well test analysis - fanarco will be felt at the well reservoir models a reservoir model is the...

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APP

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987-

00Well Test Analysis

Well Test Analysis, Theory

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987-

00

Well Testing is Integral and Indispensable partof Reservoir Description and Management

Petrophysics/ LogsSeismic PVTCores/Geology

RESERVOIR MODEL ECONOMIC MODEL+DEVELOPMENT STRATEGY AND MANAGEMENT

=

RESERVOIR MODEL

INTERPRETATION

Well Tests

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APP

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987-

00Well Test Objectives

Well Tests are conducted to :

•identify produced fluids and determine their respective volume ratios•measure dynamic reservoir pressure and temperature•obtain samples suitable for PVT analysis•determine well deliverability

PRODUCTIVITYWELL TESTING

DESCRIPTIVEWELL TESTING

evaluate reservoir parameters•evaluate completion efficiency•characterise well damage•evaluate workover or stimulation treatment•characterize reservoir heterogenities•access reservoir extent and geometry•determine hydraulic communication between wells

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APP

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987-

00Interpretation Methodology

No

Yes

No

Yes

Data Validation

OKDifferentialPressure Analysis

DifferentialPressure Analysis

Identificationof

Validity

Choice of Flow period

Data Loaded

Generation of Diagnostic Plot

Pattern Recognition

Identification of Interpretation

Model

Consistency

Finalised

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APP

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987-

00Types of Test

Drawdown Test

4 5 0 0

0 2 4 6 8 1 0 1 2

9 0 0

History plot (Pressure, Liquid Rate vs Time)

Description Well static, stable, shut in, open at constant flow rate

Advantages Good for limit testing as flow rate fluctuation is lesssignificant over long term

Disadvantages Difficult to make well flow at constant rateIf recently drilled, well may not be static or stable

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APP

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987-

00Types of Test

Build-up Test

4 5 0 0

0 2 4 6 8 1 0 1 2

9 0 0


Description Well flowing at constant rate, shut in

Advantages Constant flow rate easily achieved as it is zero

Disadvantages Difficult to make well flow at constant rateIf recently drilled, well may not be static or stable

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APP

A 1

987-

00Types of Test

Injection Test

5 6 0 0

0 2 4 6 8 1 0 1 2

-1 0 0 0

0


Description Well static, stable, shut in, injection at constant rateConceptually identical to drawdown test

Advantages Better injection rate control than production rates

Disadvantages Complicated multiphase analysis unless reservoir fluid injected

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APP

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987-

00Types of Test

Falloff Test

5 6 0 0

0 2 4 6 8 1 0 1 2

-1 0 0 0

0


Description Well injection at constant rate, shut inConceptually identical to build-up test

Advantages Constant flow rate easily achieved as it is zero

Disadvantages Complicated multiphase analysis unless reservoir fluid injected

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APP

A 1

987-

00Types of Test

Interference Test

4 9 6 5

4 9 8 5

3 8 1 3 1 8 2 3


Description One well is subjected to a drawdown, build-up,injection or falloff and pressure is observed in a different well or wells

Advantages Evaluates reservoir properties over a greater area

Disadvantages Requires very sensitive pressure recorders and maytake a long time

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APP

A 1

987-

00Types of Test

Isochronal Test

4 9 8 2

4 9 9 2

1 0 3 0 5 0 7 0

1 8 0 0

History plot (Pressure, Gas Rate vs Time)

Description For low productivity gas wells. Well flowed at four different rates of equal duration.Between each flow period, the well is shut in till staticconditions are reached.The last flow period is extended till stabilized flowing conditions are reached

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APP

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987-

00Types of Test

Modified Isochronal Test

4 9 6 5

4 9 8 5

5 1 5 2 5 3 5 4 5 5 5 6 5

1 8 0 0

History plot (Pressure, Gas Rate vs Time)

Description For low productivity gas wells. Well flowed at four different rates of equal duration.Between each period, the well is shut in for the same duration as the flow periodThe last flow period is extended till stabilized flowing conditions are reached

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APP

A 1

987-

00Well Test Analysis Principles

Reality Model

Reservoir Rock

Hydrocarbons

Faults

Pressure Plots

Mathematicalequations

Analysis

R&D

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APP

A 1

987-

00

Fluid flow in a porous medium is governed by:

Diffusivity Equation

Combining:

Darcy’s LawConservation of mass Diffusivity EquationEquation of state.

