SINGLE EQUATION SIMPLIFIES HORIZONTAL, DIRECTIONAL DRILLING PLANS

Michael L. Wiggins
University of Oklahoma
Norman, Okla.

Jonggeun Choe, Hans C. Juvkam-Wold
Texas A&M University
College Station, Tex.

The development of a single equation has simplified the planning of a two-dimensional well bore trajectory.

The basic relationship can be modified easily for the various types of directional and horizontal wells. The particular geometry of each type of well bore defines the variables used in the equation.

Directional drilling techniques have been used for many years to reach subsurface objectives overlain by inaccessible or difficult to reach surface locations. Economic and environmental considerations have increased the number of directional and horizontal wells drilled in recent years.

Drilling numerous wells from a single location ca also greatly simplify the installation of gathering and production facilities.

Generally, horizontal and extended reach wells are drilled for economic reasons. These applications often involve consolidated, naturally fractured reservoirs where the well bore may intersect multiple fracture systems. Horizontal and extended reach wells have been drilled to reduce coning problems in reservoirs that have large gas caps or strong water drives.

In some reservoirs the horizontal well may improve drainage by increasing the area of the well bore in contact with the reservoir.

WELL PLANNING

Deviating a well bore involves many factors that must be considered individually. Thus, careful planning is the key to successful directional drilling.

One of the first steps in planning a directional well should be the design of the well bore trajectory. Although drilling is a three-dimensional operation, many directional wells are planned in two dimensions, especially in the early stages of well planning.

Well bore trajectories can be categorized into two classes: the directional well and the horizontal well. Fig. 1 shows the three basic directional well bore configurations (Type 11 Type 2, and Type 3).1

A Type 1 well has a build and hold trajectory. The well bore is deflected from vertical at some kickoff point, the angle is built until a maximum angle is reached, and then the angle is held until the target is intercepted.

A Type 2 well has a build, hold, and drop or "S" trajectory. The well bore is deflected to some angle, the angle is held, and then the well drops in a manner such that the target is penetrated vertically. A modified Type 2 well differs in that the well bore is not returned to vertical in the drop portion.

A Type 3 well has a continuous build trajectory. The well bore inclination continues to increase through the target.

The second class of well bore trajectories is the horizonal well. This class includes the horizontal or near horizontal well (angles greater than about 80) and the extended reach well.

These geometries may have one or two build sections (Fig. 2).

In the past, there have been two major methods used in planning directional wells:

The use of build-up or composite build-up charts
The use of several directional well planning equations.

Each method depends on the particular well bore geometry desired.1-4 The use of buildup charts is tedious and often yields inaccurate results because it requires the use of preplotted graphs that need interpretation and interpolation.

The equation approach is often confusing to use because of the similarity of the various equations in which the selection of the proper relationship depends on the desired well bore geometry.

To assist and simplify the planning of a two-dimensional well bore trajectory, a single planning equation has been developed (see Equation 8 in box). This basic relationship can be modified for the directional or horizontal well by the proper definition of the equation variables. These definitions are based on the particular geometry of the well bore. In this equation, 0 is the inclination angle of the well bore in the first hold section.

Table 1 presents this relationship along with the definition of the variables for both classes of well bore trajectories. Modifications required to use the planning equation are also shown. With this information and the correct relations for the desired geometry, one can quickly, simply, and accurately plan the well bore trajectory for any directional well.

EQUATION DEVELOPMENT

Fig. 3 shows a typical configuration for a directional well that can be modified for any two-dimensional well geometry. From this figure, a two-dimensional planning equation can be developed based on basic geometric relationships. The equation is based on the Type 2 or S-shaped directional well and can be modified for the other directional well geometries.

The bottom hole location can be represented by coordinates D4 and X4, which are the vertical and horizontal displacements of the well bore from the surface location. These locations can be written as shown in Equations 1 and 2.

In these equations, L is the length of the hold portion of the well bore and r1 and r2 are the radiuses of curvature in the build and drop portions of the well bore. These radiuses can be determined from the desired build and drop rates by use of Equation 3.

Equations 1 and 2 can be rewritten as Equations 4 and 5, which represent a system of equations with two unknowns, L and 0. In general, the kickoff point, the vertical and horizontal displacements, and the radiuses of curvature in the build and drop portions of the well will be known.

