Stephen E. Silliman

A transient laboratory method for determining the hydraulic properties of ‘tight’ rocks—I. Theory

Abstract Transient pulse testing has been employed increasingly in the laboratory to measure the hydraulic properties of rock samples with low permeability. Several investigators have proposed a mathematical model in terms of an initial-boundary value problem to describe fluid flow in a transient pulse test. However, the solution of this problem has not been available. In analyzing data from the transient pulse test, previous investigators have either employed analytical solutions that are derived with the use of additional, restrictive assumptions, or have resorted to numerical methods. In Part I of this paper, a general, analytical solution for the transient pulse test is presented. This solution is graphically illustrated by plots of dimensionless variables for several cases of interest. The solution is shown to contain, as limiting cases, the more restrictive analytical solutions that the previous investigators have derived. A method of computing both the permeability and specific storage of the test sample from experimental data will be presented in Part II.

Analysis of time-series measurements of sediment temperature for identification of gaining vs. losing portions of Juday Creek, Indiana

Abstract Identifying locations at which groundwater and surface water interact can be important in a variety of contexts including the development of water budgets, the determination of migration pathways for nutrients and contaminants, and the estimation of thermal budgets along small water bodies. Some difficulty has been experienced in identifying locations of inflows over large regions when detailed groundwater information is not known. A method is suggested herein for the identification of the location of inflows-outflows in both creeks and streams. This method, applied to a small creek in northern Indiana, involves measurement over time of both the sediment temperature and the temperature at the base of the overlying water column. The results are used to define reaches along the creek which are gaining and reaches along the creek which are losing. These results are supported by independent measurements of groundwater elevations at several points along the stream. It is argued that this method provides an excellent screening tool for the identification of gaining and losing reaches of small waterways.

Quantifying downflow through creek sediments using temperature time series: one-dimensional solution incorporating measured surface temperature

Several authors have addressed the question of identifying the location of and quantifying the volume of groundwater/surface water interaction. Utilizing measurement of temperature within the sediments has been suggested as a means of estimating water flux through the sediments. A qualitative version of this method has been applied to identifying locations of communication between a creek and groundwater based upon temperature time series measurements in the water column and the sediments. The discussion presented earlier is extended to a more general solution which allows for incorporation of measured surface temperature (rather than an assumed surface temperature). The mathematical formulation presented is targeted on the quantification of flux across the sediment for conditions of one-dimensional downflow with a constant flux over periods of days to weeks. This technique, based on the assumption that the temperature of the surface water is the primary thermal influence on sediment temperature, is shown to provide an estimate of fluid flux (volume of water per area per time) for flux rates above a critical threshold. The flux threshold depends on a number of factors including thermal diffusivity of the sediments and depth of burial of the temperature measuring device. Based on use of three simple forcing functions for temperature at the surface of the sediments, it is shown that the amplitude of the temperature response in the sediments to a change in temperature in the overlying water column decreases with increasing depth and decreasing flux. Further, the timing of the peak response in the sediments becomes increasingly delayed as depth increases and flux decreases. These observations, combined with consideration of the assumption of one-dimensional flow, lead to suggesting that field design be based on relatively shallow burial (e.g. 5–15 cm) of the device to measure sediment temperatures. Based on these observations, one of the data sets presented earlier is reanalyzed to derive flux estimates. It is shown that a reasonable fit is obtained with a downflow flux of less than 0.03 cm day−1. Independent measurement of hydraulic gradient and hydraulic conductivity provide a range of flux estimates for the same site (although measured in a different year) which are consistent with this value. Based on these results, it is argued that this technique is a reasonable screening tool for use in situations where relatively inexpensive, point estimates of water flux are required within losing reaches of a creek.

A transient laboratory method for determining the hydraulic properties of ‘tight’ rocks—II. Application

Abstract In Part I a general analytical solution for the transient pulse test was presented. Part II presents a graphical method for analyzing data from a test to obtain the hydraulic properties of the sample. The general solution depends on both hydraulic conductivity and specific storage and, in theory, analysis of the data can provide values for both of these hydraulic properties. However, in practice, one of two limiting cases may apply in which case it is possible to calculate only hydraulic conductivity or the product of hydraulic conductivity times specific storage. In this paper we examine the conditions when both hydraulic parameters can be calculated. The analyses of data from two tests are presented. In Appendix I the general solution presented in Part I is compared with an earlier analysis, in which compressive storage in the sample is assumed negligible, and the error in calculated hydraulic conductivity due to this simplifying assumption is examined.

