FLUID FLOW AND THE PRODUCTION SYSTEM

INFLOW PERFORMANCE RELATIONSHIP (IPR)

Gilbert (1954), the father of modern production engineering, who worked for many years at Shell Oil, was the first to realize the full significance of the decreasing productivity index at pressures below the bubble point.

He plotted the bottomhole flowing pressure, pwf, versus the flow rate, q, and referred to this curve as the inflow performance relationship or IPR ( Figure 1 ).An individual curve is drawn for a given average reservoir pressure.

Figure 1

Because reservoir pressure will generally be depleted by production, the IPR, over the life of a well, may be shown by a family of curves shrinking toward the origin. Each curve represents the pressure-rate relationship at a given average reservoir pressure ( Figure 2 ).

Figure 2

The endpoints of the IPR curve are the average reservoir pressure, R at a flow rate of zero and the maximum potential flow rate, q’, at a bottomhole flowing pressure of zero ( Figure 1 ). q’ called the "pumped-off potential" or "open-flow potential" of the well represents the "ideal" maximum flow rate that would occur if we could reduce the bottomhole flowing pressure to zero. In practice, it is not possible to achieve this rate because the bottomhole flowing pressure must always have some finite value.

Above the bubble point the IPR curve is a straight line because only one phase is flowing, and permeability is a constant equal to the absolute permeability. The productivity index is equal to the inverse slope of the IPR curve. It, too, must be a constant above the bubble point. Below the bubble point, as gas comes out of solution and begins to interfere with flow, the IPR curve trends downward and the productivity index continues to decrease ( Figure 3 ).

Figure 3

This particular shape of the IPR curve is characteristic of reservoirs with a solution gas drive. Reservoirs with other drive mechanisms such as water drive, gas cap expansion, gravity segregation, or a combination of mechanisms will have IPR curves of a different shape or perhaps a straight line ( Figure 3 ).

To be accurate about the specific pressure being discussed, you will note that the vertical axis label pwf changes to p when pressure-rate curves other than IPR’s are presented.

Formation stratification has a marked influence on the shape of the IPR curve, particularly if multiple zones, each with different permeabilities, produce into the same wellbore. Horizontal wells can also have a significant effect on the IPR curve. Under certain conditions (e.g., thin formations where the permeability anisotropy favors vertical flow), horizontal wells show a much higher productivity index than vertical wells. Thus, they can produce at higher rates for a given pressure drawdown (i.e., for a given pwf), or they can produce at a constant rate with a much lower drawdown.

Flow Regimes

A number of different flow regimes may occur during natural flow in vertical tubing. In order to describe each, let us assume that the pressure at the base of the tubing is above the bubble point. In such a case the flow regime at that point will consist of liquid flow ( Figure 1 ).

Figure 1

Upward movement of the liquid is accompanied by reduced pressures and, as the pressure drops below the bubble point, gas bubbles begin to form. These bubbles slip upward through the rising column of liquid, with the larger ones rising more rapidly than the smaller. This is referred to as bubble flow.

Further up the tubing, as pressure continues to drop, more gas is released from solution and the larger bubbles grow steadily by overtaking and coalescing with the smaller ones. Eventually a stage may be reached at which the larger gas bubbles fill almost the entire cross section of the tubing and, as they move upward, carry between them slugs of oil containing small gas bubbles. This is referred to as plug or slug flow. It is the most efficient natural lift regime because it uses the gas to full effect rather than losing its potential lifting power to the slippage that occurs during bubble flow.

Higher in the tubing, at even lower pressures, the gas may break through and form a continuous channel in the center of the string, with oil moving slowly upward in an annular ring on the inside wall of the tubing. Such annular flow is clearly inefficient.

Finally, if the tubing is of considerable length so that a large pressure drop exists from the bottom to top, the annulus of liquid may almost disappear, leaving only the

flow of gas carrying a mist of liquid droplets. Now we have what is called mist flow and it is characteristic of many wet gas wells or condensate producers.

The description of tubing flow regimes and pressure losses that occur is an extremely complex subject. In practice not all of these flow regimes are present simultaneously in a single tubing string. On the other hand, two, three or even more may occur at the same time. In any case, identifying the flow regime is the first step in determining the tubing pressure drop.

Correlations

In order to analyze and design our production system it is necessary to be able to calculate the pressure drop which exists between the bottomhole and the surface during natural flow. The calculation of this pressure drop for all possible conditions is so complex that we are forced to rely on empirical or semi-empirical correlations. These correlations take into account the seven important variables that affect the pressure losses of a flowing well. These variables are tubing size, flow rate, fluid viscosity, fluid density, gas-liquid ratio (GLR), water-oil ratio (WOR), and, finally, the effect of slippage. Another variable, vertical well deviation, is receiving more attention because of the many directional wells being drilled offshore.

Since the first published work of practical significance by Poettmann and Carpenter (1952), numerous additional studies have been undertaken. Investigators have analyzed the effect of each of the above variables on the vertical pressure profile of a well. From their work a number of correlations have been developed, many of which have been incorporated into computer programs, which may be used with specific well data in order to calculate the pressure losses during flow.

In addition, a number of pressure gradient or pressure traverse curves, such as the one shown in Figure 1 , have been published for use in the field.

Figure 1

These curves show depth on the vertical axis and pressure along the horizontal axis.

Since a separate curve is needed for each set or well and flowing conditions, there are a large number of published curves. Nowadays, most engineers have access to computer programs which use the most appropriate correlation for the specific problem that is to be solved.

Our objective in calculating pressure losses during natural flow through tubing is to predict the performance of our production system under various equipment and operating conditions and thereby develop an optimal design. One convenient way of presenting the results of vertical pressure loss is to incorporate it into our IPR diagram.

We start with the IPR curve ( Figure 2 ).

Figure 2

Using the value of the bottomhole flowing pressure at a specific production rate, we subtract the vertical pressure loss obtained from vertical profile curves or computer programs for that production rate ( Figure 3 ).

Figure 3

Subtracting the vertical pressure loss from the bottomhole flowing pressure at that flow rate gives the value for the tubing head pressure at that rate. The appropriate value of tubing head pressure, referred to as ptf, is now plotted on the graph as shown in Figure 4 .

Figure 4

Another flow rate is then assumed, the calculation repeated, and a second tubing head pressure is determined. As we continue in this way, a tubing head pressure curve is built ( Figure 4 ). The difference vertically between the IPR and the tubing head pressure curve is the pressure loss in the tubing at each production rate. We shall refer to this as the THP curve. The procedure is quite straightforward and, for given flow conditions, may be repeated for larger or smaller tubing size until an optimum design is found.

Surface Control

Now that the topics of reservoir performance, IPR curves, vertical flow in the tubing, and various pressure loss correlations have been introduced, we should turn to the third element of our flowing well system - the wellhead choke which provides control at the surface ( Figure 1 ).

