COURSE: WELL TESTING PRINCIPLES

Ebrahim

Piramoon

Ahwaz

1387

IN THE NAME OF GOD

PRINCIPLES OF WELL TESTING

•

The aim of well testing is to get information about a well and a reservoir.

•

During a well test a transient pressure response is created by a temporary change in production rate.

•

The well response is usually monitored during a relatively short period of time compared to the life of the reservoir.

•

For well evaluation, tests are frequently achieved in less than two days. But in the case of reservoir limit testing, several months of pressure data may be needed.

•

In most cases, the flow rate is measured at surface while the pressure is recorded down-hole.

•

During flow time, the drawdown pressure response ΔP is expressed: ΔP =Pi-P(t)

•

During shut-in time, the build-up pressure change ΔP is estimated from the last flowing pressure P(Δt=0): ΔP =P(t)-P(Δt=0)

The pressure response is analyzed vs. the elapsed time Δt, from the start of the period

(time of opening or shut-in)

drawdown Build-up

Pi

Δp(Δt=0)

Well test objectives:•

Associated to the geology and geophysics, well test results are used for prediction of the field behavior and fluid recovery to different operating scenarios.

1)Exploration well: on initial wells, well testing is used to confirm the exploration assumptions and to establish a first production forecast: nature and rate of produced fluids, initial pressure (RFT, MDT), reservoir properties.

2)Appraisal well: the previous well and reservoir description can be refined (well productivity, bottom hole sampling, drainage mechanism, heterogeneities, reservoir boundaries, etc.)

3)Development well: on producing wells periodic tests are made to adjust the reservoir description and to evaluate the need of a well treatment such as work over, perforation strategy etc..

Communication between wells, monitoring of the average pressure are some usual objectives of development well testing.

Objectives of Well Testing•

Investigate the Response of a Reservoir subjected to Production/Injection in a Well

•

Objectives fall under four categories–

Reservoir Evaluation

–

Reservoir Management–

Reservoir Description

–

Rate Prediction

•

The type of test selected and its duration often depend upon the objectives.

Information obtained from well test:•

Test provide a description of a reservoir in dynamic condition, as opposed to geological and log data.

•

Reservoir description:-permeability (Kv

and Kh)

-reservoir heterogeneities (natural fracture, layering, change of characteristics)

-boundaries (distance and shape)-pressure (initial Pi and average P)

•

Well description:-production potential (PI, Skin)-well geometry

Types of tests•

Drawdown test: the flowing bottom hole pressure is used for analysis. The well should be producing at constant rate.

•

Build-up test: the increase of bottom hole pressure after shut-in is used for analysis. Before the build-up test, the well must have been flowing long enough to reach stabilize rate.

•

Injection test / fall-off test:

when fluid is injected into the reservoir, the bottom hole pressure increases and after shut-in it drops during the fall-off period.

•

Interference test and pulse test: The bottom hole pressure is monitored in shut-in observation well some distance away from the producer. Interference tests are designed to evaluate communication between wells. With pulse tests, the active well is produced with a series of short flow/shut-in periods, the resulting pressure oscillations in the observation well are analyzed.

•

Gas well test: specific testing methods are used to evaluate the deliverability of gas wells (Absolute open flow potential, AOFP) and possibility of non-darcy

flow condition. The

usual procedures are back-pressure test (flow after flow), Isochronal and modified isochronal test.

•

Production test: such as PLT,RFT,MDT, GOR test

•

Drill stem test (DST): the well is completed temporarily with a down-hole shut-in valve. Frequently the well is cased but DST can be made also in open hole.

s law’Darcy

Darcy’s law:•

k/μ

: mobility of the fluid, s: sample

cross-sectional •

Darcy’s law valid within a time interval when the flow rate and other parameters are constant.

•

Darcy’s law assumptions:1)Laminar flow 2) steady state flow 3)incompressible fluid

4) homogenous formation•

Filtration rate : u=V/φso

pSk

q ∇−=μ

S

qV =

•

Darcy’s law in radial flow:

•

Compressibility:•

All the information from a well test is obtained because the rock and the fluids are compressible.

