Induced Fractures Modelling in Reservoir Dynamic Simulators

7/21/2019 Induced Fractures Modelling in Reservoir Dynamic Simulators

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Induced Fractures Modelling inReservoir Dynamic Simulators

Khaldoon AlObaidiInstitute of Petroleum Engineering

MSc Petroleum EngineeringProject Report 2013/2014

SupervisorHeriot Watt University

This study was completed as part of the Masters of Science in Petroleum Engineering at the Heriot Watt University.


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Declaration

I, Khaldoon AlObaidi, confirm that this work submitted for assessment is my own and is

expressed in my own words. Any uses made within it of the works of other authors in any form

(e.g. ideas, equations, figures, text, tables, programs) are properly acknowledged at the point of

their use. A list of the references employed is included.

SignedK.A...

Date 27 August 2014


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Dedication

To my family for their support.

To my uncle for his continuous encouragement.


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Acknowledgments

Thanks also go to NSI Technologies Inc. for providing me with StimPlan software and licenses

which helped me establishing the basis of this work.


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Abstract

Since the middle of the twentieth century, hydraulic fractures and fractures created due to

injection under fracturing conditions have been proven to be effective in increasing the

productivity and injectivity factors of wells considerably. In this work, an algorithms for

determining the optimum hydraulic fracture dimensions, the growth of induced fractures

created due to injection under fracturing conditions and modelling fractures in dynamic

reservoir simulators are introduced. Additionally, the optimum dimensionless conductivity is

derived to be 1.6363 and is used in addition to practical limitations and economic considerations

to determine the optimum hydraulic fracture dimensions result in maximum folds of increase

in production. Also in this work, an algorithm adopting Perkins-Kern-Nordgren- (PKN-) and

Ahmed and Economides notation after Simonson analysis is adopted to determine the

dimensions of the induced fractures created due to injection under fracturing conditions. The

induced fractures are implemented in reservoir dynamic simulators using gridblocks refinement

and properties multiplications to increase net to gross, porosity and permeability to mimic the

fracture properties. For two simple box models, only approximately 42% increase in run time

due to implementing this algorithm in reservoir dynamic simulator is resulted. Therefore, the

algorithm presented provides a good approximation for modelling induced fractures growth

with reduced simulation run time and storage capacity compared to three-dimensional fracture

models. Also, it provides more accurate results compared to the simple two-dimensional models

that assumes fixed fracture height. The advantage of the algorithms presented is they combine

the fracturing physics with the reservoir dynamic simulator constraints. Therefore,

implementing this work provides robust reserves estimation and forecasts for wells with

induced fractures warning of fractures propagation into unintended with relatively fast running

simulation models.


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Table of Contents

Declaration ................................................................................................................................ ii

Dedication .................................................................................................................................iii

Acknowledgments .................................................................................................................... iv

Abstract ..................................................................................................................................... v

Table of Contents ..................................................................................................................... vi

List of Figures ........................................................................................................................viii

List of Tables ............................................................................................................................ ix

Nomenclature ............................................................................................................................ x

1 Project Scope and Objectives .......................................................................................... 1

1.1 Induced Fractures ....................................................................................................... 1

1.1.1 Fracture Orientation ................................................................................................ 3

1.1.2 Leak-off Test .......................................................................................................... 4

1.1.3 Hydraulic Fracturing Procedure ............................................................................. 6

1.1.4 Analytical and Numerical Models for Estimating Fracture Dimensions,

Propagation and Recession ................................................................................................. 7

1.1.4.1 PKN- Model ................................................................................................. 8

1.1.4.2 Fracture Height Growth .................................................................................. 9

1.2 Dynamic Simulation of Hydraulic Fractures ............................................................ 10

1.3 Objectives ................................................................................................................. 11

2 Methodology .................................................................................................................... 12

2.1 Determining the Optimum Hydraulic Fracture Dimensions for a Well ................... 12

2.1.1 Cases Input for Testing the Procedure Used to Determine the Optimum Hydraulic

Fracture Dimensions for a Well........................................................................................ 19


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| Table of Contents

2.1.1.1 Low Permeability Reservoir Case ................................................................ 20

2.1.1.2 High Permeability Reservoir Case ............................................................... 20

2.2 Incorporating Hydraulic Fractures in Reservoir Dynamic Simulators ..................... 21

2.3 An Algorithm for Modelling Induced Fractures Created During Injection under

Fracturing Conditions in Reservoir Dynamic Simulators .................................................... 24

