dda book 05 wellbore models

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Dynamic Data Analysis - v4.12.03 - © KAPPA 1988-2012 Chapter 5 – Wellbore models - p167/558

5 – Wellbore modelsOH – OSF – DV

5.A Introduction

Until we are able to beam the fluid directly from the pore space into the ship cargo bay we will

need to use this route called the wellbore. Wellbore effects are seen very differently,

depending where you stand:

For the Pressure Transient Analysts anything related to the wellbore is a nuisance. Wellbore

effects will spoil the early part of the pressure response, and may even persist throughout

the whole test or shut-in survey.

So to the PT-Analyst Wellbore Effects = BAD.

Production Analysts are a little luckier, because they work on a time scale where transient

effects are not that important, and addressing wellbore effects amounts to connecting a lift

curve. In fact, playing with the lift curves and implementing ‘what if’ scenarios is part of

their jobs.

So to the Production Analyst Wellbore Effects = OK.

This Manichean split can be presented another way:

The steady-state component of wellbore effects is a key element of the well productivity. It

may be modeled using lift curves, or VLP curves, and this in turn requires flow correlationsthat are present in both Production Logging and Well Performance Analysis, a.k.a. Nodal

Analysis™ (Trademark of Schlumberger).

Correction to datum may be either applied to the data in order to correct the real pressure

to sandface, or integrated in the model in order to simulate the pressure at gauge level.

Correction to datum and integration of VLP curves are detailed in the PTA (QA/QC) and the

Well Performance Analysis chapters of this book.

The transient component of wellbore effects often ruins the life of the PT-Analyst. The

action at the origin of a sequence of flow (opening and shut-in of a valve, change of a

choke) is occurring at a certain distance from the sandface, and any wellbore volume

between the operating point and the sandface acts as a cushion. This induces a delay

between what we want to see and what effectively occurs at the sandface.

In welltest operations, it is highly recommended to reduce this nuisance as much as

possible by means of downhole shut-in tools.

In Production Analysis it is not much of an issue, as transient wellbore effects occur at a

time scale of little interest for rate decline.

This chapter deals with the modeling of some of the simplest transient wellbore models,

and is mainly applicable to Pressure Transient Analysis only.




5.B Constant Wellbore storage

The simplest wellbore model is the constant wellbore storage

As introduced in the ‘Theory’ chapter, the wellbore storage introduces a time delay between

the rate we impose at the operating point (typically the choke manifold at surface) and the

sandface rate. The wellbore storage equation was introduced in the ‘Theory’ chapter:

Wellbore storage equation:t

pC qBq

wf

sf

24

Not surprisingly, the constant wellbore storage model assumes that the wellbore storage factor

C is constant. The below figure illustrates the behavior of the sandface rate during the opening

and shut-in of a well.

Fig. 5.B.1 – Wellbore storage

5.B.1 Loglog analysis

Fig. 5.B.2 with various constant wellbore storage constants is illustrated below. Pure wellbore

storage is characterized by the merge of both Pressure and Bourdet Derivative curves on the

same unit slope.

At a point in time, and in the absence of any other interfering behaviors, the Derivative will

leave the unit slope and transit into a hump which will stabilize into the horizontal linecorresponding to Infirnite Acting Radial Flow. The form and the width of the hump is governed

by the parameter group S Ce2

, where S is the Skin factor.

The horizontal position of the curve is only controlled by the wellbore storage coefficient C.

Taking a larger C will move the unit slope to the right, hence increase the time at which

wellbore storage will fade. More exactly, multiplying C by 10 will translate the curve to one log

cycle to the right.




Fig. 5.B.2–

Wellbore storage loglog response

5.B.2 Specialized analysis (Cartesian plot)

A unique slope on the loglog plot corresponds to a linearity of the pressure response on a

Cartesian plot. This Cartesian plot may show either P or P vs. t.

Below is shown a Cartesian plot of pressure versus time.

Fig. 5.B.3 – Cartesian plot of pressure vs. elapsed time

The early time straight line corresponding to the pure wellbore storage is given by:

Wellbore Storage Straight line: t mt C

qB p

24

So one can get the wellbore storage constant with:

Specialized plot result:m

qBC

24




5.B.3 Sensitivity analysis on the wellbore storage coefficient

The figure below presents the response with wellbore storage values, C of 0.001, 0.003, 0.01,

0.03 and 0.1 (stb/psi).

The value of C has a major effect, which is actually exaggerated by the logarithmic time scale.

You can see on the linear history plot that all responses seem to be the same, however.

When the influence of wellbore storage is over both the pressure change and the derivative

merge together. Wellbore storage tends to masks infinite acting radial flow on a time that is

proportional to the value of C. Wellbore storage will also tend to mask other flow regimes that

can be present in a well test. Early time well responses such as linear, bi-linear, spherical and

hemispherical flow will disappear if the storage effect is considerable. Effects of heterogeneous

reservoirs can also be masked by wellbore storage. The wellbore storage effect on other well

and reservoir models are covered in the individual chapters of these models.

Wellbore storage does not affect the late time pseudo steady state response.

Fig. 5.B.4 – Effect of wellbore storage, loglog plot

Fig. 5.B.5 – Effect of wellbore storage, semilog and history plot




5.C Changing wellbore storage

The most frequent case of changing wellbore storage is related to the compressibility change

of the wellbore fluid.