Tp

v

vc )(1

∂∂

−= Te pc )(1

∂∂

−=ρ

ρor

)(2r

prhkq

∂∂

=μπ

e

w

ew

r

rLn

ppkhq

−=

μπ2

•

By expansion of; Oil: Δvo

=-co

so

vp

ΔpWater: Δvw

=-cw

sw

vp

Δp•

During decompression (expansion), the fluid pressure decreases while the overburden pressure remains constant. The pore volume decreases: Δvp

=-cp

vp

Δp•

Overall compressibility of a pore volume unit:ct

=co

so

+cw

sw

+cp

;

φct

=compressible capacity

Equivalent compressibility:

o

pwwoo

scscsc ++

=ec

Diffusivity Equation:The diffusivity equation governs the variation in pressure

in the reservoir vs. time. It is based on two laws and one EOS:

1)Fluid flow equation:It is assumed that Darcy’s law governs fluid flow.2)Material balance or continuity equation:The variation in the mass of fluid contained in the

reservoir volume unit is equal to the difference between the amount of fluid input and output during the time interval:

3)EOS:

0)()( =∂

∂+

t

svdiv oρϕρr

Te pc )(1

∂∂

−=ρ

ρ

)()()()( ρφρρρ ozyx st

vz

vy

vx ∂

∂−=

∂∂

+∂∂

+∂∂

•

With consider three above equation we will get the following pressure equation:

•

Providing two assumptions:1) fluid is low and constant compressible (liquid)2) pressure gradients are low (flow is low)

is small, therefore:

•

for radial flow:

0)( 2 =∂∂

−∇+∇t

p

k

cpcp t

e

ϕμr

2)( pce ∇r

01=

∂∂

−∇t

p

Kp

ydiffusivithydraulicc

kK

t

−==ϕμ

0112

2

=∂∂

−∂∂

+∂∂

t

p

Kr

p

rr

p

)()/(

tc

kK

φμ

=

•

Solving the diffusivity equation:•

Boundary condition:

1)the pressure at beginning of the test2)the reservoir boundaries 3)the well condition

in infinite homogeneous reservoir solution suppose that the reservoir is:

1)homogenous and isotropic 2)constant thickness and limited by impermeable3)the well penetrates the total reservoir thickness 4)the fluid compressible and viscosity are constant and uniform•

B.C.:

-uniform initial pressure; Pi-infinite reservoir-constant flow rate in the well with very small radius

Variation in pressure vs. time and distance:

)4(

4),(

2

Kt

rE

kh

qBtrpp ii

−−=−

πμ

Compressible zone:the flow at a distance r from the well at time t:

q=wellhead flow rate qB=bottomhole flow rate

duu

exE

x

u

∫∞ −

=−− )()

4(

21 2

D

DiD t

rEP −−=

pqB

khPD Δ=

μ2.141 wD r

rr =

2000264.0

wt

Drc

tkt

ϕμΔ

=

Kt

r

qBetrq 4

2

),(−

=

•

Between t and t’

the pressure drop between an infinite distance and the well is therefore mainly due to what is occurring between r1

and r2

’. In this area the reservoir compressibility allows the flow to go from 0 to qB. This area is called the compressibility zone.

•

The pressure drop in the well mainly reflects the reservoir properties in the compressible zone

•

At the beginning of the test pressure drop reflects the reservoir properties in the vicinity of the well. Later on the test reaches areas that are farther away.

Radius of investigation:•

The pressure variations at the well give an indication of the properties of the part of the reservoir involved in the compressible zone. It is important to locate the compressible zone and this is what is involved in the concept of radius of investigation.

•

There is a large number of different definitions of radius of investigation in literature.