Equations 4 and 5 can be solved simultaneously for L by squaring each equation, summing the resulting products, and simplifying to yield Equation 6. Equations 4 and 5 can also be solved simultaneously for 0. This solution is accomplished by multiplying Equation 4 by (r1 + r2) and Equation 5 by L, adding the resulting products, and solving for 0 to obtain Equation 7. This equation can be simplified further to Equation 8 by defining D, R, X, and L, as shown in Equations 9-12.

With the proper definitions of the variables, Equation 8 can be used to plan any type of directional well. For a Type 1 well, Equation 8 is used by setting r2 equal to 0. For a modified Type 2 well, Equation 8 must be modified. From the modified Type 2 geometry in Fig. 4 and basic geometric relationships, D4 and X4 can be determined from Equations 13 and 14. In these equations, f is the desired angle in the final hold portion of the well bore. Once D4 and X4 are determined from these equations, Equation 8 can be used to complete the planning of the modified Type 2 well.

D.9 and X5 in Equations 13 and 14 represent the vertical and horizonal distances to the beginning of the final hold portion of the well bore; this position corresponds to the end of the drop portion of the well bore and not to the vertical and horizontal displacements of the bottom hole objective.

Planning the Type 3 well is more complicated than for other directional wells because the angle of inclination constantly changes after it deviates from vertical. Although the engineer will know the desired bottom hole location, he must ensure that the build rate and kickoff point are compatible to reach the objective. To do this, one must specify the kickoff point and determine the proper build rate, or one must specify the build rate and determine the kickoff point.

If the kickoff point is specified, the radius of curvature in the build section is determined from Equation 15. Once r1 is determined, the build rate can be calculated with Equation 3. If the build rate is specified, then the kickoff point is calculated with Equation 16. Once the proper kickoff depth and build rate have been determined, the planning equation can be used to determine the maximum angle obtained at total depth. Note that r2 and L both equal 0 in a Type 3 well.

Occasionally, an engineer may know the maximum angle allowed in the hold portion of the well, and he wishes to determine the kickoff point. This calculation can be accomplished by solving Equations 1 and 2 for D1 to obtain Equation 17.

HORIZONTAL WELLS

Fig. 5 presents a typical geometry for a horizontal well that can be modified to represent any two-dimensional extended reach or nearly horizontal well. The derivation of the planning equation is very similar to that for the directional well equation.

The vertical and horizontal displacements of the well bore can be represented by D4 and X4, respectively, as shown in Equations 18 and 19. These equations can be rewritten as Equations 20 and 21.

Equations 20 and 21 can be solved simultaneously for L by summing the squares of each equation, yielding Equation 22. If Equation 20 is multiplied by (r1 - r2) and Equation 21 by L, the equations can be solved for 0. This solution is Equation 23, which can be rewritten as Equation 8 where D, R, X, and L are given by Equations 24-27.

Equation 8, with the proper definition of the variables, can be used to plan a two-dimensional horizontal well with one or two build sections. If the horizontal well will have only one build section, r2 should be set to 0.

In extended reach or near horizontal well plans, D4 and X4 no longer represent the coordinates of the end of the build section in the actual well bore; D5 and X5 are now the coordinates.

Fig. 6 represents these types of wells. With basic geometric relationships, D4 and X4 can be determined with Equations 28 and 29. The angle f is the desired angle in the final hold portion of the well bore. Once the proper D4 and X4 are determined from Equations 28 and 29, Equation 8 can be used as written.

As in the directional case, D5 and X5 represent the displacement to the beginning of the second hold portion of the well bore and not the total displacement to the bottom hole objective.

REFERENCE

Eastman Whipstock, "Introduction to Directional Drilling," Houston, pp. 4-6.
Bourgoyne, A.T. Jr., et al., Applied Drilling Engineering, SPE Textbook Series, Richardson, Tex., 1968, pp. 354-357.
Inglis, T.A., Directional Drilling, Graham & Trotman, London, 1987, pp. 47-57,
Wiggins, M.L., and Juvkam-Wold, H.C., "Simplified Equations for Planning Directional and Horizontal Wells," SPE paper 21261 presented at the 1990 SPE Eastern Regional Meeting, Columbus, Ohio, Oct. 31-Nov. 2, 1990.