Theis Solution Research Articles

Drawdown in a finite circular aquifer with constant well discharge

An analytical solution, which was obtained previously by Muskat, is derived for the groundwater head in a confined, homogeneous, isotropic aquifer with a constant thickness and a circular impermeable boundary at radial distance a from a well penetrating the entire depth of the aquifer and discharging at a constant rate. The solution is compared with the Theis solution, which is valid for an aquifer of unlimited extent. The comparison shows that the two solutions agree very closely at radial distance r when Tt/Sa2 < 0.25 and r/a ≤ 0.1, where t denotes the time since the well discharge began, T and S are the transmissivity and storage coefficients of the aquifer, and a is the radial distance to the circular impermeable boundary. The two solutions agree poorly when Tt/Sa2 becomes larger than 1.0 or r/a ⋍ 1.0.

A Method of Analyzing Transient Fluid Flow in Multilayered Aquifers

A new approach to transient fluid flow in multilayered aquifers has been developed using the finite element method. This is a numeric technique in which an initial boundary value problem is converted to a variational problem and applied to a descretized system of elements. Details for the case of isotropic systems have been presented elsewhere. This paper extends the treatment to anisotropic systems. The method has been used to investigate potential distributions in multilayered aquifers of finite radial extent being pumped at constant rate using completely penetrating wells. An analysis of two‐layer systems with permeability contrasts of up to 100:1 indicates that at early time the drawdown behavior can vary significantly from the Theis solution, depending on where observations are made. As time increases, however, the results eventually converge on the Theis solution regardless of the permeability contrast. A 13‐layer aquifer containing either isotropic or anisotropic layers has been examined, and the results are in general agreement with the behavior for two‐layer systems.

Flow into a finite well with arbitrary discharge

The problem of determining the drawdown in response to flow from a finite radius well fully penetrating an infinite confined aquifer, where the rate of flow from the well is an arbitrary function of time, is solved exactly. In the limit where the well radius is small and the flow rate is constant the solution is shown to generate the Theis solution.

Theory of flow in aquicludes adjacent to slightly leaky aquifers

The theory of flow in aquicludes that are adjacent to slightly leaky aquifers is presented for two cases: (a) aquicludes of finite thickness and (b) aquicludes of semi‐infinite thickness. By ‘slightly leaky’ we simply mean that when the aquifer is being pumped at constant rate, the drawdowns in the aquifer will be predicted by the Theis solution, even though some slight movement of water through adjacent confining beds occurs. It is assumed that flow is essentially horizontal in the aquifer and vertical in the aquiclude. On this basis, analytical expressions for drawdown in the aquiclude are developed for each case. The results are presented in the form of type curves that can be used for aquiclude evaluation. The effect of multiple‐well fields is considered in each case.

A method of determining the permeability and effective porosity of unconfined anisotropie aquifers

The equations of unsteady flow toward a partially penetrating well in an unconfined aquifer of finite thickness are solved by linearization. It is assumed that the aquifer is nondeformable and that the effective porosity at the water table is constant. It is also assumed that the aquifer is anisotropic (the principal axes of the permeability tensor being horizontal and vertical, respectively), that the pumping discharge is constant, and that the drawdowns are small. The vertical component of the flow velocity is not neglected. The solution is, therefore, equally valid in the vicinity and at large distances from the pumping well. In the latter case, it coincides with the Theis solution. A method of matching the theoretical solution with drawdown measurements during pumping tests is outlined. By the matching method the horizontal and vertical hydraulic conductivities and the effective porosity of the aquifer may be determined. The method is illustrated by analyzing data from pumping tests carried out in three anisotropic aquifers.

Pressure Interference Effects Within Reservoirs and Aquifers

Abstract For the case of an infinite radial system operating at constant terminal rate, the reservoir engineer often uses the "point source" solution of the diffusivity equation to study pressure interference effects. At early times and at short distances from the inner boundary these solutions are invalid. The amount of error is often not precisely defined because of mathematical difficulties. Work presented here shows that, if dimensionless time is defined appropriately, previous solutions of the pressure equation can be displayed as a family of curves on one chart. These solutions include the point source solution (referred to in the field of hydrology as the Theis solution) and other solutions obtained with digital computer methods. With these curves, an exact evaluation of the pressure drop within a reservoir or an aquifer can be made by the engineer. Examples of field problem solutions are presented. In most reservoirs the error involved when the Theis solution is employed is often negligible; whereas, in the calculation of interference effects in an aquifer, a substantial error can occur through such an approach. Introduction Flow equations are used in petroleum engineering to study the behavior of individual wells and reservoirs. In the case of wells, the pressure response at the wellbore face is the major point of interest; whereas, in the case of reservoirs, the pressure response at the interface of the aquifer boundary is sought. To aid in such studies, the flow equations have been solved in terms of the behavior at these two inner boundaries. Only limited work has been published in regard to the pressure conditions away from these points, i.e., within the reservoir or aquifer. Theis and Mortada are among the few who have reported on this problem. The Theis approach employs the exponential integral and is valid for pressure conditions that occur some distance away from the flow disturbance. It is derived from the concept of a point source, as opposed to a flow across some finite area. The Mortada results, on the other hand, are valid at all points within the reservoir or aquifer. They are presented in terms of dimensionless ratios of the radius where the pressure is desired to the radius where the flow rate is measured. Their main use, in the past, has been in aquifer studies. The published results are presented in the form of graphs that are limited to a maximum radius ratio of 64. These graphical results are cumbersome to interpolate at non-integral radius ratios, so that one may be forced to utilize a rather involved analytical expression presented by Mortada. BASIC EQUATIONS The solutions of Mortada and Theis are both based on the diffusivity equation as applied to the case of an infinite radial system subject to a constant terminal rate. The equation is obtained by combining the material balance equation with Darcy's flow equation. The assumptions implicit in the use of this equation are as follows: a single fluid is present that occupies the entire pore volume; the reservoir is horizontal, homogeneous, uniform in thickness, and of infinite radial extent; compressibility and viscosity of the fluid remain constant at all pressures; and fluid density obeys the equation (1) Using the diffusivity equation in situations where the above conditions do not hold will result in errors. These errors (not discussed here) but only the errors which arise in the solution of the equation itself. The diffusivity equation for the homogeneous reservoir conditions cited above can be written in cylindrical coordinates, as (2) To obtain a dimensionless equation, so a single solution can be used for applications of different porosity, permeability and fluid properties, the following transformations are usually made: (3) (4) (5) After these transformations are made, Eq. 3 can be written in dimensionless form, as (6) JPT P. 471ˆ