Randall M. Roberts

Well testing in fractured media: flow dimensions and diagnostic plots

Hydraulic tests in heterogeneous media, particularly fractured media, are difficult to analyze because of the absence of radial flow. The theory of flow dimensions introduced by Barker in 1988 (Water Resour. Res. 24(10). 1988. 1796) provided a method of analyzing pumping (constant-rate) tests in non-radial systems, and this approach was later extended to constant-pressure tests. However, little use seems to be made of the flow-dimension approach to well-test analysis, perhaps because no easily applied method has been presented for determining, at the initial stage of an analysis, if such an approach would be productive. Depending on the distribution of heterogeneities within an aquifer, flow to a well may have almost any dimension (not limited to linear, radial, or spherical), or no constant dimension at all. Any well-test analytical solution requires that hydraulic properties be stable on some scale before those properties can be uniquely quantified. For each type of hydraulic test (constant-rate, constant-pressure, or slug/pulse), we suggest that a diagnostic plot of the scaled first or second derivative of the pressure or flow-rate response be created to determine, first, if a stable flow dimension was reached during the test and. second, what the value of that flow dimension is. If a stable flow dimension was reached, the scaled derivative will exhibit a constant value (scaled to be equal to the flow dimension). If the scaled derivative does not stabilize at a constant value, then no flow dimension can be specilied and no unique hydraulic properties can be inferred analytically from the test. In all cases, the scale of testing must be appropriate to the scale of underlying interest.

Confidence region estimation techniques for nonlinear regression in groundwater flow: Three case studies

[1] This work focuses on different methods to generate confidence regions for nonlinear parameter identification problems. Three methods for confidence region estimation are considered: a linear approximation method, an F test method, and a log likelihood method. Each of these methods are applied to three case studies. One case study is a problem with synthetic data, and the other two case studies identify hydraulic parameters in groundwater flow problems based on experimental well test results. The confidence regions for each case study are analyzed and compared. Although the F test and log likelihood methods result in similar regions, there are differences between these regions and the regions generated by the linear approximation method for nonlinear problems. The differing results, capabilities, and drawbacks of all three methods are discussed.

Hydraulic Testing of Salado Formation Evaporites at the Waste Isolation Pilot Plant Site: Final Report

This report presents interpretations of hydraulic tests conducted in bedded evaporates of the Salado Formation from May 1992 through May 1995 at the Waste Isolation Pilot Plant (WIPP) site in southeastern New Mexico. The WIPP is a US Department of Energy research and development facility designed to demonstrate safe disposal of transuranic wastes from the nations defense programs. The WIPP disposal horizon is located in the lower portion of the Permian Salado Formation. The hydraulic tests discussed in this report were performed in the WIPP underground facility by INTERA inc. (now Duke Engineering and Services, Inc.), Austin, Texas, following the Field Operations Plan and Addendum prepared by Saulnier (1988, 1991 ) under the technical direction of Sandia National Laboratories, Albuquerque, New Mexico.

Generalized radial flow in synthetic flow systems.

Traditional analysis methods used to determine hydraulic properties from pumping tests work well in many porous media aquifers, but they often do not work in heterogeneous and fractured-rock aquifers, producing non-plausible and erroneous results. The generalized radial flow model developed by Barker (1988) can reveal information about heterogeneity characteristics and aquifer geometry from pumping test data by way of a flow dimension parameter. The physical meaning of non-integer flow dimensions has long been a subject of debate and research. We focus on understanding and interpreting non-radial flow through high permeability conduits within fractured aquifers. We develop and simulate flow within idealized non-radial flow conduits and expand on this concept by simulating pumping in non-fractal random fields with specific properties that mimic persistent sub-radial flow responses. Our results demonstrate that non-integer flow dimensions can arise from non-fractal geometries within aquifers. We expand on these geometric concepts and successfully simulate pumping in random fields that mimic well-test responses seen in the Culebra Dolomite above the Waste Isolation Pilot Plant.

Reply to comment by Chia‐Shyun Chen and I. Y. Liu on “Flow dimensions corresponding to hydrogeologic conditions”

[1] We appreciate the comments of Chen and Liu [2007] regarding our paper [Walker and Roberts, 2003] and are grateful for the opportunity for further discussion regarding interpreting the flow dimension of a hydraulic test. The comments of Chen and Liu (hereinafter referred to as CL) indicate that some aspects of Walker and Roberts (hereinafter referred to as WR) were unclear, and we hope to use this reply to reduce the confusion we have apparently caused. [2] We should first note that the flow dimension as a parameter of the generalized radial flow (GRF) model differs somewhat from our use of the apparent flow dimension as a diagnostic tool for hydraulic test interpretation. The GRF interpretive model fits the flow dimension as a parameter to account for the radial change in the crosssectional area of flow through an irregular network of homogeneous fractures [Barker, 1988]. The GRF estimates of hydraulic conductivity (KGRF) and specific storage (SGRF) are applicable to only that part of the domain conducting flow. That is, the GRF estimates depend on the fitted value of the flow dimension, similar to the parameters of any other interpretive model, e.g., the leakance parameter is specific to the leaky aquifer model and its value constrains the estimates of storage coefficient and transmissivity. In contrast, the apparent flow dimension is a diagnostic statistic that may be estimated from the late time slope of any observed hydraulic test. System geometry, heterogeneity [Doe, 1991; Walker et al., 2006], boundary conditions, and leakage [Walker and Roberts, 2003] have specific effects on the apparent flow dimension, thus it is a useful diagnostic for inferring conceptual models. In this sense, the apparent flow dimension is an extension of the use of the slope of the drawdown derivative as a diagnostic [Acuna and Yortsos, 1995; Horne, 1995]. The apparent flow dimension is not necessarily unique to a particular conceptual model [Doe, 1991;Walker and Roberts, 2003], thus all conceptual models producing the apparent flow dimension of a hydraulic test should be considered plausible until eliminated on the basis of other site characteristics. Using the apparent flow dimension as a diagnostic does not require using the GRF approach to interpret a hydraulic test, and the GRF model of homogeneous fractures is not necessarily a plausible conceptual model for an aquifer. Therefore we respectfully disagree with initial premise of CL that ‘‘if the apparent flow dimensions obtained are appropriate for the idealized systems, their application to [the GRF model] ought to reproduce the drawdown variations of these idealized systems.’’ We regret that that WR was not more explicit in this regard, and apologize for the confusion that this has created. [3] CL present useful calculations illustrating the ambiguities of forcing a site into the GRF model, and we would like to here add several comments. Much of the ambiguity of the GRF model arises from the definition of the extent of the flow zone, b, which Barker [1988] noted was difficult to describe for nonintegral flow dimensions, n. The extent of the flow zone cannot be separated easily from the GRF estimates, thus the estimates are sometimes reported as a generalized parameters, e.g., the generalized transmissivity is KGRF b 3 n [L /T] [Barker, 1988; Geier et al., 1996; Marechal et al., 2004]. We do not know how CL treated the flow zone in their analysis, but as we noted above, KGRF is the hydraulic conductivity of the space occupied by flow, so that KGRF should not be expected to match the hydraulic conductivity estimated using an interpretive model that assumes flow covers the entire Euclidean space. [4] For our analyses, we use a radial finite difference approach for interpreting hydraulic tests in fractured media. Recall that the flow area (A) in Barker’s [1988] formulation is given by