well test mod sol

8/9/2019 Well Test Mod Sol

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Model Solutions to Examination

1

Date:

1. Complete the sections above but do not seal until the examination is finished.

2. Insert in box on right the numbers of the questions attempted.

3. Start each question on a new page.

4. Rough work ing should be confined to lef t hand pages.

5. This book must be handed in entire with the top corner sealed.

6. Additional book s must bear the name of the candidate, be sealed and be affixed to

the first book by means of a tag provided

Sub ject:

INSTRUCTIONS TO CANDIDATES

8 Pages

PLEASE READ EXAMINATION REGULATIONS ON BACK COVER

No. Mk.

N A M

E :

R E G

I S T R

A T

I O N N O . :

C O U

R S E

:

Y E A R :

S I G N

A T U

R E : C o m

p l e t e t h i s s e c

t i o n b u t d o n o

t

s e a l u n

t i l t h e e x

a m

i n a t i o n

i s f i n i s h e d

Reservoir Engineering - Well Test Analysis


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3

ANSWER 1

The general solution is:

P tD Dxf = π

Convert dimensionless variables to actual real variables:

PPi Pff kh

qt

kt

x

Pi Pkh

q

kt

x

D Dxf t f

ff t f

= − =

− =

( )

( )

2

2

2

2

πµ ϕµ

πµ

πϕµ

C

C

Now rearrange for Pff:

( )

( )

Pi Pq

kh

kt

x

Pi Pq

kh

kt

x

ff t f

ff t f

− =

− =

µ

π

π

ϕµ

µ

π

π

ϕµ

2

2

2

2

C

C

This can be easily rearranged to the following:

P Piq

kh

k

xtff

t f

= −

−µ

ππ

ϕµ2 2

2

C

Which is the equation of a straight line:


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y = mx + c

m = -C

q

kh

k

xt f

µπ

πϕµ2 2

Thus, if a plot of Pff, measured by the downhole pressure transducer,

against t , the square root of the flowing time, is made, the gradient

m, can be evaluated.

P Pq

kh

k

xtff

t f

=

−1 22

µπ

πϕµ C

Pff - measured by pressure transducer

t - recorded over flow period

Pi - (not important) but measured by RFT or early DST

Plot Pff v t

Evaluate gradient through data, m

mq

kh

k

xt f = −

µπ

πϕµ2 2C


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5

q - held constant through test and recorded at surface

µ - known from PVT test data

h - known from geological modelling

φ - known from petrophysics/core tests etc

t C - known from rock mechanical testing of cores

k - known - as given in question

= = −

xf

q

m kh

k

u t

µπ

πϕ2 C

The analysis of the plot of Pff v t should be carefully carried out, i.e.

select appropriate data that is not affected by early time effects e.g.

wellbore storage, or late time effects e.g. influenced by boundaries to

flow.


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ANSWER 2

a) Gas material balance is given by:

Mg = M

gi - M

gp (1)

i.e. current mass of gas = initial mass - produced mass

in reservoir of gas in reservoir of gas

From equation of state:

(PV = ZnRT)

Mg = V

gρ

g ρg

MW P

ZRT

.=

Mass = volume x density MW = molecular weight of gas

Define G = Volume of gas at standard conditions

= = = . . ..

VG

g

G M W P

Z RT

ZRT

MW Pg

sc sc

sc sc

ρρ

at standard conditions ZSC

= 1

= = .V G P TZ

T Pg

sc

sc


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7

This equation can be written, taking account of gas material balance

equation (1) as:

VG G P T

T Pg

i p sc z

sc

( )

= −

Also, by the same argument:

VG P T

T PZigi

i sc

sc i

= T = constant reservoir pressure

For volumetric depletion

V V

G G P Tz

T P

G P T

T P

g gi

i p sc

sc

i sc zi

sc i

( )

=

= −

=

= − =

= = −

= = −

( )

( )

G GZ

PG

Z

P

P

Z

G G

G

P

Z

P

Z

P

Z

P

G ZG

i p i

i

i

i p

i

i

i

i

i

i

i i

p

Thus, plotting - against Gp


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where P = measured reservoir pressure

Z = compressibility of gas at P

Gp = cumulative gas produced at standard conditions

yields a straight line:

Pi

Zi

P

Z

0

The data points can be plotted, as above, and the line

extrapolated to = 0.

At this point, Gp = G

i i.e. the initial gas-in-place.

(b) In most situations the plot will not be valid since

depletion will not be entirely volumetric. The expansion of rock

and connate water will lead to errors in the process, sometimes

of 100%.


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9

This is most apparent in abnormally pressured reservoirs where

over-pressure results in gas compressibility and rock and water

compressibility being of similar magnitude. Under such

circumstances, these effects cannot be ignored.

