Xavier Sánchez-Vila

An evaluation of Jacob's Method for the interpretation of pumping tests in heterogeneous formations

Most pumping tests are interpreted using the classical Theis assumption of large-scale homogeneity with various corrections to account for early time behavior of drawdown curves. When drawdowns are plotted versus log time, late time data often delineate a straight line, which is consistent with Jacobs approximation of Theis solution but may seem surprising in view of the heterogeneity of natural media. The aim of our work is to show that Jacobs method leads to a good approximation of the effective transmissivity of heterogeneous media when constrained to late time data. A review of several multiwell pumping tests demonstrates that when drawdown curves from each observation well are interpreted separately, they produce very similar transmissivity T estimates. However, the corresponding estimates for storativity span a broad range. This behavior is verified numerically for several models of formation heterogeneity. A very significant finding of the numerical investigation is that T values estimated using simulated drawdown from individual observation wells are all very close to the effective T value for parallel flow. This was observed even in nonmultiGaussian T fields, where high T zones are well connected and where the effective T is larger than the geometric average of point values. This implies that Jacobs method can be used for estimating effective T values in many, if not most, formations.

Scale effects in transmissivity

Heterogeneity accounts for several paradoxes in groundwater flow and solute transport. One of the most striking observations is the emergence of scale effects in transmissivity, that is, the increase in effective transmissivity (or hydraulic conductivity, for that matter) with increasing scale of observation. Traditional stochastic approaches, where transmissivity is treated as a multilog-normal random function, lead to a large-scale effective transmissivity equal to the geometric average of local measurements. n nWe present several field cases in which large-scale transmissivities are indeed larger than the geometric average of local tests. This suggests that the assumption of multilog-normality may not be valid in many cases, even if point T values display a log-normal distribution. We conjecture that scale dependence of T may, in part, be a consequence of high T zones being better connected than average or low T zones, a feature which may occur in many geological environments, but which is not consistent with multinormal log-T fields. We go on to generate a suite of log-T fields with a normal distribution for point values but non-multinormal spatial correlation. In all our fields, high T zones show longer correlations than average of low T zones. By simulating flow through these synthetic fields under simple boundary conditions, and estimating their effective transmissivity values, we conclude that these types of departures from the multilog-normality assumption lead consistently to scale effects.

Pumping tests in heterogeneous aquifers: An analytical study of what can be obtained from their interpretation using Jacob's Method

Interpretation of pumping tests to estimate hydraulic parameter values is typically based on the assumption of aquifer homogeneity. The applicability of the traditional methods of interpretation in real aquifers can be questioned, since the evaluation of the drawdown curves observed at different locations in a single test may not result in one consistent set of hydraulic parameters. Thus most hydrogeologists tend to look at estimated transmissivities (T) as some average property of the medium, while estimated storativities (S) are disregarded in some cases, particularly when they are obtained from data measured at the pumping well. An analytical study of drawdown under radially convergent flow toward a single point in heterogeneous aquifers shows that large time drawdown values form a straight line on a drawdown versus log time plot. Jacobs method consists of obtaining estimates for T and S from the slope and intercept of this line. We find that even in a heterogeneous field, these estimates provide valuable information about the aquifer. Estimated T values for different observation points tend to converge to a single value, which corresponds to the effective T derived under parallel flow conditions. Estimated storativities, however, display higher variability, but the geometric mean of the Sest values can be used as an unbiased estimator of the actual S. Thus it appears that although Jacobs method was originally derived for homogeneous media, it can provide valuable information in real aquifers.

A Synthesis of Approaches to Upscaling of Hydraulic Conductivities

Simulation of flow through heterogeneous media often requires discretizing the flow domain into blocks and assigning an equivalent block conductivity value to each one of them. The process of defining block conductivities from point values is termed upscaling. A number of approaches to upscaling are available, most of which consider the uncertainty associated with any natural property, so that they cast the problem in a stochastic frame. Recently, Indelman and Dagan (1993a, b) provided a general stochastic methodology to upscaling in heterogeneous anisotropic formations by means of the dissipation energy function; unfortunately, they did not provide any “practical” method to compute block values from point ones. The objective of this work is twofold: First, we analyze different practical approaches to compute block conductivities and find that all of them provide very similar results in terms of actual computed values; second, we check that all approaches verify approximately a number of conditions stated by Indelman and Dagan (1993a). Specifically, we show analytically that for regular blocks, the methodologies of both Rubin and Gomez-Hernandez (1990) and Desbarats (1992) (which we call “practical” methodologies) satisfy the condition that the effective conductivity obtained from a field where the elementary conductivities are defined over a certain support (we call this the actual formation) is identical to that obtained from the same field with conductivities defined at a larger support (upscaled formation). The analysis is carried out by working with the logarithm of block conductivities and using a small-perturbation expansion and thus is strictly valid for small variances. On the other hand, we show numerically that the two methodologies satisfy approximately an important condition stated in terms of the dissipation energy: that block-averaged dissipation values computed are indeed very close to the true dissipation values in each block. The agreement is even better if we consider statistical moments instead of point values. As an important conclusion we should note that all practical methodologies considered in this work perform equally well and, more important, constitute a simple way to treat an otherwise very complex problem.

