Derivative Of Drawdown Research Articles

A well test analysis model of generalized tube flow and seepage coupling

“Generalized mobility” is used to realize the unification of tube flow and seepage in form and the unification of commonly used linear and nonlinear flow laws in form, which makes it possible to use the same form of motion equations to construct unified governing equations for reservoirs of different scales in different regions. Firstly, by defining the generalized mobility under different flow conditions, the basic equation governing fluid flow in reservoir coupling generalized tube flow and seepage is established. Secondly, two typical well test analysis models for coupling tube flow and seepage flow are given, namely, pipe-shaped composite reservoir model and partially open cylindrical reservoir model. The log-log pressure draw-down type-curve of composite pipe-shaped reservoir model can show characteristics of two sets of linear flow. The log-log pressure drawdown plot of partially opened cylindrical reservoir model can show the characteristics of spherical flow and linear flow, as well as spherical flow and radial flow. The pressure build-up derivative curves of the two models basically coincide with their respective pressure drawdown derivative curves in the early stage, pulling down features in the late stage, and the shorter the production time is, the earlier the pulling down feature appears. Finally, the practicability and reliability of the models presented in this paper are verified by three application examples.

Unsteady flow to a partially penetrating pumping well with wellbore storage in a dual-permeability confined aquifer

In this article, a new model is presented for imparting unsteady flow to a partially penetrating pumping well with wellbore storage placed in double-permeability confined aquifers to investigate the wellbore drawdown transient behavior. In this model, both the fracture and matrix systems are connected directly with the wellbore as flow pathways, while comprehensively considering the crossflow between the matrix and the fractures. The top and bottom boundaries are assumed to be impermeable, and the outer side boundaries could be infinite. Subsequently, Laplace transformation and finite Fourier cosine transformation are employed to obtain analytical solutions of the dual-permeability flow model in the Laplace domain. The derived analytical solution is general and can be reduced to several other previous results. The numerical inversion algorithm of Stehfest is used to solve the wellbore drawdown in the real time domain. Subsequently, the model is validated by comparing it with previous results. Groundwater flow characteristics are analyzed in detail and divided into five flow stages: wellbore storage flow, early-radial flow, quasi-spherical flow, crossflow between the matrix and the fractures, and quasi-radial flow. The different flow stages show some special values on the drawdown and its derivative log-log plots. There is a hump/dip feature in the semi-logarithmic curve of the drawdown derivative, which are inherent characteristic for dual-permeability flow behavior. The effects of some important parameters of the aquifer or pumping wells on drawdown and its derivative, such as wellbore storage coefficient, penetration ratio, crossflow coefficient, storativity ratio, transmissivity ratio, dimensionless aquifer thickness of anisotropy and skin factor, are analyzed and discussed in detail.

An Alternative BEM for Simulating the Flow Behavior of a Leaky Confined Fractured Aquifer With the Use of the Semianalytical Approach

AbstractThis study developed an alternative boundary element method (BEM) to simulate the transient flow behavior of groundwater induced by well extraction in a confined fractured aquifer containing a network of discrete or connected fractures. The matrix flow, network‐fracture flow, and matrix‐fracture fluid exchange were considered. The aquifer was treated as a heterogeneous whole that consisted of fracture and matrix blocks with locally homogeneous hydraulic properties. The fractures were explicitly represented to be of true finite volume rather than nonrepresentational line sources. A semianalytical solution was developed based on the theory of a BEM in the Laplace transform domain, but the analytical Green's function was used for the bounded domain rather than the free‐space Green's function in a conventional BEM. Case studies were presented in order to investigate the flow exchange behavior between the matrix and fractures and the corresponding transient drawdown response. The results showed that (1) exchange flux distribution calculated with the classical infinitesimal fracture model was consistent with the difference of the normal drawdown derivative values on both sides of the fracture body in our model. (2) When the well was in the matrix, the fractures acted as both highly conductive conduits and leaky faults, and the drawdown derivative behaviors resembled the characteristics of a dual‐porosity reservoir model. (3) When the well was in the network fracture and when the volume of fracture was of the same order of magnitude as the matrix, the drawdown derivative might exhibit the look‐alike behavior of a dual‐porosity model.

