Well Performance
Two interrelated components are key to understanding flow regimes – the potential for injection into the reservoir (reservoir performance) and the restrictions imposed by the completion and wellbore system hydraulics.
Reservoir Performance
Designing a successful injector begins with careful consideration of the target formations themselves. Consider the simplest flow situation – a nonfractured reservoir, single phase, injecting/producing under pseudosteadystate, radial flow conditions. Darcy’s Law indicates that:
_{} 
(1) 
where:
q 
= 
flow rate (STB/D or Mscf/D at 14.7 psia and 60°F, 1Mscf/D=1000 scf/D), q is negative for injection 
k 
= 
permeability (md) 
h 
= 
net thickness (feet) 
p_{wf} 
= 
flowing bottomhole pressure (psi) 
_{} 
= 
static (average) reservoir pressure (psi) 
p_{i} 
= 
initial pressure (often assumed to be equal to _{} (psi))_{} 
B_{o} 
= 
oil formation volume factor, which converts reservoir barrels to stock tank 


barrels (RB/STB) 
r_{e} 
= 
drainage radius (feet) 
r_{w} 
= 
wellbore radius (feet) 
s 
= 
skin (dimensionless) 
D 
= 
turbulent flow factor 
µ 
= 
viscosity (cP) 
_{} 
= 
average compressibility factor (dimensionless) 
_{} 
= 
average temperature (°R, °R = °F + 459.67) 
Consider an example. You are planning an injection operation in an aquifer that is (or will be) into a twentyfoot thick zone with a permeability of 1000 md. The average reservoir pressure is 2000 psi. Suppose that the bottomhole pressure during injection is 2100 psi, the viscosity is 1 cP, the formation volume factor is 1.0, the porosity is 20%, the total compressibility is 1.0 x 10^{5} psi^{1}, and the well spacing is 20 acres. The drainage radius is usually estimated from the well spacing. You can inscribe a circle of radius 467 feet into a square that has an area of 20 acres. One acre is 43,560 ft^{2}. Twenty acres is 8.712 x 10^{5 }ft^{2}. The inscribed circle will have a radius of _{}. Assume that there is no skin (damage) or turbulence and that the well is drilled to 6 inches in diameter. What sort of flow rates should you anticipate?
_{}
Why do you need to know these flow rate equations? It is essential to be able to anticipate flow rates under various bottomhole conditions that can arise during the injection operations. Looking at these rates helps to determine if the well has been fractured, the degree of damage, etc.
Estimating rates is complicated by the fact that injection is transient (is not constant) up to the time where the well reaches pseudosteady state or steadystate conditions. Unlike production situations, steadystate behavior is a real possibility with injection wells.
Infiniteacting flow regime: At early times wells do not sense reservoir boundaries. Flow during this period is transient.
Pseudosteadystate flow regime: Infiniteacting flow ends. After a transition period, a well in a closed system feels the effects of the boundaries of the reservoir. After this transition period, pseudosteadystate flow develops. The pressure changes linearly with time. The change in pressure, dp/dt, is constant at all points in the reservoir. This flow regime occurs in all closed reservoirs.
Steadystate flow regime. Steadysate flow occurs when the pressure at every point in a system does not vary with time. This situation can only occur in reservoirs that are completely recharged by a strong aquifer or where injection and production are balanced (i.e., in certain flooding and injection programs). If there are situations where steadystate flow exists, the relationships are only slightly different from those for pseudosteadystate flow (the 0.75 in Equation 1 is replaced by 0.50).
It may be necessary to estimate the injection rates before pseudosteady state conditions develop. It is possible to estimate the time to pseudosteadystate production and the flow conditions during transient behavior. For an injected fluid identical to the reservoir fluid, the time needed to reach pseudosteadystate can be estimated as:
_{} 
(2) 
where:
t_{DApss} 
= 
function of the reservoir shape (t_{DApss} = 0.1 for a well at the center of a bounded circular reservoir; (see Earlougher, 1977, Table C1) 
A 
= 
area, for a circular reservoir, A = pr_{e}^{2 }(feet^{2}) 
t_{pss} 
= 
time to pseudosteadystate flow (hours) 
k 
= 
permeability (md) 
f 
= 
porosity (fraction) 
c_{t} 
= 
total system compressibility (psi^{1}) 
Prior to pseudosteadystate, for infiniteacting radial flow, the injection can be estimated from the following equation (ignoring turbulence):
_{} 

where:
t 
= 
time (hours) 
Using the above equations for a circular reservoir, the approximate time for pseudosteady state flow to develop is given by:
_{} 
(4) 
Before pseudosteadystate (Equation 2 is an estimate of the time needed to reach a pseudosteadystate regime), estimate flow conditions using Equation 3. After pseudosteadystate conditions have developed, estimate flow using Equation 1 or its steadystate equivalent.
RECAP (ignoring turbulence)
Flow Regime 
Description 
Liquid 
Gas 
Infinite Acting Radial Flow IAR 
Occurring at early times and boundaries are not "felt." 
_{} 
_{} 
t_{pss} 
Time for pseudosteadystate flow to develop. 
_{} _{} 
_{} _{} 
PseudoSteadyState 
IAR flow has ended and the change in pressure with time, dp/dt, is constant everywhere in the reservoir. 
_{} 
_{} 
SteadyState 
This regime only occurs when there is pressure support at the outer boundary _{}. 
_{} 
_{} 
_{}
Injection Performance Relationships – Single Phase
As you can see in the previous equations, pressure change in the reservoir and injection are related by Darcy’s Law. The injection into a reservoir, as a function of the reservoir parameters and the pressure differential can be characterized over a range of pressure differences (difference between the bottomhole pressure and the reservoir pressure). In its simplest form, this is a straight line (assuming single phase flow and using Darcy’s Law for steadystate conditions), until fracturing occurs, as shown in Figure 1.
Figure 1. The performance relationship for a characteristic well for single phase, laminar flow of a slightly compressible liquid. This example is for production. Injection is analogous.
By assuming pseudosteadystate or steadystate radial flow, the performance curve for an injector above the bubble point (the pressure is high enough that gas is in solution) can be developed from (same as Equation 1):
_{} 
(5) 
Figure 1 represents a well with water injection. Similar relationships are available for gas.
The curve shown (Figures 1) does not reflect restrictions imposed by the completion itself and any tubulars or damage/stimulation (skin). These curves provide a starting point for determining the flow and pressure conditions that you will encounter during your injection operations.
The Effect of Skin on Performance
You can see from the basic equations that performance relationships are affected by skin – damage or flow restriction or flow improvement. According to Earlougher, “There are several ways to quantify damage or improvement in operating (producing or injecting) wells. A favored method represents the wellbore condition by a steadystate pressure drop at the wellface in addition to the normal pressure drop in the reservoir. The additional pressure drop, called the “skin effect,” occurs in an infinitesimally thin “skin zone” [for mathematical purposes even though damage may extend deep into the formation]. In the flow equation …, the degree of damage (or improvement) is expressed in terms of a “skin factor" (s) which is positive for damage and negative for improvement. It can vary from about –5 for a hydraulically fractured well to + ¥ for a well that is too badly damaged to produce” (Earlougher, 1977).
The skin factor itself is dimensionless, but you can have an indication of the supplementary pressure drop caused by skin (relative to an undamaged well) for radial, nonturbulent liquid flow using:
_{} 
(6) 
where:
Dp_{s} 
= 
pressure drop due to skin (psi) 
It is important to consider the skin factor that the completion will develop, how much skin can be alleviated by manipulating the pressure during completion, and in the longer term, deciding when a well will need to be worked over or abandoned. Alternatively to Equation 6, you can calculate the skin by specifying a finite depth of damage in the formation using:
_{} 
(7) 
where:
k_{s} 
= 
permeability of the damaged zone (md) 
r_{s} 
= 
radial extent of the damaged zone (feet) 
Unfortunately, the depth of damage and the damaged zone permeability are hard to infer. You can also assess the influence of skin by considering skin in terms of an apparent (or effective) wellbore radius:
_{} 
(8) 
where:
r_{wa} = apparent wellbore radius. Positive skin implies that the well has been damaged and the apparent wellbore radius is reduced, and vice versa for a stimulated well (feet).
This relationship implies a reduced apparent wellbore radius (and consequently less production) for positive skin. Figure 2 shows skin schematically.
Figure 2. Schematic representation of the variation of skin as a pressure drop through the reservoir, for a particular flow rate. With positive skin more pressure is required to achieve a particular rate. Note that this shows the “real” variation of damage/improvement away from the well. Some of the mathematical treatments represent this pressure change as occurring in an infinitely thin zone at the wellbore. This figure is for production. Similar figures are possible for injection.
Skin can develop in a number of different ways, some depending on the damage done to the formation during drilling and/or completion, some due to the completion itself, and some even due strictly to the inclination of the well. The total skin is the combination of mechanical and pseudoskins (total skin is determined from well testing analysis).
Mechanical skin is associated with formation damage due to drilling fluids, cementing, etc. Mechanical skin is mathematically defined as an infinitely thin zone that creates a steadystate pressure drop at the formation face. A primary goal is to minimize turbulence, multiphase effects, perforation losses and restrictions due to completion hardware.
The total skin can be considered as a summation of skins from various mechanisms, some of which can be impacted by underbalanced operations. For example:
_{} 
(9) 
where:
s 
= 
total skin (all “skins” are dimensionless) 
s_{M} 
= 
mechanical skin (due to damage) 
s_{PP} 
= 
partial penetration skin (only part of a zone is completed) 
s_{Q} 
= 
ratedependent skin (due to turbulence) 
s_{PERF} 
= 
skin associated with perforating or completing 
Mechanical Skin (Drilling/Completion Damage)
These skins develop during drilling, workover or injection/production and are associated with the formation near the wellbore. Undamaged, naturally fractured reservoirs can have a slightly negative skin (anything beyond –3 is probably induced). These reservoirs are extremely susceptible to drilling and completion damage.
Consider the situation shown in Figure 4, where concentric radial damage has been created by drilling and/or completing overbalanced. How do you characterize the skin?
Ø Let’s suppose that we are dealing with permeable flow alone and that there is no damage or improvement around the well.
Ø The mechanical skin for such a case is 0.
Ø The only pressure drop is due to flow through the porous medium. Denote this as _{}.
Ø If there is skin present, the bottomhole injection pressure, for the same rate as the undamaged case, will be lower or higher (stimulated or damaged, respectively). Let’s denote this as p_{wfs}.
Ø When there is skin, the pressure drop from the undamaged part of the formation (through the "damaged" zone) into the wellbore would be _{}.
Ø You can see that the difference between Dp_{k}_{ }and Dp_{ks} indicates the pressure drop due to mechanical skin, giving Dp_{s }= Dp_{ks}  Dp_{k}_{ }
Ø If you use Equation (6) you can estimate the skin as (laminar):
_{}
Ø If you know, or can estimate, r_{s}, you can estimate the permeability in the damaged zone by using Dp_{s }in a steadystate version of Darcy’s law:
_{} 
(10) 
Ø If you know, or can estimate, r_{s }, Equation 7 can be used.
_{}
Ø
Finally, you can consider an equivalent wellbore
radius, using Equation 8, r_{wa }= r_{w}^{s}
Partial Penetration
Partial penetration or partial completion skin is associated with drilling through only part, or completing only part, of a formation. Though it may be most important during early time, it can be an issue through the lifetime of a reservoir. This skin depends on the deviation of the well through the target zone, the amount of formation penetrated during drilling and/or the amount of formation open to the wellbore.
RateDependent Skin
Ratedependent skin is associated with turbulent flow. Turbulent flow can be important in highrate wells – particularly for gas but also oil in some high rate situations. If this is the case, the flow equation is modified to represent an additional pressure drop. To account for turbulence in the previously developed relationships (see Equation 1):
For single phase liquid flow: _{} 
(11) 
where:
r_{o} 
= 
oil density at reservoir temperature and average pressure_{ .} 
For additional reading, refer to:
Jones, L.G., Blount, E.M., and Glaze, O. H.: “Use of Short Term Multiple Rate Tests to Predict Performance of Wells Having Turbulence,” SPE 6133, SPE 51^{st} Annual Fall Meeting, New Orleans, LA (1976). 
Figure 3. Schematic representation of mechanical skin in a generic reservoir
Perforation/Completion Friction
Perforation friction can also impair injection. Perforation friction  pressure drop due to flowing through the perforated, or other, completion  can definitely be reduced by appropriate underbalanced completion, in addition to an intelligent perforating program. According to Suman, “Perforating is always a cause of additional damage in formation rocks.”^{4} Whether it is perforated overbalanced or underbalanced, perforating compacts the formation around the perforations. This compacted zone probably has an average thickness of 0.5 inches and the permeability reduction averages 80%.
While compaction damage during perforating is anticipated
(regardless of the balance conditions), underbalanced perforating offers
significant advantages for removal of gun debris and damaged material and may
reduce the extent of the damaged zone. You can estimate pressure drop
associated with a perforated completion and infer the corresponding skin factor
using the methods described in Completion
Hydraulics.
Typically, pressure drop and skin estimates are based on the premise that all guns fired properly and that all perforations are open. You can infer the damage due to alone by measuring the total skin with some form of well testing, isolating the skin due to turbulence and then subtracting the skin due to perforating, partial penetration and wellbore geometry. Perforation friction is often not adequately considered. Though it is often rather small, there can be important exceptions, and you should include it in your completions' design.
For now, consider the implications of the relationship shown in Figure 4. This is the most basic perforation design plot, highlighting the role of spacing. The trends are as you would expect – greater perforation density generally leads to higher rates. Design methodologies are available for spf, phasing, charge size, gun characteristics and balance. One of the more difficult characteristics of a perforating program is to estimate how much the nearperforation permeability is damaged and to what extent. In a well consolidated formation, the damaged depth can be 0.5 inches and permeability may be reduced to 10 to 25% of the virgin formation permeability (McLeod, 1983).
Performance relationships are impacted by skin. It is important to minimize formation damage during all phases of drilling, completion, and injection. For example, Figure 1 is based on no skin, and Figure 5 shows the influence of skin on performance.
Figure 4. Schematic relationship showing the variation of total inflow with perforation density. This curve is schematic. The particular curve shape depends on the perforation and formation specifics and the degree of underbalance during shooting.
Figure 5. Example IPR curves for an oil well (ignoring dissolved gas for now) with different values of total skin. Even modest skins can seriously impact productivity and overall economics (i.e., the bottomhole flowing pressure must be lower to achieve the same rate as the skin increases). This example is for production.
Gas Flow and Turbulence
Flow mechanics in the reservoir, in the completions, and in the production string are greatly complicated when turbulent flow occurs. For a dry gas or for a situation with a GOR > 30,000, recall that pseudosteadystate gas flow can be expressed as:
_{} 
(12) 
where:
q_{g} 
= 
gas flow rate (Mscf/D), 
k 
= 
permeability (md), 
h 
= 
thickness (feet), 
= 
average temperature (°R, °R = °F + 459.67), and, 

