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# Fluid flow with formation damage

Formation damage caused by drilling-fluid invasion, production, or injection can lead to positive skin factors and affect fluid flow by reducing permeability.

## Permeability reduction and formation damage

When mud filtrate invades the formation surrounding a borehole, it will generally remain in the formation even after the well is cased and perforated. This mud filtrate in the formation reduces the effective permeability to hydrocarbons near the wellbore. It may also cause clays in the formation to swell, reducing the absolute permeability of the formation. In addition, solid particles from the mud may enter the formation and reduce permeability at the formation face.

The production process may also reduce permeability and introduce a positive skin factor. For example, in an otherwise undersaturated oil reservoir, pressure near the well may be below the bubblepoint pressure, causing a free-gas saturation and reducing the effective permeability to oil. In a retrograde gas reservoir, the pressure near the wellbore may drop below the dewpoint and an immobile liquid phase may form and reduce the effective permeability to gas near the wellbore.

Injection can also cause damage. The water injected may be dirty; that is, it may contain fines that may plug the formation and reduce permeability. In other cases, the injected water may be incompatible with the formation water, causing solids to precipitate and plug the formation. In still other cases, the injected water may be incompatible with clays in the formation (e.g., fresh water can destabilize some clays, causing fines to migrate and plug the formation).

## Altered zone and skin effect

A two-region reservoir model (**Fig. 1**) is a convenient representation of a damaged well (and some stimulated wells with radially symmetric permeability alteration around the wellbore). In this model, the altered zone around the wellbore is assumed to have uniform permeability *k*_{s} out to a radius *r*_{s}, beyond which the formation permeability, *k*, is unaltered.

For a damaged well, the reduced permeability in the altered zone causes an additional pressure drop, Δ*p*_{s} (**Fig. 2**). The dimensionless skin factor, *s*, and the additional pressure drop across the altered zone are related by

For a well with a known skin factor, *s*, **Eq. 1** provides a method of translating the somewhat abstract dimensionless skin factor into a more concrete characterization of the practical effect of damage or stimulation.

In a two-region reservoir model, the skin factor, *s*, is related to the properties of the altered zone:

Rearrange **Eq. 2** and solve for the permeability of the altered zone:

Rearrangements of **Eq. 2** provide a second method of translating skin into a more concrete characterization of a well with altered permeability near the wellbore. If the depth of damage can be estimated for a well with a known skin factor, *s*, the permeability of the altered zone can be estimated. Even if the depth of permeability alteration, *r*_{s}, is estimated **Eq. 3** can still provide a reasonable estimate of altered zone permeability because *r*_{s} appears in a logarithmic term. Alternatively, an estimate of the permeability reduction ratio (for example, from laboratory tests on cores) can produce an estimate of the depth of damage from another rearrangement of **Eq. 2**,

## Apparent wellbore radius

A third method of translating skin to a more concrete characterization of near-well conditions is to calculate apparent or effective wellbore radius, *r*_{wa}. Apparent wellbore radius is defined as

For a stimulated well, the pressure drawdown at the wellbore is the same as it would be in a formation with unaltered permeability but with wellbore radius equal to the apparent wellbore radius. This concept has value in some simulation applications. Note that *r*_{wa} can be calculated from the actual wellbore radius and skin factor.

**Eqs. 5** and **6** are also useful to illustrate the minimum (i.e., the most-negative possible) skin factor. This minimum skin, *s*_{min}, occurs when the apparent wellbore radius is equal to the drainage radius of the well:

For a well with a circular drainage area of 40 acres for which *r*_{e} is 745 ft and a wellbore radius of 0.3 ft, the minimum skin (maximum stimulation) is *s*_{min} = - ln(*r*_{e}/*r*_{w}) = −(745/0.3) = −7.82. Such a skin implies increasing the permeability throughout the entire altered zone to infinity—clearly an idealistic "upper limit." More realistically, research^{[1]} has shown that the half-length, *L*_{f}, of a highly conductive vertical fracture is related to *r*_{wa} by

Thus, for *L*_{f} = *r*_{e} = 745 ft, *s* = −7.12 is a more realistic minimum (for the given drainage radius and wellbore radius).

## Flow efficiency

A fourth way to characterize a well with nonzero skin is to calculate the flow efficiency of the well. Flow efficiency, *E*_{f}, is defined as the ratio of the actual productivity index of the well (including skin) to the ideal productivity index if the skin factor were zero. Because the productivity index is the ratio of stabilized flow rate to pressure drop required to sustain that stabilized rate,

For a well with neither damage nor stimulation, *E*_{f} = 1; for a damaged well, *E*_{f} < 1; and for a stimulated well, *E*_{f} > 1.

## Geometric skin

When the area open to flow decreases, the pressure drop is greater than when the area is unchanged all the way to the formation face. Examples include flow converging to perforations (**Fig. 3**), partial penetration (**Fig. 4**), and an incompletely perforated interval (**Fig. 5**).

**Fig. 3** illustrates flow converging into perforations. When the perforation spacing is too large, this converging flow results in a positive skin factor. The skin increases as vertical permeability decreases and increases as shot density decreases.

### Partial penetration

**Fig. 4** illustrates flow converging into an interval that is only partly penetrated by perforations. When a well is completed in only a fraction of the productive interval, the flow must converge through a smaller area, increasing the pressure drop near the well (compared to a fully completed interval). The additional pressure drop near the well results in a more positive skin. It increases as the vertical permeability decreases and as the perforated interval as a fraction of the total interval decreases. Formation damage (reduced permeability) near the completion face can significantly increase the additional pressure drop and thus the calculated skin factor.