Figure 1

The choke or bean is used to ensure that the flow from the well is reasonably steady. The size of the orifice is usually chosen so that variations in wellhead pressure do not affect the pressure of separators, lines and other surface equipment. Also, we want to ensure that fluctuations of pressure in the gathering system (caused, for example, by the action of a dump separator) do not affect well performance.

To ensure that downstream pressure variations are not transmitted to the upstream side of the choke the flow through the orifice must attain critical flow velocity. In practice we have found that this critical flow velocity is achieved under most circumstances when the upstream or the tubing head pressure is at least double the downstream or flow-line pressure. This condition is one that the petroleum engineer must design into his flowing well system.

During the critical flow of fluids through an orifice, the tubing head pressure is a linear or almost linear function of the liquid-flow rate.

This means that if we plot the tubing head pressure (Ptf) on the vertical axis and the flow rate (q) corresponding to critical flow on the horizontal axis, the choke performance plots as a straight line through the origin ( Figure 2 ). This is limited by the fact that as the tubing head pressure approaches the downstream line pressure the flow rate goes to zero.

Figure 2

The larger the orifice size, the larger the flow rate for a given tubing head pressure.

Integrating the IPR, THP and Choke Performance

To determine flow conditions for the well on the downstream side of the choke, we must integrate the performance of the three components of the flowing well system: the formation, the vertical wellbore, and the choke itself.

To do this we begin with the IPR curve, then add first the THP curve as we did earlier and finally the choke performance line ( Figure 1 ).

Figure 1

The intersection of the THP and choke lines gives the production rate attainable from the well under these flow conditions. With a larger choke size the flow rate will be higher, but the tubing head pressure at the higher rate will be lower. We may say that the flow rate is controlled by the choke size.

How can we include the limitations on the production rate which may be imposed by facilities downstream of the wellhead choke? In order to analyze this situation, we must consider the pressures at the following points within our system: (1.) the wellbore, (2.) the tubing head, (3.) immediately downstream of the choke; and (4.) at the entrance to the separator. To graphically illustrate the pressure response at these points, we draw the IPR and THP curves, the first two points in our system ( Figure 2 ).

Figure 2

Now we add curve 3 which represents the pressure at the point immediately downstream of the choke. Each point on this curve has a pressure which is one-half that of the tubing head pressure. From the discussion on chokes we recognize that this is the maximum allowable pressure immediately downstream of the choke under critical flow conditions. Subtracting the pressure losses in the gathering system between the choke and the separator at the various flow rates we obtain a fourth curve, curve 4, which represents the pressures at the downstream side of the gathering system immediately before the separator. Now we add curve 5, the pressure-rate curve for the separator.

The rate defined by the intersection of the last two curves is the maximum that can be produced from the well under critical flow conditions with the reservoir performance and equipment specified (see Figure 2 ). The choke size would have to be chosen so that the intersection of the choke performance and THP lines give an equal or lower flow rate. If the choke is not selected in this manner then it is the performance of some downstream element of the gathering system that controls the well’s output, not the choke ( Figure 3 ).

Figure 3

If, for example, as shown in Figure 4 , the choke is chosen so that the well produces at rate q1 then the pressure downstream of the gathering system will be p1.

Figure 4

The separator, however, operates at the lower rate, q2, at that pressure, therefore, it is the separator that becomes the controlling factor in the overall production rate. Anytime there is a flow-rate fluctuation in the separator the well reacts to it. In order for the choke to control production, it must have a smaller diameter and a performance line that intersects the THP to the left of the vertical line as shown in Figure 3 .

Of course as more fluids are produced from the well the average reservoir will decrease and the IPR will change. This, in turn, will require a redrawing of Figure 4 and, in all likelihood, a change of choke size.

This form of choke analysis, modified to reflect actual field equipment, may be used to analyze pressure losses, identify bottlenecks, and, with revised designs, obtain higher flow rates. A similar approach can be taken in the analysis of the effect of a down-hole choke on a well’s performance. We will study choke performance in much greater detail in PE 104.

IPR curves can also be used to forecast the flowing performance of a well, the timing of artificial lift installation, and to design the size. and type of such installation that would be appropriate to the well.

In order to make a proper analysis, we need to know the shape of the IPR curve for each well draining the pool. The IPR curves must be known for different levels of

average reservoir pressure, water-oil ratio, and gas liquid ratios. The average reservoir pressure as a function of cumulative production must also be known.

With this information the engineer may predict future performance for each of the wells and recommend future producing strategies for the field as a whole. For example, he may recommend when enhanced recovery operations should be initiated and decide on the type and capacity of artificial lift installations.

Inflow Performance Relationship

Introduction

The inflow performance relationship is the production engineer’s shorthand description or the performance potential or a reservoir at a given average reservoir pressure. It is the relationship between the bottomhole flowing pressure and flow rate, and is the starting point in the analysis or a well’s behavior.

Inflow performance is the major contributor to the production process, yet our knowledge or the factors that control it, and the way it varies over time, is inexact because or the numerous variables that can affect it. While it is the first link in the chain or our production system, it is unfortunately, the weakest link.

We will look at some or the techniques currently used for calculating IPR’s, learn some or the basic assumptions involved, and see how IPR curves are applied in practice. A flowing well never achieves its maximum pumped-off potential flow rate. Pressure losses in the tubing, chokes, and other surface equipment, make it impossible to get the pressure opposite the formation down to zero. The bottomhole flowing pressure is equivalent to the backpressure exerted by the flowing column of fluid as it moves to the surface. This backpressure is usually quite large.

The inflow rate that may exist against this backpressure is not a true reflection of what the flow rate of the well might be after installation of artificial lift because artificial lift unloads the fluid column, reduces the bottomhole pressure, and with it, the backpressure on the formation. It is important in the analysis of a well for an engineer to know the relationship that exists between the bottomhole flowing pressure and flow rate even down to a very low pressure. For this reason the engineer must define the IPR and predict how it changes with time. The following bottomhole pressure could come close to zero with a pump. However, there are other very important reasons why a very low bottomhole pressure may not be desirable. These include such problems as production, and water or gas coning, which preclude a low pwf.

INFLUENCE OF DRIVE MECHANISM ON IPR

IPR curves have different shapes for different reservoirs, depending primarily on the drive mechanism of the reservoir ( Figure 1 ). A reservoir with a strong water drive, or a solution gas drive above the bubble point will have a straight line IPR. In the

special case where the IPR is a straight line, J equals the reciprocal of the slope of the IPR and is constant.

For a solution-gas-drive reservoir, the straight-line portion above the bubble point reflects the dynamic flow characteristics of single-phase liquid flow through the formation ( Figure 2 ).

Figure 2

However, when the flowing pressure in the formation falls below the bubble point, Pb, gas comes out of solution, reduces the permeability to the oil phase, decreases the productivity index, and reduces the oil flow rate within the formation. Remember, the relative permeability to the oil phase is dependent on the oil-phase saturation.