2.3.1 Fracture Height Growth ........................................................................................ 24

2.3.2 Fracture Width and Half-Length .......................................................................... 27

3 Results and Discussion ................................................................................................... 33

3.1 Determining the Optimum Hydraulic Fracture Dimensions for a Well ................... 33

3.1.1 Low Permeability and High Permeability Case Results ....................................... 33

3.1.1.1 Low Permeability Reservoir Case ................................................................ 34

3.1.1.2 High Permeability Reservoir Case ............................................................... 36

3.2 Incorporating Hydraulic Fractures in Reservoir Dynamic Simulators ..................... 38

3.3 Algorithm for Modelling Induced Fractures Created During Injection under

Fracturing Conditions in Reservoir Dynamic Simulators .................................................... 40

4 Conclusions and Recommendations.............................................................................. 42

References ................................................................................................................................ 43

Appendices ................................................................................................................................ I

A.1 Fracture Height Equations ...........................................................................................I


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List of Figures

Figure 1. Principal stresses (Anon.g2014)................................................................................ 3

Figure 2. Extended leak-off results (Crain 2013) ....................................................................... 5

Figure 3. Fracture geometry (a) PKN type (b) KGD type (c) Pseudo 3D cell approach (d) Global

3D, parameterised (e) Full 3D, meshed (Yang 2011)................................................................. 8

Figure 4. Warpinski and Smith analysis and notation (Valko and Economides 1995) ............ 10

Figure 5. Cinco-Ley and Samaniego graph for Hydraulic Fracture Performance with f = sf + ln

(xf/rw) (Valko 2005) ................................................................................................................ 13

Figure 6. Prats dimensionless effective wellbore radius (Valko 2005)................................... 17

Figure 7. Low permeability case model ................................................................................... 22Figure 8. Hydraulic fracture for low permeability case ............................................................ 22

Figure 9. High permeability case model ................................................................................... 23

Figure 10. Hydraulic fracture for high permeability case ........................................................ 23

Figure 11. Ahmed and Economides notation for fracture height (Valko and Economides 1995)

.................................................................................................................................................. 24

Figure 12. Algorithm for estimating fracture dimensions created due to injection under

fracturing conditions ................................................................................................................. 32

Figure 13. Folds of increase and fracture volume vs. folds of increase for low permeability case

.................................................................................................................................................. 35

Figure 14. Folds of increase and fracture volume vs. folds of increase for high permeability case

.................................................................................................................................................. 37

Figure 15. Increase in cumulative oil production for low permeability case ........................... 38

Figure 16. Increase in cumulative oil production for high permeability case .......................... 38


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List of Tables

Table 1: Low permeability case input ...................................................................................... 20

Table 2: High permeability case input ...................................................................................... 20

Table 3: Model dimensions summary ...................................................................................... 21

Table 4: Low permeability case results .................................................................................... 34

Table 5: High permeability case results ................................................................................... 36

Table 6: Run time and storage capacity requirement for base cases and cases with induced

fracture modelling ..................................................................................................................... 40


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Nomenclature

A = the fracture surface area at any instant during injection, ft2

Ae= the fracture surface area at the end of pumping, ft 2

Bo= the oil formation volume factor, rb/STB

CL= leak-off coefficient ft/s0.5

E = strain modulus, psi

Fcd= the fracture dimensionless conductivity, dimensionless

FOI = folds of increase, dimensionless

h = the flow unit height, ft

hd= the lower height growth, ft

hds= the dimensionless thickness, dimensionless

hf= fracture height, ft

hp= the perforation interval length, ft

hs= the thickness of a symmetry element, ft

hu= the upper height growth, ft

i = half injection rate, ft3/s

k = permeability, mD

k00= the pressure at the middle of the crack, psi

k1= the slope of net pressure, psi

K(C,2)= fracture toughness in the upper layer, psi.ft0.5

K(C,3)= fracture toughness in the lower layer, psi.ft0.5

kf= the fracture permeability, mD

kh= the horizontal permeability, mD

kv= the vertical permeability, mD


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Nprop= the dimensionless proppant number, dimensionless

pbhpf= the bottom-hole flowing pressure, psi

pcp= the pressure at mid perforation, psi

pn,w= the net wellbore pressure, psi

re= the drainage radius, ft

rw= the wellbore radius, ft

rw is the effective wellbore radius, ft

Sf= the skin factor due to fracture, dimensionless

Sp= spurt loss coefficient, ft

t = time, second

vf = the total fracture volume of both wings, ft3

vL= leak-off velocity, ft/s

w = average fracture width, ft

wf = fracture width, ft

ww,0= the maximum fracture width at the wellbore, ft

xf = fracture half-length, ft

= the exponent of fracture length growth (constant), dimensionless

= the viscosity, cP

= density, lb/ft3

1= the minimum horizontal stress in the targeted layer, psi

2= the minimum horizontal stress in the upper layer, psi

3 = the minimum horizontal stress in the lower layer, psi

min= the minimum horizontal stress, psi


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1 Project Scope and Objectives

