Log Transmissivity Research Articles

Stochastic analysis of spatial variability in unconfined groundwater flow

In this paper, spatial variability in steady one-dimensional unconfined groundwater flow in heterogeneous formations is investigated. An approach to deriving the variance of the hydraulic head is developed using the nonlinear filter theory. The nonlinear governing equation describing the one-dimensional unconfined groundwater flow is decomposed into three linear partial differential equations using the perturbation method. The linear and quadratic frequency response functions are obtained from the first- and second-order perturbation equations using the spectral method. Furthermore, under the assumption of the exponential covariance function of log hydraulic conductivity, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the linear system are derived. The results show that the variance derived herein is less than that of Gelhar (1977). The reason is that the log transmissivity is linearized in Gelhar’s work. In addition, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the quadratic system are derived as well. It is found that the correlation scale and the trend in mean of log hydraulic conductivity are important to the dimensionless variance ratio.

Type‐curve estimation of statistical heterogeneity

The analysis of pumping tests has traditionally relied on analytical solutions of groundwater flow equations in relatively simple domains, consisting of one or at most a few units having uniform hydraulic properties. Recently, attention has been shifting toward methods and solutions that would allow one to characterize subsurface heterogeneities in greater detail. On one hand, geostatistical inverse methods are being used to assess the spatial variability of parameters, such as permeability and porosity, on the basis of multiple cross‐hole pressure interference tests. On the other hand, analytical solutions are being developed to describe the mean and variance (first and second statistical moments) of flow to a well in a randomly heterogeneous medium. We explore numerically the feasibility of using a simple graphical approach (without numerical inversion) to estimate the geometric mean, integral scale, and variance of local log transmissivity on the basis of quasi steady state head data when a randomly heterogeneous confined aquifer is pumped at a constant rate. By local log transmissivity we mean a function varying randomly over horizontal distances that are small in comparison with a characteristic spacing between pumping and observation wells during a test. Experimental evidence and hydrogeologic scaling theory suggest that such a function would tend to exhibit an integral scale well below the maximum well spacing. This is in contrast to equivalent transmissivities derived from pumping tests by treating the aquifer as being locally uniform (on the scale of each test), which tend to exhibit regional‐scale spatial correlations. We show that whereas the mean and integral scale of local log transmissivity can be estimated reasonably well based on theoretical ensemble mean variations of head and drawdown with radial distance from a pumping well, estimating the log transmissivity variance is more difficult. We obtain reasonable estimates of the latter based on theoretical variation of the standard deviation of circumferentially averaged drawdown about its mean.

A synthetic groundwater modelling study of the accuracy of GLUE uncertainty intervals

Synthetic groundwater flow models with one unknown parameter, the average log transmissivity of the flow domain, and with Gaussian log-transmissivity error structure were used to study the nature and the accuracy of Generalized Likelihood Uncertainty Estimation (GLUE) intervals. The uniform prior distribution of log10 transmissivities was sampled uniformly and 1000 values per log10-transmissivity cycle were required to produce unbiased GLUE results. Because the errors in hydraulic head resulting from the log-transmissivity errors are known to be Gaussian for a linear model for heads, the Gaussian likelihood function was used as the GLUE goodness-of-fit function in most cases studied. The GLUE interval computed for the hydraulic head at different locations within the domain has the characteristics of a confidence interval for the hydraulic heads computed using the spatial average log transmissivity. The GLUE interval does not have the characteristics of a prediction interval, which is a probability interval for an uncertain observation of some variable such as the hydraulic head. The goodness-of-fit function can be corrected so that the resulting GLUE interval has the characteristics of a prediction interval. However, neither the original nor the corrected GLUE interval account for the uncertainty caused by small-scale model errors, which is always present in practical groundwater flow modelling. It is therefore concluded that one should be careful with using the GLUE interval to evaluate the validity of a model's structure. The structure may be valid even though observations fall significantly outside the GLUE interval if the observations are uncertain and the goodness-of-fit function is not corrected to account for this, or if small-scale model error is significant. If small-scale model error does not significantly bias model predictions, the predictions will be useful although uncertain. Small-scale model error did not bias the predictions in the examples studied here except near a strong sink. Changing the goodness-of-fit function from the Gaussian likelihood function to a similar but different function made the resulting GLUE intervals very inaccurate. Changing to a third function based on a fitted model rejection value produced results that were somewhat better than those obtained with the Gaussian likelihood function. However, the results were sensitive to the model rejection value, which in the ideal case should be adjusted for each predicted variable individually. Thus, the third function is not attractive for practical applications with the GLUE methodology.

