WO2015014800A1

WO2015014800A1 - Method of determining well productivity along a section of a wellbore

Info

Publication number: WO2015014800A1
Application number: PCT/EP2014/066190
Authority: WO
Inventors: Jens Henrik Hansen; Kristian Mogensen
Original assignee: Mærsk Olie Og Gas A/S
Priority date: 2013-08-01
Filing date: 2014-07-28
Publication date: 2015-02-05
Also published as: DK201370421A1

Abstract

By the method of determining well productivity along a section (2) of a wellbore (1), a gel (7) is injected into the section, so that it gradually fills up the section of the wellbore from a first end (4) to a second end (5). The viscosity of the gel is sufficiently high in order for the gel to form a gel front (6) which travels in a piston-like manner from the first end (4) to the second end (5) and in order for the gel (7) to at least partly plug up pores of the formation (3). A pressure representative of the bottom hole pressure (BHP) in the wellbore is monitored as the gel front (6) travels, and an injectivity index (II) distribution is estimated or determined as a function of the longitudinal position of the wellbore on the basis of the monitored pressure and a gel injection rate.

Description

Method of determining well productivity along a section of a wellbore

The present invention relates to a method of determining well productivity along a section of a wellbore passing through a formation on the basis of pressure measurements, the section of the wellbore having a first end and a second end, whereby a gel is present in the section of the wellbore during the pressure measurements.

US 6,058,771 discloses a method of determining fluid flow into or out of a borehole during drilling of the borehole using a gelling drilling fluid. However, this method requires stopping the drilling, rotation and pumping operation, and, while keeping the drill string stationary for a period of time, measuring a downhole differential pressure between two points spaced along the longitudinal axial orientation of the borehole. Thereby, a fluid flow in the immediate vicinity of the drill bit may be determined. Consequently, this method is unsuitable for the estimation or determination of variations in well productivity over a longer section of a wellbore. Furthermore, it is a disadvantage that the determination of fluid flow into or out of a wellbore takes place between drilling operations when the wellbore has not been cleaned up. Consequently, the fluid flow determination cannot be taken to reflect the productivity of the wellbore during production of oil.

RU 2 105 871 C1 discloses a method of adaptation of the general injectivity index of specific wellbores to an average injectivity index of a formation by means of the injection of a polymer solution having a certain viscosity in these wellbores.

US 2010/0126717 A1 discloses a method whereby the permeability of the formation of a wellbore may be characterised by pumping a fluid into a sec- tion of the wellbore and measuring resistivity by means of electromagnetic waves. It is mentioned that the section of the wellbore may be isolated by means of a dual packer assembly or by means of a viscous gel. The fluid used for measuring resistivity as well as the viscous gel may be pumped into the wellbore by means of a downhole tool deployed via a wireline or a drill string, or it may be pumped into the wellbore from surface equipment.

It is furthermore known to pump logging tools down into a wellbore in order to perform registration of several parameters, such as pressure and temperature. It has in addition been suggested to deploy self-propelled drones provided with different sensors into a wellbore. These methods, however, rely on being able to convey the surveillance tools to the end of the wellbore and back, either with coiled tubing or with wireline.

Flow profiling is important for optimizing well performance, especially for long horizontal wells.

The object of the present invention is to determine variations in well productivity in the longitudinal direction of the wellbore in a simple manner.

In view of this object, the gel is injected into the section of the wellbore, pref- erably at a constant rate, so that it gradually fills up the section of the wellbore from the first end to the second end of the section of the wellbore, the viscosity of the gel is sufficiently high in order for the gel to form a gel front which travels in a piston-like manner from the first end to the second end of the section of the wellbore and in order for the gel to at least partly plug up pores of the formation, a pressure representative of the bottom hole pressure in the wellbore is monitored as the gel front travels from the first end to the second end of the section of the wellbore, and an injectivity index distribution is estimated or determined as a function of the longitudinal position of the wellbore on the basis of the monitored pressure and a gel injection rate. In this way, as the gel reaches the formation, it may continuously at least partly plug up the pores thereby reducing the overall injectivity index, thereby giving rise to an increase in injection pressure that may be quantified at downhole conditions. During injection, the gel front may be tracked based on knowledge of the diameter of the section of the wellbore and the injection rate. The injectivity reduction during injection may then be determined as a function of the longitudinal position of the wellbore on the basis of the monitored injection pressure and the original injectivity distribution over the section of the wellbore may then be determined on the basis thereof.

