Solution For Radial Flow Research Articles

How borehole wall elasticity affects wellbore storage capacity in periodic hydraulic tests

Characterizing fluid transport and pore pressure diffusion is key for monitoring and understanding many natural (e.g., seismically active zones and volcanic systems) and engineered environments (e.g., enhanced geothermal reservoirs and CO2 underground storage). Borehole hydraulic testing allows for inferring relevant properties of the probed subsurface volume, such as, for example, its transmissivity and diffusivity, assessing the governing flow regime as well as detecting the presence of hydraulic boundaries. Periodic hydraulic tests (PHT) achieve these objectives using an oscillatory fluid injection procedure while measuring the fluid pressure response in monitoring boreholes. The relevant information on the pressure diffusion process occurring in a probed formation is retrieved from the phase shifts and amplitude ratios between the flow rate and the interval pressure. In general, the interpretation of PHT data relies on the assumption that the pressure diffusion process is largely unaffected by the deformation of the probed rock volume. To assess this assumption, we present a conceptual PHT model based on the theory of poroelasticity to investigate the role played by hydromechanical coupling (HMC) effects during PHT and to assess whether and to what extent additional mechanical information can be extracted from these tests. These effects include the influence of the borehole wall deformation on the wellbore storage coefficient Sw, which quantifies the difference between the injected flow rates and those actually entering the porous formation. We show that, in homogeneous formations, the commonly taken approach based on the uncoupled diffusion solution for radial flow reproduces the results of our HMC model. However, neglecting the deformation effect of the borehole wall on the storage capacity of the system during injection can lead to underestimations of both transmissivity and diffusivity, particularly at shorter oscillation periods. We present analytical expressions for the borehole wall deformation contribution to Sw, which depends on the shear modulus and does not change with the oscillatory period. We show that it is possible to obtain the effective Sw along with the transmissivity and diffusivity using observations at various oscillatory periods. In the presence of hydraulic boundaries, such as no-flow or constant-pressure boundaries, at a given distance from the injection borehole, Sw becomes period-dependent, introducing apparent variations in the inferred shear modulus of the formation if the associated HMC effects are not accounted for. Our results thus highlight the importance of accounting for HMC effects in PHT data interpretation, not only to reliably and accurately characterize and monitor hydraulic properties but also to fully exploit the potential of PHT to provide additional information on the mechanical behavior of the probed rock volume.

Period‐ and Pressure‐Dependent Hydraulic Properties of a Fractured Granite Inferred From Periodic Pumping Tests

AbstractFractures can greatly impact fluid flow and pore pressure distribution in the upper crust. By conducting borehole hydraulic experiments, we can determine the ability of fractures to transport fluids and affect pressure diffusion under in situ conditions. We performed a series of periodic hydraulic tests in a borehole at the Bedretto Underground Laboratory for Geosciences and Geoenergies (Switzerland). We investigated the period‐ and pressure‐dependent response of a fractured granite by controlling the flow rate of harmonic and non‐harmonic oscillations, covering a wide range of periods as well as mean‐ and amplitude‐pressure values. By analyzing the phase and amplitude relation between the flow rate and fluid pressure in the injection interval, we found that the intersected fracture zone exhibits a period‐dependent hydraulic response close to radial flow perturbed by fracture‐related hydromechanical coupling effects. The main features of the interval's hydraulic response were reproduced by an uncoupled diffusion solution for radial flow, which was used to derive apparent hydraulic properties. The largest deviations from this model occur for the phase shifts and correlate with the pressure amplitude. This non‐linear expression of hydromechanical coupling associated with deformable fractures was interpreted in terms of a pressure‐dependent effective compliance of the fractures that affects their effective storativity. The period dependence is likely related to the spatial heterogeneity expected in fractured rocks and the varying sensitivity of the hydraulic response for different periods. Accounting for these effects is necessary for a correct hydraulic characterization in fractured environments and is valuable for inferring fracture mechanical behavior.

The special solutions of two-dimensional drift-flux equations for the two-phase flow

This paper considers the special solutions of two-dimensional isentropic drift-flux equations for the two-phase flow. The nonlinear system is transformed into a second order linear partial differential equation by the hodograph and the polar coordinate transformations. Specific solutions (circulatory flow, radial flow, and spiral flow) are obtained for steady and irrotational systems. Moreover, the special solution of radial flow for the pseudo-steady two-phase flow system is obtained.

