geological conditions. Recommendations on selecting an optimal NMR model were provided for a highly heterogeneous glaciofluvial environment.

Background

Borehole NMR Basics

A net nuclear spin angular momentum is possessed by the hydrogen protons in water molecules, which is used in the geophysical application of NMR methods (see review by Behroozmand et al. ²⁰¹⁵). In terms of borehole NMR measurements, an NMR probe is lowered down an open bedrock hole or a PVC‐lined monitoring well. A permanent magnet built into the probe is utilized to generate a static magnetic field, B ₀. Proton spins become polarized in an equilibrium state aligned parallel to B ₀, which is defined as the longitudinal direction. Hydrogen protons precess at the Larmor frequency in the B ₀ field, with the precession frequency and the magnitude of the magnetization vector proportional to the B ₀ field strength. A pulsed oscillating magnetic field, B ₁, tuned to a specific Larmor frequency, is applied in the plane transverse to B ₀ by the NMR probe to rotate the magnetization of the hydrogen protons at a desired radial distance from the NMR probe. A cylindrical shell of deposits is sampled during this process. When proton spins return to the equilibrium state (i.e., relaxation), a spin‐echo decay is detected by the receiving coil. For multiple pores with varying geometry, the spin‐echo decay curve, S(t), can be described as a multiexponential decay function:

$$S(t)={\sum }_i{A}_{0i}{e}^{-t/T_{2i}}$$ (1)

where i is the pore index, A _0i is the initial amplitude for each proportion of T _2i, and T _2i is a characteristic time (i.e., the relaxation time) for pores with different T ₂. A ₀ can be normalized by the initial amplitude of the spins in pure water to quantify water content. The S(t) signal can be processed using a Laplace inversion to obtain the multiexponential decay time distribution, showing the amplitudes as a function of T ₂.

The measured T ₂ is a linear combination of three relaxation mechanisms (Kleinberg and Horsfield 1990):

$$T_2^{-1}=T_{2S}^{-1}+T_{2B}^{-1}+T_{2D}^{-1}$$ (2)

where T _2S is the surface relaxation time, T _2B is the bulk fluid relaxation time, and T _2D is the diffusion‐gradient relaxation time, since refocusing pulses have a reduced efficiency, which can be mitigated using a sufficiently short echo time (Walsh et al. 2013). The surface relaxation time, T _2S, ranges from several milliseconds to a second, while T _2B is around 1–3 seconds for water. The surface relaxation time typically dominates the relaxation response in unconsolidated materials, corresponding to the whole pore that is sampled during relaxation. In such a fast diffusion regime, $T_{2S}^{-1}$ (with a unit of 1/s) can be related to the geometry of the pore in the following manner (Brownstein and Tarr 1979):

$${T_{2S}}^{-1}=\rho \frac{S}{V}$$ (3)

where ρ is a material‐specific surface relaxivity (m/s), governed by the presence of paramagnetic ions on the pore surface, measuring its ability to enhance relaxation, and S/V is the surface area to volume ratio of the pore space (1/m).

Models for K Estimation

Empirical relationships of NMR models follow the Kozeny–Carman type of equations. One form of the K‐C equation can be written as follows to estimate hydraulic conductivity, K (Seevers 1966):

$$K=\frac{\mathit{\varrho g\varphi }}{\mu \tau {\left(\frac{S}{V}\right)}^2}$$ (4)

where $\varrho$ is water density, g is the gravitational acceleration, $\varphi$ is porosity, μ is dynamic viscosity, and τ is tortuosity. The relationship between T ₂ and S/V in Equation (3) is utilized to estimate K.

Therefore, T ₂ is the main indicator of K in the SDR model (Kenyon et al. 1988), which can be written as:

$$K_{\mathit{SDR}}=C_{\mathit{SDR}}{\varphi }^m{T}_{2\mathit{ML}}^n$$ (5)

where C _SDR, m, and n are site‐specific parameters, in which the C _SDR constant mainly accounts for mineralogical parameters (e.g., surface relaxivity). T _2ML is the mean‐log T ₂ used to characterize the T ₂ distribution:

$$T_{2\mathit{ML}}=\exp \left(\frac{\sum A_i\ln \left(T_{2i}\right)}{\sum A_i}\right)$$ (6)

Generally, n is assigned to be 2, which follows the assumption of the fast‐diffusion regime. For consolidated materials, m is originally assigned a value of 4 (Kenyon et al. 1988) to account for variations in the ratio of pore throats to pore openings. In recent studies for unconsolidated aquifers, m = 0 (Maurer and Knight 2016; Crow et al. ²⁰²²), m = 1 (Knight et al. 2016; Kendrick et al. ²⁰²³), or m = 2 (Dlubac et al. 2013) was usually adopted. Here, m = 0 was adopted such that porosity was removed from the SDR equation.

