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Frequently, wells are partially completed; that is, the producing height that is open to the formation is smaller than the reservoir height, and particularly when referring to horizontal wells, this is known as partial penetration. The effect of partial completion can occur inadvertently as a result of a bad

perforation job or poor gravel pack placement, or it may be done deliberately in an often misguided effort to retard or avoid gas or water coning.

In any of these cases, the ensuing alteration of the stream lines from those that occur in radial flow to a fully completed well would result in a skin effect denoted by s_c. The smaller the perforated interval, compared to the reservoir height and the less it is centered in the total formation height, the larger the skin effect would be. If the completed interval is 75% of the reservoir height or more, this skin effect becomes negligible.

While partial completion generates a positive skin effect by reducing the well exposure to the

reservoir, a deviated well has an opposite impact. The larger the deviation angle, the larger the negative contribution to the total skin effect because of the increased amount of reservoir contact by the

wellbore. The skin effect due to well deviation is denoted by s_θ.

The effect of partial completion on the productivity of vertical wells has been considered in numerous studies, beginning with the work of Muskat (1946), who derived an analytical solution for the flow to a well penetrating only partially into the upper part of a reservoir (a common completion practice in the time of Muskat’s work).

For a completion like that shown in Figure 6-4, Muskat presented the following inflow equation:

Figure 6-4. Well completed openhole in the top of the reservoir.

In this equation, h_w is the thickness of the completed interval. By comparing this equation with a steady-state inflow equation, including a skin factor (e.g., Equation 2-7), the partial completion skin factor from Muskat’s relationship is

Because of symmetry, Muskat’s partial completion inflow equation can also be used to derive a partial completion skin factor for a well completed in the center of a reservoir zone. Thus, for a well with a completed thickness of h_w in the middle of a reservoir of thickness h, the partial completion skin factor is

Several studies of partial completion effects have been presented since Muskat’s work, including those of Brons and Marting (1961), Cinco-Ley, Ramey, and Miller (1975), Strelstova-Adams (1979), Odeh (1980), and Papatzacos (1987). Because it is relatively simple, includes the effect of permeability anisotropy, allows for any arbitrary location of the completed interval in the reservoir zone, and

reproduces other partial completion models well, we present here the Papatzacos model. To describe the completion geometry shown in Figure 6-5, Papatzacos uses the following dimensionless variables:

Figure 6-5. Partially completed well configuration.

and

In terms of these dimensionless variables, the partial completion skin factor is

where

and

The effect of wellbore deviation through the producing reservoir has also been studied frequently. A deviated well has higher productivity than a vertical well through the same reservoir because of the longer length of wellbore in contact with the formation in the deviated well case. Thus, a skin factor accounting for this effect will always be negative. Besson (1990) presented analytical equations for the deviated well skin effect for both isotropic and anisotropic reservoirs for a wellbore deviated at an angle θ from the vertical. For the isotropic case,

For the anisotropic case,

where

This correlation for the isotropic case reproduces the work of Cinco-Ley et al. (1975).

For a partially completed, deviated well, we use the Papatzacos correlation to calculate the partial completion skin factor, but using the true vertical thickness of the completed interval for h_w, not the length of the completed interval measured along the wellbore. The Besson equation applies for the deviated well skin for such a well.

Partial completion effects (often called partial penetration) are a very important aspect of horizontal well inflow behavior, and have been discussed in Chapter 5.

Example 6-4. Partial Completion and Well Deviation Skin Effect

Two factors that are likely to have important influences on partial completion skin effects are the

reservoir anisotropy and the location of the completed interval in the reservoir, because the convergence of flow to the completed interval in the vertical direction is the cause of the positive partial completion skin. Consider a vertical well with a radius of 0.25 ft completed along 20 ft of the wellbore in a 100-ft thick reservoir. Using the Papatzacos partial completion skin model, show how the partial completion skin factor depends on the anisotropy ratio and on the location of the completed interval.

Repeat this exercise using the Besson correlation for the well deviation skin factor for wells completed in 20 vertical ft of the reservoir with deviations of 30 and 60 degrees from the vertical.

Solution

First, we calculate the dimensionless size and position of the completion:

For the completion being at the top of the reservoir,

while for a completion in the middle of the reservoir,

The dimensionless radius, r_D, is

Now, for the case of the completion in the middle of the reservoir, for example (h_1D = 0.4),

and

The results of this calculation for I_ani, ranging from 1 to 20, and for the completion being at the top of the reservoir and in the middle are shown in Figure 6-6. The partial completion skin factor depends strongly on the fraction of the reservoir thickness completed and strongly on reservoir anisotropy. The vertical location of the completion has a smaller effect.

Figure 6-6. The effect of anisotropy on the partial completion skin factor.

For the deviated well skin calculation, using Equations 6-25 and 6-24, for the 30° deviation case,

while for the 60° case,

Using these equations, the variation of deviated well skin with anisotropy ratio is generated (Figure 6-7).

Figure 6-7. Dependence of deviated well skin factor on reservoir anisotropy.