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Zhang Wen - One of the best experts on this subject based on the ideXlab platform.
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combined role of leaky and non darcian effects on the flow to a pumping well with a non uniform flux well face boundary
Journal of Hydrology, 2020Co-Authors: Qi Zhu, Zhang WenAbstract:Abstract In this study, a non-Darcian flow model was developed for a constant-rate test of a partially penetrating well with a non-uniform flux boundary in a leaky confined aquifer. The Izbash equation was applied to describe non-Darcian flow in the radial direction. Both analytical and numerical methods were employed to solve this model. It was found that the analytical solution with linearization was adequate only when non-Darcian effects were relatively small. With the finite difference method, the radial and vertical fluxes along the well screen were analyzed under four different cases involving leaky and non-Darcian effects. The leaky and non-Darcian effects on the Drawdowns with non-uniform flux (NUF) boundary were compared with those with uniform flux (UF) boundary. The results indicate that both leaky and non-Darcian effects can reduce the fluxes along the well screen, while non-Darcian effects can diminish the leakage-induced differences of flux and Drawdown between the two well screen ends. Non-Darcian effects can reduce the differences between UF and NUF Drawdowns at any elevation with the greatest reduction difference at the elevation of the top end of the screen. The UF solution can replace the NUF solution when the distance is far from the well, as this distance is smallest at the elevation of the screen midpoint and can be distinctly reduced by non-Darcian effects. In general, the results are most sensitive to the power index n , moderately sensitive to the other aquifer parameters and least sensitive to the parameters of the aquitard.
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non darcian flow to a partially penetrating well in a confined aquifer with a finite thickness skin
Hydrogeology Journal, 2016Co-Authors: Qinggao Feng, Zhang WenAbstract:Non-Darcian flow to a partially penetrating well in a confined aquifer with a finite-thickness skin was investigated. The Izbash equation is used to describe the non-Darcian flow in the horizontal direction, and the vertical flow is described as Darcian. The solution for the newly developed non-Darcian flow model can be obtained by applying the linearization procedure in conjunction with the Laplace transform and the finite Fourier cosine transform. The flow model combines the effects of the non-Darcian flow, partial penetration of the well, and the finite thickness of the well skin. The results show that the depression cone spread is larger for the Darcian flow than for the non-Darcian flow. The Drawdowns within the skin zone for a fully penetrating well are smaller than those for the partially penetrating well. The skin type and skin thickness have great impact on the Drawdown in the skin zone, while they have little influence on Drawdown in the formation zone. The sensitivity analysis indicates that the Drawdown in the formation zone is sensitive to the power index (n), the length of well screen (w), the apparent radial hydraulic conductivity of the formation zone (Kr2), and the specific storage of the formation zone (Ss2) at early times, and it is very sensitive to the parameters n, w and Kr2 at late times, especially to n, while it is not sensitive to the skin thickness (rs).
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Non-Darcian flow toward a larger-diameter partially penetrating well in a confined aquifer
Environmental Earth Sciences, 2014Co-Authors: Zhang Wen, Kai Liu, Hongbin ZhanAbstract:In this study, non-Darcian flow to a larger-diameter partially penetrating well in a confined aquifer was investigated. The flow in the horizontal direction was assumed to be non-Darcian and described by the Izbash equation, and the flow in the vertical direction was assumed to be Darcian. A linearization procedure was used to approximate the nonlinear governing equation. The Laplace transform associated with the finite cosine Fourier transform was used to solve such non-Darcian flow model. Both the Drawdowns inside the well and in the aquifer were analyzed under different conditions. The results indicated that the Drawdowns inside the well were generally the same at early times under different conditions, and the features of the Drawdowns inside the well at late times were similar to those of the Drawdowns in the aquifer. The Drawdown in the aquifer for the non-Darcian flow case was larger at early times and smaller at late times than their counterparts of Darcian flow case. The Drawdowns for a partially penetrating well were the same as those of a fully penetrating well at early times, and were larger than those for a fully penetrating well at late times. A longer well screen resulted in a smaller Drawdown in the aquifer at late times. A larger power index n in the Izbash equation resulted in a larger Drawdown in the aquifer at early times and led to a smaller Drawdown in the aquifer at late times. A larger well radius led to a smaller Drawdown at early times, but it had little impact on the Drawdown at late times. The wellbore storage effect disappears earlier when n is larger.