In an embodiment, the pressure representative of the bottom hole pressure in the wellbore is monitored continuously. Thereby, a continuous calculation of the estimated instant, reduced injectivity may be possible. In an embodiment, the gel injection rate is registered or monitored continuously. Thereby, it may be possible to vary the injection rate and still keep track of the aggregate volume of gel injected. Furthermore, the calculation of the estimated, instant, reduced injectivity may be facilitated in that it may be based on the relation between the instant injection rate and the difference between the bottom hole pressure and the average reservoir pressure.

In an embodiment, an average wellbore diameter is calculated when the gel front has reached the second end of the section of the wellbore on the basis of a total quantity of gel that has entered the section of the wellbore, and this average wellbore diameter is used for the estimation or determination of the injectivity index as a function of the longitudinal position of the wellbore. Thereby, compensation may be made for the fact that the actual diameter of the wellbore generally may be larger than the size of the drill bit used to drill the hole and furthermore for the fact that some gel may be lost to the forma- tion and consequently the gel front may move slower than predicted in the case that the gel only partly plugs the pores of the formation. In an embodiment, the viscosity of the gel is at least twice the viscosity of water at reservoir conditions, preferably at least 2 centipoises, more preferred at least 3 centipoises and most preferred at least 4 centipoises. Thereby, the gel may form a stable front travelling along the wellbore without mixing substantially with the water present in the wellbore. Thereby, a more precise registration of fractures or the like in the formation may be obtained.

In an embodiment, the weight percent of the gel is less than 15%, preferably less than 10%, more preferred less than 5% and most preferred corresponds to that of NaCI in seawater. Thereby, the gel may form a stable front travelling along the wellbore without mixing substantially with the water present in the wellbore. Thereby, a more precise registration of fractures or the like in the formation may be obtained.

In an embodiment, the gel is pumped into the section of the wellbore at a rate less than 500 cubic metres per day, preferably at a rate less than 400 cubic metres per day, and most preferred at a rate less than 300 cubic metres per day. Thereby, a more accurate flow profile may be obtained because the gel may move slower and the variations of the measured pressure signal may be easier to detect. Furthermore, the accuracy of the predicted gel rheology, i.e. viscosity, may be improved. Finally, the risk of unintentional fracturing may be reduced as a result of a lower pressure in the wellbore. In an embodiment, the gel injection rate is reduced continuously or stepwise as the gel front travels from the first end to the second end of the section of the wellbore. Thereby, it may be prevented that the pressure in the wellbore increases inappropriately as the gel front advances and more and more fractures are plugged by the gel. Consequently, the risk of unintentional fractur- ing may be reduced as a result of a lower pressure in the wellbore. Further- more, the accuracy of the predicted gel rheology, i.e. viscosity, may be improved as a result of a reduced pressure.

In an embodiment, the gel injection rate is controlled on the basis of the monitored pressure representative of the bottom hole pressure, preferably so that said monitored pressure is maintained substantially constant or is maintained below a limit value. Thereby, the above-mentioned advantages regarding reduced risk of unintentional fracturing and better accuracy of the predicted gel rheology may be even better ensured.

In an embodiment, the injectivity index is estimated or calculated on the basis of an average value of the reservoir pressure in the formation around the wellbore from the first end to the second end of the section of the wellbore. Thereby, the estimation or calculation of the injectivity index may be facili- tated. Although the reservoir pressure in reality may be non-uniform across the well, it may be sufficient to use an average value for the entire wellbore section because changes in injectivity index may be more important to detect than absolute values. Presence of a fracture may be identified regardless of the value of the reservoir pressure.

In an embodiment, the gel is adapted to decompose after it has been forced into the section of the wellbore, preferably within less than 5 days, more preferred within less than 3 days, even more preferred within less than 2 days, and most preferred within less than 24 hours. Thereby, the initial injectivity profile along the well may be restored after a period of time corresponding to the time it may take to fill the wellbore with gel. Thus, the gel may provide details about the positions where a significant amount of fluid enters or exits from the reservoir without permanently altering the flow distribution. In a structurally advantageous embodiment, the first end of the section of the wellbore is a heel of the wellbore and the second end of the section of the wellbore is a toe of the wellbore. In an embodiment, the pressure representative of the bottom hole pressure is monitored by means of a dedicated bottom hole gauge. Thereby, a more precise measurement may be achieved.