1) Jones’s definition:the radius of investigation is the point in the reservoir where the pressure variations represent 1% of the variations observed at the well:

SI2)Poettmann’s definition:The radius of investigation is the point in the reservoir

where the flow is equal to 1% of the flow rate:SI

ti c

ktr

φμ29.4=

ti c

ktr

φμ4=

3) J. Lee and Muskat’s definition:

The radius of investigation is the point where the pressure variations are the fastest.The variation is at a maximum for;

)4(

4),(

2

Kt

rE

kh

qBtrpp ii

−−=−πμ

Kt

r

kh

qB

dt

dp

4)exp(

4

2−=

πμ

14:.,02

2

2

=−=Kt

rforei

dt

pd

ti c

ktr

φμ2=

ti c

ktr

φμ032.0=

Flow regimes:Transient flow (unsteady state): until the compressible zone reaches the boundaries of the reservoir or comes under the influence of another well, the reservoir behaves as if it was infinite for testing purposes.Pseudosteady-state flow: when the compressible zone reaches a series of no-flow boundaries. This is the type of flow in a producing reservoir with no flow boundaries.Steady state flow: when the compressible zone is affected by some constant pressure outer boundaries. This is the type of flow in a reservoir producing under gas cap or Strong water drive conditions.

A well test is almost always performed in a transient flow regime even though some boundaries are reached.

tconst

pi tan)( =

∂∂

),( tift

p=

∂∂

0)( =∂∂

it

p

Principle of superposition:How can the pressure be described the reservoir when several flow rate variations occur?

The pressure variations due to several flow rates are equal to the sum of the pressure drops due to each of the different flow rates.Two flow rates:

)(2)()(

2)( 1

121 ttpkh

Bqqtp

kh

Bqtpp DDi −

−+=− π

μπμ

Multirate testing:With q=0 and t=0.

)()(2

)( 111

−−=

−−=− ∑ iDi

n

iii ttpqq

kh

Btpp

πμ

Pressure build-up test;When q2 is zero. This is case for the great majority of tests.

)]()([2

)( tpttpkh

qBtpp DpDi Δ−Δ+=−

πμ

storageWellbore•

When a well is opened, the production at surface is first due to the expansion of the fluid in the wellbore, and the reservoir

contribution is negligible. After any change of surface rate, there is a time lag between the surface production and sand face rate.

•

The period when the bottomhole

flow varies is called the wellbore

storage effect period.

•

Wellbore

storage coefficient:

C: Bbl/psiΔV : volume variation of fluid in the well under

well condition;Δp : variation in pressure applied to the well.c : liquid compressibilityVw

: wellbore

volumedt

dp

B

Cqqf

24+=

wVcp

VC =

ΔΔ

−=

)1(D

DDf dt

dpCqq −=

•

A dimensionless factor related to wellbore

storage:

in field unit:

When there is a liquid level in well (pumping well), with ;Δp=ρΔh g /gc , ΔV=Vu

Δh

ρ: liquid density , Vu

: wellbore

volume per length

In a naturally eruptive well, the variation in fluid volume depends on the compressibility of the fluid in the well; ΔV= -cVw

Δp

, C=cVw

The compressibility of the fluid in the wellbore

is very often much greater than that of oil in reservoir conditions because the oil

releases gas.Wellbore storage of pumping wells is considerably greater than wellbore storage of eruptive wells.

2

2

89.02

wt

D

wt

D

hrc

CC

hrc

CC

φ

πφ

=

=

)/(144

c

u

gg

VC

ρ=

Pressure variations: just after the well has been opened or shut-in , the

bottomhole

pressure is mainly affected by the wellbore storage effect.

Plot of the Δp vs. Δt on a linear scale. At the early time, the response follows a straight line of slope, intersecting the origin.

D

DD C

tp =

wbsm

qBC

tC

qBp

24

24

=

Δ=Δ

D

DD C

tp =

•

End of wellbore

storage effect:1)Ramey’s criterion: tD

=(60+3.5 S) CD

t=[(200000 + 12000 S) C ]/(kh/µ)

F. U.

2)Chen and Brighham’s

criterion:

tD

=50 CD exp(0.14 S)t=[170000 C exp(0.14 S) ]/(kh/µ)

F. U.

3) Rule of thumb:A straight line with Slope of 1

Skin•

The vicinity of the wellbore

has characteristics that are

different from those in the reservoir as a result of drilling and well treatment operations.