Precision in Defining Diagnostic Plots Optimizes Well-Test Results

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 206108, “Diagnostic Plots of Pressure-Transient, Rate-Transient, and Diagnostic Fracture Injection/Falloff Tests,” by David Craig, SPE, Occidental, and Thomas Blasingame, SPE, Texas A&M University. The paper has not been peer reviewed. _ All transient-test interpretation methods use diagnostic plots for the identification of wellbore or fracture storage distortion, flow regimes, and other parameters. The associated diagnostic plots are not interchangeable between such tests. The objective of this work is to clearly define the appropriate diagnostic plots for each type of transient test. Introduction In the complete paper, the authors review constant-rate drawdown solutions for a few specific cases of interest and the plotting function used to match buildup data to the drawdown solutions. They extend their review to the cases of buildup or falloff analysis following short flow periods and illustrate the differences between analysis of buildup or falloffs following short and long flow periods. After covering pressure-transient testing, they discuss rate-transient diagnostic plots and the ambiguity observed during analysis of multifractured horizontal well diagnostic plots and the uncertainty of analyst straight lines. Finally, they discuss and demonstrate the correct plotting functions and flow-regime interpretation for diagnostic fracture injection and falloff tests. Multiple examples from the literature are included as part of the discussion. Pressure-Transient Diagnostic Plots Pressure-transient-testing diagnostic plots evolved from the solutions to constant-rate drawdown problems. The authors’ intent is to review infinite-acting solutions for radial flow with wellbore storage and skin and flow through an infinite-conductivity fracture. On a log-log plot, the characteristic slopes, detailed in the complete paper, are unit slope, ½ slope, and zero slope. Many authors have proposed plotting functions for overlaying the observed pressure response from transient tests other than a constant-rate drawdown on the drawdown solutions. In many cases, the slopes identifying storage distortion and flow regimes from drawdown solutions will match and can be used to identify storage and flow regimes from pressure-buildup tests. A situation in which pressure-buildup or falloff test plotting functions fail to match the drawdown solution is when the flow period before shut-in is short. Short-flow transient tests have been reviewed by multiple authors. In lower-permeability reservoirs, pressure-transient tests with short flow periods are very common, and, in unconventional reservoirs, short-flow transient tests are the most-common transient test. Diagnostic plots constructed based on the characteristic shapes of the dimensionless pressure and the dimensionless pressure derivative with respect to the natural logarithm of dimensionless time for constant-rate drawdown solutions cannot be applied in the same manner for short-flow transient tests. With short-flow transient tests, the observed derivative of pressure difference with respect to the natural logarithm of time matches the negative product of dimensionless time and the second derivative of dimensionless pressure with respect to dimensionless time. Additionally, the commonly known impulse derivative, which is defined as the product of time and the derivative of pressure difference with respect to the natural logarithm of time, will match the negative product of dimensionless time squared and the second derivative of dimensionless pressure with respect to dimensionless time.

Limit equilibrium analysis of drilling mud loss into an inclined natural fracture

A theoretical model is presented for the worst case scenario of drilling mud loss when an inclined wellbore intersects an arbitrarily oriented natural fracture. The fracture is assumed to have a constant aperture and be bounded by rigid walls. The ultimate invasion profile when cessation of mud flow eventually occurs is solved based on limit equilibrium conditions. Effects of the fluid yield stress and the wellbore and fracture orientations upon the mud loss behaviors are analyzed. In the special case when a horizontal natural fracture is intersected by a vertical wellbore, our analysis shows that the commonly used late time limit solution for radial flow of a yield stress fluid needs to be revised. Excellent agreement is achieved when our theoretical solution is compared with experimental data in the literature in determining the fluid yield stress based on the ultimate invasion radius. The theoretical model can also be applied to provide the lower bound estimates of the fracture aperture and permeability based on the mud loss data from the field.

Analysis of Bingham fluid radial flow in smooth fractures

Abstract Solutions for radial flow of a Bingham fluid are analyzed in this paper. It aims to eliminate confusions in the literature concerning the plug flow region in different solutions for analysis and design of grouting in rock fractures. The analyses based on the force balance equation reveal that the plug flow region in Bingham radial flow is independent of the fracture radius, and is not a growth function adapted from the solution of one-dimensional (1D) slit flow according to ‘similarity’. Based on the shear stress distribution, we analytically proposed that a non-uniform plug flow region cannot exist. The Bingham fluid (grout) penetration and flowrate evolution as functions of grouting time are given using the correct expression for the plug flow region. The radius-independent plug flow region and the presented flowrate evolution equation are also verified numerically. For radial flow, the relative penetration length is equal to the relative width of plug flow region, which is the same as that for 1D channel flow. Discrepancies in analytical solutions for grout penetration and flowrate evolution were also illustrated. The clarification of the plug flow region and evaluation of discrepancies in analytical solutions presented in this work could simplify modeling and design of grouting in rock engineering applications.

