Generalized Radial Flow Model Research Articles

Characterizing the radius of influence during pumping tests using the absolute critical drawdown criterion: Cases of integer flow dimensions

Determining the radius of influence r0 of wells during pumping tests is critical for the characterization of aquifers and the management of groundwater. However, because a convenient analytical interpretative framework is lacking, this is a very difficult task during routine investigations. Practicing hydrogeologists have resorted to using semi-empirical equations developed by certain authors. Most studies aiming to characterize the radius of influence are based on radial flow models. In this study, we propose to investigate, from an analytical standpoint, the radius of influence equation for integer flow dimensions (n=1,2,3), using both Barker’s generalized radial flow model and Theis’ radial flow model. The current approach may thus be considered valid for those hydrogeological contexts (fractured or granular aquifer media) that produce the specified flow dimension. The radius of influence is defined as the maximum distance from the pumping well at which the drawdown reaches its critical value of detectability: the absolute critical drawdown criterion sc. The radius of influence is a theoretical, non-intrinsic and variable parameter that reflects the ability of drawdown recording systems to measure very small variations. Our investigations show that the radius of influence equation can be generalized as follows: r0=Ctγ where the coefficients C and γ depend not only on the flow dimension parameter n, but also on the criterion sc, the pumping flow rate Q, the hydraulic conductivity K and the aquifer thickness b. The specificities of the radius of influence equation for each flow dimension are also discussed. Finally, results obtained from this analytical approach are verified against numerical simulations.

The extended generalized radial flow model and effective conductivity for truncated power law variograms

Pumping tests are established for characterizing spatial average properties of aquifers. At the same time, they are promising tools to identify heterogeneity characteristics such as log-conductivity variance and correlation scales. We present the extended Generalized Radial Flow Model (eGRF) which combines the characterization of well flow in fractal geometry with an upscaled conductivity for pumping tests in heterogeneous media. We show that the eGRF is a generalization of previously solutions, such as that of Barker, Butler and Neuman. We derive effective conductivities for uniform and well flow conditions in heterogeneous log-normal media with a truncated power law correlation structure through the upscaling procedure Coarse Graining. The radial-dependent effective conductivity for well flow reflects the gradual change of heterogeneity impact on average pumping test drawdowns. We then combine upscaled conductivities with the eGRF model to determine the effective pumping test solution. We provide a proof of concept by comparing theoretical upscaling results with Monte Carlo well flow simulations in heterogeneous fractal fields. The eGRF and upscaling results are implemented and made freely available as python code for transport simulation as well as pumping test analysis.

Insights on pumping well interpretation from flow dimension analysis: The learnings of a multi-context field database

The flow dimension parameter n, derived from the Generalized Radial Flow model, is a valuable tool to investigate the actual flow regimes that really occur during a pumping test rather than suppose them to be radial, as postulated by the Theis-derived models. A numerical approach has shown that, when the flow dimension is not radial, using the derivative analysis rather than the conventional Theis and Cooper-Jacob methods helps to estimate much more accurately the hydraulic conductivity of the aquifer. Although n has been analysed in numerous studies including field-based studies, there is a striking lack of knowledge about its occurrence in nature and how it may be related to the hydrogeological setting. This study provides an overview of the occurrence of n in natural aquifers located in various geological contexts including crystalline rock, carbonate rock and granular aquifers. A comprehensive database is compiled from governmental and industrial sources, based on 69 constant-rate pumping tests. By means of a sequential analysis approach, we systematically performed a flow dimension analysis in which straight segments on drawdown-log derivative time series are interpreted as successive, specific and independent flow regimes. To reduce the uncertainties inherent in the identification of n sequences, we used the proprietary SIREN code to execute a dual simultaneous fit on both the drawdown and the drawdown-log derivative signals. Using the stated database, we investigate the frequency with which the radial and non-radial flow regimes occur in fractured rock and granular aquifers, and also provide outcomes that indicate the lack of applicability of Theis-derived models in representing nature. The results also emphasize the complexity of hydraulic signatures observed in nature by pointing out n sequential signals and non-integer n values that are frequently observed in the database.

Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation

The classical Darcy law is generalized by regarding the water flow as a function of a noninteger order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow. Two methods including Frobenius and Adomian decomposition method are used to obtain an asymptotic analytical solution to the generalized groundwater flow equation. The solution obtained via Frobenius method is valid in the vicinity of the borehole. This solution is in perfect agreement with the data observed from the pumping test performed by the institute for groundwater study on one of their boreholes settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rateQand monitoring the piezometric head for 350 minutes. Numerical solutions obtained via Adomian method are compared with the Barker generalized radial flow model for which a fractal dimension for the flow is assumed. Proposition for uncertainties in groundwater studies was given.