= 
average compressibility factor (dimensionless) (Average viscosity should also be considered.) 
This equation can be written as:
_{} 
(13) 
It is helpful to separate pressure drop due to turbulence from skin arising from other sources so that you can assess the specifics of the completion. The quadratic term on the right hand side of Equation 13 accounts for nonDarcy, turbulent effects. This relationship can be rearranged by dividing both sides by the flow rate:
_{} 
(14) 
This equation is sometimes a more convenient form for visualization because it can be expressed as a straight line (as shown in Figure 8), and the term on the lefthand side of this equation can be considered as a reciprocal injectivity or productivity index.
Figure 6. Variation of injectivity/productivity index for gas as a function of the gas flow rate.
This type of plot is useful for identifying the presence of turbulence. If this plot has a positive slope (as shown in Figure 6), turbulence is indicated. Figure 7 shows how this type of plot can be useful for discriminating between turbulence and other types of skin. In Figure 7, relationships are shown for eight generic well situations in a hypothetical field – different wells are indicated by the numbers on the plot. Wells 1 and 5 show no turbulent effects because b = 0 (slope is zero). Wells 2 through 4 and 6 through 8 show increasing levels of turbulence. Why are the two sets of curves different? There is greater skin (not associated with turbulence) for Wells 5 through 8.
Figure 7. Variation of injectivity/productivity index for gas as a function of the gas flow rate, with varying degrees of turbulence and skin due to damage.
Evaluating the skin due to turbulence helps you to determine whether the pressure drop is intrinsic to the formation and/or the completion and cannot be overcome without stimulation. You can use simple diagnostic plots to separate skin caused by turbulent flow from skin caused by damage and, to a certain extent, skin caused by the completion itself. Similarly, you can use simple methods to design the most effective completion (beforehand) and to assess its effectiveness or the need for remedial measures (after the fact).
You can prepare the same types of plots for oil to separate turbulence from other skins (turbulence is less common when oil is produced, but it can occur). Recall that the slope b in Figure 7 is indicative of the nature of the skin in the well. If b is high, it indicates high turbulence. A large value of the intercept a implies that the skin is high (not associated with turbulence) or that permeability is low.
If the well has been fractured, you must consider other factors.
For additional reading on this subject refer to: Beggs, H.D.: Production Optimization Using Nodal ^{TM} Analysis, OGCI Publications, Oil & Gas Consultants International Inc., Tulsa, OK (1991). 
Horizontal Wells
The relationships presented so far are for straight hole. Various relationships have been extended to approximate flow in horizontal wells. Be careful. Analytical relationships may be inadequate (A. Settari, personal communication). Some of the analysis techniques are summarized below.
· Using numerical simulation, Bendakhlia and Aziz (1989) suggested that:
_{} 
(15) 
where:
V 
= 
variable parameter in a modified Vogel’s equation 
n 
= 
exponent in Fetkovich’s equation 
To use this, recognize that there are three unknowns (q_{max, }V_{, }and n). At least three stabilized flow measurements are required to build the IPR.
· For pseudosteadystate flow, Joshi (1991) presented an approximate formulation:
_{} 
(16) 
where:
J_{H} 
= 
Productivity/Injectivity Index (PI) for a horizontal well section (BOPD/psi) 
h 
= 
true vertical thickness of a zone (feet) 
r_{eH} 
= 
drainage radius (feet) 
k_{h} 
= 
horizontal permeability (md) 
k_{v} 
= 
vertical permeability (md) 
L 
= 
horizontal length (feet) 
r_{w} 
= 
wellbore radius (feet) 
b 
= 
(k_{h}/k_{v})^{0.5} 
Thomas et al. (1996) showed calculations of nearwellbore skin and the nonDarcy flow coefficient (turbulence) for horizontal wells for various drilling and completion scenarios. They provided analytical and numerical solutions for nearwellbore skin and nonDarcy flow for horizontals drilled underbalanced or overbalanced with various completion options. In particular, they explored the effects of drilling overbalanced versus underbalanced and completing openhole with or without a slotted liner or completing cased hole. This reference is recommended.
According to Thomas et al., “Traditional formulations of wellbore skin assume radial flow into a vertical wellbore, and must be transformed to apply to horizontal wells.” In these formulations, adopt the convention that the horizontal wellbore is oriented with the xaxis and its length is L. k is the geometric mean of the two permeability components perpendicular to the direction of the well. “Implementation of these terms [skin terms] into different wellbore models results in different multiplying factors on the skin and nonDarcy flow terms depending upon how a particular well model was derived. This approach is different than the treatment for vertical wells (Thomas et al., 1992) where the effect of partial penetration on nearwellbore skin is included with the s and D terms.”
Available formulations, by various different authors, are shown below. Thomas argued that there were basically no significant predictive differences between any of the models.
Mutalik, et al., (1988) developed a relationship by using the solution for a fully penetrating infinite conductivity fracture. For pseudosteady state flow into a horizontal well:
(17) 
where:
s_{f} 
= 
skin for a fully penetrating vertical fracture (dimensionless) 
c’ 
= 
horizontal shape factor in this model (dimensionless) 
r_{e}’ 
= 
effective drainage radius (feet) 
b 
= 
a turbulence factor 
s 
= 
mechanical skin (dimensionless) 
k_{x} 
= 
horizontal permeability in the direction of the well (md) 
k_{y} 
= 
horizontal permeability normal to the direction of the well (md) 
k_{z} 
= 
is the vertical reservoir permeability (md) 
L 
= 
is the length of the completed interval (feet) 
s_{Ca,h} 
= 
is a skin associated with the drainage area’s shape (dimensionless) 
Babu and Odeh (1989) developed an equation from the classical vertical solution turned on its side, accounting for the resulting geometry. Thomas et al. (1996) added mechanical skin to this relationship to give the following equation:
(18) 
where:
s_{R} 
= 
partial penetration skin in the Babu and Odeh model (dimensionless) 
x_{e} 
= 
halflength of the drainage area in the xdirection (feet) 
A_{l} 
= 
horizontal well drainage area (ft^{2}) 
C_{H} 
= 
horizontal shape factor in the Babu and Odeh model 
Economides et al. (1994) used a semianalytical technique in which an instantaneous point source analytical solution was integrated numerically in time and space to give constant flux solutions for a horizontal well located anywhere in the drainage volume of a uniformly heterogeneous reservoir with threedimensional permeability anisotropy. They used a horizontal shape factor from Besson (1990). These solutions are used to calculate constant pressure solutions for pseudosteadystate:
(19) 
where:
k 
= 
absolute permeability (md) 
x_{e} 
= 
drainage length in the xdirection (this is the well direction)(feet) 
s_{x} 
= 
vertical skin effect (Kuchuk, 1995) (dimensionless) 
s_{e} 
= 
term to account for vertical eccentricity (dimensionless) 
s_{ew} 
= 
distance of the well from the center of the reservoir (feet) 
C_{h} 
= 
a horizontal shape factor in the Economides et al. model (dimensionless) 
There are certainly other analytical formulations. For additional information, the reader can review Goode and Kuchuk, or Kuppe and Settari, 1996.
Analytical and even numerical prediction of the behavior in horizontal wells can be quite difficult. As Teichrob et al. (1998) state: “Without some knowledge regarding inflow distribution and magnitude, accurately predicting bottomhole pressure becomes problematic at best and impossible at worst. At best, design considerations must focus on ‘dead’ or nonproductive holes, then attempt to bracket several flowing conditions recognizing differing points of inflow and magnitude.” A reasonable integrated approach can use information acquired during underbalanced drilling. According to Butler et al. (1996).^{ } “For a number of years we have postulated that sweet spots within a lateral section of a reservoir could be identified (in real time while drilling) through superposition of elemental inflow performance relationships. Using 50 meters as [an] elemental wellbore length, the concept is as follows: element #1 is drilled and inflow is measured and correlated to a given pressure drop. Injection ratios are changed, while still honoring hole cleaning and bottomhole pressure constraints. New inflow rates are measured and correlated to a different pressure drop. An inflow relationship is therefore developed for element #1, which describes inflow over a range of differing pressures. A new element is drilled. Because we have a relationship that describes inflow over a range of pressure drawdown for element #1, we are able to reconcile what the contribution of that previous element should be based on the new pressure drops across the first element. Overall, flow to surface can be measured and will be the sum of predicted inflow from element #1 plus inflow from the newly drilled element.”
Multilaterals
Following horizontal wells, the next most recent trend in reservoir exploitation has been to drill multilaterals. Some inflow performance procedures have been formulated for multilaterals and branched wells (for example, Larsen, 1996). These can be useful in determining if underbalanced completion is merited.
Larsen used pseudoradial skin factors for systems of connected wells, intervals or fractures with a common bottomhole pressure or potential. The approach assumes a certain distance between the midpoints of the well elements (e.g. branches, drainholes, fractures, the well, or multiple wells) and then assumes that the pressure development only depends on the distance. Larsen stated, “Even relatively close to a horizontal well, it is, for instance, possible to estimate the pressure development with negligible error by using a fully penetrating vertical line well. Moreover, since the linesource solution can be approximated with a simple logarithmic expression at sufficiently later times, it is possible to combine the effect of, for instance, several wells or branches by elementary methods.”
Larsen argued that you can apply the basic pseudosteady state equations (e.g. Equation 2 for a circular, bounded drainage area) to any well configuration with some minimal distance between inner and outer boundaries of the system. For cases with significant boundary effects occurring before pseudosteadystate flow (e.g., for a horizontal well approaching the length of the reservoir in the direction of the well), you should use a direct solution rather than analytical approximations. Regardless, for situations where the equivalent wellbore radius remains within the drainage area, indirect methods are acceptable.
By considering symmetrical characteristics in the well system, Larsen implemented the basic analytical relationships for wells with long horizontal branches in bounded reservoirs. For regular patterns of branches or wells treated as a single drainage system, Larsen suggested subdividing and adding the partial IIs. For n_{w} wells:
_{} 