Radial propagation of yield-power-law grouts into water-saturated homogeneous fractures

This study presents analytical solutions for single-phase radial flow of yield-power-law fluids between homogeneous fractures (smooth parallel plates), which covers the special cases of yield-power-law fluids, i.e., Herschel-Bulkley, Bingham and power-law fluids, as well as Newtonian fluids. The analytical solution for radial flow of Herschel-Bulkley fluids is given for the first time. A formula for the plug flow region created by the yield stress is deduced, showing that this is constant and independent of the radius for radial flow of yield-stress fluids, which is also verified by numerical simulations. For modeling of the rock grouting process where the yield-power-law grouts are injected into water-saturated fractures, we also established a general mathematical model for the two-phase (one yield-power-law grout and one Newtonian fluid) radial flow in homogeneous fractures using the Reynolds equation based on the derived analytical solutions for single-phase radial flow of yield-power-law grouts. By solving the two-phase radial flow problem, the evolution of pressure distribution and grout propagation length is illustrated. The results generally show the high sensitivity of the rheological parameters and the potentially important impact of water flow and time-dependent rheological properties on the radial propagation of yield power-law grouts.

Radial flow of yield-power-law fluids: Numerical analysis, experimental study and the application for drilling fluid losses in fractured formations

A theoretical solution for radial flow of Yield-Power-Law (Herschel–Bulkley) fluids between two parallel disks is presented. The fundamental equations are based on momentum balance in radial coordinates. The solution encompasses the Newtonian, Bingham Plastic and Power-Law fluids as special cases. The numerical solution has been compared with experimental data obtained from radial flow between two parallel disks. Upon application of mathematical simplifications, a closed form expression for pressure drop for radial flow of Yield-Power-Law (YPL) fluids is obtained and presented as an “approximate analytical” solution. The validity of the approximate solution is examined versus numerical solution. The application of the approximate solution to the prediction of drilling fluid losses in fractured formations is discussed.

A Stable and Efficient Numerical Algorithm for Unconfined Aquifer Analysis

The nonlinearity of equations governing flow in unconfined aquifers poses challenges for numerical models, particularly in field-scale applications. Existing methods are often unstable, do not converge, or require extremely fine grids and small time steps. Standard modeling procedures such as automated model calibration and Monte Carlo uncertainty analysis typically require thousands of model runs. Stable and efficient model performance is essential to these analyses. We propose a new method that offers improvements in stability and efficiency and is relatively tolerant of coarse grids. It applies a strategy similar to that in the MODFLOW code to the solution of Richard's equation with a grid-dependent pressure/saturation relationship. The method imposes a contrast between horizontal and vertical permeability in gridblocks containing the water table, does not require "dry" cells to convert to inactive cells, and allows recharge to flow through relatively dry cells to the water table. We establish the accuracy of the method by comparison to an analytical solution for radial flow to a well in an unconfined aquifer with delayed yield. Using a suite of test problems, we demonstrate the efficiencies gained in speed and accuracy over two-phase simulations, and improved stability when compared to MODFLOW. The advantages for applications to transient unconfined aquifer analysis are clearly demonstrated by our examples. We also demonstrate applicability to mixed vadose zone/saturated zone applications, including transport, and find that the method shows great promise for these types of problem as well.