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.

Scale dependency of fractional flow dimension in a fractured formation

Abstract. The flow dimensions of fractured media were usually predefined before the determination of the hydraulic parameters from the analysis of field data in the past. However, it would be improper to make assumption about the flow geometry of fractured media before site characterization because the hydraulic structures and flow paths are complex in the fractured media. An appropriate way to investigate the hydrodynamic behavior of a fracture system is to determine the flow dimension and aquifer parameters simultaneously. The objective of this study is to analyze a set of field data obtained from four observation wells during an 11-day hydraulic test at Chingshui geothermal field (CGF) in Taiwan in determining the hydrogeologic properties of the fractured formation. Based on the generalized radial flow (GRF) model and the optimization scheme, simulated annealing, an approach is therefore developed for the data analyses. The GRF model allows the flow dimension to be integer or fractional. We found that the fractional flow dimension of CGF increases near linearly with the distance between the pumping well and observation well, i.e. the flow dimension of CGF exhibits scale-dependent phenomenon. This study provides insights into interpretation of fracture flow at CGF and gives a reference for characterizing the hydrogeologic properties of fractured media.

Skin effect in generalized radial flow model in fractured media

SUMMARY A generalized radial flow model, treating the flow dimension as an arbitrary number and taking into account the well skin effect, is presented to describe the transient head distribution in a fractured medium under the Robin condition during constant-head tests. The Laplace-domain solution for the head distribution is first developed via the Laplace transforms; and the corresponding time-domain solution in terms of the hydraulic head is then obtained using the Bromwich integral method. In addition, based on the head solution and Darcy's law, the solution for the wellbore flux is further developed to investigate the skin effect. It is found that the skin effect on the wellbore flux is significant at early times. Under this circumstance, it would not be appropriate to neglect the skin effect in the model in determining the hydraulic parameters from the analysis of field data. Our solutions will be useful for the predictions of the spatial and temporal head distribution and wellbore flux or in investigating the skin effect on the head distribution and wellbore flux for flow with various dimensions in fractured media.

Analytical solutions for groundwater flow with arbitrary dimensionality and a finite well radius in fractured rock

The generalized radial flow model describes mathematically nonsteady flow of arbitrary dimensionality from a source in a porous medium. Closed solutions of the corresponding equation have hitherto been considered as impractical except for one simple special case. Two closed solutions of the generalized radial flow equation, corresponding to given head in or given discharge from the source have been derived. The noninteger dimensionality is the only parameter in the problem. The solutions become not valid if the time tends to infinity, such as for 1‐D and 2‐D flows. The influence of a possible noninteger dimensionality has attracted interest in connection with the flow of groundwater in fractured rock, particularly around a repository for nuclear waste or in connection with grouting. In contrast to numerical solutions, the closed solutions offer simple means for evaluation of field tests.

Influences of Aquifer Properties on Flow Dimensions in Dolomites

The paper focuses on analyses and correlations of flow dimensions in different dolomite aquifers in Slovenia. Flow dimensions are obtained through the reinterpretation of 72 pumping tests with the generalized radial flow model, based on the fractional flow dimension. The average value of flow dimensions is 2.16 for all dolomites. A study of flow dimensions in individual aquifers categorized according to their lithological properties shows that higher dimensions occur in massive late-diagenetic Cordevolian and Anisian dolomites compared with bedded Main, Baca, and especially Lower Triassic dolomites, which contain a greater proportion of noncarbonate minerals. Partially penetrating wells have higher flow dimensions than fully penetrating wells. Flow dimensions are poorly correlated with hydraulic conductivities of fractures. When comparing the quantities of major dissolved minerals, obtained by hydrogeochemical inverse modeling, with the values of flow dimensions, the Cordevolian and Anisian dolomites are found to exhibit the highest values of both dissolved dolomite and flow dimensions, indicating that greater dissolution occurs at higher flow dimensions. For other aquifers, data points are more scattered and the correlation is mostly poor. When compared with three-dimensional fractal dimensions of fracture networks, there is no correlation with flow dimensions. However, almost all the values of flow dimensions are lower than the corresponding fractal dimensions in dolomites (average D= 2.77), possibly indicating the channeling of flow within the available space of the fracture networks, consequently reducing the flow dimensions.

