The Relation Between Wellblock and Wellbore Pressures in Numerical Simulation of Horizontal Wells
- D.K. Babu (Mobil R and D Corp.) | A.S. Odeh (Consultant) | A.J. Al-Khalifa (Aramco) | R.C. McCann (Mobil R and D Corp.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- August 1991
- Document Type
- Journal Paper
- 324 - 328
- 1991. Society of Petroleum Engineers
- 1.10.1 Drill string components and drilling tools (tubulars, jars, subs, stabilisers, reamers, etc), 5.5 Reservoir Simulation
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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.
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,, = 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.
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