Summary. An analytical equation for calculating, the effective radius, r, in a wellblock is derived. The radius r, is needed to relate wellblock pressure to well pressure. The equation is general and valid for vertical and horizontal wells and for any well location, aspect ratio of the well's drainage area, and anisotropy. The equation is used to show when Peaceman's formulations are adequate and when they require modification. Generally, if the drainage area of the well is a rectangle with sides of large aspect ratio and/or the formation is highly anisotropic, Peaceman's equations need modifications. Simplified forms of the equation applicable to special cases are reported. Furthermore, an easy-to-use approximate equation for calculating r values for nearly centered horizontal wells is given. The formulas of this work, as well as those of Peaceman, are relevant only to simulations on uniform grids in homogeneous media. Introduction Reservoir simulators require a functional relation between wellblock and wellbore pressures to calculate the wellbore pressure when the flow rate is assigned or the flow rate when the wellbore pressure is given. Peaceman published his first paper on the subject in 1978. Using a repeated five-spot pattern and square gridblocks, he showed that for an isotropic, square medium containing, a producer and injector, the wellblock pressure, p, is related to the wellbore pressure, p, by This equation is applicable to a vertical well where p is considered the steady-state flowing pressure located at a radius . Here, the grid is a square and delta is the grid dimension. This relation has been accepted almost universally and has replaced the many equations used before Peaceman's publications. In 1983, Peaceman published a second paper that provided equations for calculating r values when the wellblock is a rectangle and/or the formation is anisotropic. To the best of our knowledge, no method was available in the literature to test the applicability of Peaceman's formulas to horizontal wells until recently. To do this, one needs an independent approach for calculating a reliable value for p for an assigned set of parameters. The solution given in Refs. 3 and 4 provided a means to do so. In this work, we detail a procedure for calculating the wellblock radius, r. In our formulation of the problem, the well's drainage volume is idealized as a rectangular box-shaped region with all six faces closed to crossflow. The flow domain is homogeneous but anisotropic. The well axis is idealized as a constant-rate, uniform line sink. The physical wellbore is assumed to coincide with a cylindrical surface at a radial distance of r from this line sink. Peaceman studied a steady-state flow problem with producers and injectors placed at the corners of a square pattern. The effects of rectangular patterns and rectangular drainage areas on r were not studied. By isolating the well, Peaceman eliminated the influence of boundaries on the flow near the well. The assumption was that r values were a function of only the properties of the wellblocks. Our analysis shows that the aspect ratio of the drainage area of the well has a strong effect on r values. The aspect ratio is defined as the ratio of the scaled dimensions of the area perpendicular to the well direction. A scaled dimension is defined as, where i may be x. y, or z, and li is a typical physical length in the i direction. Because the aspect ratio for a horizontal well is considerably different from unity, it became necessary to investigate the deviations from Peaceman's formulas for horizontal wells. Peaceman solved the isotropic reservoir case by specifying uniform (and constant-rate) flux q at the wellbore, We found, however, that Peaceman's equation for an anisotropic medium, regardless of the aspect ratio, is based on the assumption of constant pressure at the wellbore. Because p is a function of r, an error in r translates to an error in p. Kim also examined the effect of partial penetration on the r values calculated by Peaceman when p . He found that Peaceman's r, should be modified to obtain the correct p . Kim's work needs to be extended to the horizontal-well environment. In this work, we derive an accurate analytical equation for calculating r. This equation is general and valid for vertical and horizontal wells, for any well location, and for isotropic and anisotropic formations. We use the equation to derive simplified forms applicable to special cases and show the conditions under which Peaceman's formulation is valid. We also report an easy-to-use approximate equation for calculating r applicable to horizontal wells that are basically centrally located in the drainage volumes. Throughout this work, gravity is neglected. To account for gravity, we simply select a reference datum and replace "pressure head (p/gp)" with the "hydraulic head (p/gp+z)," paying proper attention to the units of gravitational acceleration, g, density, p, a nd elevation. Calculation of the Correct Effective Radius We present two methods for calculating r. The first is analytical, and the second is graphical. Both methods start with the general solution 3 relating pressure and flow rate for any well of arbitrary location in a box-shaped drainage volume. Analytical Method. Fig. 1 indicates a finite-difference grid n X n) in the vertical cross sections of the drainage area of a horizontal well. The following steps lead to an analytical formula for r. We assume that the well is located at (x, z,). The nodes are represented by (i, j), with i=0, l ... (n - 1), and - j = 0,1 ... (n - 1). If (i, j,) are the well node coordinates, then x, = × =, +1/2) and z =, (+ 1/2). If a is the length and thickness, it follows that a = and h = . The following system of finite-difference equations for the pressure is obtained on the grid of Fig. 1. SPERE P. 324^