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approximate analytical solution for non darcian flow toward a partially penetrating well in a confined aquifer
Journal of Hydrology, 2013Co-Authors: Zhang Wen, Kai Liu, Xiaolian ChenAbstract:Summary In this study, non-Darcian flow to a partially penetrating well in a confined aquifer was investigated. The flow in the horizontal direction was assumed to be non-Darcian, while the flow in the vertical direction was assumed to be Darcian. The Izbash equation was employed to describe the non-Darcian flow in the horizontal direction of the aquifer. We used a linearization procedure to approximate the non-linear term in the governing equation enabling the mathematical model to be solved using a combination of Laplace and Fourier cosine transforms. Approximate analytical solutions for the Drawdown were obtained and the impacts of different parameters on the Drawdown were analyzed. The results indicated that a larger power index n in the Izbash equation leads to a larger Drawdown at early times, while a larger n results in a smaller Drawdown at late times. The Drawdowns along the vertical direction z are symmetric if the well screen is located in the center of the aquifer, and the Drawdown at the center of the aquifer is the largest along the vertical direction for this case. The length of the well screen w has little impact on the Drawdown at early times, while a larger length of the well screen results in a smaller Drawdown at late times. The Drawdown increases with K r at early times, while it decreases as K r increases at late times, in which K r is the apparent radial hydraulic conductivity. A sensitivity analysis of the parameters, i.e., the specific storage S s , w , n and K r , indicated that the Drawdown is not sensitive to them at early times, while it is very sensitive to these parameters at late times especially to the power index n .
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approximate analytical and numerical solutions for radial non darcian flow to a well in a leaky aquifer with wellbore storage and skin effect
International Journal for Numerical and Analytical Methods in Geomechanics, 2013Co-Authors: Zhang Wen, Quanrong WangAbstract:SUMMARY This study investigated non-Darcian flow to a well in a leaky aquifer considering wellbore storage and a finite-thickness skin. The non-Darcian flow is described by the Izbash equation. We have used a linearization procedure associated with the Laplace transform to solve such a non-Darcian flow model. Besides, the Stehfest method has been used to invert the Laplace domain solutions for the Drawdowns. We further analyzed the Drawdowns inside the well for different cases. The results indicated that a smaller BD results in a smaller Drawdown at late times and the leakage has little effect on the Drawdown inside the well at early times, where BD is a dimensionless parameter reflecting the leakage. We have also found that the flow for the negative skin case approaches the steady-state earlier than that for the positive skin. In addition, the Drawdown inside the well with a positive skin is larger than that without skin effect at late times, and a larger thickness of the skin results in a greater Drawdown inside the well at late times for the positive skin case. A reverse result has been found for the negative skin case. Finally, we have developed a finite-difference solution for such a non-Darcian flow model and compared the numerical solution with the approximate analytical solution. It has been shown that the linearization procedure works very well for such a non-Darcian flow model at late times, and it underestimates the Drawdowns at early times. Copyright © 2012 John Wiley & Sons, Ltd.
Hund-der Yeh - One of the best experts on this subject based on the ideXlab platform.
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A Semianalytical Solution for Residual Drawdown at a Finite Diameter Well in a Confined Aquifer
JAWRA Journal of the American Water Resources Association, 2013Co-Authors: Hund-der Yeh, Chih Tse WangAbstract:After the end of pumping the water level in the observation well starts to recover and the reduced Drawdown during the recovery period is named as the residual Drawdown. Traditional approaches in analyzing the data of residual Drawdown for estimating the aquifer hydraulic parameters are mostly based on the application of superposition principle and Theis equation. In addition, the effect of wellbore storage is commonly ignored in the evaluation even if the test well has a finite diameter. In this article, we develop a mathematical model for describing the residual Drawdown with considering the wellbore storage effect and the existing Drawdown distribution produced by the pumping part of the test. The Laplace-domain solution of the model is derived using the Laplace transform technique and the time-domain result is inverted based on the Stehfest algorithm. This new solution shows that the residual Drawdown associated with the boundary and initial conditions are related to the well Drawdown and the aquifer Drawdown, respectively. The well residual Drawdown will be overestimated by the Theis residual Drawdown solution in the early recovery part if neglecting the wellbore storage. On the other hand, the Theis residual Drawdown solution can be used to approximate the present residual Drawdown solution in the late recovery part of the test.