The modification is as follows:

Vg = V

gi - ∆V

g

∆Vg = V

pi (S

wcC

w+C

f) ∆P

here Vpi is the initial pore volume∆P is the pressure depletion

Swc

is the connate water solution

Cw is the water compressibility

Cf is the formation compressibility

Vgi

= Vpi

(1-Swc

)


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= = −−

+

= = − ∆

−( ) +

= −

=

=> −

( )

( )

( )

,( )

( )

V VV

SS C C P

V VP

S

S C C

From before VG G P T

T P

VG P T

T P

G G

g gi

gi

wc

wc w f

g gi

wc

wc w f

g

i p sc z

sc

gi sc zi

sc i

i p

1

1

1

∆

ZZP

G ZP

PS

S C C

P

Z

P

SS C C

G G

G

P

Z

P

Z

P

SS C C

P

Z

P

G ZG

ii

i wc

wc w f

wc

wc w f

i p

i

i

i

wc

wc w f i

i

i

i i

p

= − ∆−

+

=> − ∆−

+

=

−

=> − ∆−

+

= −

11

11

11

( )

( )( )

( )

( )

This will give a straight line only when Cf is constant. This is rarely

the case, since Cf is a pressure dependent quantity. Selection of C

f is

therefore a difficult and important decision if this modified plot is to

be of benefit.


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(c) The problems with water influx are significant. Not only does

influx affect the material balance

Vg = V

gi - W

e,

but there are additional effects of gas trapping in the invaded region

of the reservoir/aquifer system.

The management of such fields demands that both reservoir and

aquifer behaviour are understood. The nature of gas sales is to

arrange long term contracts with specified Daily Contracted

Quantities of gas, which are, additionally, subject to swing factors i.e.the delivery must be able to meet demand above the DCQ, or below,

depending on seasoned variation. In designing such contracts, it is,

therefore, crucially important to fully understand reservoir and

aquifer characteristics, as failure to meet contracts can prove a costly

business.

The DCQ is based on the peak production rate of the reservoir, again

accounting for saving factor. The Annual Contract Quantity = 365

DCQ i.e. saving is balanced. The reservoir model must be able to

predict future performance. In this situation of Water influx, the

material balance equation can be conveniently coupled to an analysis

method which has well testing theory as its routes. The reservoir

becomes “the well”, while the aquifer becomes “the reservoir”.


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Variable rate solution of this familiar problem is required to model the

aquifer performance.

Fetkovich showed that results derived by a quasi-semi-steady state

solution to the problem generated acceptable results, saving on

computation time.

With the material balance and Fetkovich model, the production of the

system can be predicted. The aquifer inner boundary pressure is the

gas reservoir average pressure, so required depletion rates can be

modelled.

The timing of decline will be when:

P Pg− −

= swing i.e. the gas reservoir average pressure is equal to that

required to meet the maximum gas output. Both this and the peak

production rate must be well designed to honour the gas contract.

Production problems include the trapping of gas in the region of water

influx. As water invades the reservoir it displaces gas to a residual

concentration Sgr which becomes lost in terms of production.

The water influx will then be impeded due to relative permeability

effects and pressure maintenance may be affected.


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A major benefit in the management of such fields is to have good

aquifer description which may be achieved by better data gathering

from the aquifer i.e. core sampling. Many characteristic will be

different between reservoir and aquifer not least of which is

permeability, which will be greater in the aquifer due to increased

periods of diagenesis.

With good aquifer data, simulation becomes more valuable, helping the

management process.

ANSWER 3

a) Several factors influence well test design.

Assuming wellbore storage, an estimate of CD, which can be deter

mined from very early time data (for constant compressibility (in

single phase) and skin (an estimate can be determined from anearlier well test or from corey time data using type curve matching)

can give an indication of the time needed to reach the semi-log

straight line. So, to be able to set good estimates of k and S, the

straight line on the semi-log plot must be reached (out of wellbore

storage affected region) and at least 1 log cycle of data should be

measured in this region (MTR).


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In a bounded reservoir, if the geometry of the drainage region (and

- average drainage area) is the objective of the test, then time to

reach the boundaries has to be allocated to the test (reservoir limit

test). In this case, m* is calculated from slope in SSS depletion. In

Cartesian plot m q

cth* = −

φ A and can be determined. C

A Dietz

shape factor necessary for the MBH correction. If CA is already

known, the well test only needs to reach the SSS zone. The radius at

investigation r 4 t 4kt

ci

t

= =α φµ

gives an indication of the distance that

the pressure disturbance traveled, so if the distance to boundaries

(faults, etc) is known (from geophysics, etc), an estimation of time

taken to reach boundaries can also be performed.

For this, relevant data is compressibility of oil, total compressibility

(great uncertainty), estimates of k and S, oil viscosity, porosity,

thickness of formation, and flow rates.

b) A software package can import data in real time and perform an

early time analysis to determine the existence of wellbore storage,

and from type curve matching or derivative (log-log) analysis set CD,

and type curve matching an estimate of K and S can be made and CD

stated before, this can give management an indication of time

necessary to reach MTR region. Analysis of log - log (p’ vs t) plot can

give a real time indication of when the MTR region is reached(derivative plateau) and depending on objective of test, the real time


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analysis can continue for the design of the testing in the LTR as

stated before, with some knowledge of the distance to the boundaries.