The Seevers model (Seevers 1966) has a similar form to SDR. The main difference is that the contribution of bulk fluid relaxation is incorporated independently:

$$K_{\mathit{\text{Seevers}}}=C_{\mathit{\text{Seevers}}}{\varphi }^m{\left[{\left({T_{2\mathit{ML}}}^{-1}-{T_{2B}}^{-1}\right)}^{-1}\right]}^n$$ (7)

where C _Seevers is the Seevers constant, while m = 1 and n = 2 are typically utilized. The relaxation time of bulk water (in seconds), T _2B, can be estimated as follows (Dlugosch et al. 2013):

$$T_{2B}=3.3+0.044(\theta -35)$$ (8)

where θ is the temperature in degrees Celsius.

The SOE model estimates K as a linear function of the squared spin echo decay curve, S(t). The SOE signal is defined as:

$$\mathit{SOE}={\int }_0^{\mathrm{\infty }}S(t)\mathit{dt}$$ (9)

and K can be obtained by:

$$K_{\mathit{SOE}}=C_{\mathit{SOE}}{(\mathit{SOE})}^n$$ (10)

where C _SOE is the SOE constant, and n is the exponent normally assigned to be 2 (e.g., see Walsh et al. ²⁰¹³). As the integral of S(t) is used to substitute T _2ML from inversion, the SOE model is theoretically more reliable in a high‐noise environment.

In the KGM model, the material‐specific NMR parameters are utilized to replace the empirically derived factors (Dlugosch et al. 2013):

$$K_{\mathit{KGM}}=\frac{\mathit{\varrho g}}{8{\tau }^2\mu }\varphi {\left(-\frac{D}{\rho }+\sqrt{{\left(\frac{D}{\rho }\right)}^2+\frac{4D{T}_{2\mathrm{B}}{T}_2}{T_{2B}-T_2}}\right)}^2$$ (11)

where D is the diffusion constant of bulk water (in m²/s), which can be estimated as follows (Maurer and Knight 2016):

$$D=\left(1.0413+0.039828\theta +0.00040318{\theta }^2\right)\times {10}^{-9}$$ (12)

Site and Data Descriptions

The North Campus Research Site

The NMR data were collected at the North Campus Research Site (NCRS) located on the University of Waterloo campus in Waterloo, Ontario, Canada. Previous studies have shown that the shallow surface consists of glaciofluvial deposits and till ranging from gravel, sand, silt, to clay, forming alternating layers of aquifer and aquitard units (e.g., Alexander et al. ²⁰¹¹; Berg and Illman ²⁰¹¹; Zhao and Illman ^{2017, 2018}). A dense and stiff Catfish Creek Till located at approximately 16 m below ground surface (mbgs) is regarded as a hydraulic barrier at the bottom of the aquifer system of interest. A sandy‐to‐clayey silt sequence, namely the Maryhill Till, is deposited over the Catfish Creek Till. Thin erosional remnants of the Tavistock Till overlie the Maryhill Till, which is below the top organic soil. Alternating layers of aquifer and aquitard units were found based on borehole geology logs (refer to Alexander et al. ²⁰¹¹; Berg and Illman ²⁰¹¹). The primary “aquifer zone” is situated between 8 and 13 mbgs, with low K materials separating two aquifer layers. The upper aquifer layer consists of sand to sandy silt, and the lower aquifer layer is composed of sandy gravel.

The NCRS has nine multi‐level wells configured in a square pattern (Figure 1a). Four continuous multichannel tubing (CMT) wells were installed at the midpoint of each side. CMT tubes are 18 m long with seven channels to varying depths in the formation. The CMT tubing has a diameter of 0.10 m. A 0.17 m section was cut for each channel at the desired depth and wrapped with a fine mesh to build a discrete observation point with a ~2 m spacing between adjacent points.