Groundwater flow to a general well configuration in an unconfined aquifer overlying a fractured bedrock

Unconfined aquifers sometimes overlie fractured bedrocks, consisting of unconfined-fractured aquifer systems. Pumping induced hydrodynamics in such aquifer systems has not received much attention as in other types of aquifer systems. This paper aims to obtain semi-analytical solutions of flow to a general well configuration in such an unconfined-fractured two-layer aquifer system. The examples of general well configuration, specifically discussed here, include: 1) a horizontal well, 2) a vertical well, 3) a vertical well with one horizontal collector, and 4) a vertical well with two horizontal collectors which are perpendicular with each other. The methodology is general and can be extended to other types of wells with minor modifications of solutions developed in this study. The point sink/source solution is first obtained using double infinite Fourier and Laplace transformations (where sink is for pumping and source is for injection). The line sink/source solution is then obtained via integrating the point sink/source along the desired direction and following the principle of superposition. The de Hoog numerical inverse Laplace transform and Gaussian Quadrature are used to obtain time-domain dimensionless drawdown solutions. The instantaneous drainage of water table and the inter-porosity flow of the underlying fractured aquifer are taken into account. The effects of the length of horizontal well on dimensionless drawdown and dimensionless drawdown derivative, and scaled sensitivity of the hydraulic parameters of the underlying fractured aquifer are explored. The dimensionless drawdowns and drawdown derivatives of a vertical pumping well, a vertical well with a single horizontal collector and a vertical well with two perpendicular horizontal collectors are also compared. The scaled sensitivity of dimensionless drawdown to the hydraulic parameters of the underlying fractured aquifer of these different pumping well structures is explored. The results of this study can be used to evaluate the hydraulic parameters using the drawdown data collected during a pumping test performed in a general well configuration in a unconfined-fractured two-layer aquifer system. Furthermore, the presented solutions can be utilized to design the pumping well to minimize drawdown in low-permeability or thin aquifers. By eliminating the inter-porosity flow term from the underlying fractured aquifer, the solutions are applicable to groundwater flow to a pumping well in leaky aquifers.

On the Derivative of Drawdown for Single Well Pumping Tests

The communication explains the physical meaning of the derivative of drawdown for a pumping test.

Hydrogeological assessment of non-linear underground enclosures

Excavations below the water table are usually undertaken by combining the protection of retaining walls with dewatering by pumping wells. Severe difficulties may arise if the retaining walls have defects. Therefore, their state must be determined and, if needed, the defects repaired or the dewatering system redesigned. The state of underground retaining walls can be evaluated using hydrogeological methods, but these methods are well-established only for linear excavations. The objective of this work is to propose a procedure to evaluate the state of non-linear underground enclosures by analysing the groundwater response to pumping inside the enclosure. The proposed method, which is based on diagnostic plots (derivative of drawdown with respect to the logarithm of time), allows (1) determining if an underground non-linear enclosure has isolated openings or numerous defects and (2) computing its effective conductance or effective hydraulic conductivity. The methodology is tested with data collected during the excavation of a shaft required for the construction of the high speed train (HST) tunnel in Barcelona, Spain. The procedure can be applied using the wells drilled for dewatering. Although a test before the excavation is recommended to evaluate the underground retaining walls (Watertightness Assessment Test), the method can be applied using data collected at the beginning of the dewatering stage.

Semi-analytical solutions of groundwater flow in multi-zone (patchy) wedge-shaped aquifers

Alluvial fans are potential sites of potable groundwater in many parts of the world. Characteristics of alluvial fans sediments are changed radially from high energy coarse-grained deposition near the apex to low energy fine-grained deposition downstream so that patchy wedge-shaped aquifers with radial heterogeneity are formed. The hydraulic parameters of the aquifers (e.g. hydraulic conductivity and specific storage) change in the same fashion. Analytical or semi-analytical solutions of the flow in wedge-shaped aquifers are available for homogeneous cases. In this paper we derive semi-analytical solutions of groundwater flow to a well in multi-zone wedge-shaped aquifers. Solutions are provided for three wedge boundary configurations namely: constant head–constant head wedge, constant head–barrier wedge and barrier–barrier wedge. Derivation involves the use of integral transforms methods. The effect of heterogeneity ratios of zones on the response of the aquifer is examined. The results are presented in form of drawdown and drawdown derivative type curves. Heterogeneity has a significant effect on over all response of the pumped aquifer. Solutions help understanding the behavior of heterogeneous multi-zone aquifers for sustainable development of the groundwater resources in alluvial fans.