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radial groundwater flow to a finite diameter well in a leaky confined aquifer with a finite thickness skin
Hydrological Processes, 2009Co-Authors: Shaw Yang Yang, Hund-der YehAbstract:A mathematical model that describes the Drawdown due to constant pumpage from a finite radius well in a two-zone leaky confined aquifer system is presented. The aquifer system is overlain by an aquitard and underlain by an impermeable formation. A skin zone of constant thickness exists around the wellbore. A general solution to a two-zone leaky confined aquifer system in Laplace domain is developed and inverted numerically to the time-domain solution using the modified Crump (1976) algorithm. The results show that the Drawdown distribution is significantly influenced by the properties and thickness of the skin zone and aquitard. The sensitivity analyses of parameters of the aquifer and aquitard are performed to illustrate their effects on Drawdowns in a two-zone leaky confined aquifer system. For the negative-skin case, the Drawdown is very sensitive to the relative change in the formation transmissivity. For the positive-skin case, the Drawdown is also sensitive to the relative changes in the skin thickness, and both the skin and formation transmissivities over the entire pumping period and the well radius and formation storage coefficient at early pumping time. Copyright © 2009 John Wiley & Sons, Ltd.
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Analysis of well residual Drawdown after a constant-head test
Journal of Hydrology, 2009Co-Authors: Hund-der Yeh, Chih Tse WangAbstract:summary A recovery test measures the residual Drawdown after an aquifer pumping test has ended and analyzes the recovery data to determine hydrogeological parameters such as transmissivity and storage coefficient. To our knowledge, the solution for the distribution of residual Drawdown following a constanthead test has never been presented. In this paper, we first develop a mathematical model that describes the residual Drawdown taking into consideration the wellbore-storage effect and the Drawdown distribution occurring at the end of a previous constant-head test. Then, the Laplace-domain solution of the model is developed using the Laplace transforms and its time-domain solution is obtained using the Stehfest algorithm. Numerical results show that the distribution of residual Drawdown depends on the boundary condition related to the well Drawdown and the initial condition related to the aquifer Drawdown. The well residual Drawdown (i.e., the residual Drawdown at wellbore) during the early recovery period will be over-estimated by the approximate residual Drawdown solution based on the Theis-type solution and superposition principle due to the neglect of wellbore storage. For a large recovery time, the effect of wellbore storage is negligible and the approximate residual Drawdown solution is therefore applicable.
Hongbin Zhan - One of the best experts on this subject based on the ideXlab platform.
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groundwater flow to a general well configuration in an unconfined aquifer overlying a fractured bedrock
Journal of Hydrology, 2019Co-Authors: Mohammad M Sedghi, Hongbin ZhanAbstract:Abstract 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.
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Non-Darcian flow toward a larger-diameter partially penetrating well in a confined aquifer
Environmental Earth Sciences, 2014Co-Authors: Zhang Wen, Kai Liu, Hongbin ZhanAbstract:In this study, non-Darcian flow to a larger-diameter partially penetrating well in a confined aquifer was investigated. The flow in the horizontal direction was assumed to be non-Darcian and described by the Izbash equation, and the flow in the vertical direction was assumed to be Darcian. A linearization procedure was used to approximate the nonlinear governing equation. The Laplace transform associated with the finite cosine Fourier transform was used to solve such non-Darcian flow model. Both the Drawdowns inside the well and in the aquifer were analyzed under different conditions. The results indicated that the Drawdowns inside the well were generally the same at early times under different conditions, and the features of the Drawdowns inside the well at late times were similar to those of the Drawdowns in the aquifer. The Drawdown in the aquifer for the non-Darcian flow case was larger at early times and smaller at late times than their counterparts of Darcian flow case. The Drawdowns for a partially penetrating well were the same as those of a fully penetrating well at early times, and were larger than those for a fully penetrating well at late times. A longer well screen resulted in a smaller Drawdown in the aquifer at late times. A larger power index n in the Izbash equation resulted in a larger Drawdown in the aquifer at early times and led to a smaller Drawdown in the aquifer at late times. A larger well radius led to a smaller Drawdown at early times, but it had little impact on the Drawdown at late times. The wellbore storage effect disappears earlier when n is larger.