•

The skin effect reflects the difference in pressure drop that exists in the vicinity of the well between the reservoir as it is.

•

The skin is a dimensionless parameter. For a damaged well S>0, and for a stimulated well S<0.

•

Damaged well (S>0): poor contact between the well and the reservoir (mud cake, insufficient perforation density, partial penetration) or invaded zone.

•

Stimulated well (S<0): surface of contact between the well and the reservoir increased (fracture, horizontal well) or acid stimulated zone.

•

The difference in pressure drop in the vicinity of the wellbore

can be interpreted in several ways:

1)By using infinitesimal skin2)Skin of a finite thickness3)Effective radius method

•

Infinitesimal skin

(rw

≅rs

):Δpactual

=Δpideal

+ΔpsThe additional pressure drop due tothe skin effect is defined by:

α=1/2π

(in SI) , α=141.2 (in field)

Skh

qBpS

μα=Δ

•

Finite thickness skin:•

The pressure drop is located in an area with a radius rS

and permeability kS

around the well.

•

When the compressible zone leaves this area, the flow can be considered psedosteady-state and is governed by Darcy’s law:

•

The difference in pressure drop between the real reservoir and a reservoir uniform right up to the wellbore

is expressed as follows by Darcy’s law:

•

Damaged (kS

<k) corresponds to a positive skin

and improved permeability (kS

>k) corresponds to a negative skin.

w

S

S r

rLn

k

kS )1( −=

w

S

w

S

SSwSwS r

rLn

kh

qB

r

rLn

hk

qBppp

πμ

πμ

22)( 0,, −=−Δ =

•

Effective radius:•

The effective radius method consists in replacing the real well with a radius rw

and skin S by a fictitious well with a equivalent radius rw

’

and zero

skin.

•

Radius rw’

is determined to have a pressure drop

between rS

and rw’

in the fictitious well equal to the

pressure drop between rS

and rw in the real well: ΔP(rw

’

, S=0) = ΔP(rw

, S)

Expressing the pressure drop by Dary’s

law:

•

The effective well radius expresses the effect of well treatments.

)(22 ' S

r

rLn

kh

qB

r

rLn

kh

qB

w

SS

w

+=πμ

πμ S

ww err −='

Interpretation methods

Interpretation methods

•

Two main group are used for analyzed a well test:

1)conventional methods2)methods using type curves•

Inside each of the two groups the methods depend on the type of well, reservoir, and reservoir boundaries.

Conventional interpretation methods•

During a well test on an infinite homogenous reservoir two flows can be seen if the test long enough:

1) flow that related to wellbore

storage.2) radial flow over the whole reservoir thickness

•

Analysis methods will be presented for this configuration, then developed with more complex reservoir-well configuration.

•

Three flow rate conditions are examined:-

the drawdown test.

-

the build-up test, with pressure build-up preceded by one constant flow rate.

-

The test following any number and type of previous flow rates

Drawdown test•

The equation that describes pressure variations vs. time and distance from the well after opening the well at a constant flow rate;

•

K=k/φμct

, when the pressure is measured in the well where the flow rate disturbance is located, r=rw

; •

As soon as rw

2/4Kt < 0.01 , which usually occurs before the wellbore

storage effect is over , the Ei

function can be replaced by its logarithmic approx.;

)4(

4),(

2

Kt

rE

kh

qBtrpp ii

−−=−

πμ

)81.0(4

)( 2 +=−w

wfi r

KtLn

kh

qBtpp

πμ

*For small x, Ei(−x)=−ln(γ

x) : the Exponential Integral can be approximated

by a log (with γ

= 1.78, Euler's constant).

•

Taking pressure drops in the skin into account, this expression becomes;

•

It can also be written in other equivalent form (F.U.);

•

As a dimensionless eq.;

•

Above equations show that bottomhole

pressure varies logarithmically vs. time.