At increased production rates, pwf decreases and more gas comes out of solution within the formation. At higher gas saturations, the relative permeability to oil drops further. This results in a downward curving IPR and a steadily decreasing productivity index at decreasing flowing bottomhole pressure, with its antecedent phase-behavior dependence on relative permeabilities. Other factors such as

increased oil viscosity, rock compressibility, and turbulence can add to these effects as wellbore pressures fall and rates increase.

We conclude, then, that a solution-gas-drive reservoir below the bubble point has a downward curving IPR. Often a well’s IPR curvature is intermediate between a straight line and this classic solution-gas drive curve. In such cases the average reservoir pressure is receiving support from gas cap expansion or a water drive.

VOGEL’S METHOD

Vogel’s main objective was to simulate two-phase flow through a reservoir into a wellbore. By analyzing a number of different solution-gas-drive reservoirs, he established an empirical relationship which could apply to all such reservoirs.

The computer program that he prepared solved the equations of flow for somewhat idealized reservoirs. For example, he assumed that the reservoir was circular, completely bounded, and with a fully penetrating well at its center; that the formation was uniform, isotropic, and had a constant water saturation; that gravity and compressibility could be neglected and that semi-steady-state flow occurred.

Vogel simulated reservoirs covering a wide range of conditions. These conditions included differing reservoir relative permeability characteristics as well as the various effects of well spacings, fracturing geometry, and skin restrictions. Analysis was limited to flow conditions below the bubble point.

Vogel found that as depletion occurs in a solution-gas-drive reservoir, the productivity of a typical well decreases. This occurs primarily because (1.) the reservoir pressure is reduced, and (2.) because increasing gas saturation causes greater resistance to oil flow. The result is a progressive downward shift of the IPR ( Figure 1 ).

Figure 1

The values on the lines reflect the percentage of reserves produced. Vogel, then, took the important step of plotting each curve as "dimensionless" IPR’s or "type curves." He obtained these curves by plotting the bottomhole flowing pressures divided by the average reservoir pressure on the vertical axis and the production rate divided by the maximum flow rate, C’’, on the horizontal axis. When this was done for each curve, they were replotted as shown in Figure 2 .

Figure 2

It is immediately apparent with this transformation that the curves now are remarkably similar throughout most of the producing life of the reservoir.

After analyzing twenty-one different reservoirs with various crude oil properties, relative permeabilities, and wellbore characteristics, Vogel found that IPR’s generally exhibited a similar shape, as long as the bottomhole flowing pressure was below the bubble point. Extending this observation one step further, he developed a standard reference curve which can be used for all solution-gas-drive reservoirs. This standard curve is shown in Figure 3 .

Figure 3

Specific plot points for this curve are given in the table below.

The use of this curve does not imply that all reservoirs are identical, but that it may be used as a reference standard for all reservoirs within a tolerable error. This reference curve is described exactly by the following equation:

Note that q is the producing rate corresponding to a given bottomhole flowing

pressure, pwf; q’ is the well’s potential at 100 percent drawdown, and R is the average reservoir pressure or the bubble-point pressure, whichever is lower.

X-q/q’

1.00 0.000

0.95 0.088

0.90 0.172

0.85 0.252

0.80 0.328

0.75 0.400

0.70 0.468

0.65 0.532

0.60 0.592

0.55 0.648

0.50 0.700

0.45 0.748

0.40 0.792

0.35 0.832

0.30 0.868

0.25 0.900

0.20 0.928

0.15 0.958

0.10 0.972

0.05 0.988

0.00 1.000

Example:

Assume:

q = 1172 BOPD

pwf = 716 psi

R = 1420 psi

R = pb

Construct the IPR curve for this well at the average reservoir pressure. Assume that Vogel’s dimensionless standard curve describes this well’s behavior.

First, we calculate the dimensionless pressure.

With this value and Vogel’s dimensionless standard curve (or Equation 1.2), we find the dimensionless rate (see Figure 4 ).

Figure 4

= 0.696.

This gives a value of:

q’= = 1684 BOPD.

The type curve can now be made into this well’s IPR curve simply by adding the values for average reservoir pressure and C;’ at the appropriate end points. The scale of the graph is now established and any desired point can now be read ( Figure 5 ).

Figure 5

Remember that Vogel’s results are only for the curved portion of the IPR curve which exists below the bubble point. Above the bubble point the IPR curve is a straight line. We can obtain its shape by drawing the tangent to the curve at the bubble-point pressure and extending it to the original average reservoir pressure, pi. Such as extrapolation is shown in Figure 6 .

Figure 6

In order to determine the shape of the IPR curve at a future average reservoir pressure, we need to know a single bottomhole flowing pressure and its corresponding flow rate at that average reservoir pressure. Using our dimensionless curve and a known data point we would repeat what we have just done. This would yield a second curve. The difficulty is that we do not have well test data at some future, unknown average reservoir pressure.

STANDING'S EXTENSION OF VOGEL'S METHODS

With Vogel’s type curve, one flowing well test, and a value for the average reservoir pressure, we can obtain a single IPR curve for our well. But how do we calculate the IPR curve at a future average reservoir pressure?

That is the same question that Marshall B. Standing (1970) asked when he published the results of his work. His approach was as follows. We remember that the productivity index, J, is defined as:

(1.3)

If we substitute J into Vogel’s equation with the average reservoir pressure below the bubble point, we obtain this relationship:

(1.4)

J is given in terms of flow rates and pressures. If J could be calculated for some future average reservoir pressure, then with this value of J and the above equation, the pressure and flow rate values needed to find the future IPR curve could be determined. Standing suggested that, in the limiting case, that is, where there is very small drawdown, the bottomhole flowing pressure would tend to be equal to average reservoir pressure, that is:

The value of J, under these conditions, is referred to as J* and, by substituting this ratio into Eq. 1.4, we obtain:

(1.5)

The next step is to calculate how J*, changes with average reservoir pressure.

Standing suggested that J*, at different average reservoir pressures, is proportional to relative permeability and inversely proportional to the formation volume factor and the viscosity. This is referred to as the relative mobility and is written:

J* = (1.6)

With this relationship, a future value of J* referred to as, Jf*, is equal to the present value of J*, Jp* multiplied by the inverse ratio of the respective mobilities, that is:

(1.7)

Combining these relationships into the Vogel equation, (Eq. 1.2), Standing found that future IPR curves could be plotted from the following equation:

(1.8)

Finding the IPR curve is rather direct. First, we assume a value for the future average reservoir pressure at which we would like to know an IPR curve. Then we calculate a value for Jf*.

Substituting these two values into Eq. 1.8 yields an equation in q and pwf This equation give us the future IPR curve.

In substance, Vogel’s type curve is used for the well’s IPR curve at the original reservoir pressure. This gives us q’ and Jp* which we need for Standing’s method. We then use Standing’s technique to obtain IPR curves at lower pressures. There is a good example as to how this calculation proceeds on page 56 of Nind’s text (1981).