1.1 I nduced Fractures

Induced fracturing is a stimulation method used to accelerate the production and increase the

ultimate recovery of hydrocarbon reservoirs by fracturing the reservoir rock (Anon. a2013).

Fracturing the rocks creates high conductivity channels growing into the reservoir away from

the wellbore, providing communication between the two (Anon. b2013). These fractures are

called induced fractures since they are introduced to the reservoir and are not formed due to

natural causes (e.g. tectonic activities).

Since the 1940s, induced fracturing has proven to be an effective method for developing low

permeability reservoirs and increasing the commercial viability of the development of

conventional reservoirs (Taleghani et al. 2013). Also, induced fractures have made it possible

to produce hydrocarbon from shale formations (tight reservoirs) where conventional

technologies are ineffective (Anon. c2014). Progress in hydraulic fracturing technologies has

resulted in a huge increase in the oil and gas reserves worldwide by making the development

of unconventional reservoirs feasible (Anon. d2014).

There are three types of induced fractures: hydraulic fractures; fractures created by fluid

(usually water and/or polymer) injection under fracturing conditions; and thermal fracturing

(Taleghani et al. 2013). Hydraulic fractures are created by injecting specially engineered fluid

under high pressure for a short period of time to break the rocks. The created fractures are kept

open after treatment using proppant (a material similar to sand grains) of a particular size, which

is mixed with the treatment fluid (Anon. e2014).

Another type of induced fracture is created by the continuous injection of fluid under high

pressure into the reservoir (greater than the fracture initiation pressure to create the fracture,

greater than the closure pressure to keep the fracture open and greater than the fracture


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propagation pressure to extend the fracture). These fractures are closed once the fluid injection

stops or the injection pressure becomes less than the fracture closure pressure (Moreno et al.

2005). The final type of induced fracture is thermal fracturing. Thermal fractures are created

due to the difference between the temperature of the reservoir rock and that of the injected fluid,

with the latter being colder (Anon.f2013). Only the first two types of induced fractures will be

considered in this work.

Fracture dimensions are considered the most important factor in induced fracturing for three

main reasons: the incremental increase of production/injection rates is directly dependent on

the fracture dimensions; the cost of creating the fracture is directly proportional to the fracture

volume; and there is the possibility of induced fractures growing into unintended zones like

fresh water zones.

The environmental impacts associated with hydraulic fracturing are the main reason for it being

a controversial topic among the public (Shukman 2013). Therefore, it is necessary to simulate

the induced fracture propagation, dimensions and recession before the actual operations take

place (Xiang 2011). Extensive work has been done to simulate the fracture propagation and

dimensions. There are multiple analytical and numerical models available in the literature for

estimating fracture dimensions, propagation and recession with different geometries. They

include two-dimensional, three-dimensional and pseudo three-dimensional models (Yang

2011).

The advantage of simulating fracture dimensions and propagation using the algorithms and

methods introduced in this work is that they incorporate the actual reservoir dynamic simulator

constraints and pressure data for the whole life of the field. Also, they take into account the

practical limitations to estimate the optimum fracture dimensions, resulting in the maximum

possible increase in production or injection rates for the wells. Therefore, this results in robust

production and injection forecasts for reservoirs with induced fractured wells, and thus more


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representative economics for field development. Estimating the optimum hydraulic fracture

dimensions, modelling hydraulic fracture dimensions, induced fracture propagation in reservoir

dynamic simulators, and estimating the height growth of induced fractures are all covered in

this work. Over the past 70 years, extensive work has been done on induced fracturing. This

work is documented and can be found in the literature. The following sections discuss topics

related to induced fracturing available in the literature.

1.1.1 Fracture Orientation

Based on rock mechanics, there are three principal stresses acting on underground formations.

These are the overburden stress, the maximum horizontal stress, and the minimum horizontal

stress, as shown inFigure 1.