According to Larsen, “For wells with multiple branches, the pressure transient behavior can be obtained by spatial superposition of solutions for individual branches. The late time difference from the solution of a fully penetrating vertical well can then be used to determine the pseudoradial system skin factor … of the given well configuration.” This analysis is complicated for layered or heterogeneous reservoirs and at this time it should consider only single layered systems with horizontal isotropy.
Other analytical formulations for evaluating transient behavior in multiple laterals are available in the literature (for example, Ozkan et al., 1997).
Obviously, these situations are so complex that simulation is recommended. However, for backoftheenvelope evaluations, methodologies like this can be used for determining productivity, for building composite performance curves, and for evaluating the contribution of mechanical skin (which may be alleviated by underbalanced operations, if economically appropriate).
Transient Performance Curves
Let’s take a step back. We have briefly summarized some of the issues for relationships between pseudosteadystate or steadystate flow. The complexities for pseudosteadystate flow in horizontal wells have been indicated. You can see that developing performance curves for multilaterals during pseudosteadystate flow is even more complicated.
Most of the relationships shown to this point have been for pseudosteadystate or steadystate flow. When a reservoir is produced without voidage replacement, the average reservoir pressure is reduced (i.e., transient pressure behavior) and the inflows performance changes. The performance can also change with time even if the average reservoir pressure does not change significantly  the flow behavior is transient. During underbalanced drilling and completion, flow usually is transient, but you can develop IPR curves for any time in the productive life of a reservoir, as shown in Figure 8.
Wells initially start out under infinite acting radial flow (unless the perforations or completed zone are very restrictive – spherical flow). As the radius of influence approaches the boundaries, the flow regime changes until pseudosteadystate is reached. Further behavior depends on the type of boundary (e.g., no flow, constant pressure, mixed, etc.). Regardless, most drilling and completion scenarios can probably be reasonably represented as infiniteacting radial flow. This flow condition in itself is transient.
Equation 3 showed relationships for how flow and/or pressure change during this transient period. These are:
_{} 
(21) 
Figure 8. Typical transient performance curves for a reduction in average reservoir pressure. You can see that pressurerate relationships change over the producing time. There are implications. For example, behavior during a workover intervention may be different than during the initial completion. This example is for production.
Regardless of which technique you choose, you can develop IPR curves for various values of drawdown, depletion, and time. Remember that m, B, c_{t, } and _{} change with reservoir pressure and drawdown.
Summary
· Most injection operations, but not all, will be in pseudosteadystate or steadystate regime. This is all premised on the injection fluid properties being the same as the reservoir fluid properties.
· If your reservoir is high permeability the period for infinite acting radial flow (IAR) may be short.
· As soon as the pressure disturbance caused by flow from the well reaches the outer boundary, a transition period starts from infinite acting radial flow, going to a pseudosteadystate period, or to steadystate.
· The pseudosteadystate period occurs if a noflow boundary farthest from the well is reached by the pressure disturbance and the entire area of influence is affected by injection. Rate and wellbore pressure tend to stabilize.
· When does pseudosteadystate behavior start? As was indicated in Equation 2:
_{} 
(22) 
· Can I develop performance curves for infiniteacting conditions as a series of ratepressure plots just using Equation 21 and inputting different values of p_{wf}? You can only do this to provide rough indications of behavior because the transient effects are not properly accounted for when the rate changes.
Recommended Reference: Golan, M., and Whitson, C.H.: Well Performance, Second Edition, Prentice Hall, Englewood Cliffs, NJ (1991). 
What Happens if the Reservoir is Layered?
In many cases, the reservoir encountered may not be homogeneous. Consider the simple situation shown in Figure 9. The permeability can be approximated as:
_{} 
(23) 
Figure 9. Schematic
of a layered reservoir.
where:
_{} 
= 
average permeability (md) 
h 
= 
gross thickness (feet) 
i 
= 
subscript denoting an individual layer 
The situation is dramatically more complicated if there is variable water or gas cut, different rates of depletion in different zones, and crossflow. This is not to say that there are not some possible approximations. Consider commingled injection (no vertical, inreservoir crossflow) of two zones. If the reservoir pressure in Zone 2 is less than that in Zone 1, net production will not occur until the producing pressure, p_{wf } is less than a critical value, p*_{wf}. Above this pressure, fluid just goes from one zone to the other. This “critical” pressure is:
_{} 
(24) 
For this basic commingled case, you are able to construct the performance curve for each zone and add the predicted rates at various pressures less than p*_{wf}. Somehow, however, you have to know the properties of each zone (core analysis, logging, production logging, PTA…) from this well or from offsets.
The previous sections provided a brief introduction to some of the reservoir engineering considerations that are essential to effectively and safely design any injection. Note that this only covers matrix injection without a front. But the performance of the reservoir is only part of the whole story. The other half of the story is that flow to the surface can be restricted by the specific completion, the string in the hole, and the surface facilities. This relates to the hydraulic behavior of the completion itself and the tubing. Completion hydraulics are described first and then tubingrelated hydraulics are briefly covered.
Completion Hydraulics
After evaluating the pressure profile in the reservoir, the next step in estimating potential well performance is to estimate the pressure drop through the specific completion. The scenarios can range from barefoot to gravel pack completions. You can assess barefoot completions using the methods already described. This section covers both gravel pack and perforated completions. Gravel pack completions are described for completeness and for the possibility that they can be components in underbalanced interventions. You can also use these examples to infer skins associated with slotted liners, screens, etc.
CasedHole Gravel Pack Completion
The brief background in this section will help you in assessing the comparative merits of this type of completion and in estimating skin. You can also modify the basic concepts for situations where other sand exclusion hardware is run (e.g., slotted liners, expandable sand screens, etc.). For example, Kaiser et al, 2000 provide data on slotted liners and Asadi and Penny, 2000 describe pressure drop through screens.
In its simplest terms, the pressure drop that occurs across a gravel pack completion is related to the gravel permeability (and damage), the perforation density, length and damage. You can estimate the pressure drop (and ultimately convert it to a skin, if appropriate) using the following equations:
For Oil or Water:
_{} 
(25) 
where:
p_{wfs} 
= 
sandface flowing pressure determined from the IPR (psia) 
p_{wf} 
= 
bottomhole flowing pressure (psia) 
q 
= 
flowrate (BLPD) 
m 
= 
viscosity (cP) 
B 
= 
formation volume factor (RB/STB) 
L_{perf} 
= 
distance from the outer screen diameter to the tip of the perforation (feet) 
k_{gravel} 
= 
permeability of the gravel (md) (see Table 1) 
A_{perf} 
= 
available area for access to the formation (total of the nominal perforation crosssectional areas through the perforated interval) (feet^{2}) 
b_{gravel} 
= 
a turbulence term for the gravel 
r 
= 
liquid density (lbm/ft^{3}) 
h_{perf} 
= 
net perforated interval (feet) 
n_{perf} 
= 
shot density (spf) 
d_{perf} 
= 
perforation diameter (inches) (see Table 2) 
d_{screen} 
= 
screen diameter (inches) (see Table 3) 
Table 1. Typical Gravel
Permeability
Gravel Size 
Average Porosity 
Average Permeability 
4060 
39.8 
69 
2040 
40.9 
171 
1020 
40.5 
652 
812 
41.5 
1969 
610 
42.0 
2703 
Note: These are
representative numbers. Confirm all values with the appropriate service
companies. This is actually quite
important. Three different sources gave three different permeability ranges.
The values shown here can be used for default calculations. They are from
Dowell Schlumberger’s Well Analysis Manual, circa 1986. 
Table 2. Typical Perforation Information
Gun Size 
Casing Size 
Average Perforation Diameter 
Penetration 