FETKOVICH'S METHOD

Fetkovich (1973) proposed an alternative method for calculating IPR curves for solution-gas-drive reservoirs.

He made a number of assumptions including the idea that two-phase flow occurred through a uniform, circular, horizontal reservoir with a constant outer boundary pressure below the bubble point. One of Fetkovich's key assumptions was that the relative permeability to oil divided by the oil viscosity and formation volume factor varied linearly with pressure as shown in the following equation:

(1.9) The straight line passes through the origin. With this basic relationship assumed, Fetkovich was able to show:

(1.10) We may calculate Jo' at the original reservoir pressure pi using Eq. 1.11. This value of Jo' is referred to as Joi' and is a function of effective permeability to oil at the original reservoir boundary pressure, pi. Saturation is assumed to be constant for the well being analyzed.

(1.11) Joi' may be thought of as a replacement for J, the productivity index.

With these equations, it is not difficult to plot the IPR curve at a given reservoir or boundary pressure pRs

Let's now solve the same problem that we did earlier using Vogel's method.

Example: Assume:

q = 1172 BOPD

pwf = 716 psi

pi = 1420 psi

We insert these values into Eq. 1.10 to obtain:

Substituting this constant into Eq. 1.10 gives:

q = 7.793 10-4 (pi2 - pwf

2)

This equation gives us the inflow rate as a function of bottomhole pressure with it we can generate the IPR curve. For the original reservoir pressure, pi, we may now calculate the potential, q', of the well under these conditions, that is where pwf = 0.

q' = 7.793 l0-4 (14202 - 0) = 1572 BOPD

For comparison purposes, you will remember that we calculated a value of 1684 B0PD using the Vogel technique.

The agreement between these two methods of calculation is generally good in the intermediate pressure ranges, but there is often deviation at the outer ranges of pressure-rate axes. Major differences between these exist; however, either method may be used with the assurance that the results from the other will not differ dramatically.

To learn how Fetkovich's method is used for calculating future IPR curves, we must assume that Joi' will decrease in proportion to the average reservoir pressure.

When the average reservoir pressure drops below pi a new value of Joi', referred to as Jo , can be calculated using Eq. 2.11.

So in our example, if pR drops to 1000 psi, we would calculate:

Jo = 7.793 X 10-4 =5.488 X 10-4

Knowing this value of Jo' for an assumed future value of pR, we have a new IPR equation:

q = 5.488 10-4 (10002 - pwf2)

Fetkovich's method, then, yields two equations--one describing the initial reservoir performance and another describing performance, at an assumed future average

reservoir pressure. From these two equations, we can calculate values for Jo' and plot IPR curves for any future average reservoir pressure. In Figure 1 we see the two curves from the example we just solved.

Figure 1

We would proceed in the same manner if we wanted to find another IPR curve at a lower value.

IPR AND SKIN EFFECT

The skin effect is a near-wellbore phenomenon. In an ideal flowing well—one that fully penetrates the formation, where the full formation is open to flow and where no formation damage or stimulation exists—the pressure profile during flow looks like the one shown in Figure 1 .

Figure 1

In this case, the drawdown is equal to:

R - pwf.

If, however , the formation near the wellbore has been damaged (for example, by drilling fluid invasion)...or if the well only partially penetrates the formation or has limited perforations...or if there is turbulent flow in the formation near the wellbore...there will be an additional pressure drop ( Figure 2 ).

Figure 2

Because this additional pressure drop occurs near the wellbore, it is referred to as ∆pskin. The total pressure drop in a damaged well is equal to:

( R - pwf) + ∆pskin

The skin effect (s) is defined as

s = (kh)/(constant X B X pskin)

where B is the formation volume factor.

Because a damaged well causes an additional pressure drop, the skin effect is said to be positive.

If, on the other hand, the formation near the wellbore has been stimulated (say by fracturing or acidizing) rather than damaged, then the drawdown will be reduced ( Figure 3 ). The reduced pressure drop is again referred to as ∆pskin, but this time it is negative and the skin effect is negative. The total pressure drop in an enhanced well is:

( R - pwf) - ∆pskin (1.12)

Figure 3

The magnitude of the skin effect and whether it is positive or negative is obtained by conducting special well tests. These tests give us a value for ∆pskin and enable us to calculate the flow efficiency (FE) of the well. FE is defined as the drawdown of an ideal well divided by the drawdown of the well with skin effects.

Flow efficiency for a damaged well is less than one, and is equal to:

FE= (1.13a)

For an enhanced well, the skin relationship will be negative and the value of the flow efficiency will be greater than 1.0:

FE= (1.13b)

Standing prepared a series of curves which may be used by us to calculate the IPR for wells that have flow efficiencies different than 1.0. Using these curves we can calculate the IPR of a well if the damage were removed or the well stimulated. His curves are shown in Figure 1 .

Figure 1

The vertical axis is the dimensionless pressure of the flowing well and the horizontal axis is a dimensionless flow rate, specifically the flow rate of the well divided by its maximum flow rate with damage or fracturing. The curves are drawn for flow efficiencies from 0.5 to 1.5. The curves have the following relationship:

(1.14)

Where F is the flow efficiency. Neither this equation nor the curves should be extrapolated effectively to q/q’ values greater than unity.

STRATIFIED FORMATIONS

Often, the producing intervals in a well are separated by relatively thin but highly impermeable horizontal shale breaks. Production rates and fluid properties in any one layer may not be the same as those in other layers contributing to the well's overall production.

Consider a well that is completed in a horizon having three zones, in which there is no vertical communication among the zones:

• Zone 1 has an average permeability of 1 millidarcy (md) and an average pressure of 1500 psi.

• Zone 2 has an average permeability of 10 md and an average pressure of 1200 psi. • Zone 3 has an average permability of 100 md and an average pressure of 1000 psi.

Initially, the bulk of the production will come from Zone 3, and the smallest contribution will come Zone 1. Thus, after the well has been producing for several months, Zone 3 will be the most depleted and at the lowest average reservoir pressure, while Zone 1 will be the least depleted and at the highest average reservoir pressure.

The well is now tested at various production rates to establish the IPR. If the IPRs of each zone are as shown in Figure 1 , then the Gross IPR curve is the sum of all three.

Figure 1

At any given pressure, a point on the Gross IPR curve has a flow rate which is equal to the sum of the flow rates of the three individual curves.

In general, because of differential depletion, a well producing from a stratified formation will exhibit a Gross IPR as shown in Figure 2 : that is to say, an improving productivity index with increasing production at lower rates but a deteriorating productivity index at the higher rates.

Figure 2

Now consider a well completed in a two-layered horizon where water breakthrough has occurred in the more permeable, more depleted layer. In such a circumstance the watered-out zone has the higher permeability but the lower pressure of the two zones.