Determination of minimum permeability plateau and characteristic length in porous media with non-Darcy flow behavior

Abstract A mathematical model has been formulated to determine the minimum permeability plateau and characteristic length in porous media with non-Darcy flow behaviour. More specifically, two new parameters are first defined to respectively include effect of minimum permeability plateau and characteristic length on non-Darcy flow behaviour across the completion with consideration of wellbore storage and skin effects. The newly formulated mathematical model is solved by discretizing the convolution integral and material balance equation, and then validated by simplifying it to the traditional Forchheimer model. The pressure together with its corresponding derivative type curves for both buildup and drawdown tests can be reproduced to examine effects of both wellbore storage and non-Darcy flow across the completion. The pressure derivative response with non-Darcy flow across the completion has been found to have a steep slope since the non-Darcy effect is increased after the wellbore-storage-dominated period. Such steep derivatives resulting from the increased non-Darcy effect cannot be observed from the existing models based on Darcy׳s law. In particular, the non-Darcy parameter depending on flow rate is found to dominate the shape of the derivative curves. Compared to the drawdown derivative curves, there exists a larger gap in the buildup derivative curves, indicating that buildup tests are more appropriate than drawdown tests for type curve matching non-Darcy flow behaviour.

Derivative-Assisted Classification of Fractured Zones Crossing a Deep Borehole

In this study, the derivative analysis using the derivative of drawdown with respect to log-time was utilized to determine candidates for hydraulic conductor domains (HCDs). At a 500-m deep borehole in the study site, the fractured rocks crossing the borehole were first classified in fractured and nonfractured zones by core logging and geophysical loggings, such as acoustic televiewing, density, and flow loggings. After conducting the hydraulic tests such as constant head withdrawal and recovery tests at the fractured zones and the nonfractured zones, the derivative analyses were carried out, of which the results were evaluated to determine the candidates for HCDs. For the nonfractured zones, the diagnostic plot has only a big hump indicating poor connection of the background fractures to the permeable geologic media, while those of the candidates for HCDs show various flow regimes. On the basis of these results, the candidates for HCDs among the fractured zones were determined. From discussion on the results, the combination of the spacing analysis and derivative analysis following a hydraulic test is recommended for determining the candidates for HCDs rather than other geophysical loggings.

Application de la courbe diagnostique aux essais de pompage en milieu karstique

Diagnostic plots applied to pumping tests in karst systems. Pumping tests conducted on wells intersecting karst heterogeneities such as the conduit network are difficult to interpret. Nevertheless, this case can be solved by assimilating the horizontal karst conduit to a finite-conductivity vertical fracture. In this case, several flow patterns corresponding to the respective contributions of karst subsystems (fractured matrix, small conduits, and main karst drainage network) can be identified on the diagnostic plot of the drawdown derivative. This is illustrated on two examples from Mediterranean karst systems in southern France. A pumping test on a well intersecting the main karst drainage network of the Cent-Fonts karst system shows (i) a preliminary contribution of the karst conduit storage capacity followed by (ii) linear flows into the fractured matrix. A pumping test on a well intersecting a small karst conduit of the Corbières karst system shows the existence of(i) bi-linear flow within both the karst conduit and the fractured matrix at early times, followed by (ii) radial flows within the fractured matrix and (Hi) finally the contribution of a major karst cavity. The use of diagnostic plots allows identifying the various flow regimes during pumping tests, corresponding to the response of the individual karst aquifer subsystems. This is helpful for improving the understanding of the structure of the karst aquifer and flow exchanges between subsystems.

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