Average 
Longest 

Through Tubing Retrievable 

1_{} 
4_{} casing 
0.21 
3.0 
3.3 
1_{} 
5_{} casing 
0.24 
4.7 
5.4 
1_{} 
4_{} to 5_{} casing 
0.24 
4.8 
5.5 
2 
4_{} to 5_{} casing 
0.32 
6.5 
8.1 
2_{} 
2_{} tubing to 4_{} casing 
0.33 
7.2 
8.1 
2_{} 
4_{} casing 
0.36 
10.3 
10.3 
Through Tubing Expendable 

1_{} 
4_{} casing 
0.19 
3.1 
3.1 
1_{} 
2_{} tubing?? 
0.30 
3.9 
3.9 
1_{} 
2_{} tubing to 5_{} casing 
0.34 
6.0 
8.1 
2_{} 
5_{} to 7 casing 
0.42 
8.2 
8.6 
2_{} 
2_{} tubing 
0.39 
7.7 
8..6 
Retrievable Casing Guns 

2_{} 
4_{} casing 
0.38 
10.5 
10.5 
2_{} 
4_{} casing 
0.37 
10.6 
10.6 
3_{} 
4_{} casing 
0.42 
8.6 
11.1 
3_{} 
4_{} casing 
0.36 
9.1 
10.8 
3_{} 
4_{} and 5_{} casing 
0.39 
8.9 
12.8 
4 
5_{} to 9_{} casing 
0.51 
10.6 
13.5 
5 
6_{} to 9_{} casing 
0.73 
12.3 
13.6 
NOTE: These are representative numbers only. Confirm all values with the appropriate service companies. In fact, many of the perforating companies now have calculation programs for approximating diameter and depth of penetration. This table should be used as a fallback resource only. 
Table 3. Common Screen
Diameters for Inside Casing Gravel Packs
Casing 
Maximum Screen Diameter 
Commonly Used Screen Diameter* 

OD 
Weight 
ID 
Pipe OD 
Wire OD 
Pipe OD 
Wire OD 
4 
9.5 
3.548 
1 
1.815 
1 
1.815 
4_{} 
11.6 
4.000 
1_{} 
2.160 
1_{} 
2.160 
5 
18.0 
4.276 
1_{} 
2.400 
1_{} 
2.400 
5_{} 
17.0 
4.892 
2_{} 
2.875 
2_{} 
2.875 
6_{} 
24.0 
5.921 
3_{} 
4.000 
2_{} 
3.375 
7 
29.0 
6.184 
3_{} 
4.000 
2_{} 
3.375 
7_{} 
33.7 
6.765 
4 
4.500 
2_{} 
3.375 
8_{} 
36.0 
7.825 
5 
5.500 
2_{} 
3.375 
9_{} 
47.0 
8.681 
5_{} 
6.000 
2_{} 
3.375 
NOTE: These are representative numbers. Confirm all values with the appropriate service companies. This table should be used as a fallback resource only. *Different screen diameters may be required for specific circumstances such as larger diameter production tubing, multiple gravel packs, etc. 
For Gas
It is commonly assumed that most of the pressure drop occurs in the packed perforations (at least as a starting point). As will be discussed later, this is likely not completely true. Nevertheless, assuming that this is the case, linear flow in the perforation tunnels can be represented as:
(26) 
where:
p_{wfs} 
= 
sandface flowing pressure determined from the IPR or deliverability equations (psia) 
p_{wf} 
= 
bottomhole flowing pressure (psia) 
q 
= 
volumetric flowrate (MscfD) 
m 
= 
gas viscosity at the mean pressure (cP) 
_{} 
= 
mean pressure, _{}=_{} (psia) 
_{} 
= 
real gas deviation factor (dimensionless) at T, _{} 
T 
= 
temperature (°R = °F + 460) 
L_{perf} 
= 
distance from the outer screen diameter (feet) to the tip of the perforation (feet) 
d_{screen} 
= 
screen diameter (inches) (see Table 3) 
k_{gravel} 
= 
permeability of the gravel (md) (see Table 1) 
A_{perf} 
= 
available area for access to the formation (the total of the nominal perforation crosssectional areas through the perforated interval) (feet^{2}) 
b_{gravel} 
= 
a turbulence term for the gravel 
h_{perf} 
= 
net perforated interval (feet) 
n_{perf} 
= 
shot density (spf) 
d_{perf} 
= 
perforation diameter (inches) (see Table 2) 
g_{g} 
= 
gas gravity (dimensionless, air = 1.0) 
Example: Calculating IPR for a CasedHole Gravel Pack Completion
The frictional pressure drop through the completion (which can be converted to a skin) can be determined as a function of the rate. Consider an example for an oil reservoir injecting into a wellbore and the injected fluid has the identical properties to the fluid in the reservoir. The relevant parameters are:
· Average reservoir pressure = 2,000 psia
· Water viscosity at reservoir temperature 1 cP
· Water formation volume factor = 1 RB/STB
_{}
· Reservoir permeability = 100 md
· This well has a drilled diameter of 9 inches
· r_{e }~ 1140 feet
Calculate Performance Relationship for this Example as Follows:
1 Suppose that size selection evaluation indicated that 10/20 gravel was appropriate. For 10/20 gravel (from Table 1):
k_{gravel} = 652 darcies, _{}.
2 Suppose that the well is vertical, the perforated interval is 30 feet, the shot density is 24 spf, and a 5inch casing gun will be used in 65/8 inch, 20 lb/ft casing. For this gun, the following specifications apply (see Table 2 for approximate penetration features):
d_{perf }» 0.73 inches
L_{perf }» {12.3 inches = 1.025 feet} [perforation penetration]
+{6.049 inches = 0.504 feet}/2 [radius of 6 5/8inch, 20 lb/ft casing]  {3.375 inches = 0.281 feet}/2 [outer screen radius]
= 1.14 feet
3 Calculate the available area for access to the formation as:
A = 5.454 x 10^{3} x 30 feet x 24 spf x 0.73^{2} in^{2} = 2.093 feet^{2}
4 Determine the linear coefficient in the flow equation as:
_{}
5 Calculate the quadratic coefficient in the flow equation as:
_{}
6 The pressure drop through the completion is:
p_{wf }_{ }p_{wfs }= 0.00074q + 1.37 x 10^{7 }q^{2}
7 Construct a performance curve for the reservoir itself using the equations presented previously. These data are shown in the first two columns in Table 3.
8 Determine the pressure drop due to the completion (p_{wfs } p_{wf} in Table 4). Substitute the rate from the second column in Table 4 into the equation in Item 6 above to calculate p_{wf }_{ }p_{wfs}
The pressure drops due to the completion itself can be plotted to indicate the contribution from the completion more specifically, as shown in Figure 10.
Figure 10. Pressure drop due to the casedhole, gravel pack completion in the example case. The baseline reservoir performance assumed no mechanical skin (no plugging), but formation damage skin can be considered. Just use a skin term when the formation performance is calculated.
Cased, Cemented and Perforated Completion
The previous gravel pack completion example included some obvious simplifications about the effectiveness of the perforations. It assumed that the perforations themselves did not damage the formation in any way. In reality, the skin associated with perforating can be surprisingly important. Perforation pressure drop (from which skin can be calculated) can be inferred. Some approximations for a casedcementedperforated completion are shown below.
For Oil or Water
The procedure is not unlike that taken for the cased hole gravel pack, with the exception of the presumed flow regime around the perforation tunnel. Assuming that there is no interaction between perforations, a firstorder assumption is that there is radial flow from each perforation. Assuming this:
_{} 
(27) 
where:
Dp 
= 
pressure differential across the completion (through the perforation and the surrounding crushed/damaged zone) (psi) 
q’ 
= 
flow rate per perforation (BOPD/perforation) 
q_{total} 
= 
total flow rate (BOPD) 
b 
= 
turbulence factor 
r_{perf} 
= 
perforation radius (feet) 
d_{perf} 
= 
perforation diameter (feet) 
n_{perf} 
= 
perforation density (spf) 
h_{perf} 
= 
net perforated length (feet). Partial penetration of a deviated well will create additional skin 
m 
= 
viscosity (cP) 
B_{o} 
= 
Formation Volume Factor (RB/STB) 
r_{c} 
= 
radial extent of the crushed zone from the centerline of the perforation tunnel (feet). The example shown is for a situation where it is assumed that the crushed zone around the tunnel is 0.5 inches thick. You can use other values, if available. This is a common default assumption. As can be seen in the equation, the crushed zone thickness is converted to feet by dividing by 12. 
k_{cf} 
= 
permeability of the perforation damaged zone (md). This is determined by multiplying the native reservoir permeability (k) by a damage factor. 
r_{o} 
= 
oil density (lbm/ft^{3}) 
The permeability of the compacted zone is commonly assumed to be a fraction of the nominal reservoir permeability. A ruleofthumb is:
k_{c }= 0.1k Overbalanced