### Incompletely perforated interval

Partial penetration is a special case of an incompletely perforated interval (**Fig. 5**). In the general case, the well is perforated starting at a distance *h*_{1} from the top of the productive interval and has perforations extending over a distance, *h*_{p}, in an interval of total thickness, *h*. The total skin for the well in this general situation is

In **Eq. 13**, *s*_{d} is the skin caused by formation damage, and s p is the skin resulting from an incompletely perforated interval. This equation is not valid for a stimulated well.

The skin factor for an incompletely perforated interval, *s*_{p}, can be quantified by^{[2]}

where ....................(15)

The most significant limitation in applying **Eq. 14** in practice is the difficulty in estimating accurately the vertical-to-horizontal-permeability ratio, *k*_{v}/*k*_{h}. Fortunately, this ratio appears only in a logarithmic term in **Eq. 14**, so errors will not seriously distort the calculated value of *s*_{p}.

### Deviated well

For a deviated well (**Fig. 6**), which penetrates the formation at an angle other than 90°, more surface is in contact with the formation. This introduces a negative skin factor, *s*_{θ}, which makes the total skin factor, *s*, more negative.

The effect increases as the vertical permeability increases and increases as the angle from the vertical, *θ*_{w}, increases. The deviated well skin factor, *s*_{θ}, is given by a correlation of simulated results^{[3]} (valid for *θ*_{w} < 75°):

where ....................(22)

### Gravel-pack skin

When a well is gravel packed (**Fig. 7**), there is a pressure drop through the gravel pack within the perforations, given by^{[4]}

where *s*_{gp} is the skin factor because of Darcy flow through the gravel pack; *h*, the net pay thickness, ft; *k*_{gp}, the permeability of the gravel in the gravel pack, md; *k*, the reservoir permeability, md; *L*_{g}, the length of the flow path through the gravel pack, ft; *n*, the number of perforations open; and *r*_{p}, the radius of the perforation tunnel, ft. **Eq. 24** does not include the effects of non-Darcy flow, which may be extremely important in high-rate gas wells.

### Completion skin

For a perforated well, any reduced permeability, *k*_{dp}, in the zone surrounding the perforations (**Fig. 8**) introduces an additional pressure drop. The additional skin is^{[5]}

where *s*_{p} is the geometric skin from flow converging to the perforations; *s*_{d}, the damage skin; *s*_{dp}, perforation damage skin; *k*_{d}, permeability of the damaged zone around the wellbore, md; *k*_{dp}, permeability of the damaged zone around perforation tunnels, md; *k*, reservoir permeability, md; *L*_{p}, length of perforation tunnel, ft; *n*, number of perforations; *h*, formation thickness, ft; *r*_{d}, radius of the damaged zone around the wellbore, ft; *r*_{dp}, radius of the damages zone around the perforation tunnel, ft; *r*_{p}, radius of the perforation tunnel, ft; and *r*_{w}, wellbore radius, ft. **Eq. 26** does not include the effects of non-Darcy flow.

### Hydraulically fractured wells

Wells are frequently fractured hydraulically to improve their productivity, especially in low-permeability formations where fractures increase the effective drained area and in high-permeability formations where they penetrate near-well damage or promote sand control. These fractures, almost always vertical (**Fig. 9**), are high-conductivity paths between the reservoir and the wellbore. If the fracture conductivity is large enough relative to the formation permeability and fracture length, the pressure drop within the fracture will be negligible. This distributes the pressure drop caused by fluid influx into the wellbore over a much larger area, resulting in a negative skin factor, which is interpreted as a geometric skin.

Dimensionless fracture conductivity, *C*_{r}, is defined by

where *w*_{f} is the fracture length, ft; *k*_{f}, the permeability of the proppant in the fracture; *k*, the formation permeability, md; and *L*_{f}, the fracture half-length, ft. Pressure drop in the fracture is negligible for *C*_{r} > 100.

## Nomenclature

## References

- ↑ Prats, M., Hazebroek, P., and Strickler, W.R. 1962. Effect of Vertical Fractures on Reservoir Behavior--Compressible-Fluid Case. SPE J. 2 (2): 87-94. http://dx.doi.org/10.2118/98-PA
- ↑ Papatzacos, P. 1987. Approximate Partial-Penetration Pseudoskin for Infinite-Conductivity Wells. SPE Res Eng 2 (2): 227–234. SPE-13956-PA. http://dx.doi.org/10.2118/13956-PA
- ↑ Cinco, H., Miller, F.G., and Ramey, H.J. Jr.: "Unsteady-State Pressure Distribution Created by a Directionally Drilled Well," JPT (November 1975) 1392; Trans., AIME, 259. SPE-5131-PA. http://dx.doi.org/10.2118/5131-PA
- ↑ Brown, K.E. 1984. The Technology of Artificial Lift Methods, Vol. 4, 134. Tulsa, Oklahoma: PennWell.
- ↑ McLeod, H.O.J. 1983. The Effect of Perforating Conditions on Well Performance. Journal of Petroleum Technology 35 (1): 31–39. SPE-10649-PA. http://dx.doi.org/10.2118/10649-PA

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## See also

Fluid flow through permeable media

Fluid flow in naturally fractured reservoirs

Fluid flow in hydraulically fractured wells

Fluid flow in horizontal wells

PEH:Fluid_Flow_Through_Permeable_Media