Nonparametric method for transmissivity distributions along boreholes

The transmissivities of individual fractures along a borehole are difficult to obtain unless each fracture is tested. To estimate a fracture-transmissivity distribution from section transmissivities, a method was developed based on fixed-interval-length transmissivities and the corresponding number of fractures for each interval. The method is nonparametric and iterative, and the fractures are viewed as two-dimensional features, in which the total transmissivity of a borehole is equal to the sum of individual fracture transmissivities. Initially, a linear a priori assumption of the transmissivity distribution is made, and from this a so-called mean transmissivity function is derived. Subsequently, the mean transmissivity of the Nj fractures within a section, j, of the borehole is estimated, and the same value of the mean transmissivity function represents Nj possible fracture transmissivities from the initial distribution. This is repeated for each borehole section, and, eventually, all fracture transmissivities are sorted to give the next iteration's transmissivity distribution and the corresponding mean transmissivity function. Finally, the distributions converge, yielding a possible fracture-transmissivity distribution. The method was verified for a synthetic data sample and then tested on a sample from a borehole at the Asp? Hard Rock Laboratory, Sweden. For the synthetic data, the method gave a distribution that was fairly close to the original one; for the Asp? data, 15% of the fractures had a transmissivity larger than the measurement limit (1 x 10(-9) m2/sec), and these transmissivities follow a log-normal distribution.     

A numerical comparison between two upscaling techniques: non-local inverse based scaling and simplified renormalization

In this paper, we face the problem of upscaling transmissivity from the macroscopic to the megascopic scale; here the macroscopic scale is that of the continuous flow equations, whereas the megascopic scale is that of the flow models on a coarse grid. In this paper, we introduce the non-local inverse based scaling (NIBS) and compare it with the simplified renormalization (SR). The latter is a classical technique that we adapt to compute internode transmissivities for a finite differences flow model in a direct way. NIBS is implemented in three steps: in the first step, the macroscopic transmissivity, together with arbitrarily chosen auxiliary boundary conditions and sources, is used to solve forward problems (FPs) at the macroscopic scale; in the second step, the resulting heads are sampled at the megascopic scale; in the third step, the upscaled internode transmissivities are obtained by solving an inverse problem with the differential system method (DS) for which the heads resulting from the second step are used. NIBS is a non-local technique, because the computation of the internode transmissivities relies upon the whole transmissivity field at the macroscopic scale. We test NIBS against SR in the case of synthetic, isotropic, confined aquifers under the assumptions of two-dimensional (2D) and steady-state flow; the aquifers differ for the degree of heterogeneity, which is represented by a normally distributed uncorrelated component of lnT. For the comparison, the reference heads and fluxes at the megascopic scale are computed from the solution of FPs at the macroscopic scale. These reference values are compared with the heads and the fluxes predicted from models at the megascopic scale using the upscaled parameters of SR and NIBS. For the class of aquifers considered in this paper, the results of SR are better than those of NIBS, which hints that non-local effects can be disregarded at the megascopic scale. The two techniques provide comparable results when the heterogeneity increases, when the megascopic scale is large with respect to the heterogeneity length scale, or when the source terms are relevant.     

A Single Packer Method for Characterizing Water Contributing Fractures in Crystalline Bedrock Wells

Water levels and water quality of open borehole wells in fractured bedrock are flow-weighted averages that are a function of the hydraulic heads and transmissivities of water contributing fractures, properties that are rarely known. Without such knowledge using water levels and water quality data from fractured bedrock wells to assess groundwater flow and contaminant conditions can be highly misleading. This study demonstrates a cost-effective single packer method to determine the hydraulic heads and transmissivities of water contributing fracture zones in crystalline bedrock wells. The method entails inflating a pipe plug to isolate sections of an open borehole at different depths and monitoring changes in the water level with time. At each depth, the change in water level with time was used to determine the sum of fracture transmissivities above the packer and then to solve for individual fracture transmissivity. Steady-state wellbore heads along with the transmissivities were used to determine individual fracture heads using the weighted average head equation. The method was tested in five wells in crystalline bedrock located at the University of Connecticut in Storrs. The single packer head and transmissivity results were found to agree closely with those determined using conventional logging methods and the dissolved oxygen alteration method. The method appears to be a simple and cost-effective alternative in obtaining important information on flow conditions in fractured crystalline bedrock wells.     