Let us assume further that the watered-out layer produces 100 percent water, while the other layer produces water-free oil. Beginning with the oil zone’s IPR and adding the water zone’s IPR, we obtain the Gross IPR ( Figure 3 ). At any given bottomhole flowing pressure we can observe the oil rate, the water rate, and the gross production rate. This allows us to calculate and plot the water cut as a function of the gross production rate.

Figure 3

The water cut is zero until we reach a bottomhole flowing pressure low enough for water to flow. Thereafter the water cut at any pressure and flow rate is equal to the ratio of the water production rate divided by the gross production rate.

When the bottomhole flowing pressure is. greater than the average reservoir pressure in the water zone, oil will enter into the water zone by inter flow taking place through the wellbore.

CONDUCTING AN INFLOW PERFROMANCE TEST

Below is a practical procedure for conducting an inflow performance test.

First: Shut the well in and conduct a pressure buildup test. This will give you the average reservoir pressure.

Second: With a recording pressure gauge on the bottom, place the well on production at a low rate and, after ample time is allowed for the rate to stabilize, record the bottomhole flowing pressure.

Third: Flow the well at two successively higher flow rates. Let each rate stabilize and note the bottomhole flowing pressure.

Fourth: After the last test is run, shut the well in and conduct another buildup test. This test will give four points on the IPR curve.

Unfortunately this test requires a great deal of time and so, for economic reasons, we often have only sufficient time to conduct a single flow test. A single-flow test and a value for the average reservoir pressure is sufficient for flowing and artificial lift well predictions; however, the complete multiple flow test provides more accuracy.

Remember that the IPR is a characteristic of an individual well, and that it is best to generate an appropriate family of IPR curves for that well based on its known reservoir and fluid properties, pressures, downhole hardware and completion data. A variety of software products are available for this purpose. You should review your company’s capabilities in this area.

VARIABLES AFFECTING TUBING PRESSURE LOSS

The variables that affect vertical pressure losses in tubing are tubing size, flow rate, density and viscosity. Because there is probably more than one phase flowing, we must add two more variables: gas-liquid ratio and water-oil ratio. Finally we should add the effect of slippage.

Tubing Size

Suppose that we increase the tubing size in a well from inches to 3 inches, leaving all other parameters constant. The result, as shown in Figure 1 , is that the total pressure loss that occurs between the formation and the surface drops from 1900 psi for the smaller tubing string to 900 psi for the larger string.

Figure 1

This substantial pressure drop reflects the reduction in friction pressure for the larger-diameter tubing. We conclude, then, that under these conditions, as the tubing size increases, the pressure losses will decrease.

Fluid Density

The second variable to consider is fluid density, which, for oil, we may express in terms of API gravity. For the well of Figure 2 we see that the pressure loss over an 8000 foot interval is approximately 1700 psi if it is flowing brine, but only 1200 psi if it is flowing a 50-degree API oil.

Figure 2

We conclude that for similar flowing conditions, pressure losses will be lower for lower fluid densities. At higher fluid densities, the hydrostatic pressure gradient becomes the dominant component of pressure loss.

Fluid Viscosity

In Figure 3 we see that higher viscosities give higher pressure losses, again due to an increase in friction pressure.

Figure 3

At a fluid viscosity of 50 cp, the total pressure loss over the 8000-ft interval is 1900 psi; at 1 cp, the total pressure loss is 1200 psi. Note that the effect is much less pronounced as the viscosity decreases from 10 cp to 1 cp. We conclude, then, that the viscosity of the flowing fluid is an important variable and that lower-viscosity oils under similar conditions will have lower friction pressure losses.

Gas-Liquid Ratio

In Figure 4 , we see that at a GLR of 250 SCF/STB, the pressure loss from the formation to the surface is about 1900 psi, whereas at a GLR of 5000 SCF/STB, the pressure loss is about 700 psi.

Figure 4

In both cases the surface pressure is assumed to be 100 psi. Thus, at a GLR of 250 SCF/STB, this well will flow if the bottomhole pressure exceeds (1900+100), or 2000 psi, while at a GLR of 5000 SCF/STB, the well will flow if the bottomhole pressure exceeds (700+100), or 800 psi.

In general, then, the higher the GLR at a given flow rate, the lower will be the tubing pressure loss—but only up to a point. While higher GLRs reduce a fluid's density, resulting in lower hydrostatic pressure, they also result in higher friction pressure losses, which offset this hydrostatic pressure decrease. The decrease in the hydrostatic pressure is overcome by the increase in the friction losses. At some limiting value of GLR, the increase in friction pressure becomes approximately equal to the decrease in hydrostatic pressure, and above this limt, the total pressure loss actually begins to increase.

Water-Oil Ratio

We see in Figure 5 that as the water-oil ratio (WOR) increases from 0 to 1000, the pressure losses in the tubing also increase.

Figure 5

This means that it will require a higher bottomhole flowing pressure to lift produced liquids that have water in them. The greater the WOR, the greater the pressure needed because water is slightly denser than oil. The magnitude of the pressure increase is not as large as those noted with other variables.

Slippage and Holdup

In order to illustrate the condition of holdup, which results from slippage, we plot the bottomhole flowing pressure at different flow rates for several GLR’s. In Figure 6 we see that for a GLR of 800 the bottomhole flowing pressure required to maintain flow increases as the flow rate increases.

Figure 6

This is as we might expect. At the higher flow rates the frictional losses increase and so will the required bottomhole flowing pressure. At lower flow rates the frictional losses are smaller and so the bottomhole flowing pressure required to maintain flow is lower. At lower GLR’s, however, we see that the curves have a point of reversal or minimum. For the 400 GLR curve we see that the required bottomhole pressure decreases as we reduce the rate until at about 150 BOPD it begins to increase again. This reversal, or holdup, is caused by slippage, a condition where liquid flow rate becomes so low that excessive fallback begins to occur. Liquid falls back around the rising gas bubble. A smaller diameter tubing, giving higher velocities, should be used in this situation.

VERTICAL FLOW CORRELATIONS

There are various methods in place for determining the pressure losses that occur in flowing wells. It is not surprising that our prediction methods are not based on the exact solution of mathematical equations but rather on empirical or semi-empirical relationships. These relationships were developed by making certain assumptions about the applicable flow equations and then collecting data from a number of flowing wells under controlled conditions. The result is the publication of one or more

correlations based on mathematical foundations and supported by observed field data.

Early theoretical work in vertical flow analysis was undertaken by Versluys (1930). This was followed by the first practical application proposed by Poettmann and Carpenter (1952). Other important contributions include the work of Gilbert (1954), Duns and Ros (1963), Hagedorn and Brown (1965), Orkiszewski (1967), Govier and Aziz (1972), and Beggs and Brill (1973). We refer you to volume 1 of Brown’s text (1977) to learn how each of the theoretical and practical developments of these individuals evolved. Let us here summarize the approach and the important contributions of each.