Figure 1.

Field site and NMR logging information. (a) A site map with locations of pumping wells, HPT logs, and NMR logs. The geographic coordinates of CMT1 are 43°28′36.4″ N, 80°32′44.6″ W. The other wells at the NCRS not involved in this study, including PW1, PW2s, PW3, PW4s, and PW5, are labeled in gray. (b) Illustration of how the Dart system works with the direct‐push technique. (c) A field photo of NMR logging with the direct‐push rig proximal to CMT1 at the NCRS.

The drilling of the CMT wells was completed using a hollow‐stem auger with a diameter of 0.26 m. Continuous samples of sediments were obtained during the drilling process of each CMT well for core logging and permeameter testing in the lab. During the installation of the CMT wells, each observation port was encased in 0.3 m to 0.6 m filter packs consisting of #0 silica sand filled above and below each observation port, with bentonite pellets used to isolate the filter packs.

NMR Logging

A slim‐hole NMR logging system (Dart™ DP140N, Vista Clara Inc.) was used to collect borehole NMR data adjacent to the CMT wells. The distance between NMR logs and existing CMT wells was <1 m. The DP140N slim‐hole probe is specifically designed to work with a Geoprobe direct‐push rig. The DP140N has a diameter of 3.56 cm and a length of 1.33 m. The vertical resolution of the DP140N is 0.22 m. Two coils are used to generate cylindrical responses of 12.7 cm and 15.2 cm radially from the center of the probe. Together, these measurements can isolate the signal from formation material within a thin cylindrical shell outside the drilling alteration zone. The measurement spacing of the four CMT logs was 0.15 m.

An illustration of how the Dart works with the direct‐push rig is shown in Figure 1b. First, hollow‐stem rods with a closed cutting shoe are advanced to the desired depth by the percussion hammer. The shoe is then opened, and the DP140N is inserted through the rods until it reaches the bottom of the hole. The rods are retracted while the probe remains fixed. As the sensor exits the bottom of the casing and becomes exposed to the formation, a specially designed key at the top of the probe lifts it upward while remaining fully exposed to the formation. NMR measurements are collected in a stepwise manner, with the retraction of the drill rods performed in pre‐defined vertical sampling increments (e.g., 0.15 m). Figure 1c is a field photo of NMR logging at CMT1. The NMR logging deployed with direct‐push rigs enables the profiling of soil conditions without the need for permanent well installations or potential impacts of well construction materials (e.g., bentonite seals). To date, the Dart is the only NMR logging system that can be deployed with the direct‐push rig.

In total, 211 discrete NMR measurements were collected at the four CMT wells. In our NMR dataset, the signal‐to‐noise ratio was always less than 10%, indicating high‐quality measurements. The NMR signals from CMT1 to CMT4 are plotted in Figure 2. Three signals, including T _2ML, SOE, and total water content (θ), are provided as potential indicators for K estimation utilized in petrophysical models. In terms of water level measurements for all pumping wells at the NCRS (see Figure 1a), the lowest water level measured on the day of the NMR logging survey was 3.78 mbgs, so we use 3.78 mbgs as the upper limit of the depth for all NMR‐derived K. The depth of exploration approached 12 mbgs for all four NMR logs. In addition, the geologic units for the four CMT wells are shown in each plot for comparison, which were compiled from the original borehole logs by Zhao and Illman (2017) and utilized in subsequent studies (Zhao and Illman 2018; Luo et al. ²⁰²³; Zhao et al. ²⁰²³).

Figure 2.

Profiles of NMR signals near CMT1 to CMT4. NMR signals include the main indicators of K in the petrophysical models (i.e., T _2ML and SOE) and total water content. Geologic units are also shown for comparison. The water table is indicated with a horizontal dashed line and an arrow. The locations of three conductive layers are indicated using colored arrows.

Hydraulic Testing Methods

Previously reported hydraulic testing results collected at the four CMT wells of the NCRS were used to assess the performance of NMR logging. The hydraulic testing data used to compare with NMR K estimates and the statistical parameters of K estimates for each hydraulic testing method are summarized in Table 1.

Table 1.