If this distance is not known, then testing should be extended to

“feel” them : normally performed after well is drilled.

c) Flow rate is dependent on type of fluid and associated conditions,

such as:

- In oil wells, qo is normally limited by test separator capacity

- In gas wells (exploration), gas rate can be limited by the capacity of

the flaring system

- In High Pressure, high temperature wells, rate can be limited bythe maximum allowable temperature at the wellhead (rating of

elastomers seals).

Production software performing a nodal analysis taking in consideration

all this (separator, chokes, tubing lift performance and reservoir in

flow - needs k and S estimates) can give a good indication of the well-test flow rate or ranges at allowable rates.


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ANSWER 4

a) This is a very useful plot. Allows the identification of 3 flow

regimes (this diagnostic role is a very important application of the

derivative response) and also allows a precise identification of the

boundaries (in time) of those regimes (another important role of the

log-log derivative diagnostic plot).

MTR) In this case, the existence of the derivative plateau

indicates the existence of a middle time region (MTR) in radial flow.

ETR) After a transition period that may be shorter or longer

(depends on geometry of system and well location within it). A periodof half slope is reached, indicating the existence of linear flow.

Following this period, another transition is reached, when the far

boundaries of the system start to affect the pressure response.

When the period of unit slope is reached, the pressure response is

dominated by the bounded reservoir model, being semi-steady-state

depletion beyond that. This transition period is also affected by the

distance to the reservoir boundaries, and if they are reached or not at

the same time. A possible geological interpretation could be a bounded

fluvial channel


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Unit slope period other

boundaries are reached

(SSS depletion)

Linear flow peridRadial flow perid

For the correct unique interpretation, input needs to be received from

other disciplines such as geology, geophysics, Petrophysics, etc →

multidisciplinary task!

b) Not counting a possible wellbore storage dominated period that

would be analysed in the log-log plot, the following specialist plots arenecessary.

Semilog-plot → to perform the analysis of the MTR region (derivative

plateau), giving the best estimates of k and S. Flow in the MTR region

(radial) is a straight line in a semilog plot.

I assume the flow periods were already defined in the log-log

derivative plot.

Linear flow plot ( ) → after the MTR period analysis, the relevant

parameters are stored (k,S) and the linear flow plot can give estimates

of the geometry of the system, L1, L

2 and W


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L1

L2

W

Cartesian flow plot _ in SSS regime, pwf

vs t is a straight line at slope

− qc hAt φ

and assuming that ct is known, an estimate of the connected

pore volume can be achieved. This allows the estimation of reserves.

Also the extrapolation of the pressure to t=0 yields a relation with the

Dietz shape factor of the drainage area.

ANSWER 5

i) The lack of uniqueness can be overcome by the input of geologists,

geophysicists, petrophysicists, etc, giving information about the

depositional environment, likely geological structure, faulting, layering

(cores, WFTs) etc.

ii) In some situations, these can be overcome using the Hawkins

equation. AC>υ the resistivity or equivalent logs can help.

iii) Analysis of cores (permeameter) or WFT measurements can allow

the estimation of vertical profiles for the permeability and identify

layering, etc.


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iv) This can be overcome using an approximation such as:

Q

qsq

tp t

tp =Q

qs

Where Q is cumulative production, qs is surface rate and T p is a

production time accounting for variation in the rate during the test.

The limitation is that before shut in a stabilised rate qs should exist.

This can overcome some variability in the rate (uncertainty). The

continuous development of more precise flowmeters (multiphase) can

also help in reducing uncertainty.

v) A material balance using the pressure decline and estimated

reserves from geology can help in identifying if there is some

pressure support from an aquifer.

b)

i) Lack of uniqueness is an important issue but assuming that there is

some previous geological or geophysical study, the effects of it can be

reduced. In new exploration areas this problem can be more serious,

where data input from other disciplines is still reduced (lack of

information).


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ii) This is also important because when designing a stimulation

treatment to reduce or overcome the damaged region (acidising,

fracturing), the knowledge of the depth of damage would help in

optimising this.

iii) As long as WFT data (MDT) and/or cores are available, this shouldbe a minor problem.

iv) This is very important as the transient well test analysis normally

assumes constant and known rate. If this is not the case, then the

analysis is incorrect. Maybe that’s why such a large effort is being put

on the development of accurate and reliable multiphase flowmeters.

v) This can be very important because in the exploration & appraisal

stage of a field, especially offshore, the knowledge of whether there

is aquifer support or not would be extremely valuable in optimising the

development cost. (Accounting for water injection support or not and

all the related facilities).

c) The fact that when there are no measurements of initial Pi by W

Ft

or MDT, some error can be introduced if the pressure transducer is

not at perforation depth. Pi = Ptranducer

+ ρh and this ρh may notbe well

known if there is some cushion fluid going down through valve.


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→ The calculation of the wellbore storage coefficient assumes

constant single phase compressibility. This can lead to serious errors

in estimating CD because prior to shut-down most oil wells would be in

Z phase flow.

→ Very important limitation → deferred production while test isbeing performed (no production at all during buildups) →maybe the

largest limitation of well testing from the managers point of view!


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well test mod sol

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