Mechanistic Rate Decline Analysis in Shale Gas Reservoirs

Dr. George Stewart Chief Reservoir Engineer

Weatherford Intl. Inc

Reservoir Engineering Technology Symposium May 11, 2012

Lfar = xe/2

W

ExternalFractures

Interior

Fractures

Dominantly

Multiple TransverseHydraulic Fractures

Horizontal WellTrajectory

xf

Approximating

Virtual No-Flow

Boundaries

N = 5frac

xe

y = We

y = W/2w

ConfiningExternal

Boundary

W = W/2far

W

ExternalFractures

Interior

Fractures

Dominantly

Multiple TransverseHydraulic Fractures

Horizontal WellTrajectory

xf

Approximating

Virtual No-Flow

Boundaries

N = 5frac

xe

y = We

y = W/2w

ConfiningExternal

Boundary

W = W/2far

Essentially ZeroPermeability

ModifiedPermeability

W

W

xf

Fracture in a Closed Channel - Image Source Solution

FirstImage

FirstImageFirst

Image

PlaneSource

VNFB LNFB

Areal View

Constant Terminal Pressure (CTP) Solution for Rate, q

q(t)

Time, t

Well produced atconstant BHP

Drawdown = p pi wf

Boundary condition of the first kind

Basis of decline curve analysis

CTP solution can be generated from the CRD solution

CRD solutions form the basis of well test analysis

CTP solutions used to analyse production data

CTP solution obtained from the CRD solution by convolution

Real time superposition method is termed forecastingLaplace space method is termed CTP convolution

pi

pwf

0 t

Dxf

3wfi

D

t4/

1

ppkh2

tqq

dwfi

D t4ln

2

1

1

ppkh2

tqq

Canonical CTP Solutions

I.-A. Radial Flow (Line Source)

I.-A. Linear Flow (Plane Source)

. . . Carslaw and Jaeger p 43

2

wt

Drc

ktt

where

2

ft

Dxfxc

ktt

where

CTP Linear Flow Plot for Gas

t

slope mT

A k clf

f t

, . 64 2881

A = hxf f

Requires knowledge of pi

Yields x kf

assuming h known

telf

0

0

S = 0fs

m(p) m(p )i wf

Q(t)

blf

fslf Skh

Tb

1422

33.0

2/Wc

kt000263679.02

t

elf

Flowing Material Balance Cartesian Plot of m(p )/Q versus t wf a

m(p)

SSS Depletion

slope, m* = 2 355. T

hA c t i b g

Rate Normalised Average Pressure Pseudotime, ta

tQ t

c Q t

p c pdta

i gi

g

t

1

0b g b g b g

pd m p Q

dta

/

Derivative of Specialised Plotb*

Intercept

VRD

t = f(A)a

straightline

Closed System

Q

Rate Normalised Pseudopressure Change

Flowing (VRD) Material Balance

Method 1: Approximate Deconvolution or equivalent constant rate (ECR)

Method 2: Material Balance Time, te

• developed by Stewart for analysing VRD data

• developed by Agarwal and Gardner following a

suggestion by Blasingame

ppcVpcVqtNVtqdN i

t

epp 0

Flowing Material Balance

Liquid material balance equation

pppwheretcVq

pie

1Rate Normalised

Average Pressure

Drop from pi

wf

tpp

qJ

wfwfi

tip

q

pp

qJ

Classical Definition of

Transient P.I.

Transient P.I.

Based on p pwf

t or t'

Analysis of Variable Rate Extended Drawdown

pwf

pwf

pwf

corr

p

p

p p m g t q t Swf i lf l l ( )

p p q m t Swf i r lf l

CRD - I.-A.

VRD-I.-A.

mh k c x

lf

t f

4 06411

2.

(field units)

ECR Method

Based on

LSTF

Approximate Deconvolution

Transformed Time, t (hr)

pwf

corr

(psia)

Approximate Deconvolution of Spanning Fracture in a Closed Reservoir

x =100 ft W = 200 ft k = 1 mdf

Exact Volume = 14,2500 bbl

Error ~ 6%

i = 1 +

p pi

iM

k

ki =

i

3

M = E1

(1 + )(1 2)

K = M

3

1 +

1

Palmer and Mansoori CBM Rock Mechanics Model

E = Young’s Modulus = Poisson’s Ratio

Constrained Axial Modulus

Bulk Modulus

Recommended by Mavor

- Based on Linear Elasticity

very sensitive to i

SPE 52607

WellborePressure

Flow-Rate, q

pw

Reservoir Pressure, pe

Locus of Roots of the Stress-Dependent Radial Flow Equation

Low Drawdown RootHigh Drawdown Roots

Only Accessible ThroughReverse Direction Integration High Drawdown Root

Fig. 18.13.5

0

500

1000

1500

2000

2500

3000

3500

4000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

y(p)

Pressure, p (psia)

Combined Pseudopressure Function(p)

(p)

(psia)

Real Gaspseudop

pseudop includingSDPP effect

g i

i i

= 0.7 T = 150 F p = 5000 psia

= 0.01 E = 500,000 = 0.25 n = 3 k = 10 md

o

p

pi

i

i

i

b

pdp1

pk

z

p

k

1

p

zp

Palmer and MansooriModel

i

Linear Steady-State Flow with Stress Dependent Permeability

Linear Steady-State Flow with Stress Dependent Permeability

dy

dppkphxq ff

4Darcy’s Law

e

f

e

f

p

p

eeff

W p

p

f

fdp

p

pkphxdppkph

xdyq

1

1)(4)()(

4

0

Integrating:

e

f

p

pe

e

efef

dpp

pk

kphkx

qW

1

1

4i.e.

Defining:

p

pe

e

b

pdp

pk

kp

1

1SDPP

Normalised

Pseudopressure

fe

efef

pphkx

qW

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1000 1500 2000 2500 3000 3500 4000 4500 5000

kf/k

i

Pressure, pwf (psia)

Permeability Ratio, kf/ki versus pwf and i

M = 600,000 psi

pi = 5000 psia

i = 0.05

i = 0.005

i = 0.01

Choked