In an embodiment, a logging tool or drone provided with data registration de- vices is propelled by means of the gel front from the first end to the second end of the section of the wellbore. Thereby, even more data about the wellbore may be collected as the gel front is advanced through the wellbore.

In an embodiment, the logging tool or drone registers the position of the gel front as a function of time and this information is used for the estimation or determination of the injectivity index as a function of the longitudinal position of the wellbore. Thereby, even more precise information on the position of the gel front at a specific point in time may be achieved and thereby an even more precise estimation or determination of the injectivity index as a function of the longitudinal position of the wellbore may be achieved.

In an embodiment, the logging tool or drone registers the pressure at the gel front as a function of time and this information is used for the estimation or determination of the injectivity index as a function of the longitudinal position of the wellbore. Thereby, even more precise information on the pressure in the wellbore at the gel front at a specific point in time may be achieved and thereby an even more precise estimation or determination of the injectivity index as a function of the longitudinal position of the wellbore may be achieved. The invention will now be explained in more detail below by means of examples of embodiments with reference to the very schematic drawing, in which

Fig. 1 illustrates a first stage of the method according to the invention, whereby gel is pumped into the well;

Fig. 2 illustrates a second stage of the method according to the invention, whereby gel reaches the reservoir and plugs pores of a formation; Fig. 3 illustrates a third stage of the method according to the invention, whereby gel reaches an end of the wellbore and pumping is stopped;

Fig. 4 is a graph illustrating calculation of injectivity index (II) reduction versus time using pressure measurements;

Fig. 5 is a graph illustrating calculation of original injectivity index (II) distribution versus length; and

Figure 6 illustrates another embodiment of the method according to the in- vention, whereby a tool is being pushed by the gel front towards an end of the wellbore, while the gel temporarily plugs pores of the formation.

Figs. 1 to 3 illustrate a method according to the invention of determining well productivity along a section 2 of a wellbore 1 passing through a formation 3 of a reservoir on the basis of pressure measurements. In the embodiment illustrated, the section 2 of the wellbore 1 has a first end 4 in the form of a heel and a second end 5 in the form of a toe. The first end 4 in the form of the heel connects to surface 1 1 by means of a vertical tubing 10 in a well- known manner. As illustrated in the figures, a gel 7 is injected into the section 2 of the well- bore 1 so that it gradually fills up the section 2 of the wellbore 1 from the first end 4 to the second end 5. The viscosity of the gel 7 is chosen sufficiently high in order for the gel 7 to form a gel front 6 which travels in a piston-like manner from the first end 4 to the second end 5 of the section 2 of the wellbore 1 . Additionally, the viscosity of the gel 7 is chosen sufficiently high in order for the gel to at least partly plug up pores of the formation 3 as it travels through the wellbore 1 . Thereby, the gel 7 may at least temporarily reduce or block the flow of wellbore fluids through the pores of the formation 3 in the part of the section 2 of the wellbore 1 filled by the gel 7, but may be designed to disintegrate again after a certain amount of time, as described below.

The viscosity of the gel 7 may for instance be at least twice the viscosity of water at reservoir conditions, preferably at least 2 centipoises, more pre- ferred at least 3 centipoises and most preferred at least 4 centipoises.

Furthermore, the weight percent of the gel 7 may be less than 15%, preferably less than 10%, more preferred less than 5% and most preferred corresponds to that of NaCI in seawater. By way of example only, the gelling agent used to form the gel 7 may be of the type SGA-II (registered trademark) from Halliburton.

In some embodiments, the gel may comprise polymers, e.g. addition polymers such as homo- and/or or copolymers of polyvinyl alcohol (PVA), poly- acrylamine (PA), polyacrylamine (PA), hydrolysed poly-acrylamine (HPAM), partially hydrolysed polyacrylamine (PHPA), polyvinyl pyrrolidone (PVP) , and the like.

In some embodiments, the gel may comprise a gelling system, e.g. an inor- ganic gelling system such as a Delayed Gelation System (DGS), for example a partially hydrolysed aluminium chloride system, or a colloidal dispersion gel (CDG).

