Pore size distributions and hydraulic conductivities of rocks derived from Magnetic Resonance Sounding relaxation data using multi-exponential decay time inversion

In hydrogeology there is a variety of empirical formulae available for determination of hydraulic conductivity of porous media, all based on the analysis of grain size distributions of aquifer materials. Sensitivity of NMR measurements to pore sizes makes it a good indicator of hydraulic conductivity. Analogous to laboratory NMR, Magnetic Resonance Sounding (MRS) relaxation data are of a multi-exponential (ME) nature due to the distribution of different pore sizes in an investigated rock layer. ME relaxation behaviour will also arise due to the superposition of NMR signals which originate from different layers. It has been shown, that both kinds of ME behaviour coexist in MRS and can principally be separated by ME inversion of the field data. Only a few publications exist that have proposed approaches to qualitatively and quantitatively estimate petrophysical parameters such as the hydraulic conductivity from MRS measurements, i.e. MRS porosity and decay times. The so far used relations for the estimation of hydraulic conductivity in hydrogeology and NMR experiments are compared and discussed with respect to their applicability in MRS. Taking into account results from a variety of laboratory NMR and MRS experiments mean rock specific calibration factors are introduced for a data-base-calibrated estimation of hydraulic conductivity when no on-site calibration of MRS is available. Field data have been analysed using conventional and ME inversion using such mean calibration values. The results for conventional and ME inversion agree with estimates obtained from well core analysis for shallow depths but are significantly improved using a ME inversion approach for greater depths.     

Influence of pore size and geometry on peat unsaturated hydraulic conductivity computed from 3D computed tomography image analysis

In organic soils, hydraulic conductivity is related to the degree of decomposition and soil compression, which reduce the effective pore diameter and consequently restrict water flow. This study investigates how the size distribution and geometry of air‐filled pores control the unsaturated hydraulic conductivity of peat soils using high‐resolution (45 µm) three‐dimensional (3D) X‐ray computed tomography (CT) and digital image processing of four peat sub‐samples from varying depths under a constant soil water pressure head. Pore structure and configuration in peat were found to be irregular, with volume and cross‐sectional area showing fractal behaviour that suggests pores having smaller values of the fractal dimension in deeper, more decomposed peat, have higher tortuosity and lower connectivity, which influences hydraulic conductivity. The image analysis showed that the large reduction of unsaturated hydraulic conductivity with depth is essentially controlled by air‐filled pore hydraulic radius, tortuosity, air‐filled pore density and the fractal dimension due to degree of decomposition and compression of the organic matter. The comparisons between unsaturated hydraulic conductivity computed from the air‐filled pore size and geometric distribution showed satisfactory agreement with direct measurements using the permeameter method. This understanding is important in characterizing peat properties and its heterogeneity for monitoring the progress of complex flow processes at the field scale in peatlands. Copyright © 2010 John Wiley & Sons, Ltd.     

Hydraulic Conductivity Calibration of Logging NMR in a Granite Aquifer,Laramie Range,Wyoming

In granite aquifers, fractures can provide both storage volume and conduits for groundwater. Characterization of fracture hydraulic conductivity (K) in such aquifers is important for predicting flow rate and calibrating models. Nuclear magnetic resonance (NMR) well logging is a method to quickly obtain near-borehole hydraulic conductivity (i.e., K_NMR) at high-vertical resolution. On the other hand, FLUTe flexible liner technology can produce a K profile at comparable resolution but requires a fluid driving force between borehole and formation. For three boreholes completed in a fractured granite, we jointly interpreted logging NMR data and FLUTe K estimates to calibrate an empirical equation for translating borehole NMR data to K estimates. For over 90% of the depth intervals investigated from these boreholes, the estimated K_NMR are within one order of magnitude of K_FLUTe. The empirical parameters obtained from calibrating the NMR data suggest that “intermediate diffusion” and/or “slow diffusion” during the NMR relaxation time may occur in the flowing fractures when hydraulic aperture are sufficiently large. For each borehole, “intermediate diffusion” dominates the relaxation time, therefore assuming “fast diffusion” in the interpretation of NMR data from fractured rock may lead to inaccurate K_NMR estimates. We also compare calibrations using inexpensive slug tests that suggest reliable K_NMR estimates for fractured rock may be achieved using limited calibration against borehole hydraulic measurements.     