Stochastic delineation of capture zones: classical versus Bayesian approach

A Bayesian approach to characterize the predictive uncertainty in the delineation of time-related well capture zones in heterogeneous formations is presented and compared with the classical or non-Bayesian approach. The transmissivity field is modelled as a random space function and conditioned on distributed measurements of the transmissivity. In conventional geostatistical methods the mean value of the log transmissivity and the functional form of the covariance and its parameters are estimated from the available measurements, and then entered into the prediction equations as if they are the true values. However, this classical approach accounts only for the uncertainty that stems from the lack of ability to exactly predict the transmissivity at unmeasured locations. In reality, the number of measurements used to infer the statistical properties of the transmissvity field is often limited, which introduces error in the estimation of the structural parameters. The method presented accounts for the uncertainty that originates from the imperfect knowledge of the parameters by treating them as random variables. In particular, we use Bayesian methods of inference so as to make proper allowance for the uncertainty associated with estimating the unknown values of the parameters. The classical and Bayesian approach to stochastic capture zone delineation are detailed and applied to a hypothetical flow field. Two different sampling densities on a regular grid are considered to evaluate the effect of data density in both methods. Results indicate that the predictions of the Bayesian approach are more conservative.

Bayesian methodology for stochastic capture zone delineation incorporating transmissivity measurements and hydraulic head observations

The paper presents a methodology to estimate the uncertainty in the prediction of capture zones using transmissivity measurements and hydraulic head observations. We use a stochastic approach to parameterise the transmissivity field. By treating the parameters of the stochastic model as random variables we account for the fact that they are unknown and that a simple deterministic process cannot model their genesis. The method requires the definition of a prior probability density function for the mean value and for the covariance parameters of the log transmissivity field. In a first phase, log transmissivity measurements are incorporated into Bayes' theorem updating the prior densities to yield the posterior densities for the model parameters. Conditional realisations of the log transmissivity field are generated using parameter sets obtained by Monte Carlo sampling from the posterior parameter distributions. The second phase consists of updating the posterior probabilities of the conditional log transmissivity fields using the hydraulic head observations through a second application of Bayes' theorem. Then, for each realisation of the log transmissivity field, the capture zone is determined using particle tracking, which is weighted by the posterior probabilities of the respective log transmissivity field. The set of weighted capture zones results in a capture zone probability distribution. We show an application to a hypothetical flow system consisting of a single abstraction well in a confined aquifer under a regional background gradient and compare the worth of log transmissivity measurements and head observations by incorporating the different data types into the procedure sequentially. Results show that head observations are more effective in reducing the spread of the predictive capture zone distribution, whereas the transmissivity measurements are more valuable in predicting the actual location of the true capture zone.

Bayesian methodology to stochastic capture zone determination: Conditioning on transmissivity measurements

A methodology to determine the uncertainty associated with the delineation of well capture zones in heterogeneous aquifers is presented. The log transmissivity field is modeled as a random space function and the Bayesian paradigm accounts for the uncertainty that stems from the imperfect knowledge about the parameters of the stochastic model. Unknown parameters are treated as random quantities and characterized by a prior probability distribution. Log transmissivity measurements are incorporated into Bayes' theorem, updating the prior distribution and yielding posterior estimates of the mean value and the covariance parameters of the log transmissivity. Conditional simulations of the log transmissivity field are generated using samples from the posterior distribution of the parameters, yielding samples from the predictive distribution of the log transmissivity field. The uncertainty in the model parameters is propagated to the predictive uncertainty of the capture zone by solving numerically the groundwater flow equation, followed by a semianalytical particle‐tracking algorithm. The method is applied to a set of hypothetical flow fields for various sampling densities and assuming different levels of parameter uncertainty. Simulation results for all the sampling densities show no univocal relation between the predictive uncertainty of the capture zones and the level of parameter uncertainty. However, in general, the predictive uncertainty increases when parameter uncertainty is taken into account.

Geostatistics and Bayesian updating for transmissivity estimation in a multiaquifer system in Manitoba, Canada.