The Role of Hand Calculations in Ground Water Flow Modeling

Most ground water modeling courses focus on the use of computer models and pay little or no attention to traditional analytic solutions to ground water flow problems. This shift in education seems logical. Why waste time to learn about the method of images, or why study analytic solutions to one-dimensional or radial flow problems? Computer models solve much more realistic problems and offer sophisticated graphical output, such as contour plots of potentiometric levels and ground water path lines. However, analytic solutions to elementary ground water flow problems do have something to offer over computer models: insight. For instance, an analytic one-dimensional or radial flow solution, in terms of a mathematical expression, may reveal which parameters affect the success of calibrating a computer model and what to expect when changing parameter values. Similarly, solutions for periodic forcing of one-dimensional or radial flow systems have resulted in a simple decision criterion to assess whether or not transient flow modeling is needed. Basic water balance calculations may offer a useful check on computer-generated capture zones for wellhead protection or aquifer remediation. An easily calculated "characteristic leakage length" provides critical insight into surface water and ground water interactions and flow in multi-aquifer systems. The list goes on. Familiarity with elementary analytic solutions and the capability of performing some simple hand calculations can promote appropriate (computer) modeling techniques, avoids unnecessary complexity, improves reliability, and is likely to save time and money. Training in basic hand calculations should be an important part of the curriculum of ground water modeling courses.

Dupuit‐Forchheimer solutions for radial flow with linearly varying hydraulic conductivity or thickness of aquifer

This paper presents new analytical solutions for steady state piezometric heads in phreatic and confined aquifers with radial flow. The new solutions utilize two constant head boundaries and a hydraulic conductivity k or thickness of aquifer H that is a linear function of radius r. Previous solutions applied constant k and H, and they were usually based on one constant head boundary, which might be superimposed on a solution that allowed a constant discharge or recharge at the center of the aquifer. The previous solutions may be difficult to apply to real aquifers because constant k and H is usually an oversimplification of nature. The Dupuit‐Forchheimer assumption implies that vertical head gradients are ignored. The magnitude of the error introduced by allowing H = H(r) is evaluated numerically in this paper.

Well hydraulics theory and data analysis of the constant head test in an unconfined aquifer with the skin effect

In a constant‐head test (CHT), groundwater is withdrawn from a well that has a fixed drawdown. Accordingly, the CHT is particularly useful in aquifers of low transmissivity because the constant‐head well would not be dewatered owing to overpumping. The problem of interest pertains to a CHT in a homogeneous and anisotropic unconfined aquifer, where water released by the water table decline is assumed to be instantaneous and the skin effect surrounding the constant‐head well is represented by a skin factor. A two‐dimensional transient well hydraulics model is developed, and its Laplace domain solutions of aquifer drawdown and the constant‐head well discharge are determined. It is found that the three distinct segments typically noted in the time drawdown history of a constant‐rate test (CRT) are also characteristic of the temporal drawdown variation of the CHT. However, the early time and late time asymptotes conform to the radial flow solution of a CHT in confined aquifers as given by Hantush [1964]. Under the late time radial flow conditions the asymptotic solution is in terms of an apparent storage coefficient equal to the sum of the storage coefficient and the specific yield, and a specific mass balance relation exists among the constant‐head well discharge, the discharge due to aquifer storage, and the discharge resulted from the water table decline. While aquifer drawdown and the constant‐head well discharge are under the influence of well skin, their ratios are free from it. A curve matching method using this ratio is developed to determine the horizontal and the vertical hydraulic conductivities, the storage coefficient, and the specific yield. When the prescribed constant drawdown is not negligibly small in comparison to the initial saturated thickness, aquifer drawdown in late times may be large, and thus a drawdown correction is needed for data analysis. It is found that the Jacob correction for large drawdown of a CRT can also be used for the late drawdown correction in a CHT.

Detection of pressure buildup data dominated by wellbore phase redistribution effects

This paper presents a new diagnostic technique that detects the presence of any type of phase redistribution pressure response and determines the true beginning of the semilog straight line for conventional analysis techniques. The method can also be used to predict the end of any wellbore effects. This greatly enhances the conventional analyses and yields more accurate estimation of the reservoir parameters. The technique is based upon existing analytical solutions for radial flow in homogenous reservoirs. The proposed method is simple and straightforward. It does not require manual or automatic type curve matching and it does not use complicated nonlinear optimization or history matching as some other methods necessitate. The applicability and accuracy of the proposed method are demonstrated through the analysis of three simulated cases and two field examples.