A comparison of two stochastic inverse methods in a field-scale application

Inverse modeling is a useful tool in ground water flow modeling studies. The most frequent difficulties encountered when using this technique are the lack of conditioning information (e.g., heads and transmissivities), the uncertainty in available data, and the nonuniqueness of the solution. These problems can be addressed and quantified through a stochastic Monte Carlo approach. The aim of this work was to compare the applicability of two stochastic inverse modeling approaches in a field-scale application. The multi-scaling (MS) approach uses a downscaling parameterization procedure that is not based on geostatistics. The pilot point (PP) approach uses geostatistical random fields as initial transmissivity values and an experimental variogram to condition the calibration. The studied area (375 km2) is part of a regional aquifer, northwest of Montreal in the St. Lawrence lowlands (southern Québec). It is located in limestone, dolomite, and sandstone formations, and is mostly a fractured porous medium. The MS approach generated small errors on heads, but the calibrated transmissivity fields did not reproduce the variogram of observed transmissivities. The PP approach generated larger errors on heads but better reproduced the spatial structure of observed transmissivities. The PP approach was also less sensitive to uncertainty in head measurements. If reliable heads are available but no transmissivities are measured, the MS approach provides useful results. If reliable transmissivities with a well inferred spatial structure are available, then the PP approach is a better alternative. This approach however must be used with caution if measured transmissivities are not reliable.     

Comparison of Two Computational Methods for Estimating Transmissivity Based on the Picking Equation

A comparison is presented of two computational methods, PICKINGmodel and PPC-Recovery, to estimate transmissivities based on the Picking equation using water-level recovery data from brief pumping tests of relatively low-yielding domestic wells. The tests were performed by the United States Geological Survey (USGS) in 50 domestic bedrock wells in south-central New York State, and USGS staff performed the analysis using PICKINGmodel based on the Picking equation. The results indicated that the estimated transmissivities ranged from 0.86 to 2900 ft²/d (0.080 to 270 m²/d) with a median of 41 ft²/d (3.8 m²/d). The same data were later analyzed using PPC-Recovery also based on the Picking equation. The two sets of estimated transmissivities were compared and statistically had the same median value at a probability of 95%. In another analysis, the PPC-Recovery method was applied to the same data that had been truncated at the point when the slope of the recovery data curve began to deviate from a straight line aligned with the middle portion of the recovery data. Comparing these resulting estimates of transmissivity with values originally obtained using the PICKINGmodel, the two had statistically the same median value for transmissivity at a probability of 95%. It was concluded that using PPC-Recovery in this manner to estimate transmissivity in low-yielding domestic wells will yield transmissivity values sufficiently close to the results had PICKINGmodel been used, and with less time and effort.     

Interpretation of transmissivity estimates from single-well pumping aquifer tests

Interpretation of single-well tests with the Cooper-Jacob method remains more reasonable than most alternatives. Drawdowns from 628 simulated single-well tests where transmissivity was specified were interpreted with the Cooper-Jacob straight-line method to estimate transmissivity. Error and bias as a function of vertical anisotropy, partial penetration, specific yield, and interpretive technique were investigated for transmissivities that ranged from 10 to 10,000 m(2)/d. Cooper-Jacob transmissivity estimates in confined aquifers were affected minimally by partial penetration, vertical anisotropy, or analyst. Cooper-Jacob transmissivity estimates of simulated unconfined aquifers averaged twice the known values. Transmissivity estimates of unconfined aquifers were not improved by interpreting results with an unconfined aquifer solution. Judicious interpretation of late-time data consistently improved estimates where transmissivity exceeded 250 m(2)/d in unconfined aquifers.     