Summary of the Number of Hydraulic Measurements at Each Location and Statistical Parameters of K Estimates from Each Hydraulic Testing Method, Including the Geometric Mean of K, the Variance of ln K, and the Maximum and Minimum K values.

	Number of Measurements					Statistical Parameters
Methods	Total	CMT1	CMT2	CMT3	CMT4	$\bar{K}$ (m/s)	${\sigma }_{\ln K}^2$	Max K (m/s)	Min K (m/s)
Permeameter	248	64	69	60	55	7.16e‐7	5.66	2.81e‐4	2.17e‐9
Slug tests	16	4	4	4	4	4.76e‐6	4.31	1.04e‐4	1.66e‐7
HPT logging	1078	539	—	539	—	1.32e‐5	1.56	1.95e‐4	4.60e‐7

Permeameter Tests

For permeameter tests, a 10‐cm diameter split spoon sampler was used to collect continuous vertical core samples at an interval of around 10 cm during the drilling of CMT wells. The core samples were delivered to the laboratory for falling‐head permeameter tests by Alexander et al. (2011). Within the depth range of this study, 248 permeameter K measurements were available.

The sampling interval of permeameter tests is 10 cm, and the sampling interval of NMR logging is 15 cm. For statistical analysis, we paired NMR‐derived K estimates with permeameter‐based K measurements, which were co‐located within 10 cm, resulting in 171 paired K measurements for CMT1, CMT2, CMT3, and CMT4.

Multi‐Level Slug Tests

The four CMT wells each have four channels within the range of NMR logging, providing 16 sampling ports for which Alexander et al. (2011) completed slug tests. A slug of water was injected into each port for falling‐head tests, which induced a 0.5 m rise in the static water level. The Hvorslev method (Hvorslev 1951) was utilized to analyze the slug test data.

CMT channels were installed at an interval of 2 m. It is assumed that the K measurements from slug tests were representative of sediment conditions over a total depth of 2 m proximal to the ~0.2 m screen and that flow was isotropic. Given the larger sampling distance of the slug tests, we upscaled the NMR‐derived K by averaging the NMR samples within the corresponding 2 m depth range. We calculated the geometric mean of the six NMR samples above and below the midpoint of each screen. In summary, 16 co‐located NMR K estimates were obtained, associated with the slug tests. Due to the limited depth range of NMR logging, fewer NMR samples were available to match the slug K measurements in some screens near the top or bottom of the downhole profiles.

HPT Logging Tests

The HPT logging test is a direct‐push injection logging technique. The HPT probe injects a flow of water into the formation while recording the injection pressure, flow rate, and the probe advance rate during the logging process (McCall and Christy ²⁰²⁰). A physically based equation modified from Borden et al. (2021) was used to estimate K (m/s):

$$K=E\left(6.113\times {10}^{-8}V{D}^2+5.891\times {10}^{-8}Q\right){P_c}^{-1.017}$$ (13)

where E is an empirically derived hydraulic efficiency factor, V is the probe advance rate (cm/s), D is the probe diameter (cm), Q is the water injection rate (mL/min), and P _c is the corrected injection pressure (kPa). The best‐fit value of E after being validated by 23 HPT profiles is 2.02 by Borden et al. (2021).

The co‐located HPT measurements to NMR logs were near CMT1 and CMT3 (see Figure 1a). The sampling interval of direct‐push HPT logging tests was 1.5 cm. Table 1 illustrates that the ${\sigma }_{\ln K}^2$ of HPT‐based K was significantly smaller than the other two methods, which may be due to the fact that both upper and lower boundaries of K estimation were limited. Liu et al. (2012) pointed out that the upper boundary of K in HPT logging tests is around 6.9 × 10⁻⁴ m/s, over which the injection rate is too small to produce a measurable pressure response. On the other hand, the maximum detected injection pressure in the HPT sensor is around 689 kPa, which also limits the HPT technique to estimate K in low‐permeable materials. In our dataset of HPT logging, the lower boundary of K is 4.60 × 10⁻⁷ m/s. Therefore, HPT logging was only adopted for qualitative comparison of K profiles with NMR logging.