In ground water flow and transport modeling, the heterogeneous nature of porous media has a considerable effect on the resulting flow and solute transport. Some method of generating the heterogeneous field from a limited dataset of uncertain measurements is required. Bayesian updating is one method that interpolates from an uncertain dataset using the statistics of the underlying probability distribution function. In this paper, Bayesian updating was used to determine the heterogeneous natural log transmissivity field for a carbonate and a sandstone aquifer in southern Manitoba. It was determined that the transmissivity in m2/sec followed a natural log normal distribution for both aquifers with a mean of -7.2 and - 8.0 for the carbonate and sandstone aquifers, respectively. The variograms were calculated using an estimator developed by Li and Lake (1994). Fractal nature was not evident in the variogram from either aquifer. The Bayesian updating heterogeneous field provided good results even in cases where little data was available. A large transmissivity zone in the sandstone aquifer was created by the Bayesian procedure, which is not a reflection of any deterministic consideration, but is a natural outcome of updating a prior probability distribution function with observations. The statistical model returns a result that is very reasonable; that is homogeneous in regions where little or no information is available to alter an initial state. No long range correlation trends or fractal behavior of the log-transmissivity field was observed in either aquifer over a distance of about 300 km.

Radial Flow in a Bounded Randomly Heterogeneous Aquifer

Flow to wells in nonuniform geologic formations is of central interest to hydrogeologists and petroleum engineers. There are, however, very few mathematical analyses of such flow. We present analytical expressions for leading statistical moments of vertically averaged hydraulic head and flux under steady-state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer. Like in the widely used Thiem equation, we prescribe a constant pumping rate deterministically at the well and a constant head at a circular outer boundary of radius L. We model the natural logarithm Y = lnT of aquifer transmissivity T as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. Perturbation of these nonlocal equations leads to a system of local recursive moment equations that we solve analytically to second order in the standard deviation of Y. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. It yields expected values of head and flux, and the variance–covariance of these quantities, as functions of distance from the well. It also yields an apparent transmissivity, T a, defined as the negative ratio between expected flux and head gradient at any radial distance. The solution is supported by numerical Monte Carlo simulations, which demonstrate that it is applicable to strongly heterogeneous aquifers, characterized by large values of log transmissivity variance. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness. Potential uses include the analysis of pumping tests and tracer test conducted in such wells, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells.

A full‐Bayesian approach to the groundwater inverse problem for steady state flow

A full‐Bayesian approach to the estimation of transmissivity from hydraulic head and transmissivity measurements is developed for two‐dimensional steady state groundwater flow. The approach combines both Bayesian and maximum entropy viewpoints of probability. In the first phase, log transmissivity measurements are incorporated into Bayes' theorem, and the prior probability density function is updated, yielding posterior estimates of the mean value of the log transmissivity field and covariance. The two central moments are generated assuming that the prior mean, variance, and integral scales are “hyperparameters”; that is, they are treated as random variables in themselves which is contrary to classical statistical approaches. The probability density functions (pdfs) of these hyperparameters are, in turn, determined from maximum entropy considerations. In other words, pdfs are chosen for each of the hyperparameters that are maximally uncommitted with respect to unknown information. This methodology is quite general and provides an alternative to kriging for spatial interpolation. The final step consists of updating the conditioned natural logarithm transmissivity (ln(T)) field with hydraulic head measurements, utilizing a linearized aquifer equation. It is assumed that the statistical properties of the noise in the hydraulic head measurements are also uncertain. At each step, uncertainties in all pertinent hyperparameters are removed by marginalization. Finally, what is produced is a ln(T) field conditioned on measurements of both hydraulic heads and log transmissivity and covariances of the ln(T) field. In addition, we can also produce resolution matrices, confidence (credibility) limits, and the like for the ln(T) field. It is shown that the application of the methodology yields good estimates of transmissivities, even when hydraulic head measurements are noisy and little or no information is specified on mean values of ln(T), variance of ln(T), and integral scales.

Estimating the variance and integral scale of the transmissivity field using head residual increments

A modification of previously published solutions regarding the spatial variation of hydraulic heads is discussed whereby the semivariogram of increments of head residuals (termed head residual increments HRIs) are related to the variance and integral scale of the transmissivity field. A first‐order solution is developed for the case of a transmissivity field which is isotropic and whose second‐order behavior can be characterized by an exponential covariance structure. The estimates of the variance σY2 and the integral scale λ of the log transmissivity field are then obtained via fitting a theoretical semivariogram for the HRI to its sample semivariogram. This approach is applied to head data sampled from a series of two‐dimensional, simulated aquifers with isotropic, exponential covariance structures and varying degrees of heterogeneity (σY2 = 0.25, 0.5, 1.0, 2.0, and 5.0). The results show that this method provided reliable estimates for both λ and σY2 in aquifers with the value of σY2 up to 2.0, but the errors in those estimates were higher for σY2 equal to 5.0. It is also demonstrated through numerical experiments and theoretical arguments that the head residual increments will provide a sample semivariogram with a lower variance than will the use of the head residuals without calculation of increments.