Quantifying uncertainty of the semivariogram of transmissivity of an existing groundwater monitoring network

A method is presented for quantifying the uncertainty of the semivariogram of transmissivity and determining the required number of measurements. In this method, the estimated semivariogram and its 95% confidence limits are first determined from a finite number of measurements. The uncertainty of the estimated semivariogram is then quantified using the random field simulation technique. For a given value of the quantitative index of uncertainty, the required number of measured data can finally be obtained. Actual transmissivity data of an existing groundwater monitoring network are used in the application of the proposed method. The required numbers of measurements of transmissivity for four different values of the quantitative index of uncertainty are provided, from which reliable semivariograms of the transmissivity can be obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

Conservative identification of nodal transmissivities: theory,computational experiments and application to the Rharb coastal aquifer (Morocco)

Abstract

In order to calculate the transmissivity from the inverse problem corresponding to the groundwater flow in an isotropic horizontal aquifer, a numerical conservative approach is tested. The method deals with triangulation of the domain and applies the conservation of mass to elements of the mesh using the harmonic mean for internodal transmissivities. An optimal sweeping algorithm is used to evaluate nodal transmissivities from one element to another with a minimal relative error accumulation. The practical importance of the method is demonstrated through two synthetic examples representing those experienced in the field, then through application to a Moroccan aquifer. The computed hydraulic head is well fitted to the reference one, which confirms the validity of the identified transmissivity model.     

A New R Program for Flow-Log Analysis of Single Holes (FLASH-R)

A new version of the computer program FLASH (Flow-Log Analysis of Single Holes) is presented for the analysis of borehole vertical flow logs to estimate fracture (or layer) transmissivities and far-field hydraulic heads. The program is written in R, an open-source environment. All previous features have been retained and new features incorporated including more rigorous parameter estimation, uncertainty analysis, and improved data import. The program has a dynamic user interface compatible with most operating systems.     

Worth of head data in well-capture zone design: deterministic and stochastic analysis

In this study, we examine the effects of conditioning spatially variable transmissivity fields using head and/or transmissivity measurements on well-capture zones. In order to address the challenge posed by conditioning a flow model with spatially varying parameters, an innovative inverse algorithm, the Representers method, is employed. The method explicitly considers this spatial variability.

A number of uniform measurement grids with different densities are used to condition transmissivity fields using the Representers method. Deterministic and stochastic analysis of well-capture zones are then examined. The deterministic study focuses on comparison between reference well-capture zones and their estimated mean conditioned on head data. It shows that model performance due to head conditioning on well-capture zone estimation is related to pumping rate. At moderate pumping rates transmissivity observations are more crucial to identify effects arising from small-scale variations in pore water velocity. However, with more aggressive pumping these effects are reduced, consequently model performance, through incorporating head observations, markedly improves. In the stochastic study, the effect of conditioning using head and/or transmissivity data on well-capture zone uncertainty is examined. The Representers method is coupled with the Monte Carlo method to propagate uncertainty in transmissivity fields to well-capture zones. For the scenario studied, the results showed that a combination of 48 head and transmissivity data could reduce the area of uncertainty (95% confidence interval) in well-capture zone location by over 50%, compared to a 40% reduction using either head or transmissivity data. This performance was comparable to that obtained through calibrating on three and a half times the number of head observations alone.     

NOTES ON THE NEW MAP OF THE ALETSCH GLACIER

ABSTRACT

The estimation of recharge and boundary flux is an important problem in deterministic groundwater modelling since these quantities are often difficult to measure directly in the field. A new method (inferred recharge) is proposed for this estimation, based upon water level and transmissivity observations; it is particularly applicable to those semiarid regions where a steady state formulation can be used and recharge is confined to certain known areas. The inferred recharge method which assumes that recharge occurs in these areas only is compared with an alternative method which does not use this information, and is found to be superior. In the example of the inferred recharge method applied to the Oman coastal plain, random normal errors have been added to both water levels and transmissivities in simulation experiments to assess the effect of errors in the data. The results of the simulations have been used to test the reliability of variance estimates derived from the theory. In addition, statistical tests have been used to examine the inferred recharge results.     