Particle transport through two-dimensional, saturated porous media: influence of physical structure of the medium

Abstract Recent work in the transport of particulate matter (e.g. bacteria, viruses, colloids, mineral grains, etc.) through saturated porous media has indicated that the physical structure of the porous medium plays an important role in both the ability of the particles to be transported and the distribution of the particles deposited within the medium. The present paper addresses the influence on particle transport within saturated porous media of high-permeability pathways and contacts between different grain size distributions. The experiments discussed in the present paper were conducted in two dimensions. The porous media were constructed from glass beads and the tracer utilized consisted of latex spheres in the size range 2–90 μm (2–30 μm in most experiments). Among the media designs investigated were simple layers in which flow was parallel to layers, simple layers in which flow was at an angle to layers, a simple form of heterogeneity including non-parallel layers and disjoint inclusions, and a complex heterogeneity involving three interconnected permeabilities. Results from these experiments demonstrated that significantly greater numbers of particles were deposited at media contacts where water moved from larger to smaller diameter glass beads. Further, it was observed that outflow concentrations of particles for media containing continuous flow paths within the coarse glass beads were substantially higher than outflow concentrations of particles for media containing a number of interfaces between different-sized glass beads. Specifically, a comparison of the results for a medium in which flow was parallel to layers with results from a medium in which flow was at an angle to the layers led to the observation of higher concentrations of latex particles in the outflow from the former medium and high concentrations of particles remaining in the region of layer contacts within the latter medium. Similar observations were made in a comparison of the simple and complex heterogeneities. Based on these results and prediction of straining based on the filtration literature, it is argued that the enhanced deposition observed near contacts results from sorption leading to increased straining in regions of lower uniformity in the glass bead distribution. These results demonstrate the importance of particle transport along high-permeability pathways and the enhanced deposition which may occur at contact surfaces within heterogeneous porous media.

Velocity dependence of dispersion for transport through a single fracture of variable roughness

Several authors have recently expressed interest in chemical transport within fractured media. The majority of these efforts have been based on a linear relationship between the dispersion coefficient and the average fluid velocity within the fracture. It is not apparent that this relationship is fully justified in all applications. For the present study, it is assumed (as suggested by authors working in porous media) that the dispersion coefficient is proportional to the velocity raised to a power, n. Further, it is assumed that transport within the fracture follows classic advection-dispersion behavior (e.g., Fickian dispersion). The present study focuses on the value of the power n in a series of artificial fractures. In particular, an experimental apparatus is utilized to run controlled tracer experiments through a single fracture. When the fracture consists of smooth parallel plates, the results from the experiments indicate that the dispersion coefficient is proportional to the velocity squared (consistent with the early work by Taylor (1953) for transport dominated by transverse diffusion). As the fracture roughness is increased through use of blockages within the fracture and/or addition of surface roughness along the fracture walls, the relationship between dispersion and velocity varied. For each fracture roughness, the results followed the general relationship in which dispersion is proportional to velocity raised to the power n. The power n, however, was strongly dependent on the fracture roughness, taking on a value of 2.0 for smooth parallel plates and decreasing to a value of approximately 1.3 for rough plates.

Comparison of Observations from a Laboratory Model with Stochastic Theory: Initial Analysis of Hydraulic and Tracer Experiments

Hydraulic and tracer tests were conducted in a flow cell containing a mixture of sediments designed to mimic a two-dimensional, log-normally distributed, second-order stationary, exponentially correlated random conductivity field. With 60 integral scales in the direction of mean flow and 25 integral scales perpendicular to this direction, behavior of flow and transport in the interior of the flow cell can be compared directly with stochastic solutions for flow and transport. Using 144 piezometers and 361 platinum electrodes, the distribution of hydraulic head and the concentrations of an ionic tracer could be monitored in substantial detail. The present discussion presents the details of the experimental equipment. Results and initial analysis of hydraulic measurements and characterization of a two-dimensional tracer plume are also presented. Analysis using first-order hydraulic theory shows that the flow through the medium was consistent with an effective conductivity equal to the geometric mean of the conductivity distribution. Further, the semivariogram of head increments as observed in the experimental results was consistent with the semivariogram predicted by theory. The chemical transport experiments are here compared with the early solutions presented by Dagan (1984, 1987). The observed rate of longitudinal spread of two tracer plumes was slightly less than that predicted using this theory. Further, the spread in the transverse dimension was observed to decline from the initial plume dimensions and then remain constant or increase slightly, but at a rate lower than predicted by the theory. The difference between the hydraulic and transport results is believed to be related to the fact that the hydraulic results were averaged over a very large portion of the flow cell such that ergodic conditions could be assumed. In contrast, the initial geometry of the plume covered only approximately five integral scales in the transverse direction such that the validity of the assumption of ergodic conditions must be questioned in the analysis of results for the chemical transport.