According to the invention, a pressure representative of the bottom hole pressure (BHP) in the wellbore 1 is monitored, preferably continuously, as the gel front 6 travels from the first end 4 to the second end 5 of the section 2 of the wellbore 1 .

Said pressure representative of the bottom hole pressure may be monitored by means of a dedicated bottom hole gauge 8 which may render a rather precise measurement of the bottom hole pressure. However, said pressure may just as well be monitored by measuring the tubing head pressure (THP) by means of a tubing head gauge 9. The pressure generated by the fluid column in the tubing 10 may then be calculated and added to the measured tub- ing head pressure in order to render the pressure representative of the bottom hole pressure.

According to the invention, on the basis of the monitored pressure and a gel injection rate, an injectivity index distribution may be estimated or determined as a function of the longitudinal position of the wellbore 1 as described in the following and illustrated in Figs. 4 and 5. The gel injection rate may be fixed during injection, for instance by setting a fixed rotational speed of a positive displacement pump, or the gel injection rate may be varied during injection and monitored, preferably continuously.

As the gel 7 is pumped into the wellbore 1 and reaches the formation, it gradually at least partially plugs up the pores of the formation 3, thereby reducing the overall injectivity index (II) as the gel 7 moves further along the wellbore 1 in a piston-like manner. From classical reservoir engineering, given that single-phase conditions prevail, we can express the reduced injectivity index (II) of the wellbore, corresponding to the injectivity index (II) of the part of the section 2 of the wellbore 1 that is still not filled up by gel 7, as follows:

Q (m³/d)

// (m³/d/Pa) = (Eq. 1)

BHP (Pa) - P_res (Pa) wherein Q is the injection or pumping rate of the gel, BHP is the bottom-hole pressure and P_res is the average reservoir pressure around the wellbore 1 .

The injection rate Q of the gel corresponds to the rate at which wellbore fluids leaves the wellbore through pores of the formation 3. The average reservoir pressure P_res around the wellbore 1 is assumed substantially constant during the job. If for instance the injection rate Q is maintained constant, the bottom- hole pressure BHP may increase over time and hence indicate a reduction in injectivity index II. On the other hand, if for instance the bottom-hole pressure BHP is maintained constant by decreasing the injection rate Q over time, this may also indicate a reduction in injectivity index II.

Figures 1 -3 illustrate the gradual plugging of the wellbore 1 by the gel 7. During the pumping, the gel front 6 can be tracked based on knowledge of the diameter of the section 2 the wellbore 1 and the pumping rate Q. It is well- known that the actual diameter of for instance an open-hole section of a wellbore is larger than the size of the drill bit used to drill the open-hole. When the gel 7 has reached the second end 5 the section 2 of the wellbore 1 , the pumping may be stopped. In the case that the second end 5 is actually the termination of the wellbore 1 , the pressure in the wellbore 1 may increase significantly as the gel 7 reaches the second end 5 and this may therefore easily be detected so that the pumping may be stopped. Simple mass balance will then indicate the average wellbore diameter, ID, as follows:

V π x ID²(m²) x L(m)

Total Gel Pumped (m^d) = ^ Q(t) x At = V_tubing H (Eq. 2) wherein Q is the injection or pumping rate of the gel 7, t is the time, L is the total length of the section 2 of the wellbore 1 and V_tUbin_g is the volume of the vertical tubing 10.

Subsequently, knowing the average wellbore diameter ID may improve the estimate of the position of the gel front 6 at any given time by isolating L in Eq. (2). The original II of a particular interval equals the drop in II while pumping the gel 7 across the given interval. Figures 4 and 5 illustrate results from such an analysis. This method is particularly suitable for detecting fractures and other heterogeneities causing uneven flow distribution.

Pumping the gel 7 at a relatively low rate, for instance 2,000 bbl/d (approximately 320 m³/day), may be beneficial in that a more accurate flow profile may be obtained because the gel may move slower and the variations of the measured pressure signal may be easier to detect. Furthermore, the accu- racy of the predicted gel rheology, i.e. viscosity, may be improved. Finally, the risk of unintentional fracturing may be reduced as a result of a lower pressure in the wellbore. Advantageously, however, the gel 7 may be pumped into the section 2 of the wellbore 1 at a rate less than 500 cubic metres per day, preferably at a rate less than 400 cubic metres per day, and most preferred at a rate less than 300 cubic metres per day.