In reviewing these contributions we find it instructive to indicate the foundation of the work done, the pipe sizes to which the work applied, the fluids considered and, finally, to comment on the work of each.

Versluys’ work was theoretically based and described vertical flow patterns.

Poettmann and Carpenter’s work was semi-empirical and applied to 2-, 2 - and 3-inch tubing. The fluids studied were oil, gas, and water. They developed practical solutions for GLR less than 1500 scf/bbl and for flow rates greater than 420 BOPD. In

1954, Gilbert used field data to investigate flow in 2-, 2 - and 3-inch tubing. He investigated oil, gas, and water flowing wells and developed a practical set of pressure profile graphs that can easily be used in the field by the engineer. Duns and Ros combined experimental laboratory work with field studies for all pipe sizes and all fluids to develop one of the best correlations for all flow rates. Hagedorn and Brown undertook both field and experimental work. They considered each of the three phases of flowing fluids in 1- to 4-inch tubing and produced a very useful generalized correlation for all ranges of flow rate. Orkiszewski reviewed all of the methods that had been published to that date and then, from his observations, prepared a single composite correlation. This correlation applies to all pipe sizes and fluids, and it may be used to predict pressure losses for all ranges of flow. It is widely used as the basis for computer programs in industry today. In 1972, Govier and Aziz, in Canada, published their correlations which were based on laboratory and field data for all pipe sizes and all fluids. Their correlations were based on a mechanistic equation which had been tested against field data. In 1973, Beggs and Brill reported on the work being conducted at the University of Tulsa. They presented

the results of laboratory studies on 1- and 1 -inch pipe for air and water. Their correlation handles all ranges of multiphase flow for any pipe angle. The practical application of this work is the prediction of pressure losses in inclined or directionally drilled wells. Many more correlations have been published and work continues today in this important research area.

The above-mentioned theoretical and empirical studies have left us numerous vertical pressure loss prediction methods, presented originally as correlations or pressure traverse curves. Brown for example, in volume 2a of his text, presented a full set of pressure traverse curves. Many computer programs have been written using one or more of their correlations to predict pressure losses during flow. The question remains as to which of these methods is most accurate under a given set of conditions. Statistical comparisons (Lawson and Brill, 1974) of several of the most

widely used methods have been undertaken in order to determine their relative accuracy over a broad range of variables and to identify the strengths and weaknesses of each technique.

No single pressure loss prediction method seems to be consistently superior under all ranges of production conditions. Comparisons of the methods of Poettmann and Carpenter, Duns and Ros, Hagedorn and Brown, Beggs and Brill, Govier and Aziz, and Orkiszewski show that the Hagedorn and Brown method has the best overall accuracy but that other methods perform better under different sets of variables and types of flow.

Despite variations in accuracy among the methods tested, they are within the range of engineering accuracy for use in sizing well equipment and designing artificial lift installations. Estimates of flow rates and bottomhole flowing pressures may also be made with reasonable accuracy by using these pressure gradient curves. The ones published by Brown or Gilbert may certainly be used with confidence. Your company may have its own internally published set of curves which you may choose to use.

Many companies have computer programs that calculate pressure losses in tubing using a combination of the various correlations. These are quite accurate because they are generally written so as to use each correlation over its range of greatest accuracy.

THE USE OF PRESSURE TRAVERSE CURVES

To better understand the basis for vertical flow calculations, and because they are quite accurate for engineering calculations, we should learn how pressure traverse or gradient curves are used.

A typical set of pressure gradient or pressure traverse curves are shown in Figure 1 .

Figure 1

Depth, on the vertical scale, runs downward from the surface to 10,000 ft. Pressure, on the horizontal scale, goes from 0 to about 2800 psi. As noted in the legend, the curves are generated using fixed values for the following parameters:

• tubing size • producing rate, • oil gravity • gas gravity • flowing fluid temperature • water-oil ratio = zero (only oil is flowing) • GLR = several selected values

If we have a well that matches these parameters, then the use of these curves is straightforward. To illustrate, assume that we have a well with the characteristics shown in Figure 1, which is producing at a GLR of 200 SCF/Bbl. The length of the tubing string is 5000 ft.

• Case 1: Determine the tubing head pressure that corresponds to a known bottomhole flowing pressure of 1600 psi:

1. First, we find the point at which the "GLR=200" curve intersects a pressure of 1600 psi.

2. We note that this intersection corresponds to a depth of about 7200 ft.

3. We continue to move upward along the "GLR=200" curve until we have moved a vertical distance of 5000 ft, which is the length of the tubing string. This puts us at a point on the curve that corresponds to 2200 ft.

4. Finally, we trace a vertical line upwards from this point, and note that the tubing head pressure is 230 psi.

• Case 2: This is the more common case, where the surface THP is known (for this example, assume 400 psi) and we wish to estimate the bottomhole flowing pressure:

1. Starting at the top, at a pressure of 400 psi, we trace a vertical line downward until we intersect the "GLR=200" curve. This takes us to a depth of 3450 ft.

2. We then move down along the curve for a vertical distance of 5000 ft (i.e., to 8450 ft).

3. We observe that the flowing bottomhole pressure at 8450 ft is about 2050 psi.

Note that, in Case 1, if the well depth had been 10,000 ft instead of 5000 ft, we would have found in Step 3 that the pressure would have gone to zero before we had moved a total vertical distance of 10,000 ft (we would actually cross the zero-pressure line at a depth of around 2400 ft). Under these circumstances we would be unable to calculate a positive tubing head pressure. This tells us that a 10,000-ft well could not flow at 1500 Bbl/D under the given conditions.

CALCULATING THE THP CURVE

The shape of the THP for a given well can be varied by changing the magnitude of such variables as tubing size and sometimes gas-liquid ratios. From an engineering design point of view, we should change the variables over which we have control until we achieve optimal flow conditions.

Example: A corroded tubing string is being removed from a well and is to be

replaced. In addition to 2 -inch tubing, we also have 1.9-inch and 3 -inch

tubing in inventory. What size tubing should be used to cause the well to flow at the maximum rate, given the following well data:

THP = 170 psi depth = 5200 ft

R = 1850 psi GLR = 400 scf/bbl

The present conditions with corroded tubing are:

q = 250 BOPD

Pwf = 1387 psi

The reservoir pressure is above the bubble point.

We begin by generating the IPR curve, in this case it is a straight line. Then, using pressure gradient curves, we calculate tubing head pressure curves for each size of tubing. The results are shown in Figure 1 .

Figure 1

At a tubing head pressure of 170 psi, the 3 -inch tubing will allow a flow rate of

about 425 BOPD, the 2 -inch tubing about 525 BOPD, and the 1.9-inch tubing about 535 BOPD. The highest flow rate is provided by the smallest tubing. In

practice, the 2 -inch tubing would probably be chosen for its strength and convenience of running tools, since its performance curve is nearly as good as that of 1.9-inch tubing. The design, then, is complete.