Figure 1. Principal stresses (Anon. g2014)

These stresses are usually anisotropic in that they differ in magnitude based on direction (Anon.

a2013). Fractures propagate in a direction which is perpendicular to the least stress (i.e. opening

in the direction of the least resistance) (Anon. h 2010). The overburden stress acting on a


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formation is due to the weight of the rocks above that formation, which depends on the

formation depth (Golf-Racht 1980).

Based on practical experience, for formations deeper than 2000 ft, the overburden stress is the

largest principal stress, followed by the maximum horizontal stress; minimum horizontal stress

is the smallest principal stress and the fractures are more likely to be vertical (Anon. h2010).

For formations shallower than 2000 ft, the maximum horizontal stress is the largest stress,

followed by the minimum horizontal stress, and the overburden stress is the smallest principal

stress (Anon. h2010). Therefore, for such formations, the fractures will be horizontal, opening

in the vertical direction with an environmental risk, since it may propagate to the surface.

It can be concluded that the magnitude and direction of the principal stresses play a major role

in determining the required pressure for fracture creation and propagation (Hudson 2005). The

interaction between the fluid pressure in the fracture and the principal stresses defines the shape,

the vertical extent and the propagation direction of the fracture (Dubey et al. 2012).

1.1.2

Leak-off Test

This is a test performed to measure the formation fracturing pressure usually carried

immediately after drilling below a new casing shoe. The test is performed by shutting-in the

well and pumping fluid, usually mud, into the wellbore to gradually increase the pressure

experienced by the formation. At some pressure, the fluid enters the formation (orleaks-off)by

fracturing the rock (Anon. e2014).

If the test is stopped just after the leak-off happens then it is called a leak-off test (LOT). If the

test is extended longer until several iterations of pumping and discontinuing pumping have been

performed then it is called an extended leak-off test (XLOT). From the XLOT, more important

parameters can be estimated and used in the propagation and recession models.Figure 2 depicts

XLOT results.
http://www.glossary.oilfield.slb.com/en/Terms/l/leak_off.aspxhttp://www.glossary.oilfield.slb.com/en/Terms/l/leak_off.aspx


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Figure 2. Extended leak-off results (Crain 2013)

From XLOT results, vital parameters for simulating the fracture propagation, opening and

closure are estimated. These include the fracture initiation pressure, fracture propagation

pressure, fracture reopening pressure and fracture closure pressure (which is synonymous with

minimum in-situ stress and minimum horizontal stress) (Anon. a2013). These data will be used

as an input to modelling induced fractures created due to injection under fracturing conditions

using the PKN- method.

Time

P friction

Bottomhole Pressure

Injection Rate1

3

5

2

3

64

1. Hydrostatic pressure

2. Breakdown pressure

3. Fracture extension pressure

4. Initial shut-in pressure (fracture gradient)

5. Fracture closure pressure (closure stress gradient)

6. Fracture reopening Pressure


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1.1.3 Hydrauli c Fracturing Procedure

The process of hydraulic fracturing consists of injecting specially engineered fluid at high

pressure to break the formations and create high permeability channels extending away from

the wellbore into the formation and establishing communication between the two. To keep the

fracture open, proppant with specific grain diameter is used.

The stages of hydraulic fracturing, as covered in the literature (Anon. h2010), include:

i. Spearhead stageor acid stagewhich consists of water mixed with acid. The purpose

of this stage is to remove the debris and clean the wellbore. This will provide a clean

wellbore and an open path for the fluid to be injected in subsequent stages.

ii. Pad stagewhich consists of slick water that is used to initiate the hydraulic fracture

in the formation. If the pressure stopped during this stage, the fractures would close

since no proppant material has yet been used.

iii. Proppant stage which consists of injecting water and proppant material into the

fractured formation to keep the fractures open. Proppant is a non-compressible

material, like sand grains, that is carried into the fractured formation to be left there

after the job has been completed. Once the pressure drops, the proppant will prevent

the fractures from closing, thus maintaining the enhanced permeability channels,

created in the pad stage, throughout the wells life.

iv. Flush stagewhich consists of fresh water being pumped into the wellbore to flush out

and remove the excess proppant from the wellbore.


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Figure 3. Fracture geometry (a) PKN type (b) KGD type (c) Pseudo 3D cell approach (d)

Global 3D, parameterised (e) Full 3D, meshed (Yang 2011)

1.1.4.1

PKN- Model

As mentioned in Section1.1.4,the PKN model assumes the fracture has constant height and

that fracture length is significantly greater than fracture height. The PKN geometry depicted in

Figure 3(a), which shows an approximately elliptical shape in the vertical and the horizontal

directions, is more interesting from the production point of view. The PKN- model assumes

the power law surface growth and Carter I leak-off to perform the material balance at any time

during injection.