Conditions:- Reservoir is a homogeneous, isotropic, porous medium- Gravity effects can be neglected- Fluid is monophasic and of small and constant compressibility- The viscosity is constant- Pressure gradients are small- µ, ct, k and Φ are independent of pressure

}

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APP

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987-

00Darcy’s law

Darcy’s Experiment

A h1

h2

h1-h2Water in

Water out

L

Manometer difference (h1 - h2)

Flow area A

Length L

Constant K

q = K Ah1 - h2

L

In Darcy’s experiment,on what did the fluid flow depend ?

more

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APP

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987-

00Darcy’s law

Darcy’s Law Linear Flow Oil Field Units

kAqµ887.2

xp −=

∂∂

Darcy’s Law Radial Flow Oil Field Units

khq

rpr µ2.141=

∂∂

Darcy, Henri-Philibert-Gaspard b. June 10, 1803, Dijon, Franced. Jan. 3, 1858, Paris

French hydraulic engineer who first derived the equation (now known as Darcy's law) that governs the laminar (nonturbulent) flow of fluids in homogeneous, porous media and who thereby established the theoretical foundation of groundwater hydrology.After studying in Paris, Darcy returned to his native city of Dijon, where he was entrusted with the design and construction of the municipal water supply system. During the course of this work, he conducted experiments on pipe flow and demonstrated that resistance to flow depended on the surface roughness of the pipe material, which previously had not been considered a factor. Planning to use the technique of water purification by filtration through sand, he also studied cases in which the pipe was filled with sand. From the data gathered, he derived the law that bears his name. The darcy is the standard unit of permeability.

Additional reading: http://www.philosophika.com/hofmann/DARCY.html

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APP

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987-

00Conservation of Mass

Mass can not be created or destroyed

What goes in = What comes out + What is left behind

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APP

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987-

00Conservation of Mass

mass in – mass out = accumulation = mass after – mass before

y∂x∂

z∂

x xx ∂+

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APP

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987-

00Equations of State

Single phase Liquid flow

Compressibility c = -

Single phase (non-ideal) Gas flow

PV = z n R T

1

V

dV

dP

The accumulation is governed by the EOS

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APP

A 1

987-

00Diffusivity Equation

Where p formation pressure, psi Φ porosityr radial distance to wellbore, ft ct total compressibility, psi-1t time, hr µ viscosity, cp∆p Laplace operator kr permeability, mD

The equation shows the influence of time and distance on the pressure

pck

tp

t

∆Φ

=∂∂

µ0002637.0General form in US field units

��

��

��

∂∂

∂∂

Φ=

∂∂

rpr

rrck

tp

t

10002637.0µ

Radial flow in US field units

2

2

0002637.0xp

ck

tp

t ∂∂

Φ=

∂∂

µLinear flow in US field units

conditions

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APP

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987-

00

Boundary

Eventually the effects of the reservoir boundaries will be felt at the well

Reservoir Models

A reservoir model is the superposition ofreservoir, inner, and outer boundary conditions

Reservoir

After the well conditions, the reservoir determines the pressure behaviour

Well

The well test pressure behaviour starts with the well conditions

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APP

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987-

00Defining the problem

��

��

��

∂∂

∂∂

Φ=

∂∂

rpr

rrck

tp

t

10002637.0µ

Initial condition ( ) iprtp == ,0

khQ

rpr

tr

µ2.141lim,0

=��

�∂∂

→

( )[ ] ir ptrp =∞→ ,lim

Well condition

Infinite condition

Line Source Well

conditions

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APP

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987-

00

Dimensionless variables are introduced

• to simplify the reservoir models• to provide model solutions independent of units

These models incorporate physical variables such as pressure, distance, time. It would be futile to solve these problems for all combinations of variables.