INTRO

A well may be produced with or without a choke at the surface to control the flow rate. Most flowing wells have surface chokes for one or more of the following reasons:

• to reduce the pressure and improve safety

• to maintain a fixed allowable production limit

• to prevent sand entry from the formation

• to produce the well and reservoir at the most

• efficient rate

• to prevent water and gas coning

• to match the surface pressure of a well into a multi-well gathering line and to prevent back flow

In addition, any situation requiring control or reduction of the well’s flow rate will normally be met by the installation of a surface choke.

The surface choke is also used to ensure that pressure fluctuations downstream from the wellhead do not affect the performance of the well. To achieve this condition, flow through the choke must be of a critical velocity. The corresponding critical flow rate is reached, when the upstream pressure is approximately twice the downstream pressure.

There are several different types of chokes currently in use. They may be divided into two broad categories: variable or adjustable chokes and positive or fixed orifice.

Positive chokes have a fixed orifice dimension which may be replaceable and is usually of the bean type ( Figure 1 ). The flow path is normally symmetric and circular. Fixed orifice chokes are commonly used when the flow rate is expected to remain steady over an extended period of time.

Figure 1

Normal beans are 6 inches long and are drilled in fractional increments of th-inch

up to -inch. Smaller bean inserts, known as X-type, are used to provide closer control. Ceramic, tungsten carbide, and stainless steel beans are used where sand or corrosive fluids are produced. Changing the size of a fixed orifice choke normally requires shutting off flow, removing and replacing the bean.

Some continuously variable or adjustable chokes operate similarly to a needle valve and allow the orifice size to be varied through a range from no flow to flow through a full opening ( Figure 2 ).

Figure 2

Flow control is obtained by turning the hand wheel which opens or closes the valve. Graduated stem markings indicate the equivalent diameter of the valve opening. Another type uses two circular discs, each of which has a pair of orifices. One disc is fixed while the other can be rotated so as to expose the desired flow area or block the flow altogether ( Figure 3 ).

Figure 3

Because of their variable size opening the calculation of flow rates through adjustable chokes may not be as accurate as through orifice chokes. However, adjustable chokes may be used to control wells where changes in the production rates may be required periodically to meet market demands or allowables.

Variable chokes are often used on water flood injection wells where variation in injection rates must be effected with minimal disruption. Variable chokes are particularly vulnerable to erosion from suspended sand particles and are not normally used in areas where this is a significant problem.

The bodies of both types of chokes are L-shaped and the end connections may be fully flanged, fully threaded, or a combination of each.

It is important in the design of the surface control system to understand the pressure versus flow rate performance of the choke at critical flow rates. Good correlations for single-phase flow of either gas or liquid through a choke are available, but they are not applicable to the multiphase flow situation we normally encounter in our wells. The performance correlations for multiphase flow through chokes are derived empirically and apply only at critical flow rates.

CHOKE PERFORMANCE RELATIONSHIPS

The equation describing the relationship between upstream pressures, gas or liquid ratios, bean size, and flow rates at critical velocities in field units is as follows:

Where:

R = GLR, Mcf/bbl q = flow rate, BOPD S = choke size, 64-th of an inch ptf = THP, psia

From the nature of this equation, we see that for a given orifice size and GLR, the tubing head pressure plots as a straight line function of flow rate q.

A typical plot is shown in Figure 1 .

Figure 1

Note that as the orifice size increases or the GLR decreases, the line shifts downward.

Gilbert (1954), while checking for choke erosion in the Ten Section Field, California, further refined the theoretical formula to yield more accurate pressure measurements, using this empirical relationship:

where ptf is in psig.

He found that these new values for the constant and the exponent agreed more accurately with empirical data. He later presented the new version of the equation as a nomogram to make it practical for field use

Ros (1961) developed a theoretical formula to account for critical flow through a restriction. The equation he developed was later adapted to oil field use and converted to graph form by Poettmann and Beck (1963). Their conversion applies to oil gravities of 20, 30, and 40 degrees, API.

Integrating the IPR, THP, and Choke Performance

We now turn to methods of calculating the flow rate attainable by a well under various operating conditions. We know that the IPR curve gives the whole range of bottomhole flowing pressures and rates possible for any given productivity index and average reservoir pressure. But what will the actual production rate be? That depends on the vertical flow performance and surface control facilities.

The most basic surface control system is one where there is no surface choke and where the wellhead and surface line pressure losses are minimal. For this condition we may analyze the well's performance by simply constructing a tubing head pressure curve.

The procedure is straightforward. For a series of bottomhole pressures and flow rates, we calculate the pressure losses in the tubing using the appropriate pressure gradient curves for the well in question ( Figure 1 ).

Figure 1

By joining the calculated tubing head pressure points we obtain the desired THP curve ( Figure 2 ).

Figure 2

For any given constant THP, then, we can use this curve to estimate the flow rate, q. In unrestricted flow, the maximum flow rate is given by the intersection of the THP curve and the surface line pressure upstream of the gathering lines ( Figure 3 ). Calculating the well's flow rate in this manner is referred to as the "bottom-up" method.

Figure 3

Another way of performing the same analysis is the "top-down" method. In this method, we start our calculation with the known value of surface pressure. We then calculate the vertical pressure differences for several flow rates ( Figure 4 ) and join the values to give the bottomhole flowing pressure needed to sustain the various rates.

Figure 4

This required BHP curve is put on a graph with the IPR ( Figure 5 ). The intersection of these curves determines the flow rate for the assumed surface pressure.

Figure 5

In both the top-down and bottom-up methods, it is possible to consider different operating or downhole equipment conditions such as different tubing sizes or GLR. In this way, we may determine optimal flowing conditions for a well by plotting several different performance curves. By analyzing a range of variables the production engineer can then choose the appropriate tubing size or, in planning a gas lift system, the optimum GLR for a particular well so as to achieve an optimal design.

Generally, the wellhead pressure must be sufficient to move oil through flow lines, separators, and other surface equipment. The pressure required at the wellhead depends upon the rate of flow and the nature of the surface equipment. To complete the analysis, we must calculate the pressure-rate relationship for the various pieces of equipment through which production must flow. By plotting in sequence such curves on our IPR diagram, we can calculate the flow potential of any system, and then learn which specific component controls the flow rate.

Example:

A well has the following data: tubing = 7000 ft of 2½-inch gathering line - 2500 ft of 2½-inch separator pressure = 150 psig

= 2000 psig

GLR = 800 Scf/bbl

q = 3000 BOPD, water cut = zero.

We are asked to estimate the well's production rate and to specify the piece of equipment that controls it.