A classic example is gas. When the well is flowing the pressure in the wellbore will decrease,

and the gas compressibility will increase. In this fixed volume this will result in an increase of

the wellbore storage parameter. The opposite will occur during the shut-in, where the increase

of pressure will result in a decrease of the wellbore storage. Though it occurs in any gas test,

this behavior will be visible, and become a nuisance, in the case of tight gas, where the high

pressure gradient in the formation results in a high pressure drop in the wellbore.

Another typical example is an oil well flowing above bubble point pressure in the reservoir. At a

stage (sometimes immediately) there will be a point in the wellbore above which the pressure

gets below bubble point. In this place the oil compressibility will be progressively dominated by

the compressibility of the produced gas, hence an increase of the wellbore storage which will

evolve in time.

In both cases, the wellbore storage will be increasing during the production and decreasing

during the shut-in.

Other sources of changing wellbore storage may be various PVT behaviors, change of

completion diameter of a rising or falling liquid level, phase redistribution, falling liquid level

during a fall-off, etc.

In some cases the wellbore effect will be so extreme that any modeling is hopeless. In this

case the engineer will focus on matching the derivative response after the wellbore effect has

faded, accepting that the early time response cannot be matched and may induce a

(cumulative) incorrect value of the skin factor.

There are three main ways today to model changing wellbore storage:

Analytical, time related wellbore storage

PVT correction using the pseudotime function and a constant storage value

Numerical, pressure dependent storage model

5.C.1 Analytical models

Most analytical formulations of changing wellbore storage involve an initial value of wellbore

storage Ci, a final value Cf , some assumption for a transition function (Hegeman, Fair, etc) and

a time at which this transition occurs. The main characteristic of these models is that thetransition occurs at a given value of t, and is NOT related to the value of the pressure.

The figures below illustrate increasing and decreasing wellbore storage as modeled by the

Hegeman model of changing wellbore storage.

The matching consists in setting the wellbore storage straight line on the FINAL value of

wellbore storage, pick a second position corresponding to the INITIAL value of storage, and

then pick the median time when the transition occurs. The initial model generation will seldom

match the response perfectly, but this model, combined with a robust nonlinear regression,

has the capacity to adjust to virtually any single trend early time response.




Fig. 5.C.1 – Increasing storage Fig. 5.C.2 – Decreasing storage

In practice, the Hegeman model is sharper and has more capabilities to match real data. Thisis related to the choice of transition function and does not mean that this model is physically

better. Actually it does not mean that ANY of these models are correct, and they should be

used with care for the following reasons:

The models are just transfer functions that happen to be good at matching real data. There

is no physics behind them. They may end up with an initial, final storage and transition

time that makes no physical sense.

These models are time related. There will be a wellbore storage at early time and a

wellbore storage at late time. This is not correct when the model is pressure related. In the

case of production, the real wellbore storage at early time will correspond to the storage at

late time of the build-up, and the reverse. So, the superposition of a time related solution

will be incorrect on all flow periods except the one on which the model was matched. This

aspect is often ignored and/or overlooked.

These models are ‘dangerous’ to the extent that they work beautifully to match ‘anything

that goes wrong’ at early time, even when the use of such model is not justified. They are

the early time version of the radial composite model at intermediate time. Actually,

combining changing wellbore storage and radial composite will match any rubbish data.

5.C.2 Combining pseudo-time and a constant storage model

In a tight reservoir, the pressure changes can be large and the assumption that ct is constant

leads to a distortion in the early time of the loglog plot. The response can in most cases be

matched using the changing wellbore storage option described above. However the changes in

ct can also be included in the diffusion equation and pseudo-time can be used during the

extraction of the period to be analyzed. Pseudo-time is defined by

Pseudo-time: d p I t t t

wf ps 0

where

pc p

p I t

1




The following figures show a loglog response before and after pseudo time correction. The useof pseudo time is detailed in the chapter on ‘Gas’.

Fig. 5.C.3 – Without pseudo-time Fig. 5.C.4 – With pseudo-time

There are two drawbacks to this approach:

This method modifies, once and for all, the data to match the model, and not the opposite.

This excludes, for example, the possibility of comparing several PVT models on the same

data. The method was the only one available at the time of type-curve matching, where

models were limited to a set of fixed drawdown type-curves.

In order to calculate the pseudotime function one needs the complete pressure history.When there are holes in the data, or if the pressure is only acquired during the shut-in, it

will not be possible to calculate the pseudotime from the acquired pressure. There is a

workaround to this: use the pressures simulated by the model, and not the real pressures.

This amounts to the same thing once the model has matched the data, and there is no

hole. However it is a bit more complicated for the calculation, as the pressure at a

considered time requires the pseudotime function, and vice versa.

5.C.3 Numerical pressure dependent wellbore storage

The principle is to use a wellbore model which, at any time, uses the pressure to define thewellbore storage parameter. In order for the model to be stable, the wellbore storage has to

be calculated implicitly at each time step. As the problem is not linear, this can only be done

using a non linear model.

This is by far the most relevant way to simulate pressure related wellbore storage. The figure

below illustrates a buildup matched with the changing wellbore storage model (Hegeman), the

extracted buildup corrected for pseudo time and matched with this model, and the match with

the non linear numerical model with pressure dependent wellbore storage.




Fig. 5.C.5 – Changing wellbore storage match

Fig. 5.C.6 – Pseudo time match

Fig. 5.C.7 – Match with non linear numerical model:Pressure dependent wellbore storage

dda book 05 wellbore models

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