Dimensionless Variables

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Permeability, viscosity, compressibility, porosity, formation volume factor and thickness are allconstant.

Dimensionless parameters are designed to eliminate the physical variables that affect quantitatively, butnot qualitatively, the reservoir response.

Dimensionless Variables

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APP

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987-

00Dimensionless Variables

Dimensionless pressure

Dimensionless time

Dimensionless radius

And others

pkhqB

pD =1412. µ

∆

tk

c rtD

t w=

0 0002642

.Φ

∆µ

rrrDw

=

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APP

A 1

987-

00Diffusivity Equation

Re-writing the radial diffusivity equation (line source)in dimensionless terms

Well condition

( )p t rD D= =0 0,Initial condition

Boundary condition

��

��

��

∂∂

∂∂=

∂∂

D

DD

DDD

D

rpr

rrtp 1

1lim,0

−=��

�

∂∂

→ trD

DD

D

rpr

( )[ ] iDDDr trp 0,lim =∞→

The rate of pressure change is a function of some parametersand the curvature of the pressure around the point

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APP

A 1

987-

00Pressure Profile, Drawdown

- The distortion of the pressure profile when a well is openedand flowing (drawdown) is initially described by Darcy’s law.

- The “bending” of the pressure profile is concave, and the“diffusivity Equation” describes how quickly it will evolve within the reservoir

B25

Distance from the well

Pressure

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APP

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987-

00Pressure profile, Build-up

- When the well is shut in, Darcy’s law shows that the profile is flat and “bent”around the wellbore, but unaltered away from the well

- This produces a pressure profile which is “convex” around the well, but still“concave” further out in the reservoir as the diffusion due to the previous drawdown continues

- For radial flow the inflexion point is a circle that moves away from the well asthe build up progresses

B25

Distance from the well

Pressure

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APP

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987-

00Infinite Acting Radial Flow

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Radial flow line

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987-

00Solving the line source problem

Solved in Laplace space and inverted analytically

( ) ��

�−−=D

DiDDD t

rEtrp42

1,2

( ) ��

��

��

Φ−−−=kt

rcEkhQptrp t

ii

21.9486.70, µµ

��

� +≈ 80907.0ln21,100

22

D

DD

D

D

rtp

rtfor

Infinite Acting Radial Flow, the Semilog Approximation

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APP

A 1

987-

00Skin Effect

Skin is an additional pressure change due to heterogenities close to the wellbore

It is a deviation from the ideal inflow.

Possible causes

Pressure drop from undamaged flowing pressure (+ve skin)•Invasion of mud filtrate or cement during drilling or completion•Non-ideal perforations - too low shot density - plugged•Limited entry - partial penetration•Limited entry - partial completion•Turbulent gas flow

Pressure increase from undamaged flowing pressure (-ve skin)•Acidization or stimulation

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APP

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00Skin Effect

Partial Penetration Partial Completion

Skin factor due to the partial penetration or partial completion depends on penetration ratio and the distance zw from the bottom of the reservoirh

hw

h hw

zw

hw

hzw

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APP

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987-

00Skin Effect

Skin Factor

Skin factor is a variable used to quantify the magnitude of the skin effect

It is a dimensionless variable

SkhqB

ps=1412. µ

∆

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APP

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987-

00Wellbore Storage

When we open the master valve at the beginning of a welltest, the well may produce at a constant rate at the surface.

However, the flow rate from the reservoir in to the wellbore may not be constant at all.

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APP

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987-

00Wellbore Storage or Afterflow

During the wellbore storage effect

Pressure is linear function of time

∆ ∆p C t=

Where C is the Wellbore storage coefficient

CC

c r hDt w

=0 8936

2

.φ

Dimensionless wellborestorage constant

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.