Reflected waves in finely layered tilted orthorhombic media

Upscaling in seismics is a homogenization of finely layered media in the zero-frequency limit. An upscaling technique for arbitrary anisotropic layers has been developed by Schoenberg and Muir. Applying this technique to a stack of layers of orthorhombic (ORT) symmetry whose vertical symmetry planes are aligned, results in an effective homogeneous layer with orthorhombic symmetry. If the symmetry planes in a horizontal orthorhombic layer are rotated with respect to vertical, the medium is referred to as tilted orthorhombic (TOR) medium, and the stack composed of TOR layers in zero-frequency limit will produce an effective medium of a lower symmetry than orthorhombic. We consider a P-wave that propagates through a stack of thin TOR layers, then it is reflected (preserving the mode) at some interface below the stack, and then propagates back through the same stack. We propose to use a special modified medium for the upscaling in case of this sequential down- and up-propagation: each TOR layer in the stack is replaced by two identical TOR layers whose tilt angles have the opposite algebraic sign. In this modified medium, one-way propagation of a seismic wave (any wave mode) is equivalent to propagation of a pure-mode reflection in the original medium. We apply this idea to study the contribution from an individual layer from the stack and show how the approach can be applied to a stack of TOR layers. To demonstrate the applicability of the model, we use well log data for the upscaling. The model we propose for the upscaling can be used in well-seismic ties to correct the effective parameters obtained from well log data for the presence of tilt, if latter is confirmed by additional measurements (for example, borehole imaging).     

井水位随气压变化之解析解

气压作用下, 井-含水层系统中地下水流是一类流体力学问题。 本文应用井壁水流通量边界条件和气压作用下井壁内外水(孔)压平衡条件, 提出了一个井水位随气压变化的解析公式。 解析式表明, 气压系数随时段长度增加而增大, 并趋于气压常数; 气压系数随时段长度的变化只依赖于导水系数与井半径平方的比值(T/r²_w), 而与气压变化过程无关; 气压常数只与含水层的一维荷载效率(B)有关, 而与导水系数和井半径无关。 解析解所反映的气压系数与时段长度的关系, 与南溪井实测序列数据分析结果具有很好的一致性。 根据气压系数随时段长度变化过程, 提出了一个参数估计方法, 应用于估计南溪井含水层气压常数和导水系数, 并对本文提出的参数估计方法进行了讨论。     

Comparison of upscaling methods to estimate hydraulic conductivity

A hydraulic field test program was performed at a hard rock laboratory (Asp? HRL) on the Swedish east coast to test upscaling theories. The test program investigated the rock volume around a borehole located at a depth of approximately 340 m below sea level. Hydraulic packer tests were performed at various scales, from 2 m to the entire borehole length of 296 m. From this set of data the predictive ability of different upscaling methods could be evaluated. The comparison of the evaluated "true" field scale hydraulic conductivity with the upscaled hydraulic conductivity yielded that the majority of the upscaling methods tested in this paper predict the large scale values with significant accuracy. However, the ability to predict rapidly decreases when the variance of the natural logarithm of hydraulic conductivity of the subsamples is larger than one. Such a variance is consistently found in the crystalline rocks at the tested site at the 2 m scale. However, at scales of 10 m and larger, a variance larger than one is uncommon. Therefore, it is concluded that there exists a smallest possible scale for use of hydraulic pumping test results for estimating the effective hydraulic conductivity at scales typical for regional flow.     

Transport upscaling using multi-rate mass transfer in three-dimensional highly heterogeneous porous media

A methodology for transport upscaling of three-dimensional highly heterogeneous formations is developed and demonstrated. The overall approach requires a prior hydraulic conductivity upscaling using an interblock-centered full-tensor Laplacian-with-skin method followed by transport upscaling. The coarse scale transport equation includes a multi-rate mass transfer term to compensate for the loss of heterogeneity inherent to all upscaling processes. The upscaling procedures for flow and transport are described in detail and then applied to a three-dimensional highly heterogeneous synthetic example. The proposed approach not only reproduces flow and transport at the coarse scale, but it also reproduces the uncertainty associated with the predictions as measured by the ensemble variability of the breakthrough curves.     

Uncertainty estimation of pathlines in ground water models

A method is proposed to estimate the uncertainty of the location of pathlines in two-dimensional, steady-state confined or unconfined flow in aquifers due to the uncertainty of the spatially variable unconditional hydraulic conductivity or transmissivity field. The method is based on concepts of the semianalytical first-order theory given in Stauffer et al. (2002, 2004), which allows estimates of the lateral second moment (variance) of the location of a moving particle. However, this method is reformulated in order to account for nonuniform recharge and nonuniform aquifer thickness. One prominent application is the uncertainty estimation of the catchment of a pumping well by considering the boundary pathlines starting at a stagnation point. In this method, the advective transport of particles is considered, based on the velocity field. In the case of a well catchment, backtracking is applied by using the reversed velocity field. Spatial variability of hydraulic conductivity or transmissivity is considered by taking into account an isotropic exponential covariance function of log-transformed values with parameters describing the variance and correlation length. The method allows postprocessing of results from ground water models with respect to uncertainty estimation. The code PPPath, which was developed for this purpose, provides a postprocessing of pathline computations under PMWIN, which is based on MODFLOW. In order to test the methodology, it was applied to results from Monte Carlo simulations for catchments of pumping wells. The results correspond well. Practical applications illustrate the use of the method in aquifers.     