Model Calibration

Model constants were calibrated using the K measurements from permeameter tests. A linear regression after log transform was performed to calibrate the model constants in the SDR, Seevers, and SOE models. As a non‐linear relationship exists between K and the model constants in KGM, a grid search analysis (Maurer and Knight 2016; Kendrick et al. ²⁰²³) was conducted. Thousands of forward model runs were performed to find the optimal constants, which corresponded to the least root‐meansquared error (RMSE). Wide ranges of ρ (i.e., 1 to 300 μm/s) and τ (i.e., 1 to 4) were applied to cover the possible values. Figure 3 shows the contour maps of RMSE distribution in the grid search analysis. We found that RMSE gradually decreased with increasing ρ and τ. Dlugosch et al. (2013) found that the KGM‐estimated K is not sensitive to the variation of τ. Here, we fixed τ to 1.5 and sought an optimal ρ for the glaciofluvial sediments at the NCRS. The optimized model constants are summarized in Table 2. As porosity is still adopted in the SDR equation for some studies, we also provided the calibrated C _SDR when m = 1. However, this scenario was not included in the comparison below.

Figure 3.

Contour maps of RMSE distribution from the grid search analysis for the calibration of the KGM model, in which the smallest RMSE leads to the optimized values for model constants. The black spot denotes the optimal ρ with the smallest RMSE when τ = 1.5.

Table 2.

Summary of Optimized Constants after Calibration with Permeameter Test Data.

	Optimized Model Constants
NMR Model	Data Domain	Well Scale	Other Parameters
SDR (m = 0)	C SDR = 3.86e‐3 m/s3	C SDR = 2.70e‐3–7.21e‐3 m/s3	n = 2
SDR (m = 1) 1	C SDR = 0.0130 m/s3	C SDR = 0.0093–0.0244 m/s3	n = 2
Seevers	C Seevers = 0.0127 m/s3	C Seevers = 0.0091–0.0241 m/s3	m = 1, n = 2
SOE	C SOE = 9.02e‐4 m/s3	C SDR = 5.71e‐4–1.43e‐3 m/s3	n = 2
KGM	ρ = 91 μm/s	ρ = 71–127 μm/s	τ = 1.5

¹ The SDR equation containing porosity.

Results and Discussion

NMR‐Derived K Profiles

Figure 4 shows the K profiles from four NMR models and three types of hydraulic testing methods. Only the K estimates in the saturated zone (i.e., below the water table) were incorporated into this figure. According to the SDR estimates, the highest and lowest K values were 1.36 × 10⁻³ m/s and 1.16 × 10⁻⁸ m/s, respectively, exhibiting a K variation of more than five orders of magnitude.

Figure 4.

Comparison of downhole K estimates between NMR‐derived K using the SDR, Seevers, KGM, SOE, and KGM models, and hydraulic measurements from permeameter, HPT logging, and slug tests. HPT logging was only conducted in CMT1 and CMT3. The vertical bars for slug tests denote their correspondence to NMR‐derived K. Red, yellow, and blue shadings stand for the conductive layers: Top, Aquifer 1, and Aquifer 2.

Based on the visual observations of vertical K trends in Figure 4, three conductive layers can be identified from matching moderate‐to‐high K segments of NMR‐derived K profiles. A moderately conductive layer was found right below the water table (defined as the Top layer) with an average K of 10⁻⁶ m/s. Furthermore, two main aquifers can be characterized: the first one (defined as Aquifer 1) was located at ~8 mbgs, with an average K of 10⁻⁵ m/s; the second aquifer (defined as Aquifer 2) was located from 10 to 12 mbgs, with an average K of 10⁻⁴ m/s. Aquifer 1 was thin (~1 m thick) but laterally continuous, dipping slightly toward the west. Aquifer 2 exhibited a more variable layer thickness, with the thickest sediments near CMT1. It dipped sharply to the northwest and eventually pinched out in CMT2. Based on the SDR estimation, the segments in the K profiles (see Figure 4) with K values over 1 × 10⁻⁵ m/s and 1 × 10⁻⁶ m/s are shaded to highlight the highly and moderately conductive layers, respectively. Hence, the Top layer, Aquifer 1, and Aquifer 2 are shaded in red, yellow, and blue in Figure 4.