Starting and steady quadrupolar flow

Planar flow induced in a viscous fluid by a small cylinder oscillating in the direction normal to its axis is modeled theoretically and reproduced experimentally. In the model, a line force dipole (force doublet) was used as the source of motion. In an initially quiescent unbounded fluid this source produces zero net momentum and generates symmetrical quadrupolar flow consisting of two dipolar vorticity fronts propagating in opposite directions from the source. For starting flow at low Reynolds numbers, a second-order unsteady solution is obtained in terms of a power series of the Reynolds number, Re=Q/4πν2, where Q is the forcing amplitude and ν is the kinematic viscosity. This solution demonstrates that, as time t→∞, the flow in the vicinity of the source becomes steady and radial. To describe this steady asymptote, the Jeffery–Hamel nonlinear solution for radial flow is used. A particular solution is derived using the nondimensional intensity Re of the force dipole as a governing parameter. It is shown that the problem permits a similarity solution for all values of Re when a mass sink of prescribed intensity q=q(Re) is added to the flow. This steady asymptote is reproduced experimentally, using a vertical porous cylinder that oscillates horizontally in the shallow upper layer of a two-layer fluid and sucks fluid through its porous walls.

Method for interpreting tracer experiments in radial flow using modified analytical solutions

The universal type curves for mass transport in radial flow established numerically by Sauty in 1978 have been modified in this paper by using different boundary conditions of high accuracy. The method to determine the hydrodispersive parameters by interpretation of tracer experiments in radial flow using modified approximate analytical solutions is simple and accurate. It is shown that the approximate analytical solutions can be applied to mass transport in radial flow thanks to the introduction of correction factors, although there are large differences between the approximate analytical solutions and numerical solutions for radial flow in certain cases. The correction factors are derived from comparison of the above two solutions. For each case, a verification of this method has been performed by interpretation of numerical data using the approach formulae with the help of estimated correction factors.

Method for interpreting tracer experiments in radial flow using modified analytical solutions

The universal type curves for mass transport in radial flow established numerically by Sauty in 1978 have been modified in this paper by using different boundary conditions of high accuracy. The method to determine the hydrodispersive parameters by interpretation of tracer experiments in radial flow using modified approximate analytical solutions is simple and accurate. It is shown that the approximate analytical solutions can be applied to mass transport in radial flow thanks to the introduction of correction factors, although there are large differences between the approximate analytical solutions and numerical solutions for radial flow in certain cases. The correction factors are derived from comparison of the above two solutions. For each case, a verification of this method has been performed by interpretation of numerical data using the approach formulae with the help of estimated correction factors.

Some exact solutions of convection‐diffusion and diffusion equations

We examine solutions of the convection‐diffusion equation during steady radial flow in two and three dimensions with diffusivity proportional to Pn, with P the Péclet number. In two dimensions, closed‐form solutions of the Laplace‐transformed equation occur only for n = 1 and n = 2; and in three dimensions only for n = 5/4 and n = 2. The inversions for n = 1 (two dimensions) and 5/4 (three dimensions) cannot be expressed in closed form. On the other hand, the value n = 2, proposed by de Gennes (1983), leads to simple exact solutions, both in two dimensions and in three. In both cases a change of variable gives a one‐dimensional equation with constant coefficients. Exact instantaneous and continuous point source solutions for radial flow in two and three dimensions are thus established. In general, finite difference methods are needed for arbitrary n. Relevant instantaneous and continuous point source diffusion solutions are developed for radial two‐ and three‐dimensional systems. These represent small‐time solutions useful for initiating the finite difference solutions for convection‐diffusion with arbitrary n.

Pressure transient response of stochastically heterogeneous fractured reservoirs

A stochastic model for flow through inhomogeneous fractured reservoirs of double porosity, based on Barenblattet al.'s continuum approach, is presented. The fractured formation is conceptualized as an interconnected fracture network surrounding porous blocks, and amenable to the continuum approach. The block permeability is negligible in comparison to that of the fractures, and therefore the reservoir permeability is represented by the permeability of the fracture network. The fractured reservoir inhomogeneity is attributed to the fracture network, while the blocks are considered homogeneous. The mathematical model is represented by a coupled system of partial differential random equations, and a general solution for the average and for the correlation moments of the fracture pressure are obtained by the Neumann expansion (or Adomian decomposition). The solution for pressure is represented by an infinite series and an approximate solution for radial flow, is obtained by retaining the first two terms of the series. The purpose of this investigation is to get an insight on the pressure behavior in inhomogeneous fractured reservoirs and not to obtain type curves for determination of reservoir properties, which owing to the nonuniqueness of the solution, is impossible. For the present analysis we assumed an ideal reservoir with cylindrical symmetric inhomogeneity around the well. Fractured rock reservoirs being practically inhomogeneous, it is of interest to compare the pressure behavior of such reservoirs, with Warren and Roots's solution for (ideal) homogeneous reservoirs, used as a routine for determining the fractured reservoir characteristic parametersλ andΩ, using the results of well tests. The comparison of the results show that inhomogeneous and homogeneous reservoirs exhibit a similar pressure behavior. While the behavior is identical, the same drawdown or a build-up pressure curve may be fitted by different characteristic dimensionless parametersλ andΩ, when attributed to an inhomogeneous or a homogeneous reservoir. It is concluded that the ambiguity in determining the fractured reservoirλ andΩ, makes questionable the usefulness of determination of these parameters. Computations were also carried out to determine the correlation between the fracture pressure at the well and the fracture pressure at different points.