Flow Dimension and Anomalous Diffusion of Aquifer Tests in Fracture Networks

Characterization and modeling of aquifers is particularly challenging in fractured media, where flow is concentrated into channels that are poorly suited to traditional approaches to analysis. The generalized radial flow model is an alternative method for hydraulic test interpretation that infers an additional parameter, the flow dimension n, to describe the flow geometry. This study was a Monte Carlo analysis of numerical models of aquifer tests in two‐dimensional fractured media, with the objective of understanding the relationships between the parameters of a discrete fracture network (DFN) with lengths distributed as a power law, the flow dimension, and the regime of hydraulic diffusion (e.g., Fickian or non‐Fickian). The diffusion regime of each realization was evaluated using 〈R2〉 ∼ tk, where t is time, 〈R2〉 is the mean squared radius of hydraulic diffusion, and an apparent value of k = 1 indicates normal (Fickian) diffusion and k < 1 indicates anomalous (non‐Fickian) diffusion. For the DFN model, the apparent flow dimension and exponent k depend on both the density and the power of the fracture length distribution, and thus also on the connectivity regime of the fracture network system. Depending on the connectivity regime, the apparent flow dimension stabilizes to less than the Euclidean dimension and the apparent value of k < 1 indicates that hydraulic diffusion is non‐Fickian. These results suggest that the flow dimension and the exponent k may be useful for characterizing flow and transport in fractured media.

Cross-borehole slug test analysis in a fractured limestone aquifer

Summary This work proposes new semi-analytical solutions for the interpretation of cross-borehole slug tests in fractured media. Our model is an extension of a previous work by Barker [Barker, J.A., 1988. A generalized radial flow model for hydraulic tests in fractured rock. Water Resources Research 24 (10), 1796–1804; Butler Jr., J.J., Zhan X., 2004. Hydraulic tests in highly permeable aquifers. Water Resources Research 40, W12402. doi:10.1029/2003/WR002998]. It includes inertial effects at both test and observation wells and a fractional flow dimension in the aquifer. The model has five fitting parameters: flow dimension n, hydraulic conductivity K, specific storage coefficient Ss, and effective lengths of test well Le and of observation well Leo. The results of a sensitivity analysis show that the most sensitive parameter is the flow dimension n. The model sensitivity to other parameters may be ranked as follows: K > Le ∼ Leo > Ss. The sensitivity to aquifer storage remains one or two orders of magnitude lower than that to other parameters. The model has been coupled to an automatic inversion algorithm for facilitating the interpretation of real field data. This inversion algorithm is based on a Gauss–Newton optimization procedure conditioned by re-scaled sensitivities. It has been used to interpret successfully cross-borehole slug test data from the Hydrogeological Experimental Site (HES) of Poitiers, France, consisting of fractured and karstic limestones. HES data provide flow dimension values ranging between 1.6 and 2.5, and hydraulic conductivity values ranging between 4.4 × 10−5 and 7.7 × 10−4 m s−1. These values are consistent with previous interpretations of single-well slug tests. The results of the sensitivity analysis are confirmed by calculations of relative errors on parameter estimates, which show that accuracy on n and K is below 20% and that on Ss is about one order of magnitude. The K-values interpreted from cross-borehole slug tests are one order of magnitude higher than those previously interpreted from interference pumping tests. These findings suggest that cross-borehole slug tests focus on preferential flowpath networks made by fractures and karstic channels, i.e. the head perturbation induced by a slug test propagates only through those flowpaths with the lowest hydraulic resistance. As a result, cross-borehole slug tests are expected to identify the hydrodynamic properties of karstic-channels and fracture flowpaths, and may be considered as complementary to pumping tests which more likely provide bulk properties of the whole fracture/karstic-channel/matrix system.

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

Interpretation of single-well tracer tests using fractional-flow dimensions. Part 2: A case study

Generalized equations using fractional-flow dimensions were derived to estimate the Darcy and groundwater-flow velocities obtained from the point-dilution and the single-well injection-withdrawal field tests. Flow velocities can only be estimated from single-well tests if the kinematic porosity or the hydraulic conductivity and hydraulic gradient are known a priori. A pumping test performed on the boreholes will yield an estimate of the fractional-flow dimension and the extent of the flow region by applying the generalized radial flow (GRF) model of Barker [Barker JA (1988) A generalized radial flow model for hydraulic tests in fractured rock. Water Resour Res 24(10):1796–1804]. These parameters are used in the generalised equations for the single-well tracer tests to estimate the flow-through area and, therewith, the Darcy or flow velocity.

Interpretation of pumping tests in the Sherwood Sandstone Group, Sellafield, Cumbria, UK

In 1995 as part of the process to assess the suitability of a site near Sellafield, Cumbria, for a deep geological repository for solid intermediate level and some solid low level radioactive waste, United Kingdom Nirex Limited (Nirex) commissioned a series of long duration pumping tests in the various stratigraphic horizons at the site. Two of the pumping tests were conducted in vertically adjacent sections of the lower Sherwood Sandstone Group (SSG) whilst pressures were monitored at discrete intervals in surrounding monitoring boreholes. Responses were observed in over twenty monitoring zones at various levels around the pumped sections during both tests. In order to arrive at an understanding of the hydraulic system in the SSG, a systematic methodology was adopted to analyse the data sets from the pumping tests. Preliminary analysis of the data from each responding zone used the standard 2D radial flow analytical approach. This indicated that, whilst responses conformed broadly to this model, there were important deviations. Three different analytical modifications were adopted to assess the significance of these deviations; a partial penetration model, a bounded aquifer model and a generalized radial flow model. The use of the different models allowed a conceptual model of flow processes occurring in the SSG to be developed.