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solutions for non darcian flow to an extended well in fractured rock
Ground Water, 2011Co-Authors: Zhang Wen, Hongbin Zhan, Guanhua HuangAbstract:We have investigated non-Darcian flow to a vertical fracture represented as an extended well using a linearization procedure and a finite difference method in this study. Approximate analytical solutions have been obtained with and without the consideration of fracture storage based on the linearization procedure. A numerical solution for such a non-Darcian flow case has also been obtained with a finite difference method. We have compared the numerical solution with the approximate analytical solutions obtained by the linearization method and the Boltzmann transform. The results indicate that the linearized solution agrees generally well with the numerical solution at late times, and underestimates the dimensionless Drawdown at early times, no matter if the fracture storage is considered or not. When the fracture storage is excluded, the Boltzmann transform solution overestimates the dimensionless Drawdown during the entire pumping period. The dimensionless Drawdowns in the fracture with fracture storage for different values of dimensionless non-Darcian hydraulic conductivity β approach the same asymptotic value at early times. A larger β value results in a smaller dimensionless Drawdown in both the fracture and the aquifer when the fracture storage is included. The dimensionless Drawdown is approximately proportional to the square root of the dimensionless time at late times.
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an analytical solution for non darcian flow in a confined aquifer using the power law function
Advances in Water Resources, 2008Co-Authors: Zhang Wen, Guanhua Huang, Hongbin ZhanAbstract:Abstract We have developed a new method to analyze the power law based non-Darcian flow toward a well in a confined aquifer with and without wellbore storage. This method is based on a combination of the linearization approximation of the non-Darcian flow equation and the Laplace transform. Analytical solutions of steady-state and late time Drawdowns are obtained. Semi-analytical solutions of the Drawdowns at any distance and time are computed by using the Stehfest numerical inverse Laplace transform. The results of this study agree perfectly with previous Theis solution for an infinitesimal well and with the Papadopulos and Cooper’s solution for a finite-diameter well under the special case of Darcian flow. The Boltzmann transform, which is commonly employed for solving non-Darcian flow problems before, is problematic for studying radial non-Darcian flow. Comparison of Drawdowns obtained by our proposed method and the Boltzmann transform method suggests that the Boltzmann transform method differs from the linearization method at early and moderate times, and it yields similar results as the linearization method at late times. If the power index n and the quasi hydraulic conductivity k get larger, Drawdowns at late times will become less, regardless of the wellbore storage. When n is larger, flow approaches steady state earlier. The Drawdown at steady state is approximately proportional to r1−n, where r is the radial distance from the pumping well. The late time Drawdown is a superposition of the steady-state solution and a negative time-dependent term that is proportional to t(1−n)/(3−n), where t is the time.
Hongzhong Zhang - One of the best experts on this subject based on the ideXlab platform.
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On Magnitude, Asymptotics and Duration of Drawdowns for Levy Models
2015Co-Authors: David Landriault, Hongzhong ZhangAbstract:This paper considers magnitude, asymptotics and duration of Drawdowns for some Levy processes. First, we revisit some existing results on the magnitude of Drawdowns for spectrally negative Levy processes using an approximation approach. For any spectrally negative Levy process whose scale functions are well-behaved at 0 , we then study the asymptotics of Drawdown quantities when the threshold of Drawdown magnitude approaches zero. We also show that such asymptotics is robust to perturbations of additional positive compound Poisson jumps. Finally, thanks to the asymptotic results and some recent works on the running maximum of Levy processes, we derive the law of duration of Drawdowns for a large class of Levy processes (with a general spectrally negative part plus a positive compound Poisson structure). The duration of Drawdowns is also known as the "Time to Recover" (TTR) the historical maximum, which is a widely used performance measure in the fund management industry. We find that the law of duration of Drawdowns qualitatively depends on the path type of the spectrally negative component of the underlying Levy process.
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On the Frequency of Drawdowns for Brownian Motion Processes
Journal of Applied Probability, 2015Co-Authors: David Landriault, Hongzhong ZhangAbstract:Drawdowns measuring the decline in value from the historical running maxima over a given period of time are considered as extremal events from the standpoint of risk management. To date, research on the topic has mainly focused on the side of severity by studying the first Drawdown over a certain prespecified size. In this paper we extend the discussion by investigating the frequency of Drawdowns and some of their inherent characteristics. We consider two types of Drawdown time sequences depending on whether a historical running maximum is reset or not. For each type we study the frequency rate of Drawdowns, the Laplace transform of the nth Drawdown time, the distribution of the running maximum, and the value process at the nth Drawdown time, as well as some other quantities of interest. Interesting relationships between these two Drawdown time sequences are also established. Finally, insurance policies protecting against the risk of frequent Drawdowns are also proposed and priced.