We may estimate the well's performance by calculating the performance of each component in our system moving upstream from the separator. This is a top-down method. We begin by assuming three arbitary flow rates and, with appropriate multiphase horizontal flow rate correlations such as those presented in Volume 1 of Brown's text (1977) we calculate the pressure losses in the gathering line. Because the pressure just upstream of the separator is 150 psig we can use these calculations to plot three tubing head pressure values ( Figure 6 ).

Figure 6

We can plot these values of pressure versus flow rate and obtain the required tubing head pressure curve as shown in Figure 7 .

Figure 7

For the same assumed flow rates, and the given tubing size, we now calculate the vertical pressure increases between the surface and the formation, and add them to the required THP's to give a plot of required bottomhole flowing pressures ( Figure 8 ).

Figure 8

This will be equal to the calculated tubing head pressure for a given rate plus the vertical pressure gain from surface to formation for that rate. Joining these points will give us the required BHP curve shown in Figure 9 .

Figure 9

It represents the effect of production through the wellbore and surface equipment for the specific case of a pressure on the upstream side of the separator equal to 150 psi.

Now we add our inflow performance curve ( Figure 10 ), which runs from our average reservoir pressure of 2000 psi to our pumped-off potential of 3000 BOPD. The point of intersection of the IPR with the required BHP curve is our system design. It represents the flowing rate for the well which will provide 150 psi at the separator. In this case it occurs at about 1800 BOPD.

Figure 10

We can also use the "bottom-up" method of calculating this flow rate. Starting with the IPR we assume flow rates and generate a THP curve for the well in the usual way ( Figure 11 ).

Figure 11

By subtracting calculated pressure losses in the gathering lines for these flow rates from the THP curve we obtain a curve representing the pressure-rate relationship at the downstream side of the gathering line. The pressure at this point is also the pressure at the inlet to the separator. The intersection of this curve and the separator pressure is the flowing rate under the assumed conditions ( Figure 12 ).

Figure 12

By changing any one variable, for example, either the separator pressure, the gathering line size, the tubing size, or the GLR, the flow rate will also change. In order to optimize the design, then, an engineer will determine the system's sensitivity to these variables and see what the most economical use of the equipment will be.

Without a choke in the line, any pressure variations on the surface will directly affect the well's ability to produce. One reason for the installation of a choke is to make it the controlling element in the system.

The installation of a choke will reduce the flow rate and increase the tubing head pressures. Effective control is achieved only when the tubing head pressure is twice the pressure at the upstream point in the gathering system. This is the critical flow requirement.

Installing a choke and using the "top-down" method, we can calculate the tubing head pressure required for criteral flow as being twice the THP that was calculated when we did not have a choke in the system.

This new curve is the pressure upstream of the choke and is the new required THP curve ( Figure 13 ).

Figure 13

Now we add the vertical pressure differences in the tubing to find the required bottomhole flowing pressures. It is the intersection of this last curve with the IPR which determines the system flow rate ( Figure 14 ).

Figure 14

The choke performance can be added to the bottom-up solution we performed earlier. The IPR and THP curves do not change because we have not yet encountered the choke in our flow system. Now we add the effect of the choke which gives a curve below the THP equal to one half of the THP at each flow rate ( Figure 15 ).

Figure 15

This difference or loss in pressure represents the pressure losses through the choke during critical flow. We add a fourth curve representing losses in the gathering line. The point of intersection of this curve with our given separator pressure value is the system production rate if the production rate is controlled by the separator ( Figure 16 ).

Figure 16

The choke size must be chosen to yield a rate equal to or less than this production rate in order for the choke to control the well's production. This limiting condition is shown in Figure 17 .

Figure 17

If the choke size selected had been larger, the choke performance line would have been lower and given a higher flow rate at its point of intersection with the THP curve q2 ( Figure 18 ). The choke calls for a higher flow rate than the separator will allow. Under these conditions, then, the separator will control flow.

Figure 18

The rates and pressures of various choke sizes for this installation can now be calculated and an optimal choke size selected.

Stable flow occurs when fluctuations of pressure and flow rate are dampened and flow rate tends to return to a stable value. We have plotted in Figure 1 the THP and choke performance curve.

Figure 1

A flow rate at point 1, that is q1, is stable because an increase in flow rate to q2, increases bean backpressure to point "A" and reduces the tubing head pressure to point "B." In essence the pressure required by the choke to sustain this flow rate is greater than the THP available at this flow rate. Because an increase in backpressure of the amount A B is imposed on the well, the flow rate tends to decrease from q2 back to q1, the stable rate. In a similar manner a reduction in rate to q3, as shown in Figure 2 , will reduce the required THP, and therefore, reduce backpressure on the formation by the amount A’ B’.

Figure 2

This will increase the flow rate back to q1 and once again the well returns to a stable flow condition.

Unstable flow is also possible. It is illustrated in Figure 3 where a slight decrease in rate below q1 reduces the tubing head pressure below that required by the choke for critical flow.

Figure 3

This causes the flow rate to decrease until the well dies. An increase in flow rate above q1 reduces the backpressure on the formation causing further rate increases until a stable flow rate is reached beyond the maximum point on the THP curve. The maximum point on the THP divides the stable flow region from the unstable region ( Figure 4 ).

Figure 4

This becomes intuitively clear if we draw the IPR curve and then add the Vertical Pressure Loss curve, or VPL. Now we subtract the Vertical Pressure Loss from the IPR and obtain the THP curve ( Figure 5 ).

Figure 5

The THP maximum occurs where the slope of the IPR is equal in magnitude to the slope of the VPL curve ( Figure 6 ).

Figure 6

To the left of that point, any decrease in rate results in increased pressure losses in the tubing due to slippage. The well gradually loses sufficient bottomhole flowing pressure to support flow to surface. To the right of the maximum point, frictional losses dominate and the flow rates stabilize ( Figure 7 ).

Figure 7

Integrated Performance of a Flowing Well

The inflow performance curve describes flow into the wellbore and allows us to predict q’, the pumped-off potential for a given average reservoir pressure. For any given tubing size we may calculate the vertical pressure losses in the tubing, and thus generate the THP curve.

Assuming the installation of a surface choke and that critical flow occurs, we may generate a third curve of pressure and rate downstream from the surface choke. A fourth curve might be added to show pressure losses in the gathering system and, finally, the pressure and rate performance of the separator can be added. The intersection of curves 4 and 5 in Figure 1 is the maximum practical flowing rate qmax for the system.

Figure 1

The choke must be chosen so as to produce at that rate or less, otherwise the separator or other downstream equipment will control production.

In Figure 2 the pressure losses throughout the system are quite apparent.

Figure 2

Starting at the average reservoir pressure and a given flow rate, q, we observe the pressure losses through the formation, through the tubing, across the surface choke and through the surface lines. In a sense the average reservoir pressure drives the whole system and is used up along the way. At each stage, however, there must be sufficient pressure to drive the subsequent systems at that flow rate otherwise flow stops at some point in the system. The component that controls or limits the flow rate determines the system capacity.