Visual analysis of the plotted NMR‐derived K profiles indicated that four models resemble each other in general trends. Particularly, the Seevers curves almost overlapped with the SDR curves. Furthermore, relatively smooth vertical K variations were obtained by the SOE model for Aquifer 2. The SOE model yielded approximately one order of magnitude lower K estimates for Aquifer 2 but consistent K estimates for the other layers, compared to the calibrated SDR model. This observation is consistent with the downhole profiles of NMR signals shown in Figure 2, where the SOE values were generally larger than the T _2ML values with a stable gap for the downhole profiles, while the detected SOE signals were smaller than the T _2ML signals for Aquifer 2 in CMT1, CMT3, and CMT4. The KGM model also yielded K profiles with similar trends and magnitude to the SDR curves, while its K estimates for Aquifer 2 were by a factor of 2 to 3 lower than the K estimates by the SDR model.

Comparison with Hydraulic Testing Methods

Based on a visual analysis, the NMR‐derived K profiles were compared with downhole K estimates from three hydraulic testing methods, respectively. First, the NMR‐derived K profiles matched reasonably with the permeameter‐based K. The K trends and magnitudes of alternating aquifers and aquitards characterized by the two methods were generally consistent across different profiles. However, discrepancies occurred in the depth correlation at ~9 mbgs, which was also found in the comparison between NMR signals and lithology in Figure 2. We noticed that the K measurements from permeameter tests were not continuous along CMT wells. For instance, records were lost from 7.5 to 8.5 mbgs in CMT1 and from 7.0 to 7.7 mbgs in CMT4. Nevertheless, the K magnitudes from NMR logging had good correspondence with permeameter measurements for each conductive layer (i.e., the Top layer, Aquifers 1 and 2) and the aquitard layers in between them, although depth misalignment existed. Borehole NMR data were obtained using in‐situ measurements, which should lead to more accurate mapping of depth ranges for aquifer layers. On the other hand, soil compression and loss may occur during the core sampling process, which was previously reported by Alexander et al. (2011) for the NCRS data. Additionally, core samples were repacked for permeameter testing. These factors may result in inaccurate depth correlation of aquifer layers. Such depth misalignment for permeameter measurements was reasonable, particularly for complex layered porous media, such as the aquifer system at the NCRS. Moreover, it has been considered in previous studies (e.g., Berg and Illman ²⁰¹¹; Zhao and Illman ²⁰¹⁸) at the NCRS that low‐K materials separating the main “aquifer zone” from 8 to 13 mbgs were discontinuous, providing hydraulic windows, for example, in CMT1. Based on the NMR‐derived K profiles (see Figure 4), we found that the main “aquifer zone” was at least fully separated into two aquifers in the CMT locations within the NCRS.

Slug‐based K was typically not consistent with the NMR‐derived K at the same depth. For instance, the slug‐based K in CMT1‐3 was more than one order of magnitude larger than the NMR‐derived K at the same depth. The mismatch is reasonable as flow in slug tests is toward both horizontal and vertical directions, sampling a larger volume with greater amounts of connected high‐K materials. Here, the location of CMT1‐3 was right below Aquifer 1, whose geometric mean of K from NMR logging was on the same order of magnitude as the slug‐based K in CMT1‐3. Likewise, most slug‐based K values (except for CMT2‐4) were close to the NMR K values of the conductive layers within the potential sampling distance of the slug tests (e.g., a depth range of 2 m). The large discrepancy between the slug‐based and NMR‐derived K values in CMT2‐4 can likely be attributed to the fact that slug testing sampled a larger scale with higher hydraulic connectivity. Still, this scale of hydraulic connection was not captured by permeameter tests or NMR logs (e.g., horizontal laminations in sedimentary structures). However, the aquifer characteristics for such a complex layered porous medium cannot be effectively reproduced at a high resolution with slug test results.

Both NMR and HPT logging can recognize a three‐layer aquifer system (i.e., the Top layer and Aquifers 1 and 2) at the NCRS, and their depth correlation was very consistent. Given that NMR and HPT logging provided consistent locations of aquifer layers, the depth correlation between various boreholes based on NMR logging was more reliable than the result based on permeameter measurements. Discrepancies between NMR and HPT logging results existed in the estimation of K for different layers. For example, HPT logging overestimated K values of the Top layer but underestimated K values of Aquifer 1 in CMT1, leading to higher K estimates for the Top layer than Aquifer 1. Also, the HPT‐characterized aquitards that intersected Aquifers 1 and 2 had an average K on the order of magnitude of 10⁻⁶ m/s, while the corresponding K estimates from permeameter testing and NMR logging were on the order of magnitude of 10⁻⁸ or 10⁻⁷ m/s. Confined by the upper and lower boundaries of measured K, HPT logging cannot adequately reproduce variable K ranging from gravel to clay for this glaciofluvial aquifer system. In comparison, NMR estimates were more representative of low K materials.