In fact, the gel 7 may be pumped at a constant rate, or the gel injection rate may be reduced continuously or stepwise as the gel front 6 travels from the first end 4 to the second end 5 of the section 2 of the wellbore 1 . Further- more, the gel injection rate may also be controlled on the basis of the monitored pressure representative of the bottom hole pressure, preferably so that said monitored pressure is maintained substantially constant or is maintained below a limit value.

Specifically it is noted that for wells that inject on vacuum (i.e. no observable THP because the reservoir pressure is lower than the hydrostatic water column), gradual plugging may cause the THP to increase above zero. In this case, in order to register the pressure variations, it is advantageous to use a well with a functioning down-hole gauge or alternatively install a wireline- retrievable gauge during the pumping.

The gel 7 may be adapted to decompose after it has been forced into the section 2 of the wellbore 1 within less than 5 days, preferably within less than 3 days, more preferred within less than 2 days, and most preferred within less than 24 hours.

As an example, a reservoir section with a total length L = 20,000 feet (approximately 6,096 metres) and an average diameter ID = 8.5" (approximately 216 mm) would contain some 1 ,500 bbl (approximately 240 cubic metres), which means that filling up the wellbore would require some 18 hours (with a pump rate of 2,000 bbl/d (approximately 320 cubic metres per day)) during which the inflow profile can be determined. This means that the gel 7 in this case should be stable for at least 24 hours.

It is preferable that neighbouring wells are shut-in during the operation to avoid interference with the pressure interpretation.

A complicating factor may arise if the gel 7 only partly plugs the pores of the formation 3. This may imply that some gel 7 will be lost to the formation 3 and that the gel front 6 may move slower than predicted. This, in turns, means that the conversion from time to position may be associated with some uncertainty. However, when the gel 7 reaches the second end 5 of the wellbore 1 , there is no more formation 3 to plug and the injectivity decrease as a function of time may change and approach zero. The volume of gel 7 required to reach this point takes care of the combined effect of chemical loss to the formation 3 and over-gauge wellbore hole size. Therefore, when isolating the the average wellbore diameter, ID, in Eq. (2) above, a theoretical average wellbore diameter is obtained that takes into account this combined effect of chemical loss to the formation 3 and over-gauge wellbore hole size, when subsequently used to estimate of the position of the gel front 6 at any given time by isolating L in Eq. (2).

Another complicating factor is the fact that the reservoir pressure may be non-uniform in the longitudinal direction of the wellbore 1 . However, it may be sufficient to use an average value for the entire wellbore section 2 because injectivity index changes may be more interesting to judge than absolute values. Presence of a fracture may be identified regardless of the value of the reservoir pressure. The gel 7 may be designed to decompose after a given amount of time, thereby restoring the initial injectivity profile along the wellbore. Thus, the gel 7 may provide details about the positions where a significant amount of fluid enters or exits from the reservoir without permanently altering the flow distribution.

As mentioned above, a pressure representative of the bottom hole pressure may be monitored by means of a dedicated bottom hole gauge 8 or by measuring the tubing head pressure (THP) by means of a tubing head gauge 9 and converting this appropriately. However, alternatively, a logging tool or drone 12 provided with data registration devices may be propelled by means of the gel front 6 from the first end 4 to the second end 5 of the section 2 of the wellbore 1 , as illustrated in Fig 6. The logging tool or drone 12 may register the pressure at the gel front 6 as a function of time, and this information may be used for the estimation or determination of the injectivity index as a function of the longitudinal position of the wellbore. Furthermore, the logging tool or drone 12 may alternatively or additionally register the position of the gel front 6 as a function of time and this information may be used for the estimation or determination of the injectivity index as a function of the longitudinal position of the wellbore 1 .