球管孔隙模型的核磁共振(NMR)弛豫特征及应用

将岩石孔隙归结为由毛细管和球形孔组成的孔隙系统,提出孔隙球管孔隙模型. 岩石孔隙按大小分组后,每一组孔隙的氢核弛豫时间可以用球管孔隙模型计算. 将此弛豫时间作为反演T2分布的时间控制点,反演岩芯的核磁共振弛豫信号. 研究表明, T2分布与特定的孔隙结构相联系. 使用不同结构的球管孔隙模型,可以使反演T2分布最大限度地拟合弛豫信号,此时的球管模型以最近似的方式模拟了岩芯的孔隙结构. 使用球管模型对实验室核磁共振资料进行了处理,对比压汞分析得到毛管力分布数据,结果证明球管孔隙模型描述了岩石孔隙的弛豫特征,而且,岩石特定孔隙结构的弛豫特征与孔隙流体有关.     

Using data assimilation method to calibrate a heterogeneous conductivity field conditioning on transient flow test data

A data assimilation method is developed to calibrate a heterogeneous hydraulic conductivity field conditioning on transient pumping test data. The ensemble Kalman filter (EnKF) approach is used to update model parameters such as hydraulic conductivity and model variables such as hydraulic head using available data. A synthetical two-dimensional flow case is used to assess the capability of the EnKF method to calibrate a heterogeneous conductivity field by assimilating transient flow data from observation wells under different hydraulic boundary conditions. The study results indicate that the EnKF method will significantly improve the estimation of the hydraulic conductivity field by assimilating continuous hydraulic head measurements and the hydraulic boundary condition will significantly affect the simulation results. For our cases, after a few data assimilation steps, the assimilated conductivity field with four Neumann boundaries matches the real field well while the assimilated conductivity field with mixed Dirichlet and Neumann boundaries does not. We found in our cases that the ensemble size should be 300 or larger for the numerical simulation. The number and the locations of the observation wells will significantly affect the hydraulic conductivity field calibration.     

Understanding of hydraulic properties from configurations of stochastically distributed fracture networks

A systematic investigation of the effect of configurations of stochastically distributed fracture networks on hydraulic behaviour for fractured rock masses could provide either quantitative or qualitative correlation between the structural configuration of the fracture network and its corresponding hydraulic behaviour, and enhance our understanding of appropriate application of groundwater flow and contaminant transport modelling in fractured rock masses. In this study, the effect of block sizes, intersection angles of fracture sets, standard deviations of fracture orientation, and fracture densities on directional block hydraulic conductivity and representative elementary volume is systematically investigated in two dimensions by implementing a numerical discrete fracture fluid flow model and incorporating stochastically distributed fracture configurations. It is shown from this investigation that the configuration of a stochastically distributed fracture network has a significant quantitative or qualitative effect on the hydraulic behaviour of fractured rock masses. Compared with the deterministic fracture configurations that have been extensively dealt with in a previous study, this investigation is expected to be more practical and adequate, since fracture geometry parameters are inherently stochastically distributed in the field. Moreover, the methodology and approach presented in this study may be generally applied to any fracture system in investigating the hydraulic behaviours from configurations of the fracture system while establishing a ‘bridge’ from the discrete fracture network flow modelling to equivalent continuum modelling in fractured rock masses. Copyright © 2007 John Wiley & Sons, Ltd.     