Solutions to bi-Maxwellian transport equations for radial solar wind beyond 28 Rs

There have been numerous attempts to model the solar wind using closed self-consistent sets of transport equations. These models have employed coupled continuity, momentum and energy equations. If stress and heat flow effects have been accounted for at all, it has usually been done in a simplistic and/or empirical fashion. Sets of transport equations that are based on a higher-order approximation to the velocity distribution function can include stress and heat flow equations in a self-consistent way, thus putting the stress and heat flow moments on an equal footing with the number density, drift velocity and temperature. Such equations are capable of describing both collisionless and collision-dominated plasmas as well as the transition between these two regimes. We present solar wind solutions for radial flow between 28 solar radii and 1 a.u. using the bi-Maxwellian based 16-moment set of transport equations. In addition to the number density, drift velocity and parallel and perpendicular temperatures, the 16-moment equations account for the transport of both longitudinal and transverse thermal energies as well as stress. Also, using the 16-moment approximation for the distribution function and assuming plasma parameter values characteristic of the solar wind, we generate contour plots for the proton velocity distribution function and show how the shape of these plots depends on various macroscopic plasma parameters.

Calculation Of Flowing Well Pressures In Reservoir Simulation Using Nine-Point Differencing

Abstract A quantity of considerable importance in flow rate predictions in reservoir simulation is the equivalent well block radius (r o) because it directly relates the numerically calculated grid block pressure to the actual well flowing pressure. Simulation of steam and miscible floods generally requires nine-point differencing. Previous work on approximating ro for nine-point differencing has either severe limitations or can give erroneous results and, hence, incorrect rate predictions. This paper presents, for the first time, a rigorous, closed form expression for ro for an isolated well, valid for any nine-point method. This expression is based on mathematical analysis and confirmed by careful numerical testing. It identically reduces to the expression for five-point differencing (obtained earlier by Peaceman) as a special case. Extension is made to the anisotropic case and confirmed numerically. Results for some additional well-geometries are also presented for the case of nine-point differencing. These include: arbitrary location of a well in the well block, multiple wells within a single block, a well in an edge or comer block. Finally, effects of heterogeneity on predicting flow rates are considered. Introduction In the numerical simulation of reservoir flow, it is usually necessary to use grid blocks whose areal dimensions are much larger than the diameter of a well. Consequently, the numerically calculated well block pressure, po is substantially different from the actual flowing bottomhole pressure of the well, pwf) as shown in Figure 1. Well inflow models are used to relate the well block pressure to the actual wellbore pressure and flow rate through the concept of an equivalent well block radius(1) (ro)), defined by Equation (1) Available In Full Paper. Equation (1) can be expressed as a rate prediction equation: Equation (2) Available In Full Paper. Equation (1) indicates that ro, may be interpreted as the distance from the well at which the steady-state flowing pressure equals the numerically calculated well block pressure, as shown in Figure 1. Peaceman(1) showed that for an isolated vertical well centred in its grid block in a homogeneous and isotropic medium, Equation (3) Available In Full Paper. when conventional five-point differencing is employed over a square grid. Peaceman obtained Equation (3) in two different ways. The first was by numerical experiments; the second by an approximate analytical approach that assumes that the pressures in the grid blocks immediately adjacent to the well block all satisfy the exact radial flow solution. For the square grid case, the first method yielded: Equation (4) Available In Full Paper. while the second method, which is henceforth referred to as the radial flow method, yielded Equation (5) Available In Full Paper. In a second paper, Peaceman(2) pointed out that the radial flow assumption is not generally valid and can result in significant error. For non-square (i.e. rectangular) grids, the radial flow method was found to predict: Equation (6) Available In Full Paper. whereas numerical experimentation showed that the correct expression is Equation (7) Available In Full Paper.