)281.0(4

)( 2 Sr

KtLn

kh

qBtpp

wwfi ++=−

πμ

)87.023.3log(log6.1622 S

rc

kt

kh

qBpp

wtwfi +−+=−

φμμ

)281.0(21

SLntp DD ++=

Skh

qBpS

μα=Δ

•

If the pressure measured at the bottomhole

is plotted on a graph vs. the logarithm of time, a straight line with a slope of m can be observed once the wellbore

storage effect has ended:

•

The skin value is usually computed using the pressure measured at 1 hr on the semi-log straight line; for this point logt=0.

kh

qBm

μ6.162=

m

qBkh

μ6.162=

)23.3log(15.1 21 +−

−=

wt

hri

rc

k

m

ppS

φμ

•

Care must be taken to read the pressure at 1

hr on the semi-log straight line and not by interpolating among the measurement points. After one hour the data may still be affected by the wellbore

storage effect. In this

case, they do not verify the semi-log straight line equation.

up test-Pressure buildHorner’s method:

•

Most of the information from a well test comes from interpreting the pressure build-ups.

•

Interpreting a drawdown test is limited by the flow rate fluctuations inherent to production. The fluctuation cause pressure variations that are greater near the end of the test than the pressure variation due to the initial change in flow rate.

•

The zero flow rate (build-up test) does not cause this type of problem.

Pwf

(t) is the flowing pressure; time is counted from when the well is opened. Pws

(Δt) is the pressure during the buildup phase; time is counted from when the well is shut–in, tp

: Pws

(Δt=0) = Pwf

(tp

) pressure build-up is analyzed using superposition

principle:)]([)]([)( tppttpptpp wfipwfiwsi Δ−−Δ+−=Δ−

•

By replacing the below Eq.;

in two right hand terms in previous Eq. , the semi-log expression of radial flow gives;

•

This equation and analysis method were presented by Horner in 1951. the equation shows the bottomhole

pressure varies linearly vs. log[(tp

+Δt)/ Δt]

)281.0(4

)( 2 Sr

KtLn

kh

qBtpp

wwfi ++=−

πμ

t

ttLn

kh

qBtpp p

wsi Δ

Δ+=Δ−

πμ

4)(

t

tt

kh

qBtpp p

wsi Δ

Δ+=Δ− log6.162)( μ

kh

qBm

μ6.162=

m

qBkh

μ6.162=

When the wellbore storage effect has

ended, a straight line with a slope of m

can

be observed.

•

The skin value is computed from the difference between pressure after 1

hr of build-up on the

semi-log straight line;

•

And the pressure at shut-in time;

•

By subtracting the two above expressions term by term;

•

The term log[(tp

+1)/tp

]

is usually negligible compared to the other term.

)1log(6.162)1( +=− pi tkh

qBhrpp

μ

)87.023.3log(log6.162)( 2 Src

kt

kh

qBtpp

wt

ppwfi +−+=−φμ

μ

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

++

−= 23.3log

1log

)(15.1 2

1

wtp

ppwfhr

rc

k

t

t

m

tppS

φμ

•

The initial pressure can be read on the pressure build-up for; Δt=∞

, (tp

+ Δt)/ Δt =1It is equal to the initial reservoir pressure in most initial tests.It is used to compute the average pressure when production is not negligible compared to the oil in place.

•

The Δpskin

can be obtained from;

)(87.0 mSpskin =Δ

Pressure build-up: MDH method• In previous section showed that in build-up the

pressure varies linearly vs. log((tp

+ Δt)/ Δt )). If tp

is large compared to Δt: tp

+ Δt # tp

•

Before equation becomes:

•

The bottomhole

pressure varies linearly vs. pressure build-up time. This means that during build-up the pressure drop due to previous production is disregarded.

•

The next fig. illustrates this interpretation method develop by Miller Dyes and Hutchinson (MDH)

)log(log6.162pwfi tt

kh

qBpp −Δ−=−

μ

•

The MDH method:-the real pressure build-up is Δp-the pressure build-up dealt

with by the MDH is ΔpMDH

•

The difference between Δp and ΔpMDH

is negligible when Δt is small compared to tp

i.e;-

at the beginning of build-up

-

after a long period at constant flow rate.