Estimation of anisotropic hydraulic conductivity using geophysical data in a coastal aquifer of Karnataka,India

Estimation of hydraulic parameters is essential to understand the interaction between groundwater flow and seawater intrusion. Though several studies have addressed hydraulic parameter estimation, based on pumping tests as well as geophysical methods, not many studies have addressed the problem with clayey formations being present. In this study, a methodology is proposed to estimate anisotropic hydraulic conductivity and porosity values for the coastal aquifer with unconsolidated formations. For this purpose, the one-dimensional resistivity of the aquifer and the groundwater conductivity data are used to estimate porosity at discrete points. The hydraulic conductivity values are estimated by its mutual dependence with porosity and petrophysical parameters. From these estimated values, the bilinear relationship between hydraulic conductivity and aquifer resistivity is established based on the clay content of the sampled formation. The methodology is applied on a coastal aquifer along with the coastal Karnataka, India, which has significant clayey formations embedded in unconsolidated rock. The estimation of hydraulic conductivity values from the established correlations has a correlation coefficient of 0.83 with pumping test data, indicating good reliability of the methodology. The established correlations also enable the estimation of horizontal hydraulic conductivity on two-dimensional resistivity sections, which was not addressed by earlier studies. The inventive approach of using the established bilinear correlations at one-dimensional to two-dimensional resistivity sections is verified by the comparison method. The horizontal hydraulic conductivity agrees with previous findings from inverse modelling. Additionally, this study provides critical insights into the estimation of vertical hydraulic conductivity and an equation is formulated which relates vertical hydraulic conductivity with horizontal. Based on the approach presented, the anisotropic hydraulic conductivity of any type aquifer with embedded clayey formations can be estimated. The anisotropic hydraulic conductivity has the potential to be used as an important input to the groundwater models.     

Evaluating the effect of using artificial pore water on the quality of laboratory hydraulic conductivity measurements of peat

Pore dilation, the compaction of humic acids on peat fibres due to the process of flocculation, causes the hydraulic conductivity of peat to increase with increasing pore water electrical conductivity. This is a reversible process and a reduction in the pore water conductivity produces a decrease in the hydraulic conductivity due to the constriction of pores. We verify how this dilation and constriction of pores, resulting from the application of artificial pore water (primarily deionized water), affects laboratory measurements of the hydraulic conductivity of peat. Repeat measurements of the hydraulic conductivity were performed on samples of Sphagnum peat. It is shown that the application of deionized water during constant head permeameter tests causes a significant decrease in the hydraulic conductivity. Between tests, the hydraulic conductivity of the peat continues to decline without an associate decrease in the pore water electrical conductivity because of a lagged pore constriction effect. We suggest that the use of artificially high or low pore water electrical conductivities, during laboratory hydraulic conductivity measurements, is likely to lead to significant errors. Experimental protocols must, therefore, be revised to take better account of the pore water chemistry. The ionic concentrations of the natural pore fluid should be replicated during hydraulic conductivity tests, either by using pore fluid extracted from the study site or by artificially replicating the major ionic composition of the natural pore fluid. In addition, prior to the hydraulic conductivity measurements, peat samples should be flushed with this solution until the hydraulic conductivity stabilizes and the samples subsequently allowed to equilibrate. Copyright © 2010 John Wiley & Sons, Ltd.     

Near Surface Electrical Characterization of Hydraulic Conductivity: From Petrophysical Properties to Aquifer Geometries—A Review

This paper reviews the recent geophysical literature addressing the estimation of saturated hydraulic conductivity (K) from static low frequency electrical measurements (electrical resistivity, induced polarization (IP) and spectral induced polarization (SIP)). In the first part of this paper, research describing how petrophysical relations between electrical properties and effective (i.e. controlling fluid transport) properties of (a) the interconnected pore volumes and interconnected pore surfaces, have been exploited to estimate K at both the core and field scale is reviewed. We start with electrical resistivity measurements, which are shown to be inherently limited in K estimation as, although resistivity is sensitive to both pore volume and pore surface area properties, the two contributions cannot be separated. Efforts to utilize the unique sensitivity of IP and SIP measurements to physical parameters that describe the interconnected pore surface area are subsequently introduced and the incorporation of such data into electrical based Kozeny–Carman type models of K estimation is reviewed. In the second part of this review, efforts to invert geophysical datasets for spatial patterns of K variability (e.g. aquifer geometries) at the field-scale are considered. Inversions, based on the conversion of an image of a geophysical property to a hydrological property assuming a valid petrophysical relationship, as well as joint/constrained inversion methods, whereby multiple geophysical and hydrological data are inverted simultaneously, are briefly covered. This review demonstrates that there currently exists an opportunity to link, (1) the petrophysics relating low frequency electrical measurements to effective hydraulic properties, with (2) the joint inversion strategies developed in recent years, in order to obtain more meaningful estimates of spatial patterns of K variability than previously reported.     