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On the Frequency of Drawdowns for Brownian Motion Processes
2014Co-Authors: David Landriault, Hongzhong ZhangAbstract:Drawdowns measuring the decline in value from the historical running maxima over a given period of time, are considered as extremal events from the standpoint of risk management. To date, research on the topic has mainly focus on the side of severity by studying the first Drawdown over certain pre-specified size. In this paper, we extend the discussion by investigating the frequency of Drawdowns, and some of their inherent characteristics. We consider two types of Drawdown time sequences depending on whether a historical running maximum is reset or not. For each type, we study the frequency rate of Drawdowns, the Laplace transform of the n-th Drawdown time, the distribution of the running maximum and the value process at the n-th Drawdown time, as well as some other quantities of interest. Interesting relationships between these two Drawdown time sequences are also established. Finally, insurance policies protecting against the risk of frequent Drawdowns are also proposed and priced.
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Stochastic Modeling and Fair Valuation of Drawdown Insurance
Insurance: Mathematics and Economics, 2013Co-Authors: Hongzhong Zhang, Tim Leung, Olympia HadjiliadisAbstract:Abstract This paper studies the stochastic modeling of market Drawdown events and the fair valuation of insurance contracts based on Drawdowns. We model the asset Drawdown process as the current relative distance from the historical maximum of the asset value. We first consider a vanilla insurance contract whereby the protection buyer pays a constant premium over time to insure against a Drawdown of a pre-specified level. This leads to the analysis of the conditional Laplace transform of the Drawdown time, which will serve as the building block for Drawdown insurance with early cancellation or drawup contingency. For the cancellable Drawdown insurance, we derive the investor’s optimal cancellation timing in terms of a two-sided first passage time of the underlying Drawdown process. Our model can also be applied to insure against a Drawdown by a defaultable stock. We provide analytic formulas for the fair premium and illustrate the impact of default risk.
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MAXIMUM Drawdown INSURANCE
International Journal of Theoretical and Applied Finance, 2011Co-Authors: Peter Carr, Hongzhong Zhang, Olympia HadjiliadisAbstract:The Drawdown of an asset is a risk measure defined in terms of the running maximum of the asset's spot price over some period [0, T]. The asset price is said to have drawn down by at least $K over this period if there exists a time at which the underlying is at least $K below its maximum-to-date. We introduce insurance against a large realization of maximum Drawdown and a novel way to hedge the liability incurred by underwriting this insurance. Our proposed insurance pays a fixed amount should the maximum Drawdown exceed some fixed threshold over a specified period. The need for this Drawdown insurance would diminish should markets rise before they fall. Consequently, we propose a second kind of cheaper maximum Drawdown insurance that pays a fixed amount contingent on the Drawdown preceding a drawup. We propose double barrier options as hedges for both kinds of insurance against large maximum Drawdowns. In fact for the second kind of insurance we show that the hedge is model-free. Since double barrier opt...
H. Jourde - One of the best experts on this subject based on the ideXlab platform.
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Hydraulic tomography in coupled discrete-continuum concept to image hydraulic properties of a fractured and karstified aquifer (Lez aquifer, France)
Advances in Water Resources, 2020Co-Authors: P. Fischer, Abderrahim Jardani, H. JourdeAbstract:We present the results of a hydraulic tomography led on a 60 × 40 m² fractured and karstic field in Southern France in order to image, in a model, its transmissivity field. The dataset employed for the tomography consists in Drawdown responses to cross-boreholes pumping tests reaching pseudo steady-state, with 8 different pumping wells and 22 measurement boreholes. The inversion of the dataset was led on a 2D model coupling a discrete network and a continuum, by following the Discrete Network Deterministic Inversion (DNDI) method. This method permits an optimization of both the transmissivity distribution and the structural geometry of the discrete network, which represents in this case the interconnected fractures and conduits in the aquifer. The optimized model obtained after inversion allows reproducing the observed Drawdown in the field, and proposes a contrasted imaging of the hydraulic properties, as awaited in such fractured site. The fracture network in the optimized model also shows coherent orientations of fracturing, compared to the orientations effectively observed on the field, even though this information was not included in the inversion. A comparison of the results obtained with this coupled model to results obtained on the same data with equivalent porous media model (without integration of a discrete network) shows that the integration of a discrete network in the model greatly improves the ability of the model to reproduce the flows existing in such fractured fields, and thus the observed Drawdowns.