•

Interpretation

:-

the pressure varies linearly vs. the logarithm of time. By plotting ΔpMDH

vs. Δt, a semi-log

straight line with a slope of m can be seen once the wellbore

storage effect has ended.

-

The m, kh

and S is computed same way as in Horner method.

The advantage of this method is that it is very simple, the major disadvantages are;

1) it can not be used to find the extrapolated pressure (p*).

2)it can be used only for values of Δt that are small compared to tp

.

•

When production time is short or close to Δt (initial tests on a well), the last build-up points are located under the theoretical semi-log straight line in the MDH representation.

•

After varying flow rates:•

A test after varying flow rates is interpreted using the flow rate superposition principle discussed before;

•

Once the wellbore

storage effect has ended, the pressure variations becomes;

∑=

−− −−=−n

iiDiiwfi ttpqq

kh

Btpp

111 )()(

2)(

πμ

∑=

−− ⎟⎟

⎠

⎞⎜⎜⎝

⎛++

−−=−

n

i w

iiiwfi S

r

ttKLnqq

kh

Btpp

12

11 281.0)()(

4)(

πμ

•

The user is interested in the pressure variations since the last change in flow rate, tn-1

. The pressure at the time when the change took place is:

•

The pressure build-up since the time when the well was shut in is expressed by:

•

Δt is the elapsed time since the last change in flow rate. It can be written as follows:

∑−

=

−−− ⎟⎟

⎠

⎞⎜⎜⎝

⎛++

−−=−

1

121

11 281.0)()(4

)(n

i w

iniinwfi S

r

ttKLnqq

kh

Btpp

πμ

⎟⎟⎠

⎞⎜⎜⎝

⎛++

Δ−−

Δ+−−

−=−Δ ∑−

=−

−−

−−−−

1

121

11

1111 )281.0)(()(

4)()(

n

i wnn

in

iniinwfws S

r

tKLnqq

ttt

ttLnqq

kh

Btptp

πμ

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+Δ−−

Δ+−−

−=−Δ ∑−

=−

−−

−−−−

1

121

11

1111 )87.023.3log)(log(log)(6.162)()(

n

i wtnn

in

iniinwfws S

rc

ktqq

ttt

ttqq

kh

Btptp

φμμ

*The slope is independent of flow rate. The results obtained with different flow rates can be compared on the same graph.

•

Interpretation:The pressure varies linearly vs. the right-hand member (between parentheses). The member is function of flow rates

and the time

and is called

superposition function. If the value of the pressure in the bottom-hole is plotted

vs. the superposition function, a straight line with a slope of m can be seen once the effect of wellbore

storage has ended.

kh

Bm

μ6.162=

•

Skin:The skin is determined based on the pressure

value read on the line 1 hr after a last flow rate variation:

•

Extrapolated Pressure (p*):If the last flow rate variation is a shut-in, the

pressure reading for infinite time, for a value of the superposition function equal zero, is used to determine the extrapolated reservoir pressure.

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−−

=−

− 23.3log)()(

15.1 21

11

wtnn

nwfh

rc

k

mqq

tppS

φμ

•

Simplification of the flow rate historyThe superposition function takes the flow rate history into

account. It is difficult when would try manual calculation even with more then two or three flow rates.

What equivalent function could be used in this case?

Equivalent time:To analyze the final build-up, the simplest method consist

in reducing the flow rate history to one single rate and using Horner’s method for actual interpreting.

1)Flow rate = the last rate2)Equivalent production time;

The production time is designed to provide a total production value identical to the production that was actually recorded.

Equivalent flow rate should not be used.

n

n

iiii

pe q

ttqt

∑−

=−−

=

1

11)(

•

Validity of the method:•

As a rule of thumb, simplifying the flow rate history can be considered to introduce negligible error when the build-up duration is less than twice the duration of the last flow rate before shut-in.

•

If substantial rate variations occurred shortly before shut-in, the simplification will introduce error. The wider the variations and the closer they occurred before shut-in, the greater the error introduced.