Bacterial transport in heterogeneous porous media: Observations from laboratory experiments

Transport of bacteria through heterogeneous porous media was investigated in small-scale columns packed with sand and in a tank designed to allow the hydraulic conductivity to vary as a two-dimensional, lognormally distributed, second-order stationary, exponentially correlated random field. The bacteria were Pseudomonas ftuorescens R8, a strain demonstrating appreciable attachment to surfaces, and strain Ml, a transposon mutant of strain R8 with reduced attachment ability. In bench top, sand-filled columns, transport was determined by measuring intensity of fluorescence of stained cells in the effluent or by measuring radiolabeled cells that were retained in the sand columns. Results demonstrated that strain Ml was transported more efficiently than strain R8 through columns packed with either a homogeneous silica sand or a more heterogeneous sand with iron oxide coatings. Two experiments conducted in the tank involved monitoring transport of bacteria to wells via sampling from wells and sample ports in the tank. Bacterial numbers were determined by direct plate count. At the end of the first experiment, the distribution of the bacteria in the sediment was determined by destructive sampling and plating. The two experiments produced bacterial breakthrough curves that were quite similar even though the similarity between the two porous media was limited to first- and second-order statistical moments. This result appears consistent with the concept of large-scale, average behavior such as has been observed for the transport of conservative chemical tracers. The transported bacteria arrived simultaneously with a conservative chemical tracer (although at significantly lower normalized concentration than the tracer). However, the bacterial breakthrough curves showed significant late time tailing. The concentrations of bacteria attached to the sediment surfaces showed considerably more spatial variation than did the concentrations of bacteria in the fluid phase. This contrast between behavior in the fluid phase and on the solids is consistent with field observations by other authors and initial modeling of these heterogeneous media.

Variation in aperture estimate ratios from hydraulic and tracer tests in a single fracture

Previous theoretical discussion of flow and transport in a single fracture has led to a difference of opinion as to whether the aperture estimated from a tracer test need always be less than (or greater than) the aperture estimated from a concurrent hydraulic test. Interpretation of previously published field results has, to our knowledge, not provided a defensible example where the ratio of these two aperture estimates has varied from less than 1.0 to greater than 1.0 in the same fracture using the same analysis techniques. In the present paper, field experiments involving concurrent hydraulic and tracer tests have been run in two different directions within an isolated fracture. Aperture estimates were derived from both the hydraulic and tracer tests. The ratio of the aperture estimated from the tracer test to the aperture estimated from the hydraulic test is shown to be greater than 1.0 in the first experiment and less than 1.0 in the second experiment. Despite noise in the hydraulic test of the first experiment, it is argued that our interpretation is reasonable. As such, this represents the first set of experimental results known to us in which a defensible interpretation leads to the conclusion that the aperture ratio varies from greater than 1.0 to less than 1.0 within the same fracture.

Laboratory study of chemical transport to wells within heterogeneous porous media

A laboratory flow cell was used to study chemical transport to wells producing water from two realizations of a heterogeneous porous medium. The wells produced water from a confined aquifer which was otherwise governed by a mean uniform hydraulic gradient. For both realizations the aquifer was composed of a two-dimensional, lognormally distributed, second-order stationary, exponentially correlated conductivity field. For the first realization, response was monitored at a single well 45 integral scales from the inflow boundary. Four wells were used in the second realization, with locations ranging from 10 to 45 integral scales from this boundary. The focus of this work was on the nature of the dispersion observed in the chemical arrival at the pumping well, and this work builds on an earlier study in which it was shown that results under mean uniform flow were relatively consistent with stochastic theories for fluid flow and chemical transport under mean uniform flow. Consistent results were obtained across the two realizations for the wells located 45 integral scales from the boundary as the breakthrough curves demonstrated an increase in dispersive behavior with an increase in the ratio of the pumping rate to the regional flow, consistent with existing theory for finite-sized sources. In contrast, results for the well located closest to the source (10 integral scales) demonstrated variability in the timing and shape of the breakthrough curves that did not correlate with the ratio of pumping at the well to regional flow. The results indicate that the shape and timing of a breakthrough curve at a well producing within a regional flow field may be strongly dependent on the distance of the well from the source. Further, the parameters (e.g., dispersivity) obtained from analysis of the breakthrough curve are shown to be functions of the ratio of pumping rate at the well to the regional flux.