Evaluation of Petrophysical Models

To evaluate the K predictive performance of NMR logging using various petrophysical models, scatterplots of NMR‐derived K using four petrophysical models versus co‐located hydraulically measured K based on permeameter and slug tests were made (see Figure 5). Due to the narrow range of K measurements from HPT logging, the quantitative comparison with HPT‐based K was not considered.

Figure 5.

Scatterplots of measured K from hydraulic testing methods versus estimated K using four NMR models. (a–d) represent K estimates using the SDR, Seevers, SOE, and KGM models. The solid line is a 1:1 line indicating the perfect match, while the two dashed lines indicate the scope in which the model fit is within one order of magnitude.

Scatter points within the scope of two dashed lines in Figure 5 were regarded as exhibiting satisfactory model fits, in which the deviation in K was no more than one order of magnitude (m/s). Most scatter points for comparison with permeameter tests were located near the 1:1 line, covering the range from 10⁻⁸ to 10⁻³ m/s. Due to the depth misalignment of permeameter measurements, some outliers with large misfits existed on either the upper‐left or lower‐right side of the plot, which were beyond the scope of satisfactory model fits. However, comparable numbers of large misfits occurred on both sides of the 1:1 line, so their impact on the calibrated model constants was reduced. For slug tests, the KGM model slightly overestimated the hydraulically measured K, while the other three models (i.e., SDR, Seevers, and SOE) conversely underestimated the co‐located K values from slug tests, with more scatter points located on the lower‐right side of the 1:1 line.

Two metrics, mean absolute error (MAE) and root‐mean‐squared error (RMSE), were computed to quantify the NMR model fits with hydraulic measurements. Figure 6 summarizes the color‐coded MAE and RMSE in the statistical analysis. We attached one more scenario where porosity was included in the SDR equation with m = 1 (see Equation 5).

Figure 6.

Summary of the color‐coded MAE and RMSE for the comparison between NMR‐derived K and hydraulic measurements from permeameter and slug tests. Color is coded column‐wise, in which maximum and minimum metric values are highlighted in red and green, respectively. The change in color is linear to the change in metric values.

As the calculated MAE and RMSE were not reduced for both comparisons with permeameter and slug tests (see Figure 6), incorporating porosity in the SDR model did not improve the prediction of K. This result was consistent with the visual observations of total porosity profiles for the CMT wells (see Figure 2), in which relatively smooth vertical variations of total porosity were found, unable to indicate the locations of aquifer layers. Likewise, the stress on bulk relaxation did not improve the estimation of K, as the model fits of Seevers always lagged behind the SDR results. These findings corroborated previous studies (e.g., Maurer and Knight ²⁰¹⁶; Kendrick et al. ²⁰²³) for the evaluations of the SDR and Seevers models at other sites.

Furthermore, the calibrated SDR constants in this study resembled the calibrated values for other glacial deposits (e.g., Crow et al. ²⁰²²; Kendrick et al. ²⁰²³). Table 3 lists calibrated C _SDR for glacial deposits given n = 2. The soil descriptions and the hydraulic testing methods used for K calibration datasets are also provided. Here, our calibrated values resembled the value by Crow et al. (2022) given m = 0 (i.e., porosity is not included) and the value by Kendrick et al. (2023) given m = 1. The differences in the calibrated C _SDR were less than 20%. In a previous study, Knight et al. (2016) calibrated C _SDR for alluvial deposits at three different sites and found stable C _SDR ranging from 0.08 to 0.09 m/s³. Although the calibrated constants can still vary between different geological environments (glacial vs. alluvial), our findings confirmed that calibrated SDR constants were reasonably consistent for a given unconsolidated sediment, such as the glacial deposits at different sites.

Table 3.

Compilation of the Calibrated SDR Constants for Glacial Deposits Given n = 2.