Below figures indicates the direction of the error takes on Horner plot.

Simplification before (tp

-Δtr

) usually introduces only a slight degree of error.

•

Build-up radius of investigationIt is depends only on duration of the pressure build-up. (Field unit)

It is theoretically independent of the duration of the drawdown period.

ti c

tkr

φμΔ

= 032.0

•

Ideal pressure build-up test:In an ideal situation, we assume that the test is conducted in an

infinite acting reservoir in which no boundary effects are felt during the entire flow and later shut-in period.

The reservoir is homogeneous and containing in a slightly

compressible, single-phase fluid with uniform properties so that Ei

function and its logarithmic approximation apply. Horner’s

approx. is applicable. Flow into the wellbore

ceases immediately at shut-in.

•

Actual build-up tests:Instead the single straight line for all time, we obtain a curve

with a complicated shape, which indicates the effect of afterflow. We can logically divide a build-up curve into three regions:

1)Early-Time region (ETR): in this region , a pressure transient is moving through the formation nearest the wellbore.

2)Middle-time region (MTR):in

this region, the pressure transient has moved away from the wellbore

into the bulk formation.

3)Late-time region (LTR): in this region, the pressure transient has reached the drainage boundaries well.

MTR is a straight line. This is the portion of the build-up test curve that we must identify and analyze. Analysis of this portion only will provide reliable reservoir properties of the tested well. The reasons for the distortion of the straight line in the ETR and LTR are as follows:

In the ETR the curve affected by; 1)altered permeability near wellbore. 2)wellbore storage effect

In the LTR, the pressure behavior

is influenced by boundary configuration, interferences from nearby wells, reservoir heterogeneities, and fluids contacts.

•

Plots Δpws

=(pws- pwf(Δt=0)

) vs. Δt on log-log graph to identify wellbore

effects, identify ETR and

beginning of MTR which can be found using type curves.The MTR ends when the radius of investigation begins to detect the drainage boundaries of the tested well. At this time the buildup

curve starts

to deviate from straight line.

Productivity index•

The productvity

index of a well;

Two cases can be distinguished, depending on whether the pressure is measured in the transient or in the psedosteady-state flow.

Transient corresponds to initial well test and psedo

to measurements made during production.PI during the infinite acting period:

The average pressure in drainage area of the well close to the pi

.

pi

-pwf

is calculated based previous eq.; (PI will be decreased with time during transient flow);

wfpp

qPI

−=

wfiwf pppp −− #

)87.023.3log(log6.162 2 Src

ktB

khPI

wt

+−+=

φμμ

PI during the psedosteady-state flow:

The variation at the well in the psedosteady-state regime:

Therefore:

The PI is constant during the psedosteady-state flow.A: reservoir sizeCA

: reservoir-well geometry

)()( pppppp iwfiwf −−−=−

⎟⎟⎠

⎞⎜⎜⎝

⎛+++=− S

Cr

A

kh

qBt

hAc

qBpp

Awtwfi 87.02458.2loglog6.162234.0

2μ

φ

thAc

qBpp

ti φ

234.0=−

)87.02458.2log(log6.162 2 SCr

AB

khPI

Aw

++=

μ

•

Real and theoretical PI:The theoretical PI is the one the well would have if its

skin were zero. It is used to estimate the gain in PI by well stimulation,

•

Flow efficiency: the flow efficiency is also known as productivity ratio, condition ratio, and/or completion factor.

•

The FE shows that the well is producing at FE% of rate it would have been producing if the well was not damaged.

swf

idealppp

qPI

Δ−−=

)(S

kh

qBps

μ2.141=Δ

Ideal

actual

PI

PIFE =

wf

swf

pp

pppFE

−Δ−−

=

•

Damage Ratio:When FE subtracted from unity it gives the

damage factor which is also a relative indicator of wellbore

condition.

Damage ratio is the inverse of FE:

The DR shows the production rate would have been how many times of present rate if the well was not damaged.

FEDR

1=