Variations in annular geometry, eccentricity, and pipe rotational speed strongly affect pressure loss of a fluid flowing in the narrow annulus of a slimhole well. Due to these factors, accurately calculating and controlling pressures in slimhole wellbores is difficult. Accurate pressure calculations are crucial for safely controlling formation pressures and protecting wellbore integrity. Attempts to model non-Newtonian fluid flow in narrow annuli with high-speed pipe rotation have been hampered by the lack of quality data. The results of numerous annular flow experiments presented herein partially correct this deficit. These results supplement annular pressure data from a 2500-ft slimhole test well and standpipe pressure data from a slimhole exploration well. Sensitive pressure measurements were used to characterize fluid flow in concentric narrow annuli created by a 1.25-in diameter (Dp) steel shaft inside clear acrylic tubes with 1.375-in to 1.75-in inside diameter (Dh). Similar tests were conducted in a fully eccentric annulus formed by the steel shaft inside an acrylic tube with Dh =1.50 in. Maximum shaft rotational speed was 900 rpm and maximum fluid flow rate was 12 gpm. Test fluids included water, glycerin solutions, viscosified clear brines, and several slimhole drilling muds. Models selected from the public domain were used with varying success to calculate results from the hydraulics tests. Simple models typically used by the drilling industry calculated annular pressure loss for non-rotating cases with reasonable accuracy. However, the simple models seldom calculated absolute effects of pipe rotation even though calculated trends correctly match those in measured data. For turbulent flow, annular pressure loss increased with increasing pipe rotation. For lamininar flow, annular pressure loss decreased with increasing pipe rotation. In all cases, annular pressure loss increased with increasing mud rheology and decreased with increasing eccentricity. Introduction Drilling or coring with high-speed pipe rotation requires excellent lateral stability for the pipe in order to avoid destructive vibrations. One means of providing this lateral stability is a narrow gap around the pipe. In this case, "narrows means Dp/Dh 0.80. This criteria distinguishes slimhole wells from reduced-bore or conventional wells. Small variations in annular gap, wellbore eccentricity, and pipe rotational speed strongly affect pressure loss of fluid flowing in the narrow annulus of a slimhole well . These factors, usually negligible in conventional drilling, significantly increase the difficulty of calculating and controlling pressures during slimhole drilling. Accurate calculations of pressure loss in the wellbore are necessary to control the well, optimize bit hydraulics, and avoid excessive pressure against the formation.

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Buckling and post-buckling lock-up place a limit on the reach of extended-reach and horizontal wells. Although buckling has received considerable theoretical attention in the past, no serious attempt has been made to study this process experimentally. This paper describes results from experiments on buckling, unbuckling and post-buckling behavior of rods laterally constrained in a cylindrical enclosure, with particular emphasis on the effects of friction. The experimental apparatus, procedures, and uncertainty analysis are described. Results indicate that friction significantly delays the onset of buckling (both sinusoidal and helical buckling), and causes noticeable hysterisis in the post-buckling behavior, As a result of this hysterisis, the unbuckling loads are always less than the corresponding buckling loads. Mitigation of friction reduces the hysterisis. Friction is also a cause of post-buckling snapping and reversals in the direction of the helix. As expected, the effects of friction become less significant as the inclination decreases. For inclinations (from vertical) less than 15 degrees, the effects of friction are negligible for the initiation of sinusoidal buckling, but are significant once the rod has buckled into a full helix. Curvature also delays the onset of buckling, and both curvature and friction have a stabilizing effect on the tubulars. Current theory is re-visited and re-interpreted irr the light of these experimental observations, and its limitations are discussed with an emphasis toward operational practices and field applications. It is shown that currently used theory actually predicts unbuckling and not buckling. When friction is significant, current theory underestimates the compressive loads at which sinusoidal and helical buckling of tubulars occur in straight wellbore intervals. Ignoring friction limits the weight on bit well below the safe load that can be used in many drilling and completion operations in extended reach or horizontal wells. Moreover the hysterisis effect of friction means that once buckling has occurred, the compressive loads must be reduced to values much below the buckling initiation loads to fully straighten the buckled pipe. INTRODUCTION Buckling of tubulars such as drill string, casing, or tubing places operational limits on extended reach wells. In particular, buckling and subsequent "lock-up" are limiting factors to the reach of horizontal wells. Buckling occurs when the compressive load on a tubular exceeds some critical value. This critical load is the load beyond which the equilibrium shape is no longer stable, and small increases in load lead to large lateral deflections. Typically, compressive load is provided by the weight on bit (WOB) or the slack-off friction. However, unlike Euler buckling of bars, the lateral deflection of buckled tubulars in wells is limited by the outer constraint of the wellbore. As a result, buckling progresses differently. When the compressive load reaches the critical value, the straight shape of the string is no longer stable. A slight increase in the load from this value causes the string to deflect (buckle) into a sinusoidal shape along the lower portion of the hole. This is referred to as sinusoids/buckling.

The effects of friction and curvature on buckling, post-buckling, and unbuckling behavior of rods laterally constrained within an enclosure are studied experimentally. The experimental apparatus, measurement procedures, and uncertainty analysis are described. Results indicate that friction significantly delays the onset of buckling, and causes noticeable hysteresis in the post-buckling behavior. As a result, the unbuckling loads are always less than the corresponding buckling loads. The drag-related loss, which eventually leads to lock-up, is also measured and reported in this work. Friction is also a cause of post-buckling snapping and reversals in the direction of wrap of the helix. As expected, the effects of friction become less significant as the inclination decreases. It is shown that predictions of current theory agree with experimentally measured unbuckling rather than buckling loads. When friction is significant, current theory underestimates the compressive loads at which buckling occurs in straight or curved wellbores. Ignoring friction or curvature limits the estimated weight on bit well below the safe load that can be used in many drilling and completion operations in extended reach or horizontal wells. Moreover, the hysteresis effect of friction means that once buckling has occurred, the compressive loads must be reduced to values much below the buckling initiation loads to fully straighten the buckled pipe.