An analysis of the pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields

An analysis of the pilot point method for automated calibration of an ensemble of conditionally simulated transmissivity fields was conducted on the basis of the simplifying assumption that the flow model is a linear function of log transmissivity. The analysis shows that the pilot point and conditional simulation method of model calibration and uncertainty analysis can produce accurate uncertainty measures if it can be assumed that errors of unknown origin in the differences between observed and model‐computed water pressures are small. When this assumption is not met, the method could yield significant errors from overparameterization and the neglect of potential sources of model inaccuracy. The conditional simulation part of the method is also shown to be a variant of the percentile bootstrap method, so that when applied to a nonlinear model, the method is subject to bootstrap errors. These sources of error must be considered when using the method.

Upscaling transmissivity under radially convergent flow in heterogeneous media

Most field methods used to estimate transmissivity values rely on the analysis of drawdown under convergent flow conditions. For a single well in a homogeneous and isotropic aquifer and under steady state flow conditions, drawdown s is directly related to the pumping rate Q through transmissivity T. In real, nonhomogeneous aquifers, s and Q are still directly related, now through a value called equivalent transmissivity Teq. In this context, Teq is defined as the value that best fits Thiem's equation and would, for example, be the transmissivity assigned to the well location in the classical interpretation of a steady state pumping test. This equivalent or upscaled transmissivity is clearly not a local value but is some representative value of a certain area surrounding the well. In this paper we present an analytical solution for upscaling transmissivities under radially convergent steady state flow conditions produced by constant pumping from a well of radius rw in a heterogeneous aquifer based upon an extension of Thiem's equation. Using a perturbation expansion, we derive a second‐order expression for Teq given as a weighted average of the fluctuations in log T throughout the domain. This expression is compared to other averaging formulae from the literature, and differences are pointed out. Teq depends upon an infinite series which may be expressed in terms of coefficients of the finite Fourier transform of the log transmissivity function. Sufficient conditions for convergence of this series are examined. Finally, we show that our solution agrees with existing analytical ones to second order and test the solution with a numerical example.

Hydrogeologic and hydrochemical framework of the shallow groundwater system in the southern Voltaian Sedimentary Basin, Ghana

The southern Voltaian Sedimentary Basin underlies an area of about 5000 km2 in east-central Ghana. Groundwater in the basin occurs in fractures in highly consolidated siliciclastic aquifers overlain by a thin unsaturated zone. Aquifer parameters were evaluated from available aquifer-test data on 28 shallow wells in the basin. Hydraulic-conductivity values range from 0.04–3.6 m/d and are about two orders of magnitude greater than the hydraulic conductivity calculated using Darcy's Law and the average groundwater velocity estimated from carbon-14 dating. Linear-regression analysis of the transmissivity and specific-capacity data allowed the establishment of an empirical relationship between log transmissivity and log specific capacity for the underlying aquifers.

Effect of Conditioning Randomly Heterogeneous Transmissivity on Temporal Hydraulic Head Measurements in Transient Two-Dimensional Aquifer Flow

We investigated the effect of conditioning transient, two-dimensional groundwater flow simulations, where the transmissivity was a spatial random field, on time dependent head data. The random fields, representing perturbations in log transmissivity, were generated using a known covariance function and then conditioned to match head data by iteratively cokriging and solving the flow model numerically. A new approximation to the cross-covariance of log transmissivity perturbations with time dependent head data and head data at different times, that greatly increased the computational efficiency, was introduced. The most noticeable effect of head data on the estimation of head and log transmissivity perturbations occurred from conditioning only on spatially distributed head measurements during steady flow. The additional improvement in the estimation of the log transmissivity and head perturbations obtained by conditioning on time dependent head data was fairly small. On the other hand, conditioning on temporal head data had a significant effect on particle tracks and reduced the lateral spreading around the center of the paths.

Stochastic analysis of solute transport in heterogeneous aquifers subject to spatially random recharge

This paper derives and solves approximate unconditional moment equations for a concentration plume in a two-dimensional spatially random transmissivity field subject to spatially random recharge and non-uniform (linear trending) velocity. Closed-form expressions are derived for the unconditional, non-stationary auto-covariances for head and velocity, and the cross-covariances between velocity, head, recharge, and log transmissivity based on a system of coupled first-order partial differential moment equations and assumed hole-type input covariance functions for log transmissivity and recharge. The unconditional flow related moment equations are incorporated into a system of solute transport moment equations, which is solved using a Galerkin finite element algorithm. A constant positive mean recharge yields a mean velocity gradient, which enhances the ensemble mean plume spreading in the longitudinal direction similar to results found by Rubin and Bellin (1994) [Water Resour. Res., 30(4) 939–948]. Spatial variability in recharge further enhances the spreading of the ensemble mean plume, especially in the lateral direction. Uncertainty in the spatial distribution of recharge also increases the extent of the ensemble standard deviation plume, particularly in the lateral direction, and increases the coefficient of variation of solute concentration. The uncertainty in the prediction of solute transport increases with increasing recharge variance and spatial correlation scale.