Radially convergent flow in heterogeneous porous media

We present an analytical solution for apparent effective transmissivity under radially convergent steady state flow conditions, produced by constant pumping from a single well of finite radius, rw. Apparent effective transmissivity, Te, is defined as the value that relates the expected values of flow and head gradient at a certain location. The domain is two-dimensional, of annular shape, and the size of the pumping well is explicitly taken into account. The solution for the steady state heads is obtained by solving the perturbed flow equation and substituting it into Darcys law to obtain a consistent second- order expansion for Te. We show that apparent effective transmissivity is a scalar for any choice of isotropic covariance model, with an expression given in integral form. Our main result is that Te in a heterogeneous, statistically isotropic random field, under radial steady state flow conditions, is a monotonie increasing function of r (distance from the well) that rises from the harmonic mean of the point transmissivity values (close to the well) and tends asymptotically towards the geometric mean (far from the well). The asymptotic value is reached at a distance of a few integral scales (1.5–2 for the Gaussian model and 3–5 for the exponential one). The apparent effective transmissivity versus normalized r curves are in excellent agreement with previously published numerical work carried out using Monte Carlo 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 Thiems 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 Thiems 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.

Conditional moments of the breakthrough curves of kinetically sorbing solute in heterogeneous porous media using multirate mass transfer models for sorption and desorption

[1]xa0A methodology is presented for evaluating the temporal moments of solutes undergoing linear rate-limited mass transfer processes based on a Lagrangian approach to solute transport in heterogeneous media. The temporal moments of sorbing solutes are written as a function of those of conservative tracers. The general continuous diffusion rate model that has recently appeared in the hydrologic literature is used to model the rate-limited mass transfer processes. The methodology is also applied to desorption from an initially uniformly contaminated aquifer, and the concentration expected value and variance are found quasi-analytically. The conditional temporal moments of sorbing solutes can be written as a function of the conditional moments of conservative tracers. Conditioning results in a reduction of the variance of travel time. The amount of reduction depends on the chemical model selected.

Convergent-flow tracer tests in heterogeneous media: combined experimental-numerical analysis for determination of equivalent transport parameters

In modeling transport within naturally heterogeneous aquifers, it is usually assumed that the transport equations valid at local scales can also be applied at larger scales. At larger scales, the heterogeneous domain is represented by an equivalent homogeneous medium. Convergent-flow tracer tests constitute one of the most frequently used field tests to estimate effective input parameters of equivalent homogeneous aquifers. Traditionally, statistical approaches applied to groundwater flow and solute transport have provided tools to estimate these equivalent parameters. These approaches are based on a number of simplifications including the assumption that the point transmissivity values follow a multilog-normal random function. Several investigators have found that this assumption may not be valid in many field cases. In order to study the applicability of the equivalent homogeneous formulation in a nontraditional stochastic field, a number of experimental and numerical studies were conducted. The results are used to determine the apparent values of porosity and dispersivity that would be obtained if convergent-flow tracer tests were conducted in a deterministically generated heterogeneous transmissivity field displaying anisotropy in the correlation structure. It is shown that in this particular heterogeneous media, apparent porosity strongly depends on connectivity rather than on transmissivity. This dependence on connectivity questions the theoretical results obtained in continuum equivalent fields to estimate effective porosity.

Directional effects on convergent flow tracer tests

Convergent flow tracer tests constitute a convenient way of characterizing hydraulic parameters in an aquifer. Interpretation of tracer breakthrough curves from convergent flow tests normally is made under the assumption of radial symmetry. Nevertheless, these curves may display directional dependence; that is when tracers are injected at several points located at the same distance, both arrival times and estimated dispersivities may be significantly different. This result is why some authors attribute a tensorial nature to porosity or, equivalently, talk about directional porosity when trying to explain the variations in computed porosity depending on the relative orientation of pumping and injection wells. Our main ponit is that this directional effect is nothing but an artifact of an inappropriate selection of a conceptual model, where anisotropy (local of statistical) in hydraulic conductivity is not properly characterized. To illustrate this point, we first consider the situation of a simple homogeneous and anisotropic model of the medium. We prove analytically that this model leads to arrival time being proportional to the square root of directional hydraulic conductivity. Using a stochastic approach, we determine the same directional behavior of arrival time for a locally isotropic hydraulic conductivity field with statistical anisotropy caused by an anisotropic correlation structure. A statistical anisotropic covariance model for hydraulic conductivity is consistent with field evidence.

Field tracer experiment in a low permeability fractured medium: results from El Berrocal site

A modelling experimental activity was developed to characterise the hydraulic behaviour of water-bearing fractures in crystalline rocks. n nThree conservative tracers were injected into two packed-off sections of the same well, 69 m deep in a granite formation, and recovered by pumping from an isolated section of a second borehole, 46 m deep, 14 m apart. The concept of this design is to characterise separately an isolated fracture zone intersecting the lower parts of both wells from the fracture network intersecting the bulk of the rock. n nBefore using the tracers in the field, their behaviour was studied in the laboratory under controlled conditions. Fluorescein, eosin and iodide were finally chosen as the best spikes. n nThe field experiments were developed under strictly controlled conditions, such as (1) checking the hydraulic pressure in the packed-off sections and in other parts of circuits; (2) mixing the tracer solutions during the injection and checking their homogeneity; (3) performing a continuous and automatic monitoring of tracer concentration in the arrival well. n nIodide and fluorescein were injected in one section, and eosin in the other section. The pumping rate was maintained at 2 1 min−1. The test lasted 27 d after which from 40 to 60% of the injected masses were recovered. n nBreakthrough curve analysis considered two conceptual models: radial advective-dispersive transport with and without matrix diffusion: the first model returns thickness-porosity values around 0.2 × 10−1 m and dispersivity around 4 m. Parameters are remarkably consistent for iodide and fluorescein, although fittings can be improved. The matrix diffusion model provides much better fittings by decreasing thickness porosity to 0.8 X 10-Z m. Dispersivities range from 0.5 to 0.9 m and the molecular diffusion term differentiates the behaviour of conservative tracers such as fluorescein and iodide.