stimulation 7

Hydraulic Fracturing Basic Relations

Basic Relations

• Necessary to understand to apply to fracturing pressure analysis and design parameters

• Material balance – provides basic design requirements for fluid volumes and proppants

• Fluid flow in a fracture

Interaction of fluid and formation

• Rock deformation

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Material Balance expressions

At the end of pumping,

(1)

Qi

Qout

Qf

Flow rate In = Flow Rate Out + Accumulation

Lpfpi VVV

formation to

lost fluid

of volume

fracture

created

of volume

injected

volume

cumulative

Lost VLp

Stored Vfp

VLs (Dt)

Vf (Dt)

VLs (Dt)

Vprop

Pumping, tp

Closure, Dt


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Material Balance expressions

At any time during shutin,

(2)

At closure, Dt = Dtc

(3)

Combining (1) and (3)

(4)

Lost VLp

Stored Vfp

VLs (Dt)

Vf (Dt)

VLs (Dt)

Vprop

Pumping, tp

Closure, Dt

)()(

p tfrom

lost fluid

of volume

fracture

created

of volume

fracture

theof

volume

tLsfptf VVV

t

DD

D

propVctfV D

)(

)space poresolids(

proppant

of volume

fracture

theof

volume

)( ctLsVLpVpropViV D


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At the end of pumping,

Lost VLp

Stored Vfp

pii tqV *

wAV ffp *

pt

fA

pr

LC

LK2

LpV

Lpfpi VVV

formation to

lost fluid

of volume

fracture

created

of volume

injected

volume

cumulative

Af = area of one face of fracture w = average created fracture width,in CL = fluid loss coefficient, ft/(min)1/2

rp = fluid loss area to fracture area KL = fluid loss multiplier

qi = injection rate tp = injection time


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Proppant scheduling

Define efficiency as:

Or

time for the fracture to close defines the efficiency

Amount of pad and proppant scheduling depends on h

Pad fraction:

Lost VLp

Stored Vfp=hVi P

rop

Co

nc,

c/c

f

1

0

0 fpVi Vi

i

fp

V

Vh

hD

p

c

t

t

h

h

1

1pf


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Example


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Example


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pad

Pre-pad flush

Pumping

Pressure geometry

Closure

Pressure CL

Time h

Design

Model

Volume required

Proppant schedule

Pumping

Pressure geometry

Closure

Pressure CL

Time h

Design

Model

Volume required

Proppant schedule


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p

tp

rL

CL

K2wf

Ap

ti

q

Rock deformation • Compliance of fracture describes the ease of fracture deformation

• Principle of crack advance and stresses at the crack tip

• The strain in the formation created by hydraulic fracturing is minor.

• As a result, formation deformation is linear elastic

• Based on the linear elastic assumption, the behavior of a fracture can be modeled using Sneddon’s classical solutions: 2D crack or radial crack Both are: • 2D with one-dimension infinite in extent • Elliptic shaped cracks • Inversely proportional to E’…plain strain modulus • Proportional to a characteristic dimension and net pressure

21

EE


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Rock deformation Stress intensity factor, KI, - characterizes the magnitude of the stresses near the crack tip

- f (geometry of body, loading parameters) - LEFM states a fracture will advance when KI reaches a critical value.

Stress concentration near the tip of the crack. Fracture toughness – measure of the resistance of the rock to crack, i.e., propagate.


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Fluid flow in fracture Pressure gradient exists along the fracture Local pressure gradient is given by the fluid rheology, velocity, and fracture width. where k’ and n’ are consistency and behavior indices, respectively, for a power law

model. If k’ = m and n’ = 1, then this equation reduces to Newtonian fluid.

Pnet

xf

Pnet=0

tiprw

closureP

fractureP

netP D

n

f

i

n h

q

w

k

dx

dp

12


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Fluid flow in fracture: pressure gradient correction Classical fracture models assume pressure in the fracture is constant. However, the fluid flow relation indicates a gradient from pwf to pc. Thus define, where pf is average pressure within the fracture. Consequently, substitute for pf – smin in width equations with, • Includes the pressure gradient effect from flow and fluid rheology

along the fracture • Includes wellbore pressure

f

f

cw

cf

p

p

pp

pp

D

D

fff ppp DD s min


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Rock deformation

Pc smin ??

• Fracture closure pressure is a global parameter which defines the fluid pressure for which the fracture effectively closes. It is the average of formation heterogeneities.

• The minimum stress is a local parameter which generally varies over the plane of the fracture.


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stimulation 7

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