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The power law surface growthassumes that the fracture surface grows according to a power

law relating the area of the fracture at any time during injection to the area of the fracture at the

end of the injection with an exponent, , that is constant during the period of injection (No lte

1986). Carter introduced the leak-off velocity by relating a leak-off coefficient to the elapsed

time since the start of the leak-off process and spurt loss based on the concept of Howard and

Fast (Howard 1957). Equations related to both assumptions are discussed in details in chapter2.

1.1.4.2 Fracture Height Growth

The two-dimensional models suggested in the previous sections are simplified approximation

of the fracture dimensions and geometry. These models assume constant fracture height and

leave the half-length and fracture width to be estimated from injected fluid volume. However,

practical experience showed that fractures in some cases grow into unintended zones up and

down the targeted interval (Valko and Economides 1995).

This observation triggered the attempts to develop models able to simulate the fracture height.

Since there are many variables in the system of equations, simplifying the approach is essential

to result in an acceptable approximation used the simplest case by neglecting hydrostatic

pressure inside the fracture and using similar properties for the upper and lower layers for

approximating fracture height growth (Simonson et al. 1978). Another analysis that is widely

acceptable in oil and gas industry is Warpinski and Smith analysis with a more complex case

(Warpinski and Smith 1989). The notation used by them is shown in Figure 4. Alternative

notation is used by Ahmed and Economides which is discussed in chapter2.


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Figure 4. Warpinski and Smith analysis and notation (Valko and Economides 1995)

The assumptions of Warpinski and Smith analysis (Valko and Economides 1995) are:

i. The minimum horizontal stress of the upper and lower layers can be different, but

higher than minimum horizontal stress of the targeted layer.

ii.

The critical stress intensity factor (stress intensity near the tip) can be different for the

upper and lower layers.

iii. The density of the fluid is considered in the analysis

The notation will be used in this work is Ahmed and Economides notation and the set of the

equations used to determine the fracture height growth is presented in the methodology chapter.

1.2 Dynamic Simulation of H ydraul ic F ractures

As for modelling the effects of induced fractures in the reservoir dynamic simulators, there are

multiple approaches, which include: using a negative skin factor, creating channels of enhanced

permeability gridblocks in the direction of fracture orientation, using non-neighbourhood

connections and/or a local increase of absolute permeability near the wellbore (Carlson 2006;

Owen1983).


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These approaches are helpful to a certain extent for increasing productivity, but they do not

incorporate the physics behind fracture propagation and recession (Carlson 2006; Owen1983).

Also, some of the routines used for 3D pseudo models result in long run times, making them

impractical for large reservoir models (Economides 2000).

1.3 Objectives

The objectives of this work are:

i. To develop an algorithm for determining the optimum hydraulic fracture dimensions of

a well and incorporating hydraulic fractures in reservoir dynamic simulators.

ii. To model induced fractures in fluid (usually water and/or polymer) injectors created

during injection under fracturing conditions using reservoir dynamic simulators.

iii.

To develop an algorithm for modelling the height growth of induced fractures during

injection under fracturing conditions using reservoir dynamic simulators.

These algorithms and methods are simple and easy to incorporate in the reservoir dynamic

simulators to provide robust production and injection forecasts. This work is of great economic

and environmental benefit because the fracture dimensions are the single most important factor

which determines the increase of production/injection rates, the volume and cost of used

fracturing material, and the zones into which fractures propagate.


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| Chapter 2Methodology

2 Methodology

This chapter represents the methodology followed to achieve the stated objectives. Derivation,

calculations and Eclipse modelling are shown in this chapter for each objective.

2.1 Determin ing the Optimum Hydraul ic Fracture Dimensions for a Well

For the first objective of determining the optimum hydraulic fracture dimensions, the optimum

fracture conductivity for pseudo-steady state and steady state flow conditions is calculated. For

conventional reservoirs, since most wells spend the majority of their lifetime in a pseudo-steady

state flow regime, the solution reached should represent the optimum hydraulic fracture

dimensions of a well (Richardson 2000).

The analysis starts with the use of Darcy law for pseudo-steady state flow conditions, as shown

in Eq.1:

2/

34 1 where: k is permeability, h is the flow unit height, is the viscosity, Bo is the oil formationvolume factor, reis the drainage radius, rwis the wellbore radius and Sfis the skin factor due to

fracture.