AN EXERCISE IN GROUND-WATER MODEL CALIBRATION AND PREDICTION

Abstract. For a classroom exercise, nine groups of graduate students calibrated a numerical ground-water flow model to a set of perfectly observed hydraulic head data for a hypothetical phreatic aquifer. All groups used exactly the same numerical model and identical sets of observed data. After calibration, the students predicted the hydraulic head distribution in the aquifer resulting from a modification in one boundary condition. A quantitative analysis of the results of this calibration-prediction exercise vividly demonstrates some of the difficulties in parameter identification for ground-water flow models. Group predictions differed significantly. Successful prediction was strongly correlated with successful estimation of conductivity values, and was essentially unrelated to successful estimation of aquifer bottom elevations or with the number of trial-and-error simulations required for calibration. Most importantly, success in prediction was unrelated to success in matching observed heads under premodification conditions. In this sense, good calibration did not lead to good prediction.     

Comments on “Steady- and transient-state inversion in hydrogeology by successive flux estimation” by P. Pasquier and D. Marcotte

Pasquier and Marcotte [Pasquier P, Marcotte D. Steady- and transient-state inversion in hydrogeology by successive flux estimation. Adv Wat Res 2006;29:1934–52] propose some modifications to the Comparison Model Method (CMM), in order to apply it to transient 3D ground water flow data for conductivity identification. We present some remarks on that paper to improve the comprehension of the basic features of the CMM and of the real value of the novelties introduced by Pasquier and Marcotte.     

Comparison of deterministic ensemble Kalman filters for assimilating hydrogeological data

Groundwater models are critical decision support tools for water resources management and environmental remediation. However, limitations in site characterization data and conceptual models can adversely affect the reliability of groundwater models. Therefore, there is a strong need for continuous model uncertainty reduction. Ensemble filters have recently emerged as promising high-dimensional data assimilation techniques. Two general categories of ensemble filters exist in the literature: perturbation-based and deterministic. Deterministic ensemble filters have been extensively studied for their better performance and robustness in assimilating oceanographic and atmospheric data. In hydrogeology, while a number of previous studies demonstrated the usefulness of the perturbation-based ensemble Kalman filter (EnKF) for joint parameter and state estimation, there have been few systematic studies investigating the performance of deterministic ensemble filters. This paper presents a comparative study of four commonly used deterministic ensemble filters for sequentially estimating the hydraulic conductivity parameter in low- and moderately high-dimensional groundwater models. The performance of the filters is assessed on the basis of twin experiments in which the true hydraulic conductivity field is assumed known. The test results indicate that the deterministic ensemble Kalman filter (DEnKF) is the most robust filter and achieves the best performance at relatively small ensemble sizes. Deterministic ensemble filters often make use of covariance inflation and localization to stabilize filter performance. Sensitivity studies demonstrate the effects of covariance inflation, localization, observation density, and conditioning on filter performance.     

A geostatistically based inverse model for three-dimensional variably saturated flow

We present a geostatistically based inverse model for characterizing heterogeneity in parameters of unsaturated hydraulic conductivity for three-dimensional flow. Pressure and moisture content are related to perturbations in hydraulic parameters through cross-covariances, which are calculated to first-order. Sensitivities needed for covariance calculations are derived using the adjoint state sensitivity method. Approximations of the conditional mean parameter fields are then obtained from the cokriging estimator. Correlation between parameters and pressure – moisture content perturbations is seen to be strongly dependent on mean pressure or moisture content. High correlation between parameters and pressure data was obtained under saturated or near saturated flow conditions, providing accurate estimation of saturated hydraulic conductivity, while moisture content measurements provided accurate estimation of the pore size distribution parameter under unsaturated flow conditions.     