Once the wellbore storage effects are over,

and before outer boundary effects are detected,

the reservoir acts as if it were infinite

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The Semi-log Approximation

( ) [ ]p t t SD D D= + +12

0 80907 2ln .

Replacing dimensionless variables

p pqB

kht

kc r

Swf it w

= − + + −� �162 6

08686 3 22752

.log log . .

µµΦ

( )p pqB

kht

kc r

Si wft w

− = + + −� �162 6

08686 322752

.log log . .

µµΦ

This is the equation of a straight line with slope = 162 6. qB

khµ

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Infinite Acting Radial Flow MDH Analysis

MDH Plot - used for analyzing 1st drawdown in a test sequence

( )p pqB

kht

kc r

Si wft w

− = + + −� �162 6

08686 322752

.log log . .

µµΦ

∆ p m

mqBkh

=162 6. µ

khqBm

=162 6. µ

Sp pm

kc r

i hr

t w=

−− +� �1151 32271

2. log .Φµ

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APP

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987-

00Principle of Superposition

Principle of Superposition

The response of a system to a number of perturbations is exactly equal to the sum of the responses to each perturbation as if they were present by themselves

In well test analysis, the Principle of Superposition in Time allows us to determine the reservoir response to a well flowing at a variable rate by using only constant rate solutions

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APP

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987-

00Principle of Superposition – Build-up

Build-up Superposition

4 2 0 0

5 1 5 2 5 3 5

9 0 0


5 7 0 0

5 1 5 2 5 3 5

-1 0 0 0

0

History plot (Pressure, L iquid Rate vs Time)

4 5 0 0

5 1 5 2 5 3 5

9 0 0


+ =

1000 STB/DFlow rate

For all 36 hours

1000 STB/DInjection

For last 24 hours

1000 STB/DFlow for 12 hours

Build-up for 24 hours

The well is producing at q until time tp, we want to find the pressure at time (tp+∆t)- As equations are linear (not normally true) we can usethe principle of superposition

The pressure change due to a combination of previous production periods is equal to the superposition of individual changes due to each production phase

- The shut-in pressure at time (tp+ ∆t) is mathematically equivalent to a continuation of the drawdown at rate q, combined with an injection at rate -q from time tp

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987-


flowrate

pressure

pi

∆PDD

∆t∆PBU

timeinjection (-q) from time tptp

tp

drawdown (q) from time 0

-q

q





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APP

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987-



flowrate

pressure

pi

∆PDD

∆t∆PBU

timeinjection (-q) from time tptp

tp

drawdown (q) from time 0

-q

q ( ) ( ) ( )p p t p t t p tD D pD D pD D D DBU= − + −[ ]∆ ∆

( )p p

qBkh

k t tc r

k tc rWS i

p

t W t W= −

+−�

��

�

�

��

��

1412 12 2 2

.ln ln

µµ µ

∆Φ

∆Φ

( )p p

qBkh

t ttws i

p= −

+162 6.log

µ ∆∆

This is the equation of a straight line with slope =162 6. qB

khµ

tp




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987-

00

Infinite Acting Radial FlowHorner Analysis

Horner Plot - used for analyzing a build-up after a constant rate Dd

mqBkh

=162 6. µ

khqBm

=162 6. µ

( )p p

qBkh

t ttws i

p= −

+162 6.log

µ ∆∆

( )log

t tt

p + ∆∆

Sp p

mtt

kc r

hr wf p

p t w=

−+

+�� − +

�

��

��

11511

3 22712. log log .

Φ µ

If reservoir is truly infinite, extrapolated p*=pi

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APP

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987-

00Principle of Superposition – Multi-rate

( )p t t pqBkh

q qq q

t t tmr n ii i

n nj

j

n

i

n

( ).

log log+ = −−−

−� �−−

− =

−

=

−

∆ ∆ ∆ ∆162 6 1

1 1

1

1

1µ

Expanding the build-up superposition technique for multiple rate changes, the equation with the semi-log approximation is

m Superposition Function

well test analysis - fanarco will be felt at the well reservoir models a reservoir model is the...

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