Izbash Equation Research Articles

Approximate analytical solution for non-Darcian flow toward a partially penetrating well in a confined aquifer

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 Kr at early times, while it decreases as Kr increases at late times, in which Kr is the apparent radial hydraulic conductivity. A sensitivity analysis of the parameters, i.e., the specific storage Ss, w, n and Kr, 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.

Approximate analytical and numerical solutions for radial non‐Darcian flow to a well in a leaky aquifer with wellbore storage and skin effect

SUMMARYThis 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.

A numerical solution for non-Darcian flow to a well in a confined aquifer using the power law function

In this study, we have obtained numerical solutions for non-Darcian flow to a well with the finite difference method on the basis of the Izbash equation, which states that the hydraulic gradient is a power function of the specific discharge. The comparisons between the numerical solutions and the Boltzmann solutions and linearization solutions have also been done in this study. The results indicated that the linearization solutions for both the infinitesimal-diameter well and the finite-diameter well agree very well with the numerical solution at late times, while the linearization method underestimates the dimensionless drawdown at early and moderate times. The Boltzmann method works well as an approximate analytical solution for the infinitesimal-diameter well. Significant differences have been found between the Boltzmann solution for a finite-diameter well and the numerical solution during the entire pumping period. The analysis of the numerical solution implies that all the type curves inside the well for different dimensionless non-Darcian conductivity kD values approach the same asymptotic value at early times, while a larger kD leads to a smaller drawdown inside the well at late times. A larger kD results in a larger drawdown in the aquifer at early times and a smaller drawdown in the aquifer at late times. Flow approaches steady-state earlier when kD is larger.

Non-Darcian flow to a well in an aquifer–aquitard system

In this study, we use a linearization procedure and a finite difference method to solve non-Darcian flow to a well in an aquifer–aquitard system. The leakage effect is considered. Flow in the aquifer is assumed to be non-Darcian and horizontal, whereas flow in the aquitard is assumed to be Darcian and vertical. The Izbash equation [Izbash SV. O filtracii V Kropnozernstom Materiale. USSR: Leningrad; 1931 [in Russian]] is employed to describe the non-Darcian flow. The wellbore storage is also considered in this study. An approximate semi-analytical solution has been obtained by the linearization procedure, and a numerical solution has been obtained by using a finite difference method. The previous solutions for Darcian flow case and non-Darcian flow case without leakage can be described as special cases of the new solutions. The error caused by the linearization procedure has also been analyzed. The relative error caused by the linearization procedure is nearly 100% at early times, and decreases to zero at late times. We have also compared the results in this study with Wen et al. [Wen Z, Huang G, Zhan H. A numerical solution for non-Darcian flow to a well in a confined aquifer using the power law function. J Hydrol, 2008d [in revision]] in which the leakage effect is not considered, and Hantush and Jacob [Hantush MS, Jacob CE. Non-steady radial flow in an infinite leaky aquifer. Trans Am Geophys Union 1955;36(1):95–100] who investigated a similar problem in Darcian flow case. The comparison of this study and Wen et al. (2008d) indicates the dimensionless drawdown in the aquifer with leakage is less than that without leakage, and the leakage has little effect at early times. The comparison between the results of this study and that of Hantush and Jacob (1955) indicates that the dimensionless drawdown in the aquifer for non-Darcian flow is larger at early times and smaller at late times, than their counterparts for Darcian flow. A larger dimensionless non-Darcian conductivity k D results in a smaller dimensionless drawdown in the aquifer at late times, and leads to a larger dimensionless drawdown in the aquifer at early times. A smaller dimensionless leakage parameter B D results in a smaller drawdown at late times, and the leakage does not affect the early-time drawdown. The analysis of the dimensionless drawdown inside the well has also been included in this study when the wellbore storage is considered.

Non-Darcian Flow Toward a Finite-Diameter Vertical Well in a Conflned Aquifer

Non-Darcian radial flow toward a flnite-diameter, fully penetrating well in a conflned aquifer was analyzed on the basis of the Izbash equation with consideration of the wellbore storage efiect. We derived semi-analytical solutions of drawdown by using the Boltzmann transform, and obtained approximate analytical solutions of the drawdown at early and late times. MATLAB programs were developed to facilitate computation of the semi-analytical solutions. The turbulence factor v which was directly related to the pumping rate appeared to have negligible influence upon the wellbore well function at early times, but imposed signiflcant infl;uence at intermediate and late times. However, the turbulence factor v imposed non-negligible influence upon the aquifer well function during the entire pumping period, provided that the observation point was not sufficiently close to the wellbore. Sensitivity analysis indicated that the power index n in the Izbash equation had less influence on the type curves at the face of the pumping wellbore, but had much greater influence upon the well function in the aquifer. As the n values increased, the drawdown in the aquifer decreased at early times and increased at late times. The Boltzmann transformation could only be used in an approximate sense for radial non-Darcian flow problems. This approximation would provide accurate solutions at early times, and introduce small but consistent discrepancies at intermediate and late times for the wellbore well function.

Non-Darcian flow in a single confined vertical fracture toward a well

Non-Darcian flow towards a well which fully penetrates a confined single vertical fracture is presented in this paper on the basis of the Izbash equation. We have obtained semi-analytical solutions for non-Darcian flow by using the Boltzmann transform and developed the non-Darcian flow well functions for cases with and without the effect of wellbore storage. The results show that the non-Darcian flow type curves are more or less deviated from the Darcian flow type curve. The non-linear effect is mainly attributable to the turbulent factor, v, a dimensionless parameter related to the pumping rate, the fracture aperture, the fracture thickness, and two constants k′ and n used in the Izbash power-law. The non-linear effect appears to be less sensitive to the power-law index, n. When excluding wellbore storage, the well function at early times is proportional to v−1/(n−1)u−n/(n−1), where u is a dimensionless term inversely proportional to time; whereas the well function at late times is approximated as v(n-1)uv[2+A0(n)], where A0(n) is a finite term depending on n. When considering wellbore storage, drawdowns inside the well with different v values approach the same asymptotic value at small times, and the effect of wellbore storage is only found at the early stage of pumping.

Comment on Izbash's equation

Izbash's equation, u = M( Δh L ) n , expressing the relationship between the fluid flow velocity and the hydraulic gradient in a granular bed, is given a theoretical background arrived at from a drag model, taking into account the voidage function and the drag coefficient correlation. The exponent n in Izbash's equation has been obtained over a wide range of Reynolds' numbers.