Descripción de la Política de nombramiento de los miembros de la Junta Directiva

In the Babu and Odeh (1988, 1989) model, the physical system is a box-shaped drainage area with a horizontal well with radius r_w and length L = (x₂–x₁), placed parallel to the x-direction, as shown in Figure 5-5. The reservoir has a length in the x-direction of b, a width in the horizontal direction perpendicular to the well (y-direction) of a, and a thickness of h. The well can be in any arbitrary location in this reservoir, except that the well must lie in the x-direction and cannot be too close to any boundary. The well location is defined by specifying the location of the heel of the well as being at x₁, y_o, and z_o, relative to the origin located at one corner of the reservoir.

Figure 5-5. Schematic of the Babu and Odeh model.

The Babu and Odeh model is based on radial flow in the y-z plane, with the deviation of the drainage area from a circular shape in this plane accounted for with a geometry factor, and flow from beyond the wellbore in the x direction accounted for with a partial penetration skin factor. Note that the Babu and Odeh geometry factor is related inversely to the commonly used Dietz shape factor (Dietz, 1965). Thus, the Babu and Odeh inflow equation is

In Equation 5-23, A is the cross-section area (ah, Figure 5-5), C_H is the shape factor, s_R is the partial penetration skin, and s includes any other skin factors, such as completion or damage skin effects. The shape factor, C_H, accounts for the deviation of the shape of the drainage area from cylindrical and the departure of the wellbore location from the center of the system (Figure 5-5). The partial penetration skin, s_R, accounts for the flow from the reservoir located beyond the ends of the well in the x-direction, and is equal to zero for a fully penetrating horizontal well.

The heart of the Babu and Odeh model are procedures for calculating the shape factor and the partial penetration skin factor. These parameters were obtained by simplifying the solution of the diffusivity equation for the parallelepiped reservoir geometry and comparing it with the assumed inflow equation (Equation 5-23). Babu and Odeh solved the 3-D diffusivity equation with a wellbore boundary

condition of constant flow rate (uniform flux) at the well and no flow across the reservoir boundaries using the Green’s function approach. In this manner, the following correlations for the shape factor and partial penetration skin factor were obtained.

or in terms of the anisotropy ratio, I_ani,

s_R is evaluated for two different cases, depending on the horizontal dimensions of the reservoir. The first case is for a reservoir that is relatively wide [i.e., the reservoir extends farther in the horizontal direction perpendicular to the well than in the direction of the wellbore trajectory (a > b)]. The second case is for a long reservoir (b > a). The particular criteria for Case 1 are

then where

and

where x_mid is the x-coordinate of the midpoint of the well,

and

F((4x_mid + L)/2b) and F((4x_mid – L)/2b) in Equation 5-28 are evaluated as follows, taking the argument, (4x_mid + L)/2b or (4x_mid – L)/2b, as X. If the values of X are less than or equal to 1, F(X) is calculated by Equation 5-30 with the argument of L/2b replaced by X. Otherwise, if X is greater than 1, then F(X) is calculated by

with X either (4x_mid + L)/2b or (4x_mid – L)/2b.

The criteria for Case 2 are

For this case, where

and

where P_xyz in Equation 5-32 is the same as defined in Equation 5-27.

Example 5-3. Well Performance with the Babu and Odeh Model

Consider again the 4000-ft long reservoir described in Examples 5-1 and 5-2. For a 2000-ft long

horizontal well centered in the reservoir as in Example 5-1, and with width, a, equal to 1414 ft, what is the production rate predicted by the Babu and Odeh model if average reservoir pressure is 4000 psi and the bottomhole flowing pressure is 2000 psi? Assume all other parameters are the same as in Examples 5-1 and 5-2.

Solution

For the well centered in the Babu and Odeh box-shaped reservoir of the dimensions given, the length of the reservoir, b, is 4000 ft; the width of the reservoir, a, is 1414 ft; the height of the reservoir, h, is 100 ft; the ends of the well are at x₁ = 1000 ft and x₂ = 3000 ft, x_mid = 2000 ft, z₀ = 50 ft, and y₀ = 707 ft.

Other necessary data from the previous examples are horizontal permeability, which is 10 md (k_x = k_y) and vertical permeability (k_z) of 1 md, the lateral is 6 in. in diameter, the oil viscosity is 5 cp, and the formation volume factor is 1.1. I_ani is 3.16.

First, we calculate the shape factor, ln C_H, using Equation 5-25:

Checking for which case to use for calculating the partial penetration skin factor, a is 1414 ft and b is 4000 ft, thus

and therefore Case 2 applies (long reservoir).

Using Equations 5-27, 5-33, and 5-34, then

so, from Equation 5-32,

The flow rate for the given conditions is then calculated with Equation 5-23:

There are more parameters in horizontal well performance calculations than for vertical wells, such as the wellbore length and permeability anisotropy. One common question about horizontal well design is how long the wellbore length should be (notice that we are comparing producing length, which is the formation thickness in the vertical well case). The length of a horizontal well should be designed based on the geometry of the drainage area (length, width, formation thickness), the type of reservoir fluid, and the well structure (new well or reentry sidetrack).

In general, for an effectively unlimited drainage area, the longer the wellbore, the higher the flow rate. However, this conclusion could be misleading if the pressure drop along the horizontal well is large compared to the pressure drop from the formation to the well. For large-permeability formations, selecting horizontal well length should consider the flow diameter in the horizontal well. If the well length is too long, the production rate will be choked back by the pressure drop in the tubing. Detailed discussion about pressure drop in pipe flow is presented in Chapters 7 and 8, and methods to estimate the conditions for which the pressure drop in the wellbore are important relative to reservoir pressure drop is presented in Section 8.5.1.

Horizontal wells sometimes may not be the optimal plan to develop a field. For example, if the formation payzone is relatively thick, then a vertical well or a hydraulically fractured vertical well may be more beneficial, especially when vertical connectivity is low (small vertical permeability).

Productivity indices for horizontal, vertical, and hydraulically fractured wells should be compared to select the well architecture.

Example 5-4. Comparison of Productivity Index for Horizontal and Vertical Wells

For the same reservoir that was presented in Example 5-3, calculate the ratio of productivity index for a 2000-ft long horizontal well to a vertical well. Horizontal permeability, k_x, is 10 md, vertical

permeability, k_z, is 1 md, wellbore radius is 0.25 ft, viscosity is 5 cp, and the formation volume factor is 1.1.

Solution

For pseudosteady state, the vertical well productivity index is

And using the Babu and Odeh model, the horizontal well productivity index is

For the horizontal well, using the result from Example 5.3, the productivity index is the flow rate, 1527 STB/d, dividing by the pressure drawdown, 2000 psi, which gives 0.76 STB/d/psi. For the vertical well, the equivalent drainage radius, r_e, can be calculated from the drainage area, 4000-ft by 1414-ft. Thus, the productivity index for the vertical well is

The productivity index ratio is 4.6, showing that drilling a horizontal well in this case is beneficial from a production point of view.