(28) 
This in itself shows a key reason for perforating underbalanced – the degree of damage in the compacted zone can be significantly less than when perforating is done balanced or overbalanced.
For Gas
The approximations shown above are a generic and ideal approximation for liquid flow through perforations. Similar concepts can be applied for gas. If the perforations are considered to not interfere with each other, a firstorder assumption is that there is radial flow from each perforation. Based on this assumption:
_{} 
(29) 
where:
q’ 
= 
flow rate per perforation (MscfD/perforation) 
q_{total} 
= 
total flow rate (MscfD) 
n_{perf} 
= 
perforation density (spf) 
h_{perf} 
= 
net perforated length (feet) 
m 
= 
viscosity (cP) 
b 
= 
turbulence factor 
r_{perf} 
= 
perforation radius (feet) 
d_{perf} 
= 
perforation diameter (feet) 
r_{c} 
= 
radial extent of the crushed zone from the centerline of the perforation tunnel (feet). The example shown is for a situation where it is assumed that the crushed zone around the tunnel is 0.5 inches thick. You can use other values, if available. This is a common default assumption. As can be seen in the equation, the crushed zone thickness is converted to feet by dividing by 12. 
L_{perf} 
= 
perforation length (feet) 
k_{c} 
= 
permeability of the perforation damaged zone (md). This is determined by multiplying the native reservoir permeability (k) by a damage factor. The same default assumptions for the damage factor are made as for oil. 
g_{g} 
= 
gas gravity (dimensionless, air = 1) 
_{} 
= 
mean real gas deviation factor (dimensionless) 
T 
= 
Temperature (°R = °F + 460) 
Calculating Performance for a Cased and Perforated Oil Well
Consider the example (an injector) shown in Table 4. Assume that this was shot underbalanced.
Table 4. Basic Reservoir and
Completions Data
Parameter 
Value 
Calculations 
Average reservoir pressure 
2000 psi 

Water viscosity 
1 cP 

Water formation volume factor 
1 RB/STB 

Water Density 
54.67lbm/ft^{3} 

Well Orientation 
vertical 

Perforation Density 
12 spf or 4 spf 

Perforated Interval 
30 feet 

Casing 
6.625" 