Composite model with fractional flow dimensions for well test analysis in fractured rocks

The represented analytical model is addressed to fracture-zone aquifers in which well test responses can be commonly matched with models involving fractional flow dimensions. The model assumes a composite structure representing the fractures opening into the pumping well and the fracture system comprising the actual aquifer. The aim is to assess the hydraulic properties for the near-well zone and the aquifer in terms of concrete parameters even if flow dimensions have fractional values. Dimensionless drawdown in pumping and observation wells was solved by applying Laplace transforms. The early-time approximation for the pumping well can be used to assess the transmissivity ( T 1) if the flow dimension in the inner zone ( n 1) is greater than about 1.7. The long-time solutions indicate that the inner zone induces a pseudo-skin effect that can be estimated by comparing the pumping well and observation well data. If K 1 is known, the extent of the inner zone ( r′) can be estimated from the pseudo-skin factor. The responses tend to form log–log straight lines with slopes of 1− n 2. By plotting drawdown vs. t (1− n 2) on the double-logarithmic scale, the product of the hydraulic conductivity and the through-flow area at the wellbore radius ( KA 2) and the hydraulic diffusivity ( D 2) of the aquifer can be estimated from the slope and the intercept. The hydraulic conductivity of the aquifer ( K 2) is estimated by considering the through-flow area as a projection of an n 2-dimensional sphere with a radius of r wb through three-dimensional space by an amount r′ (3− n 2) . The analyzed field example comprises a pumping test of a fracture zone aquifer in south–central Finland. For the pumping well, simultaneous matching of drawdown and derivative type-curves with observed responses can be used to verify the results of the early-time analysis and the proposed straight-line procedure. In addition, type-curve matching yields estimates of the hydraulic diffusivity ratio and a value of the dimensionless wellbore storage. The analyzed observation wells are considered to be sufficiently far from the inner zone. Consequently, type-curves are produced based on the dimensionless drawdown solution that is identical to generalized radial flow model. Type-curve matching of the observation well responses produces estimates for KA 2, K 2 and D 2. The obtained values are similar to previous studies that included models relying on integer flow dimensions. However, models relying on the fractional flow dimensions are capable to simulate the observed responses more accurately.

Fractional flow dimensions and hydraulic properties of a fracture-zone aquifer, Leppävirta, Finland

In order to characterize the hydraulic properties of an aquifer in Finland comprising two subvertical fracture zones, observation-well responses were matched with generalized radial flow (GRF) type curves. The responses in six wells out of seven are consistent with the GRF model. The fractional flow dimensions (1–1.2 and 1.5) were determined by regression analysis of straight-line slopes and type-curve matching. In each test, the flow dimensions in the neighboring fracture zone range from 2–2.25. Comparisons of the late-time responses with the asymptotic GRF solution and the flow dimensions obtained by reversing the pumping and observation points suggest homogeneous hydraulic properties. Deviations in responses can be explained by flow-path tortuosity. After assessments of the extent of the flow and radial distances along the fracture system, hydraulic conductivities and storativities were determined from the results of the type-curve matching procedure. The obtained hydraulic conductivities are 1.3×10–5 to 7.9×10–5 m/s and 5.0×10–6 to 2.5×10–5 m/s for the western and the eastern fracture zones, respectively. The results were verified by applying them to analytical solutions for pumping wells. The calculated pumping-well responses are consistent with the observations. The analysis of flow dimension also enhances qualitative interpretations on the hydrogeology of fracture zones.

Generalized Radial Flow model for hydraulic tests in fractured rock

Generalized Radial Flow model for hydraulic tests in fractured rock

A generalized radial flow model for hydraulic tests in fractured rock

Models commonly used for the analysis of hydraulic test data are generalized by regarding the dimension of the flow to be a parameter which is not necessarily integral and which must be determined empirically. Mathematical solutions for this generalized radial flow model are derived for the standard test conditions: constant rate, constant head, and slug tests. Solutions for the less common, sinusoidal test are contained within the general solutions given. Well bore storage and skin are included and the extension to dual‐porosity media outlined. The model is presented as a model of fractured media, for which it is most likely to find application because of the problem of choosing the appropriate flow dimension.