Subrat,

Good question. Cheng (1990) presented a form of Vogel’s equation for horizontal wells a method to estimated PI that is based on the results from a numerical simulator. Which I recommend you use since it has less variables. 

Bendakhlia and Aziz (1989) used a reservoir model to generate IPRs for a number of wells and found that a combination of Vogel and Fetkovich equations. As far as I am aware, the equation are not suitable for a predictive mode (i.e for new reservoir pressure after declining) due to the fact that the parameters for estimating PI (n and V) for a horizontal well are function of pressure and RF. 

The below discussion is for productivity index calculation in general for your reference.

For IPR construction, there are two regions: Above the bubble point pressure and below the bubble point. For the region above the bubble point, the curve is a straight line: J = Q/(Pr-Pwf). For horizontal and vertical wells, IPR is mainly Darcy law with eliminating the constant parameters and focusing on the parameters that change during pseudo-steady state flow (for pseudo-steady state flow, the draw-down decline results in production rate decline and they are related to each other with a constant that is J i.e PI). You can start calculating J using the permeability, thickness, viscosity, Bo (i.e using Darcy law with trying to solve for: J=Q/(Pr-Pwf) = KA/viscosity*L). However, since this data has to be accurate and geological uncertainty is high, usually J is estimated from production tests and/or PLT data in addition to shutin pressure value for Pr. One caution is you need to be little careful with PLT for horizontal wells since data usually is not as accurate as for the vertical/deviated wells. I am adding SLB link below.

As for below the bubble point, you can use Vogel's IPR for two phase (oil + gas). If you have three phase flow then you can use: Wiggins which is similar to Vogel but with new equation for the water (third component). There are different conditions to be applied: i.e (both Pwf and Pr above bubble point, both Pr and Pwf at or below bubble point (i.e. saturated reservoirs) and Pr above bubble point while Pwf is below or at bubble point).

As for your question about multi-layered reservoir, the approach is: plotting bottomhole pressure vs. rate for each layer of the system, however you need to correct the pressures to a common datum. You know that the total flow rate has to equal the summation of flow rates from the layers. Apply the same above principles and you should obtain the composite IPR curve.

More can be found in Reservoir Engineering Handbook by Tarik Ahmad in Chapter: Oil Well Performance. 

I think you can find more about this in the following resources:

1. http://petrowiki.org/Reservoir_pressure_data_interpretation

2. http://petrowiki.org/Oil_well_performance

3. Horizontal well: Generalized Inflow Performance Relationship (IPR) for Horizontal Wells SPE-165691-MS.

4. Composite IPR curves are discussed in more details here:

file:///C:/Users/Khaldoon/Downloads/production-II-%20composite%20IPR-multilatral%20TPR.pdf

5. PLT in horizontal wells: http://www.slb.com/services/production/production_logging/flow_scanner.aspx

I hope this helps.

All the best and please let me know if you need anything else in this.

Sir, thanks for answering my question. Sir,I have a doubt - you said in multi layered reservoir you said pressures are corrected to a common datum. Sir, could please explain this concept. Are these pressures corrected to remove the induced pressure gradients around the wellbore ?

Subrat,

So if you follow the following steps, I think you should be able to generate the composite IPR:

1. For a vertical well, for each layer, based on PLT/Test data, find PI (J) and call that J* based on: J=Q/(pr-pwf) or Vogel's equations depending on your flow case (the 3 cases I have mentioned where everything above bubble point, saturated reservoir or if Pwf is below bubble point while Pr is above the bubble point). As for horizontal well, use: Cheng (1990) which is a form of Vogel’s equation. You have to estimate Qomax from a test and then you calculated Qo at assumed pwf values.

2. For each layer, try to find the rate that results from an assumed Pwf. Example, assume Pwf = 1000 psi, you have the Pr (layer average pressure) and you can plug in J and solve for Q depending on the flow case you are in as mentioned in 1. (For vertical well use Vogel, Fetkovich, Standing .. etc and for horizontal use Cheng (1990))

3. Add the rates based on the conservation of mass and the assumption density is not changing much within the well. Mass1 + Mass 2 = Mass @ Tubing head so density1*q1+density2*q2 = density@th*q@th which will result if we eliminate density by: q1+q2=q@th. 

4. Draw for that Pwf what you have for q from the calculations in the above 3 steps.

Now you should have a composite IPR curve. 

What I meant by correcting pressure to one datum, you should always have one datum from beginning of the reservoir life till its end. Because imagine if you change that datum and/or you have different measurements at different depths, you will have different Pr and thus your J will be changing which supposed not to be the case. So, fix one datum, and preferably take it at the middle of highest perforation. Then, correct all the layers pressure to that datum and use it as your Pr1@datum, Pr2@datum ..etc When you get pressure data from tests and/or reservoirs models, you will have them at different depths and thus you should correct them all the time to that datum using the oil density (or gas density depending on your reservoir fluid).

Also, your pwf we assumed of 1000 psi in the above example should be assumed at the same datum you assumed for Pr. During your PLT, pwf is measured directly at each layer and you should correct everything to the same datum you have chosen for Pr. As you can imagine that if your Pr is at different depth from your pwf then you are not comparing like to like. Thus, your pwf should be corrected to the same datum as well and you calculate J for each layer (or reservoir) and then generate the composite curve.

In case of horizontal well, you have one depth anyway (if it is perfectly horizontal so the correction should be spot on and easy).

By this, I hope that we covered all possibilities by assumption of pwf at certain datum and using Pr estimated from shut-in test at the same datum.

One thing I can recommend in additon to reading Oil Well Performance in Reservoir Engineering Handbook is reading through chapter 4 of Well Productivity Handbook. It is available on google books, however I am not sure if it discuses the datum thing, but for sure it discusses the composite IPR.

https://books.google.com.my/books?id=DbVEAwAAQBAJ&pg=PA11&dq=Well+Productivity+Handbook&hl=en&sa=X&ei=JzCEVfDXGZGLuATxs5qQBg&ved=0CBwQ6AEwAA#v=onepage&q=Well%20Productivity%20Handbook&f=false

I hope this helps and please let me know if you need any more help in this issue.

All the best Mr. Subrat.

Kindly go through the link

https://www.onepetro.org/conference-paper/SPE-165691-MS

Dear Kaldoon, 

Thanks a lot for your extensive answers, with nice references. You mention the following reference, but I think you made a mistake: 

file:///C:/Users/Khaldoon/Downloads/production-II-%20composite%20IPR-multilatral%20TPR.pdf

Googling for the file name also does not return any results. Can you please update this link? 

Thanks.