Gun 
5"
casing gun 
_{} 
Permeability 
100 md 

Drainage Radius 
1140 feet 

Bubble Point Pressure 
1200 psi 

Drilled Diameter 
9 inches 

1 From these data, determine the linear coefficient in the flow equation as:
_{}
2 Calculate the turbulence factor as:
_{}
3 Calculate the quadratic coefficient in the flow equation as:
_{}
4 Determine the pressure drop through the completion using:
p_{wf } p_{wfs } = 2.97q¢ + 6.31 x 10^{3}q¢^{2}
5 Don’t forget that the performance is calculated with the total flow rate and that the relationship for the perforations is based on flow per perforation. Pressure drops for 12 and 4 spf are shown.
The pressure drops due to the completion itself can be plotted to indicate the contribution from the completion more specifically, as shown in Figure 11.
Figure 11. The pressure drops due to the completion in the example case. The baseline reservoir performance assumed no skin, but formation damage skin can be considered.
Other Completions
Methods were shown for estimating the pressure drop for two generic completion types  a cased and perforated completion and a gravel pack completion. The gravel pack calculations shown were for an ideal pack without any associated damage to the perforations. For the gravel pack, the basis for the calculation was to assume that there was linear flow along the perforation tunnel and through the gravel. This analysis did not incorporate any damage to the pack or to the perforations. To accommodate damage to the gravel pack itself (either due to placement or fines, etc.) the gravel pack permeability can be reduced. To account for perforation damage, split the problem into two parts – first, calculate the pressure drop for linear flow through the gravel in the pack and in the perforation tunnel. Then, calculate the pressure drop through a compacted zone around a perforation as shown in the second example. Then combine the pressure drops or skins.
These conceptual methods can provide guidelines for determining pressure drops through other types of completions. Consider an openhole gravel pack. This situation could be considered as a twopart reservoir with radial flow (one permeability zone is the gravel and the second is the formation itself). This problem can be solved by using some of the formulas presented earlier. One approximation is to use:
_{}
For example, let k be the formation permeability with a value of 1000 md. Let k_{s} be the gravel pack permeability – assume that the pack has been placed perfectly and the gravel has a permeability of 650 darcies. At this point, be a little careful  use r_{w} as the outer diameter of the screens. Use r_{s} as the drilled diameter of the wellbore. Let’s assume a 9inch hole, giving r_{s} = 9/(2 x 12) = 0.375 feet. If a 3 3/8inch diameter screen wire wrapped screen is used (and pressure drop is ignored through the screen – for demonstration purposes at least), the artificial wellbore radius can be taken as 3.375/(2 x 12) = 0.141 feet. The skin would be:
_{}
Granted, this demonstration shows negligible skin. It becomes more significant if the formation permeability is higher, if the gravel is damaged with placement or injection fluids, if the formation has been damaged by fluids, solids and inadequately cleaned up, if plugging or other mechanisms are impairing the screens themselves – and so on.
Other types of completions provide specific pressure drops. Kaiser et al., 2000, provide an excellent presentation of the methodologies that can be used for optimizing slotted liner selection.
“The primary factors considered in [slotted liner] design are sand control, inflow resistance and cost. Inflow performance is usually considered to be controlled by the open area exposed to the reservoir, and sand control governed by slot opening size. These become competing considerations in reservoirs with fine sands because slot density must be increased to maintain open area if slot size is reduced to control sand.”
The pressure drop through this type of completion is not only a function of the slot, but also the flow convergence in the sand that packs around the slots. “In fact, the flow loss through an open slot is negligible compared with that induced by the flow disturbance associated with the slot.” You can visualize the difference between uniform flow into a barefoot completion, with a slight convergence into the hole, with the much more extreme convergence that concentrates flow into a slot at similar rates. These convergence effects can lead to pressure drops that bear consideration during planning stages. You might think that fewer larger slots would have less flow resistance than more smaller slots for the same open area. This presumption ignores what Kaiser et al. indicate as the most important component of slotinduced loss – this being the convergence of flow in the sand that is packed around the slots. In fact, they indicate that this is why wire wrapped screen are so effective; because the “slots” are very close together and this minimizes the extent of flow convergence and its associated pressure loss. [Note that Kaiser was commenting on production.]
Kaiser and his colleagues outlined a combined empirical, analytical and numerical program where they optimized liner characteristics for Marathon’s South Bolney field. Their analysis model combined the radial Darcy flow relationships, a mechanical skin factor that is associated with flow convergence and flow along the completion itself (see the section on Wellbore Hydraulics). They called the slot component of the skin factor the slot factor and provided some illustrations of its relative contribution. “The slot factors calculated from the flow convergence analysis are dimensionless numbers that are used with the nearwellbore permeability to calculate the pressure loss induced by the slots.”
This paper is recommended reading. These authors also discussed the standard engineering considerations of optimizing the completion to avoid slot plugging and the considerations that may be taken for evenly distributing flow along a long slotted liner section in a horizontal or extended reach well.
Convergence through limited access in the completion hardware is the essential issue. In effect, this is analogous to partial penetration. These effects can be compounded if blank sections are run within the liner section. Blanks can be used for load control devices. The inflow performance is impacted by the reduction in the net production interval. There is local flow convergence at the slots themselves and larger scale convergence to the slotted intervals. Rough approximations of the “partial coverage” skin from this effect (for example, ignoring axial flow) were provided as:
_{}
where:
B = the coverage ratio is B.
Manufacturers should be able to provide you with the skin that is associated with a specific completion. Of course, plugging of the screens themselves compounds these skins. We all know that this is not a trivial issue. A recent paper by Asadi and Penny, 2000 demonstrates some of these issues. One of the most important aspects of this paper related to the methodologies and effectiveness of cleanup techniques on screens that were damaged while being run – this brings us back to some of the advantages of underbalanced operations where it may be possible to mitigate the damage by running in certain less damaging fluids.
Standard sand control considerations are published elsewhere (for example, Penberthy and Shaughnessy, 1992.) The work of Asadi and Penny is valuable because largescale empirical measurements were made on commercial products (20/40 PMF – porous fiber material – equivalent to a 20/40 sand, 20/40 Stratapack, and 0.008in gauge wire wrap Prepacked with 40/60 mesh resin coated sand). These components were exposed to a formulated drillin fluid containing drill solids. Various cleanup techniques were then attempted (10% HCl; enzyme breaker followed by 10% HCl; acid activated breaker, followed by 10% HCl, with and without water backflow. The efficiency of some of the evaluated hardware was low and cleaning was never 100% effective – sometimes much less. The products were vulnerable to plugging and cleanup under ideal laboratory conditions was never ideal. Consider all methodologies for reducing the degree of plugging in the first place. Also remember that the pressure drop even in clean screens will be elevated when turbulent flow exists. Asadi and Penny, 2000, indicated pressure drops of between 0.05 and 0.1 psi at an injection rate of between 9 and 18 BLPD/ft of length. Turbulent flow will elevate these values even for clean screens.
The issue of turbulent pressure losses applies to all types of completions, including, as we have seen cased and perforated situations. The reader may wish to review numerical work by Behie and Settari, 1993. Numerical models such as these can also be used to evaluate the influence of collapsed and filled perforations – another potentially significant skin component – even if shot underbalanced or maybe because they were, collapsed and infilled perforations can add significantly to pressure drops that are experienced.
It all boils down to calculating skin. This section was intended to illustrate concepts. These days in most cases, you would have software available. However, without understanding underlying concepts, using commercial or internal software is inefficient at best, and possibly dangerous.
Using the concepts shown, you can estimate the degree of damage due to the completion and the skin due to the completion itself. For an excellent summary of skin associated with perforating, refer to Chapter 6.5.3 in Bell et al., 1995. For a complete discussion of sand control issues, refer to Penberthy and Shaughnessy, 1992. For cogent discussions of the reservoir engineering aspects of various completions, refer to Beggs, 1991, and/or Golan and Whitson, 1991).
There are methods for estimation of the pressure drop through the reservoir and the completion. To complete the analysis of well performance you must next consider the effects of wellbore hydraulics.
Wellbore Hydraulics
There are methods for estimating pressure changes in the reservoir and across the completion. You will need to complete the final component of the well performance analysis – estimating the pressure drop in the wellbore and at the surface. The pressure regime in the tubing or annulus is governed by hydrostatic, frictional, and acceleration forces. This section shows you how to approximate these effects for single phase and multiphase flow in the wellbore.
The key to estimating pressure drop is determining an accurate bottomhole pressure. Consider the simplest situation – single phase flow. You can estimate bottomhole pressure, (p_{BH }or p_{wf}) from surface pressure p_{surface} using the following equation:
p_{BH} = p_{surface }+ p_{Hydrostatic }– p_{Friction} 
(30) 
p_{Hydrostatic} is the hydrostatic head of the fluid in the tubing and/or annulus and p_{Friction }is the loss in pressure due to friction loss during flow.
The bottomhole pressure is important for:
· Analyzing current performance
· Predicting future performance
· Assessing formation damage due to completion
· Evaluation of DSTs and other well tests
· Workover implementation
· Indication of scale deposits
· Design of artificial lift systems
· Formations which have undergone pressure depletion may not withstand a static column of fresh water
· Basic completion design and installation in any well under any balance condition.
Hydrostatic Pressure
The hydrostatic component is the pressure exerted by the column of liquid/gas in the wellbore. This pressure is a function of the fluids present and consequently can vary, from a static bottomhole value if the well is shutin (or not flowing), to a hydrostatic pressure reflecting changing fluid ratios during flow. Under static conditions, without flow, there are no frictional contributions and a static bottomhole pressure is calculated from a summation of surface pressure and the hydrostatic head.
Calculating the static bottomhole pressure for a liquidfilled wellbore involves simply multiplying the fluid gradient (density) times the true vertical depth.
For a vertical well filled with gas, the static condition is given by integrating the density over the depth interval. The form of the integration is given by:
_{} 
(31) 
where:
Z 
= 
gas deviation factor (dimensionless). Z can be regarded as a term by which pressure must be corrected to account for departure from the ideal gas equation pV = ZnRT 