A geostatistically based inverse model for three-dimensional variably saturated flow

We present a geostatistically based inverse model for characterizing heterogeneity in parameters of unsaturated hydraulic conductivity for three-dimensional flow. Pressure and moisture content are related to perturbations in hydraulic parameters through cross-covariances, which are calculated to first-order. Sensitivities needed for covariance calculations are derived using the adjoint state sensitivity method. Approximations of the conditional mean parameter fields are then obtained from the cokriging estimator. Correlation between parameters and pressure – moisture content perturbations is seen to be strongly dependent on mean pressure or moisture content. High correlation between parameters and pressure data was obtained under saturated or near saturated flow conditions, providing accurate estimation of saturated hydraulic conductivity, while moisture content measurements provided accurate estimation of the pore size distribution parameter under unsaturated flow conditions.     

水力压裂对速度场及微地震定位的影响

水力压裂是页岩气开发过程中的核心增产技术,微地震则广泛用于压裂分析、水驱前缘监测和储层描述.微地震反演过程中,用于反演的速度模型往往基于测井、地震或标定炮资料构建,忽略了压裂过程中裂缝及孔隙流体压力变化对地层速度的影响.本文首先基于物质守恒、渗流理论和断裂力学模拟三维水力压裂过程,得到地下裂缝发育特征和孔隙压力分布.继而根据Coates-Schoenberg方法和裂缝柔量参数计算裂缝和孔隙压力对速度场的影响,得到压裂过程中的实时速度模型.最后利用三维射线追踪方法正演微地震走时和方位信息,并采用常规微地震定位方法反演震源位置及进行误差分析.数值模拟结果表明,检波器空间分布影响定位精度,常规方法的定位误差随射线路径在压裂带中传播距离增加而变大,且不同压裂阶段的多点反演法与单点极化法精度相当.     

Use of steady-state concentration measurements in geostatistical inversion

In geostatistical inverse modeling, hydrogeological parameters, such as hydraulic conductivity, are estimated as spatial fields. Upon discretization this results in several thousand (log-)hydraulic conductivity values to be estimated. Common inversion schemes rely on gradient-based parameter estimation methods which require the sensitivity of all measurements with respect to all parameters. Point-like measurements of steady-state concentration in aquifers are generally not well suited for gradient-based methods, because typical plumes exhibit only a very narrow fringe at which the concentration decreases from a maximal value to zero. Only here the sensitivity of concentration with respect to hydraulic conductivity significantly differs from zero. Thus, if point-like measurements of steady-state concentration do not lie in this narrow fringe, their sensitivity with respect to hydraulic conductivity is zero. Observations of concentrations averaged over a larger control volume, by contrast, show a more regular sensitivity pattern. We thus suggest artificially increasing the sampling volume of steady-state concentration measurements for the evaluation of sensitivities in early stages of an iterative parameter estimation scheme. We present criteria for the extent of artificially increasing the sampling volume and for decreasing it when the simulation results converge to the measurements. By this procedure, we achieve high stability in geostatistical inversion of steady-state concentration measurements. The uncertainty of the estimated parameter fields is evaluated by generating conditional realizations.     

Magnetic resonance sounding applied to aquifer characterization

Magnetic resonance sounding (MRS) is distinguished from other geophysical tools used for ground water investigation by the fact that it measures a magnetic resonance signal generated directly from subsurface water molecules. An alternating current pulse energizes a wire loop on the ground surface and the MRS signal is generated; subsurface water is indicated, with a high degree of reliability, by nonzero amplitude readings. Measurements with varied pulse magnitudes then reveal the depth and thickness of water saturated layers. The hydraulic conductivity of aquifers can also be estimated using boreholes for calibration. MRS can be used for both predicting the yield of water supply wells and for interpolation between boreholes, thereby reducing the number of holes required for hydrogeological modeling. An example of the practical application of MRS combined with two-dimensional electrical imaging, in the Kerbernez and Kerien catchments area of France, demonstrates the efficiency of the technique.     