Claims

1 . A method of determining well productivity along a section (2) of a wellbore (1 ) passing through a formation (3) on the basis of pressure measurements, the section (2) of the wellbore (1 ) having a first end (4) and a second end (5), whereby a gel (7) is present in the section (2) of the wellbore (1 ) during the pressure measurements, characterised by that the gel (7) is injected into the section (2) of the wellbore (1 ), preferably at a constant rate, so that it gradually fills up the section (2) of the wellbore (1 ) from the first end (4) to the second end (5) of the section (2) of the wellbore (1 ), by that the viscosity of the gel (7) is sufficiently high in order for the gel (7) to form a gel front (6) which travels in a piston-like manner from the first end (4) to the second end (5) of the section (2) of the wellbore (1 ) and in order for the gel (7) to at least partly plug up pores of the formation (3), by that a pressure representative of the bottom hole pressure (BHP) in the wellbore (1 ) is monitored as the gel front (6) travels from the first end (4) to the second end (5) of the section (2) of the wellbore (1 ), and by that an injectivity index (II) distribution is estimated or determined as a function of the longitudinal position of the wellbore (1 ) on the basis of the monitored pressure and a gel injection rate.

2. A method of determining well productivity according to claim 1 , whereby the pressure representative of the bottom hole pressure (BHP) in the wellbore (1 ) is monitored continuously.

3. A method of determining well productivity according to claim 1 or 2, whereby the gel injection rate is registered or monitored continuously.

4. A method of determining well productivity according to any one of the preceding claims, whereby an average wellbore diameter (ID) is calculated when the gel front (6) has reached the second end (5) of the section (2) of the wellbore (1 ) on the basis of a total quantity of gel (7) that has entered the section (2) of the wellbore (1 ), and whereby this average wellbore diameter (ID) is used for the estimation or determination of the injectivity index (II) as a function of the longitudinal position of the wellbore (1 ).

5. A method of determining well productivity according to any one of the preceding claims, whereby the viscosity of the gel (7) is at least twice the viscosity of water at reservoir conditions, preferably at least 2 centipoises, more preferred at least 3 centipoises and most preferred at least 4 centipoises.

6. A method of determining well productivity according to any one of the preceding claims, whereby the weight percent of the gel (7) is less than 15%, preferably less than 10%, more preferred less than 5% and most preferred corresponds to that of NaCI in seawater.

7. A method of determining well productivity according to any one of the preceding claims, whereby the gel (7) is pumped into the section (2) of the wellbore (1 ) at a rate less than 500 cubic metres per day, preferably at a rate less than 400 cubic metres per day, and most preferred at a rate less than 300 cubic metres per day.

8. A method of determining well productivity according to any one of the preceding claims, whereby the gel injection rate is reduced continuously or stepwise as the gel front (6) travels from the first end (4) to the second end (5) of the section (2) of the wellbore (1 ).

9. A method of determining well productivity according to any one of the claims 2 to 8, whereby the gel injection rate is controlled on the basis of the monitored pressure representative of the bottom hole pressure (BHP), preferably so that said monitored pressure is maintained substantially constant or is maintained below a limit value.

10. A method of determining well productivity according to any one of the preceding claims, whereby the injectivity index (II) is estimated or calculated on the basis of an average value of the reservoir pressure in the formation (3) around the wellbore (1 ) from the first end (4) to the second end (5) of the sec- tion (2) of the wellbore (1 ).

1 1 . A method of determining well productivity according to any one of the preceding claims, whereby the gel (7) is adapted to decompose after it has been forced into the section (2) of the wellbore (1 ), preferably within less than 5 days, more preferred within less than 3 days, even more preferred within less than 2 days, and most preferred within less than 24 hours.

12. A method of determining well productivity according to any one of the preceding claims, whereby the first end (4) of the section (2) of the wellbore (1 ) is a heel of the wellbore and the second end (5) of the section (2) of the wellbore (1 ) is a toe of the wellbore (1 ).

13. A method of determining well productivity according to any one of the preceding claims, whereby the pressure representative of the bottom hole pressure (BHP) is monitored by means of a dedicated bottom hole gauge (8).

14. A method of determining well productivity according to any one of the preceding claims, whereby a logging tool or drone (12) provided with data registration devices is propelled by means of the gel front (6) from the first end (4) to the second end (5) of the section (2) of the wellbore (1 ).

15. A method of determining well productivity according to claim 14, whereby the logging tool or drone (12) registers the position of the gel front (6) as a function of time and that this information is used for the estimation or deter- mination of the injectivity index (II) as a function of the longitudinal position of the wellbore (1 ).

16. A method of determining well productivity according to claim 14, whereby the logging tool or drone (12) registers the pressure at the gel front (6) as a function of time and that this information is used for the estimation or deter- mination of the injectivity index (II) as a function of the longitudinal position of the wellbore (1 ).