p 
= 
absolute pressure (psia) 
¡_{g} 
= 
specific gravity of the gas (dimensionless, air = 1.0) 
T 
= 
absolute temperature (°R) 
L 
= 
length (feet) 
If the temperature and pressure are taken at an average value (indicated by the bar in the following equation), the following equation accounts for gas compressibility:
_{} 
(32) 
where:
p_{ws} 
= 
static bottomhole pressure (psia) 
p_{ts} 
= 
static wellhead pressure (psia) 
H 
= 
true vertical depth (feet) 
The temperature is commonly taken as the arithmetic mean of the bottomhole and wellhead temperatures, assuming a linear variation with depth. _{}is taken as the value at the arithmetic mean temperature and the arithmetic mean pressure. Because of the interrelationships between parameters, this bottomhole pressure equation is generally solved by trialanderror. While you likely have a numerical model or spreadsheet to do these calculations, it is instructive to understand some of the background. The iterative steps are:
1 Guess a reasonable value for p_{ws}.
2
Determine _{}at _{} and _{}.
3 Calculate p_{ws} from Equation 32. This is an estimate of the bottomhole pressure. The hydrostatic component equals the surface pressure minus the bottomhole pressure.
4 Presuming that the value of p_{ws} differs substantially from its original estimate, refine the assumed value for p_{ws} and repeat the process until adequate agreement occurs.
If you use the method of Sukkar and Cornell, 1955, trial and error procedures are not required. For even more precise calculations, you can use the method proposed by Cullender and Smith (1956). Good references on this topic are Bradley, 1991 and Ikoku, 1980.
Friction
Using the methods in the previous section, you can estimate the hydrostatic head for a well containing liquid, gas, or multiphase fluids during a completion operation. The next component to consider in calculating the bottomhole pressure is the friction that exists when fluids are moving. The complex frictional forces of fluids flowing in a wellbore depend on the type of fluid flow regime. This section considers frictional effects for:
· Single Phase Liquid Flow
· Single Phase Gas Flow
· Multiphase Flow
Single Phase Liquid Flow
For liquid flow in the wellbore, the pressure drop per unit length (for friction effects only) is given by the following equation:
_{} 
(33) 
where:
dP 
= 
pressure change (psi) 
dL 
= 
length of the segment over which the pressure drop is calculated (feet) 
p 
= 
density (specific gravity x 62.4) (lbm/ft^{3}) 
f 
= 
Moody friction factor (dimensionless) 
v 
= 
velocity (ft/second) 
D 
= 
equivalent diameter (inches) 
The term “equivalent diameter” is used to discriminate between pipe flow and annular flow. For conventional pipe flow, Equation 33 is used and D is the internal diameter of the pipe over the length being evaluated. For annular flow, the effective diameter is:
_{} 
(34) 
For annular flow, use D = D_{outer }– D_{inner }in the second form of Equation 33. In the first form of Equation 33, replace D^{5} with (D_{outer }– D_{inner})^{3} x (D_{outer }+ D_{inner})^{2}.
The length term in Equation 33 is based on measured depth because friction acts over the full length of the string (unlike hydrostatic head, which is based on true vertical depth). If there are changes in string dimensions and/or inclination, it is common practice to divide the well up into individual increments and sum the friction over the individual discrete lengths. If the wellbore contains compressible fluids and you use fundamental friction equations, you must discretize the wellbore (divide it into discrete lengths).
As reflected in Equation 33, during flow, irreversible energy losses occur. With the exception of completely laminar flow, these energy losses cannot be predicted theoretically and are usually accounted for by using the friction factor (as shown in Equation 33). For consistency, only the Moody friction factor will be used (the Moody friction factor is four times the Fanning friction factor (f_{F})). The friction factor (f) and the relative roughness (e/D) are related to the Reynolds’ Number. The Reynolds’ Number is calculated by:
_{} 
(35) 
where:
e 
= 
absolute roughness (feet) 
D 
= 
diameter (feet) 
S.G. 
= 
specific gravity (dimensionless) 
v 
= 
velocity (feet/second) 
N_{Re} 
= 
Reynolds Number (dimensionless) 
m 
= 
viscosity (1bm/ftsec or cP) 
The friction factor depends on the specific flow regime in the wellbore, as shown in Table 5. With the exception of laminar flow, many empirical representations of the friction factor are available for turbulent and transitional flow regimes. Most of these equations yield approximately the same results (see Figure 12 for a relative roughness of 0.0006).
Table
5. Summary of Various Friction Factor
Formulas
Flow Regime 
Reynolds Number 
Moody Friction Factor 
Comments 
Laminar 
< 2000 
_{} 
Independent of roughness 
Critical ^{26} 
2000 < N_{Re} < 4000 
_{} 
_{} 
Transition^{26} 
4000 < N_{Re} < (200D/e)^{1.16} 
_{} 
Colebrook (1939) 
Turbulent^{26,28} 
N_{Re} > 4000 
_{} 
Nikuradse 
Figure 12. Comparison of Moody friction factors for various calculation methods for a relative roughness of 0.0006. Woods’ curve is an explicit approximation of Colebrook’s relationship.
Though roughness is quite difficult to determine, particularly for tubulars that have been in service for some time, some guidelines are available. Figure 13 shows an example. Using these guidelines, you can estimate frictional pressure loss using Equation 33 and Table 5. It is usually appropriate to discretize the flow path and solve over various increments, using the previous increment to provide an upstream boundary condition. Commercial codes are available for these calculations.
Figure 13. Relative
roughness of various pipes.
Single Phase Gas Flow
For steadystate flow, the energy balance is given by (notice that this contains the hydrostatic head component).
_{} 
(36) 
where:
r 
= 
fluid density (lbm/ft^{3}) 
p 
= 
pressure (psia) 
v 
= 
average fluid velocity (ft/sec) 
a 
= 
correction factor to compensate for variation of velocity over the tubing crosssection, varying from 0.5 for laminar flow to 1.0 for fully developed turbulent flow. A value of 0.90 is commonly used for practical gas flow problems. 
H 
= 
distance in the vertical direction (feet) 
f 
= 
Moody friction factor (dimensionless) 
D 
= 
inside pipe diameter (feet) 
L 
= 
length of the flow string (feet), L = H for a vertical section 
_{} 
= 
pressure drop due to kinetic effects, 
_{} 
= 
pressure drop due to friction, 
w_{s} 
= 
mechanical work done on or by the gas (w_{s }= 0) 
g 
= 
acceleration due to gravity (ft/s^{2}) 
g_{c} 
= 
conversion factor (32.17 lbm  ft/1bf s^{2}) 
Ignoring the kinetic energy term and integrating over an interval from point “1” to point “2” gives:
_{} 
(37) 
Combining using oilfield units and assuming constant temperature over an interval of interest:
_{} 
(38) 
where:
p 
= 
pressure (psia) 
Q 
= 
gas flow rate (MMscfD) 
_{} 
= 
average temperature over the distance L, (°R) 
L = H 
= 
vertical distance (feet) 
D 
= 
inside diameter of the pipe (inches) 
Unfortunately, without assumptions or numerical solution, this equation cannot readily be integrated. Smith (1950) proposed an approximate solution (based on the Weymouth equation):
_{} 
(39) 
where:
Q 
= 
volumetric flow rate measured at 14.656 psia and 60°F (scfD) 
T 
= 
effective average temperature (°R) 
Z 
= 
effective compressibility factor for the gas 
d 
= 
pipe ID (inches) 
p_{2} 
= 
sandface pressure at depth X (psia) 
p_{1} 
= 
wellhead pressure (psia) 
g_{g} 
= 
gas gravity (dimensionless, air = 1.0) 
X 
= 
difference in elevation between points 1 and 2 (feet) 
f 
= 
Moody friction factor (dimensionless) 
There are numerous other approximate or numerical formulations. Most of these are available in or superceded by commercial software packages. For example:
· Poettmann (1951) derived an expression accounting for the variation of the compressibility factor. It can be applied to deviated wells.
· You can use a modified version of the Sukkar and Cornell Method (see Katz et al., 1959) for straight or inclined holes. Their relationship assumes steadystate, single phase flow, negligible kinetic energy, constant temperature at some average value, and a constant friction factor over the length of the conduit.
· Twostep approximations can be done using the Cullender and Smith method.
· There are other relationships: the Panhandle formula; the Ford, Bacon and Davis formula; and the Clinedinst flow equation.
Multiphase Flow
The procedures for evaluating multiphase flow are substantially more complicated than those for singlephase flow. The level of complexity for multiphase flow is so high that numerical analysis and/or correlations are commonly used for frictional evaluations. Some of the possible correlations that you can use for forecasting pressure drop are:
· Duns and Ros (1963) can be used for certain vertical flow situations.
· Orkiszewski (1967) can be used for certain vertical flow situations. Orkiszewski outlined methods for addressing flow of discrete bubbles, slug flow where the gas phase is more pronounced, a transition flow (transitional from continuous liquid to continuous gas) and mist flow where the gas phase is continuous and most of the liquid is present as entrained droplets. Computational versions of this method are widely used, with some specific limitations.
· Hagedorn and Brown, 1965, is probably recommended for most vertical flow situations. It was developed from experimental measurements in an instrumented 1500ft deep well. Neither liquid holdup or flow pattern was measured but correlations were developed by assuming that the twophase friction factor could be obtained from the Moody diagram based on a twophase Reynolds Number. Refer to Appendix A in Beggs 1991. This is recommended reading on this topic.
· Beggs and Brill, 1973, deals with any inclination, twophase flow. Liquid holdup and pressure gradients were measured for flow of air and water in a laboratory environment, for inclinations between horizontal and vertical. Liquid holdup is the fraction of an element of pipe that is occupied by liquid at some instant.
· Mukherjee and Brill, 1983, developed a procedure for wells of any inclination with twophase flow, and,
· Dukler, 1959, provided correlations for horizontal flow.
The Hagedorn and Brown correlation
commonly
gives good results for most situations.
There are some simpler methods as well. You can use the PoettmannCarpenter method for vertical multiphase flow. Recognize that other methods, such as Hagedorn and Brown, are more (and sometimes much more) accurate. PoettmannCarpenter determined an f’ factor empirically and correlation charts were prepared for various tubing sizes on which were plotted the total massproducing rate Q_{M} versus the pressure gradient dP/dh.
Computational routines are commercially available (and should be used) for multiphase flow for high profile situations. For rapid calculations, in addition to the Poettmann and Carpenter procedures, an extensive set of gradient curves has been generated and published by Brown (1984). Gradient curves have been generated for various tubing/annulus dimensions, rates and the specific fluid properties.
Restrictions
Inevitably, there will be pressure losses in inline restrictions, including subsurface safety valves, surface or bottomhole chokes, valves and fittings, etc., Beggs provides a clear discussion. Some of the key points are summarized below.
Single Phase Gas Flow through
Surface Chokes
Flow through a restriction can be critical or subcritical (sonic or subsonic). When the flow is critical, any downstream disturbance will not affect the upstream pressure or the rate. Since chokes are intended to control rate, they usually are designed for critical flow. “A ruleofthumb for distinguishing between critical and subcritical flow states that if the ratio of downstream pressure to upstream pressure is less than or equal to 0.5, then the flow will be critical. This is a closer approximation for singlephase gas than for twophase flow.”
_{} 
(40) 
where:
q_{sc} 
= 
volumetric gas flow rate (Mscf/D) 
d 
= 
I.D. of bore open to gas flow (inches) 
g_{g} 
= 
gas specific gravity (dimensionless, air = 1.0) 
k_{} 
= 
ratio of specific heats = C_{p}/C_{v} (dimensionless) 
p_{1} 
= 
upstream pressure (psia) 
p_{2} 
= 
downstream pressure (psia) 
