Pressure Transient Analysis

Basic concepts

Hassan Bahrami

2013

Well Models

Vertical well Deviated well

Horizontal well

Well Models

Hydraulic fractured well

Longitudinal fracture

Transverse fracture

Reservoir Flow Regimes

• Radial flow:

• Spherical flow:

• Linear flow:

&

Bilinear flow

• Radial flow:

(1/r) d/dr[ r dp/dr ] = (phi*mu*c/k) [dp/dt]

• Spherical flow:

(1/r^2) d/dr[ r dp/dr ] = (phi*mu*c/k) [dp/dt]

• Linear flow:

[d^2p/dx^2] = (phi*mu*c/k) [dp/dt]

Diffusivity equations for the reservoir

flow regimes

Reservoir models

Radial flow

Reservoir models

Spherical flow

Reservoir models

Linear flow in a channel

Elliptical Flow

Reservoir models

Wellbore Storage Effect

Early Time Data are affected by WBS effect

Question:

- What flow regimes may be observed with

passage of time, in a horizontal well ?

Answer:

Question:

- What flow regimes may be observed in a

horizontal well with a single hydraulic fracture

along the wellbore?

Welltest Analysis Basics

Diffusivity Equation for Radial Flow:

(1/r) d/dr[ r dp/dr ] = (phi*mu*c/k) [dp/dt]

Solution for constant rate

drawdown test See chapter 7 of L.P.Dake,

reservoir engineering book

Radial flow equation

Pressure Draw-Down Test Analysis

Time

Flow rate

Pressure Pwf

Pi

P = m (log[t]) + b

Slope: m

Radial flow analysis: Semi-Log Plot

Pi-Pwf

t-t0

Pressure Build-Up Analysis Example

Semi-Log

Plot

Logarithmic

curve fit ! (Not linear)

To make the plot, do

not take log of the

numbers, just show

the time values as

they are, on the log

scale !

Early

Time

Late

Time

Pressure Draw-Down Exercise

Calculate the slope (m) and ‘Pi-P1hr’ for the data

Pi=5000 psia

Answer

A

B

M= -181

psi/cycle

Answer

P1hr = 4080 psia

Pi – P1hr = 5000 – 4080

= 920 psia

P1hr

Question: Considering the following points A and B,

the slope of radial flow straight line (m) is defined as

follows:

m=PB-PA (Psi/Cycle) P

ressure

, psia

∆t,

hrs

Semi-Log Plot

1 10 100

A B

P: Pressure, t: Time

1. True

2. False

Question:

• For radial flow equation in the case of constant

flow rate (pressure declines with time), how can

permeability be determined from the transient

data?

DP = (log[Dt] + n ) kh

BQ o6.162

Question:

• For radial flow equation in the case of constant

bottom-hole pressure (rate decline with time),

how can permeability be determined from the

transient data?

DP = (log[Dt] + n ) kh

BQ o6.162

Question:

• For radial flow equation in the case that bottom-

hole pressure and flow rate are not constant and

change with passage of time, how can

permeability be determined from the transient

data?

DP = (log[Dt] + n ) kh

BQ o6.162

P = m (log[t]) + b

Slope: m

Question: Which ‘m’ should be

considered for K and S calculation?

?

m1

m2

m3

Pressure Derivative Method / Drawdown Test

P = m (log[t]) + b

m*3.2t)(ln d

P)( d

]*3.2log[][ log0]t)(ln d

P)( d[ mtLog

P’

P

))tln(-)tP1)/(ln(-(P2:P' 12

RF

t

P

&

P’r

“0 slope on derivative curve”

Pressure Derivative Method / Drawdown Test

m IARF

hm

BQK

RF **3.2

***6.162

P

&

P’r

t

Question: Using “m” for each test, reservoir around

which well has higher permeability?

m=7E+07

m=7E+05

hm

BQK

RF **3.2

***6.162

Well A

Well B

P

&

P’r

Identifying the Reservoir Flow Regimes using

pressure derivative method:

Log-Log plot of P’, d[p]/d[ln(t)] , versus time function

• Radial flow: Slope: 0 • Spherical flow: Slope: -1/2

• Linear flow:

Slope: +1/2

• Bi-linear flow:

Slope: +1/4

• Elliptical flow:

Slope: +1/3

• Wellbore storage effect:

Slope: +1

Slopes on Log-Log plot

+1 +1/2 +1/3 +1/4

10

100

1000

0.1 1 0.01

-1/2

Estimate slope of the lines

10

100

1000

0.1 1 0.01

Question: Identify the flow regimes

What flow regimes can be detected

on this plot?

A- Linear flow regime

B-Radial flow regime

C-Spherical flow regime

D-Elliptical flow regime

What is the well model?

A- Vertical well

B- Partial perforated vertical well

C- Hydraulically fractured vertical well

D- Horizontal well

E- Cannot be identified

Nomenclature

• P Pressure

• K Permeability

• Q Flow rate

• C Compressibility

• t Time

• h Layer thickness

• r Radius

• Porosity

• Density

• Viscosity

• B Formation volume factor

• NTG Net to Gross ratio

• HC Hydrocarbon

• STOIIP: Stock tank oil initially in place

• GIP: Gas in place

M: 1000

MM: 10^6

MMM: 10^9

1 Acres: 43,560 ft^2

FVF units are rbbl/stb (oil)

and r.cu.ft/scf (gas)

(r for reservoir and s for

standard)