Location (Source)	SDR Model	Calibrated C SDR (m/s3)	Well‐Scale Range (m/s3)	K Calibration Datasets	Geologic Materials
Ontario, Canada (this study)	m = 0	3.86e‐3	2.70e‐3–7.21e‐3	Falling‐head permeameter tests	Gravel to clay
	m = 1	0.0130	0.0093–0.0244
Ontario, Canada (Crow et al. 2022)	m = 0	3.2e‐3	1.4e‐3–4.6e‐3	Multi‐level slug tests	Gravel to silt
Wisconsin, USA (Kendrick et al. 2023)	m = 1	0.0155	0.0132–0.0213	Direct‐push permeameter tests	Sand to silt

Compared to the SDR model, the SOE model result fitted slightly better with permeameter‐based K, while their predictions for slug‐based K were comparable. The relatively improved performance of the SOE model may be associated with the reduced noise of the SOE signal compared with T _2ML, which was previously reported by Walsh et al. (2013).

In terms of the summarized MAE and RMSE in Figure 6, the KGM model yielded the top two model fits with permeameter‐based K and the optimal model fits with slug‐based K. However, the locally optimized ρ varied strongly across sites with similar sedimentations: the calibrated ρ in this study was 91 μm/s given τ = 1.5, while the calibrated ρ was 172 μm/s given τ = 1.143 by Kendrick et al. (2023). Therefore, whether the KGM constant remains consistent across similar geological environments still requires further investigation.

Conclusions

In this study, we evaluated the performance of NMR petrophysical models to estimate hydraulic conductivity (K) in a highly heterogeneous glaciofluvial aquifer system. Hence, we first calibrated NMR‐derived K profiles at four well locations with permeameter test data. The optimized model constants suitable for glaciofluvial deposits were listed. To highlight the performance of NMR logging, we compared the NMR‐derived K with K measurements from various hydraulic testing methods, including (1) permeameter tests that provided accurate K measurements in the laboratory; (2) multi‐level slug tests that sampled a larger extent local to the screen locations; and (3) direct‐push HPT logging survey, which is an in‐situ physical method to measure K at a dense sampling interval. We evaluated four popular NMR models, including Schlumberger‐Doll Research (SDR), Seevers, Sum‐of‐Echoes (SOE), and Kozeny‐Godefroy (KGM), to estimate K by quantifying the model fits with K measurements from permeameter and slug tests. Our study resulted in the following findings and conclusions:

1. NMR logging can yield K estimates over five orders of magnitude for the glaciofluvial deposits at the NCRS, which consist of gravel, sand, silt, and clay. The main aquifer characteristics for this complex layered porous medium can be reproduced through the cross‐hole comparison of four NMR‐derived K profiles, including the shape and volume of alternating aquifers and aquitards with their geometric mean of K.

2. Site characterization performance of NMR logging is highlighted after comparing it with various hydraulic testing methods featured on their respective advantages. Meanwhile, NMR logging can avoid the shortcomings of these testing methods found in this study. Specifically, NMR logging yields comparable K estimates to permeameter‐based K, and its high‐resolution delineation of hydrostratigraphy is similar to HPT logging. After upscaling, the NMR‐derived K estimates can generally account for the slug‐based K estimates at a larger scale.

3. Through quantitative comparison with permeameter and slug tests, all four petrophysical models can produce reliable K estimates. The SDR model always yields slightly better model fits than the Seevers model, and including porosity does not improve the predictive performance of the SDR model. The SDR constant is demonstrated to be stable across different sites for glacial deposits. After adopting NMR signals with better noise control, the SOE model yields slightly improved fits compared to the SDR model. The KGM model leads the predictive performance, especially on the model fits with slug tests, which may contribute to the calibration of surface relaxivity in this physically derived NMR estimator.

Based on these findings, we provide the following recommendations for selecting suitable NMR models to estimate K in glacial deposits: (1) when there is limited information about lithology, the SDR model without porosity can be sufficient for basic site characterization; (2) when the background noise is high, the SOE model is recommended given the improved noise control of the SOE signal; (3) the KGM model can provide more optimal K estimates if hydraulic calibration points are available at the site.

Authors' Note

The authors do not have any conflicts of interest or financial disclosures to report.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.