T_{1} 
= 
upstream temperature (°R) 
Z_{1} 
= 
compressibility factor at p_{1} and T_{1} (dimensionless) 
C_{D} 
= 
discharge coefficient (empirical, dimensionless) 
T_{sc} 
= 
standard absolute temperature base (°R) 
p_{sc} 
= 
standard absolute pressure base (psia) 
y_{} 
= 
p_{2}/p_{1} 
y_{c} 
= 
critical pressure ratio (dimensionless) 
When no information is available, the restriction is short with slightly rounded openings and critical flow conditions exist, an approximation is (using a generic default value of C_{D} = 0.82):
_{} 
(41) 
SinglePhase Liquid Flow through Surface Chokes
The following equation may be used (assuming subcritical flow, if necessary, a default value for the discharge coefficient can be used, C_{D} = 0.85):
_{} 
(42) 
where:
q_{L} 
= 
liquid flow rate (STB/day) 
d_{ } 
= 
choke diameter (inches) 
p 
= 
psi 
S.G. 
= 
liquid specific gravity 
TwoPhase Flow through Surface Chokes
An empirical relationship in the critical regime is:
_{} 
(43) 
where:
p_{i} 
= 
upstream pressure (psia) 
q_{L} 
= 
liquid flow rate (STB/day) 
R 
= 
gas/liquid ratio (scf/STB) 
d 
= 
choke diameter (inches) 
Values of a, b and c have been proposed by different investigators. For very rough approximations, try using a = 1.89, b = .00386 and c = 0.546. Sachdeva (1986) presented a more rigorous model.
Gas Flow Through a Subsurface Safety Valve
Flow will generally be subcritical  minimal downhole restriction is desirable. You can iterate and solve using the API relationship:
_{} 
(44) 
where:
p_{1} 
= 
upstream pressure (psia) 
p_{2} 
= 
downstream pressure (psia) 
g_{ } 
= 
gas gravity (dimensionless, air = 1.0) 
Z_{1} 
= 
gas compressibility at p_{1} and T_{1} 
T_{1} 
= 
upstream temperature (°R) 
q_{sc} 
= 
gas flow rate (Mscf/D) 
b 
= 
d/D 
d 
= 
bean diameter (inches) 
D 
= 
pipe inside diameter (inches) 
C_{D} 
= 
discharge coefficient (API suggests 0.9) 
Y 
= 
expansion factor (dimensionless). The expansion factor determination is iterative and may be calculated from Equation 44. Its value ranges between 0.67 and 1.0. For quick estimates, a default value of 0.85 is often used. 
K 
= 
the ratio of specific heats of the gas. 
TwoPhase Flow Through a Subsurface Safety Valve
For twophase flow through a subsurface safety valve, you can use:
_{} 
(45) 
where:
r_{n} 
= 
noslip density (lbm/ft^{3}) 
q_{L} 
= 
"insitu" total liquid rate 
q_{g} 
= 
"insitu" gas rate 
n_{m} 
= 
mixture velocity through the choke (ft/sec) 
p_{1} 
= 
upstream pressure (psi) 
p_{2} 
= 
downstream pressure (psi) 
C_{D} 
= 
discharge coefficient 
N_{n} 
= 
q_{g}/q_{L} = (1l_{L})/l_{L} 
l_{L} 
= 
q_{L}/(q_{L}+q_{g}) 
b 
= 
d/D 
d 
= 
choke diameter 
D 
= 
tubing inside diameter 
All the fluid properties necessary for calculating the density and velocities are evaluated at upstream conditions of pressure and temperature. p_{1} is iteratively determined from a known p_{2}.
Pressure Losses Through Valves and Fittings
The energy losses associated with bends and other restrictions are considered to be supplemental to those predicted by Bernoulli's basic equation and its standard representation using Moody's friction factor. To account for these additional pressure losses, the concept of a loss coefficient is used. Typically, the loss coefficient is indicated as K. In simplified terms the pressure loss for flow with bends and with other inline obstructions can be expressed as:
_{} 
(46) 
where:
d 
= 
diameter of pipe 
f 
= 
friction factor for pipe flow 
L 
= 
length of pipe 
K 
= 
resistance coefficient depending on the type or size of fitting 
An equivalent length, L_{e}, can be calculated for each fitting by using the friction factor calculated for flow in the pipe. All the L_{e }values can then be added to the actual pipe length for the pressure drop calculation. K values have been determined for singlephase flow and they are similar to values for two phase flow (for example, 0.15 for a gate valve, 0.2 0.3 for elbows, 3.0  5.0 for globe valves, 6.0  8.0 for check valves). The pressure drops will commonly be small enough to be ignored. A more basic fluid mechanics approach is provided by Granger, 1985. Granger indicated the loss coefficient for a rounded elbow could be determined from empirical relationships.
_{} 
(47) 
If no values of r/a are available, use: k_{45°} = 0.35 and k_{90°} = 0.75. Most manufacturers can provide relevant information.
The degree of pressure loss due to bending is generally taken to be a function of R (the radius of curvature and r the inner radial dimension of the pipe. The term R/r is known as the relative radius. For large values of R/r, such as those which would be associated with the transitional segment of a horizontal well, Rankin et al., 1989, proposed a loss coefficient which is empirically correlated to the relative radius, a friction factor (f), bend angle in degrees and a correction coefficient which depends on both the bend angle and R/r. These relationships were empirically developed using data for water in fully developed turbulent flow and are therefore relevant to conventional fluids (as opposed to lightened fluids). The transition from horizontal to vertical flow in the annulus is particularly sensitive to momentum flux – “the tendency of the fluid to surge as head is overcome.” This is atttributable to the transition from a somewhat static pressure regime in the horizontal segment to a sudden reduction in momentum as the fluid is forced to travel upward.
Additional pressure drops that can be considered are associated with tapered strings, tubing restrictions, the wellhead and any horizontal lines or obstructions at the surface. Tapered strings can be evaluated numerically or published gradient curves can be used. The gradient curves for the specific tubular or casing dimensions are used in combination by manipulating the effective depth (using the bottom run depth of an upper tubing as the top of the section below).
Various restrictions to flow can be present  e.g., safety valves, tubing nipples, downhole chokes. Some manufacturer information is available. Similar information can be acquired for wellhead/BOPRBOP etc. restrictions. Horizontal surface lines can be represented using one of the methods that have been described above.
Using Tubing Intake Curves (Injection or Production)
Regardless of which method you use to estimate pressure drops, you will need to determine hydrostatic pressure and friction at various flow rates. Using your selected method, you must generate tubing intake curves to represent the flowing bottomhole pressure versus rate for the particular tubing/annulus situation. Your tubing intake curves will be based on the specific set of parameters that you define (e.g., wellhead pressure, diameters, lengths, fluid type, etc.). Once you have defined these parameters, you will construct a table of bottomhole pressure versus rate. Finally, you will plot the tubing intake curves. An example is shown in Figure 14.
Figure 14. An example tubing intake curve.
Tubing intake curves depend on a number of environmental parameters. For example, different tubing intake curves will exist for different wellhead pressures (refer to Figure 15).
Figure 15. Various tubing intake curves for different flowing wellhead pressures. As p_{wf} decreases, the wellhead pressure also decreases.
The tubing intake curve (flow rate that the tubing can accommodate) is also influenced by the size of the tubulars (refer to Figure 16).
Figure 16. Schematic change in tubing intake curves for different size tubing.
You will recall that there are methods for inferring the drop in pressure for flow through the formation, and through the completion. You can also see from the example tubing intake curves that there can be further restrictions associated with flow in the wellbore. Because the formation, completion, and wellbore pressure drops are not independent, you must consider them using a systems approach. You can use Nodal Analysis^{TM} principles and techniques to evaluate the individual and combined effects of these parameters on well performance.
Nodal Analysis
Nodal Analysis^{TM} was originally developed by Flopetrol Johnston. Nevertheless, the concepts are universally applicable and used by all service and operating companies. In this method, you represent the reservoircompletionstringsurface components by nodes that can be found in different parts of the well. A node is any point at which a pressure drop is evidenced. Next, you evaluate the pressure drops at all points and determine an equilibrium situation. Finally, you can determine the maximum potential injection.
The performance relationships indicate the flow rates that a formation can deliver and they can account for the effect of a particular completion. The tubing intake curve accounts for the flow rate the string/annulus can accommodate. You must superimpose the performance relationship and tubing relationships to determine rate and pressure.
A simple example of using Nodal Analysis^{TM} or related concepts is shown in Figure 17. This schematic example is for a hypothetical injector. The effect of the perforation density has been exaggerated for illustrative purposes. A tubing intake curve developed for a particular combination of fluids is superimposed on the IPR curves. Using these data and the bottomhole pressure range, injection rate is predicted. Expected wellhead pressures can be determined by varying input into the calculation of the tubing intake curve.
Figure 17. Hypothetical injection well. Several performance curves are shown for various perforating programs. Several tubing intake curves are superimposed for varying wellhead pressures. Intersections indicate anticipated pressures and flow rates.
You can use Nodal Analysis^{TM} or equivalent methods to evaluate potential well performance. If these programs are not available, do the appropriate calculations in a spreadsheet.
References
1. Good, P.A. and Kuchuk, F.J.: “Inflow Performance of Horizontal Wells,” paper SPE 21460 (1991).
2. Kuppe, F. and Settari, A.: “A Practical Method for Determining the Productivity of MultiFractured Horizontal Wells,” CIM Edmonton 96, Edmonton, Canada (April28May 2, 1996) and JCPT 1998.
3. Kaiser, T.M.V., Willson, S., and Venning, L.A.: “Inflow Analysis and Optimization of Slotted Liners,” paper SPE 65517, 2000 SPE/PSCIM Int’l Conf on Horizontal Well Technology, Calgary, Canada, November 68.
4. Asadi, M. and Penny, G.S.: “Largescale Comparative Study of PrePacked Screen Cleanup Using Acid and Enzyme Breaker,” 2000 IADC/SPE Asia Pacific Oil and Gas Conf. Exhib, Brisbane, Australia, October 1618.
5. Penberthy, Jr., W.L. and Shaughnessy, C.M.: Sand Control, SPE Monograph Series, Richardson, TX (1992).
6. Behie, G.A. and Settari, A.: “Perforation Design Models for Heterogeneous Multiphase Flow,” paper SPE 25901 presented at the 1983 SPE Joint Rocky Mountain Regional/Low Permeability Reservoir Symposium, Denver, CO, April 2628.
7. Duns, H., Jr. and Ros, N.C.J.: Sixth World Petroleum Congress, Section 2, Paper No. 22, Frankfurt, Germany (1963).
8. Orkiszewski, J.: JPT., 19, 829 (1967).
9. Hagedorn, A.R., and Brown, K.E.: JPT, 16, 203 (1964); 17, 475 (1965).
10. Beggs, D.H. and Bril, J.P.: “A Study of TwoPhase Flow in Inclined Pipes,” SPE 4007, JPT (1973).
11. Mukherjee, H. and Brill, J.P.: “Liquid Holdup Correlations for Inclined TwoPhase Flow,” SPE 10923, JPT (1983).
12. Dukler, A.E.: Chem. Eng. Progress, 55(10), 62 (1959).
13. McLennan, J.D., Carden, R.S., Curry, D., Stone, C.R. and Wyman, R.E.: Underbalanced Drilling Manual, Gas Research Institute, Chicago, IL (1997).
14. Brown, K.E.: “The Technology of Artificial Lift,” Volume 4, Pennwell, Tulsa, OK (1984).
15. Sachdeva, R., Schmidt, Z., Brill, J.P. and Blais, R.M.: “TwoPhase Flow Through Chokes,” SPE 15657, paper presented at the 1986 (61^{st}) Annual Tech. Conf. Exhib., New Orleans, LA.
16. Granger, R.A.: Fluid Mechanics, Dover Publications, New York, NY (1995).
17. Rankin, M.D., Friessenhahn, T.J. and Price W.R.: “Lightened Fluid Hydraulics and Inclined Boreholes,” SPE 18670, paper presented at 1989 SPE/IADC Drilling Conf., New Orleans, LA, February 28 – March 3.
18. Supon, S.B. and Adewumi, M.A.: “An Experimental Study of the Annulus Pressure Drop in a Simulated AirDrilling Operation,” SPE Drill. Eng. (March 1991) 7480.
19. Wang, Z., Rommetveit, R., Bijleveld, A., Maglione, R. and Gazaniol, D.: “Underbalanced Drilling Requires Downhole Pressure Management,” Oil & Gas J. (June 16, 1997) 5460.