Impact of Concentration Measurements upon Estimation of Flow and Transport Parameters: The Lagrangian Approach

Transport of a conservative solute takes place in a heterogeneous formation of spatially variable conductivity. The latter is modeled as a random space function of stationary lognormal distribution. As a result the velocity field and the concentration are also random. Assuming that measurements of concentration of an existing plume are available, the problem addressed here is to assess their effect upon identification of log conductivity and flow transport variables. The solution is sought in a Lagrangian framework in which transport is represented in terms of the random trajectories of particles originating from the initial plume. A concentration measurement is equivalent to the passage of a trajectory through the measurement point at a given time. The impact of measurements is achieved by conditioning any variable of interest on realizations for which at least one trajectory satisfies the requirement. It is shown that cokriging of concentration and another flow or transport variable leads to the correct conditioned mean of the latter. In contrast, the conditional variance based on cokriging is erroneous. The procedure is illustrated for two‐dimensional flow under a few simplifying assumptions. The effect of a concentration measurement upon the expected value and variance of log transmissivity and plume centroid are examined in a few particular cases. The procedure may improve the solution of the inverse problem and the prediction of transport of existing plumes.

Stochastic analysis of head spatial variability in bounded rectangular heterogeneous aquifers

Groundwater flow through bounded rectangular aquifers is analyzed in a stochastic framework. New analytical expressions of the head covariance (and variance) and the log transmissivity head cross covariance are derived for the simplified first‐order problem. Effects of different boundary conditions, the aquifer size, and the log transmissivity correlation scale upon head moments are studied. The results are compared to analytical half‐plane solutions, demonstrating significant differences in several cases. Finally, the problem is solved numerically using the Monte Carlo simulation method. The realizations of the log transmissivity field are generated in two different ways. Through comparisons with the analytical results the accuracy of the numerical methods is shown to be excellent.

Eulerian-Lagrangian analysis of transport conditioned on hydraulic data. 2. Effects of log transmissivity and hydraulic head measurements : Dongxiao Zhang & S. P. Neuman, Water Resources Research, 31(1), 1995, pp 53–63

Eulerian-Lagrangian analysis of transport conditioned on hydraulic data. 2. Effects of log transmissivity and hydraulic head measurements : Dongxiao Zhang & S. P. Neuman, Water Resources Research, 31(1), 1995, pp 53–63

Quasi‐Linear Geostatistical Theory for Inversing

A quasi‐linear theory is presented for the geostatistical solution to the inverse problem. The archetypal problem is to estimate the log transmissivity function from observations of head and log transmissivity at selected locations. The unknown is parameterized as a realization of a random field, and the estimation problem is solved in two phases: structural analysis, where the random field is characterized, followed by estimation of the log transmissivity conditional on all observations. The proposed method generalizes the linear approach of Kitanidis and Vomvoris (1983). The generalized method is superior to the linear method in cases of large contrast in formation properties but informative measurements, i.e., there are enough observations that the variance of estimation error of the log transmissivity is small. The methodology deals rigorously with unknown drift coefficients and yields estimates of covariance parameters that are unbiased and grid independent. The applicability of the methodology is demonstrated through an example that includes structural analysis, determination of best estimates, and conditional simulations.

Continuous Time Stochastic Analysis of Groundwater Flow in Heterogeneous Aquifers

The problem of depth‐averaged groundwater flow in heterogeneous aquifers is looked at from a stochastic point of view. The Galerkin finite element version of the linearized stochastic equation governing the flow is solved analytically in time using an eigenvalue‐eigenvector technique. The stochastic solution, which is valid for small variance of aquifer log transmissivity, relates spatial and temporal variabilities of aquifer head to aquifer heterogeneity, stochastic recharge, and random initial head. The computational effort requires the evaluation of integrals of matrices whose elements are linear functions of the nodal mean heads. The solutions are obtained for the exact (i.e., continuous) and quasi‐steady approximation of the mean head. Aquifer‐head temporal covariances are evaluated using standard matrix operations once the storage matrix is inverted and the associated generalized eigenvalue problem is solved. Implication of the level of spatial discretization on the performance of the solution is examined, and the influence of aquifer heterogeneity on spatial and temporal variances of hydraulic heads is investigated in the presence of a semipervious boundary.