Conceptualization and Calibration of Anisotropic Alluvial Systems: Pitfalls and Biases

Physical properties of alluvial environments typically feature a high degree of anisotropy and are characterized by dynamic interactions between the surface and the subsurface. Hydrogeological models are often calibrated under the assumptions of isotropic hydraulic conductivity fields and steady-state conditions. We aim at understanding how these simplifications affect predictions of the water table using physically based models and advanced calibration and uncertainty analysis approaches based on singular value decomposition and Bayesian analysis. Specifically, we present an analysis of the information content provided by steady-state hydraulic data compared to transient data with respect to the estimation of aquifer and riverbed hydraulic properties. We show that assuming isotropy or fixed anisotropy may generate biases both in the estimation of aquifer and riverbed parameters as well as in the predictive uncertainty of the water table. We further demonstrate that the information content provided by steady-state hydraulic heads is insufficient to jointly estimate the aquifer anisotropy together with the aquifer and riverbed hydraulic conductivities and that transient data can help to reduce the predictive uncertainty to a greater extent. The outcomes of the synthetic analysis are applied to the calibration of a dynamic and anisotropic alluvial aquifer in Switzerland (The Rhône River). The results of the synthetic and real world modeling and calibration exercises documented herein provide insight on future data acquisition as well as modeling and calibration strategies for these environments. They also provide an incentive for evaluation and estimation of commonly made simplifying assumptions in order to prevent underestimation of the predictive uncertainty.     

Steady- and transient-state inversion in hydrogeology by successive flux estimation

A calibration method to solve the groundwater inverse problem under steady- and transient-state conditions is presented. The method compares kriged and numerical head field gradients to modify hydraulic conductivity without the use of non-linear optimization techniques. The process is repeated iteratively until a close match with piezometric data is reached. The approach includes a damping factor to avoid divergence and oscillation of the solution in areas of low hydraulic gradient and a weighting factor to account for temporal head variation in transient simulations. The efficiency of the method in terms of computing time and calibration results is demonstrated with a synthetic field. It is shown that the proposed method provides parameter fields that reproduce both hydraulic conductivity and piezometric data in few forward model solutions. Stochastic numerical experiments are conducted to evaluate the sensitivity of the method to the damping function and to the head field estimation errors.     

Semiautomatic and Automatic Cooperative Inversion of Seismic and Magnetotelluric Data

Natural source electromagnetic methods have the potential to recover rock property distributions from the surface to great depths. Unfortunately, results in complex 3D geo-electrical settings can be disappointing, especially where significant near-surface conductivity variations exist. In such settings, unconstrained inversion of magnetotelluric data is inexorably non-unique. We believe that: (1) correctly introduced information from seismic reflection can substantially improve MT inversion, (2) a cooperative inversion approach can be automated, and (3) massively parallel computing can make such a process viable. Nine inversion strategies including baseline unconstrained inversion and new automated/semiautomated cooperative inversion approaches are applied to industry-scale co-located 3D seismic and magnetotelluric data sets. These data sets were acquired in one of the Carlin gold deposit districts in north-central Nevada, USA. In our approach, seismic information feeds directly into the creation of sets of prior conductivity model and covariance coefficient distributions. We demonstrate how statistical analysis of the distribution of selected seismic attributes can be used to automatically extract subvolumes that form the framework for prior model 3D conductivity distribution. Our cooperative inversion strategies result in detailed subsurface conductivity distributions that are consistent with seismic, electrical logs and geochemical analysis of cores. Such 3D conductivity distributions would be expected to provide clues to 3D velocity structures that could feed back into full seismic inversion for an iterative practical and truly cooperative inversion process. We anticipate that, with the aid of parallel computing, cooperative inversion of seismic and magnetotelluric data can be fully automated, and we hold confidence that significant and practical advances in this direction have been accomplished.