Transmissivity and storage coefficient estimation by coupling the Cooper–Jacob method and modified fuzzy least-squares regression

Traditionally the Cooper–Jacob equation is used to determine the transmissivity and the storage coefficient for an aquifer using pump test results. This model, however, is a simplified version of the actual subsurface and does not allow for analysis of the uncertainty that comes from a lack of knowledge about the heterogeneity of the environment under investigation. In this paper, a modified fuzzy least-squares regression (MFLSR) method is developed that uses imprecise pump test data to obtain fuzzy intercept and slope values which are then used in the Cooper–Jacob method. Fuzzy membership functions for the transmissivity and the storage coefficient are then calculated using the extension principle. The supports of the fuzzy membership functions incorporate the transmissivity and storage coefficient values that would be obtained using ordinary least-squares regression and the Cooper–Jacob method. The MFLSR coupled with the Cooper–Jacob method allows the analyst to ascertain the uncertainty that is inherent in the estimated parameters obtained using the simplified Cooper–Jacob method and data that are uncertain due to lack of knowledge regarding the heterogeneity of the aquifer.     

Probabilistic assessment of travel times in groundwater modeling

A Monte Carlo approach is described for the quantification of uncertainty on travel time estimates. A real (non synthetic) and exhaustive data set of natural genesis is used for reference. Using an approach based on binary indicators, constraint interval data are easily accommodated in the modeling process. It is shown how the incorporation of imprecise data can reduce drastically the uncertainty in the estimates. It is also shown that unrealistic results are obtained when a deterministic modeling is carried out using a kriging estimate of the transmissivity field. Problems related with using sequential indicator simulation for the generation of fields incorporating constraint interval data are discussed. The final results consists of 95% probability intervals of arrival times at selected control planes reflecting the original uncertainty on the transmissivity maps.     

Probabilistic assessment of travel times in groundwater modeling

A Monte Carlo approach is described for the quantification of uncertainty on travel time estimates. A real (non synthetic) and exhaustive data set of natural genesis is used for reference. Using an approach based on binary indicators, constraint interval data are easily accommodated in the modeling process. It is shown how the incorporation of imprecise data can reduce drastically the uncertainty in the estimates. It is also shown that unrealistic results are obtained when a deterministic modeling is carried out using a kriging estimate of the transmissivity field. Problems related with using sequential indicator simulation for the generation of fields incorporating constraint interval data are discussed. The final results consists of 95% probability intervals of arrival times at selected control planes reflecting the original uncertainty on the transmissivity maps.     

A New Rapid Method for Measuring the Vertical Head Profile

This study describes a new technique for measuring the head profile in a geologic formation. The technique provides rapid, low cost information on the depth of water‐producing zones and aquitards in heterogeneous aquifers, yielding estimates of hydraulic heads in each zone while identifying any potential for cross contamination between zones. The measurements can be performed in a typical borehole in just a few hours. The procedure uses both the continuous transmissivity profile obtained by the installation (eversion) of a flexible borehole liner into an open borehole and the subsequent removal (inversion) of the same liner from the borehole. The method is possible because of the continuous transmissivity profile (T profile described by Keller et al. 2014) obtained by measuring the rate of liner eversion under a constant driving head. The hydraulic heads of producing zones are measured using the reverse head profile (RHP) method (patent no. 9,008,971) based on a stepwise inversion of the borehole liner. As each interval of the borehole is uncovered by inversion of the liner, the head beneath the liner is allowed to equilibrate to a steady‐state value. The individual hydraulic heads contributing to each measurement are calculated using the measured transmissivity for each zone. Application of the RHP method to a sedimentary bedrock borehole in New Jersey verified that it reproduced the head distribution obtained the same day in the same borehole instrumented with a multilevel sampling system.