It needs to be noted that the assumption here is that the skin due to damage is not a part of the

optimum fracture dimension calculations since it happens due to drilling, production and/or

completions. However, skin will be used later as a check that the wellbore radius, due to damage

(rs), is less than half the length of the fracture (xf), to confirm that the hydraulic fracture bypasses

the damage zone.


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In order to maximise the production rate, the denominator of Eq. 1 has to be minimised.

Defining function G as the denominator of Eq.1, as shown in Eq.2, which has to be minimised

to increase rate.

34 .2Now, defining function A, as shown in Eq.3.

. . 3 Based on Figure 5, A is the y-axis of the Cinco-Ley and Samaniego graph (Valko and

Economides 1995).

Figure 5. Cinco-Ley and Samaniego graph for Hydraulic Fracture Performance with f =

sf+ ln (xf/rw) (Valko 2005)


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As shown in Eq. 4, function G becomes:

34 ..4Simplifying function G, as shown in Eqs. 5 through 7.

34 . . 5 34 . . 6

3

4 . . . . . 7

Two functions should be introduced here: the fracture dimensionless conductivity and the

fracture volume as shown in Eqs. 8 and 9.

8 where Fcd is the fracture dimensionless conductivity, kfis the fracture permeability, wfis the

fracture width, k is the matrix permeability and xfis the fracture half-length.

The fracture dimensions are related to each other by the fracture volume, as shown in Eq. 9.

2 . . 9 where vf is the total fracture volume of both wings and hfis the fracture height.

By combining Eqs. 8 and 9, the fracture half-length can be estimated using Eq. 10.

2 . 1 0 By substituting Eq.10 in Eq. 6, as shown in Eq. 11, the G function becomes:

2 34 . . 11


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where A is defined in Eq. 12, as shown inFigure 5 as:

1.65 0.328 0.116 1 0.18 0.064 0.005 . 12To determine the Fcdvalue that will result in the minimum function G, the function is derived

with respect to Fcd. Substitution of function A in function G and the derivation is shown in Eqs.

13 and 14.

2 1.65 0.328 0.116 ln1 0.18 0.064ln 0.005ln

34 . 13 12

23.2 5.6552 29.521 35.862 1077.59

12.8 36 200 .14

By setting the derivative equal to zero and solving for the fracture dimensionless conductivity,

Fcdthat results in minimum value of the G function can be found;

Fcd= 1.6363.

Thus, the optimum fracture dimensionless conductivity value for a fracture in a well flowing

under pseudo-steady state flow conditions is 1.6363.

The partial penetration skin is the function of two parameters; the penetration ratio and

dimensionless thickness (b and hds) (Brons et al. 1961). The penetration ratio (b) is assumed to

be set dependent upon given facts of a specific reservoir to ensure reduction in water and/or gas

production and that the best part of the reservoir is targeted. Thus, the only variable to be

considered for the calculation of the optimum hydraulic fracture dimensions is the


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dimensionless thickness (hds). Brons and Marting defined the hdsof a fractured well as shown

in Eq. 15.

. 15where hdsis the dimensionless thickness, hsis the thickness of a symmetry element, rw is the

effective wellbore radius, khis the horizontal permeability and kv is the vertical permeability

(Brons et al. 1961).

By analysing Eq. 15, the goal of minimising hdcan be achieved by increasing rw. Since Sfis

inversely proportional to rw, determining the minimum Fcdusing Sf, as above, results in the

maximum rw.

The method described in this work includes starting from the minimum additional economic

value gain required from a hydraulic fracturing project. Let us assume that the screening

criterion of a small project for a company is a net present value of x $. Based on the oil price,

the production forecast without hydraulic fracturing and hydraulic fracturing job cost, the

additional increase in oil production rate results in a net present value of x$ due to accelerated

production can be estimated.

Thus, the minimum required folds of increase (FOI) for the project to pass the screening

criterion are calculated. To estimate the optimum fracture dimensions, relating FOI to the

fracture half-length would provide a tool to directly determine the optimum fracture dimensions

from a known or targeted FOI value. This is shown by relating the effective wellbore radius to

FOI and using Prats dimensionless effective wellbore radius, as explained below.

FOI and skin due to fracture are related, as shown in Eq. 16.

. . 1 6


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where the skin due to fracture is related to the effective wellbore radius, as shown in Eq. 17.

1 7 Thus, FOI can be related to the effective wellbore radius, as shown in Eqs. 18 and 19.

1 8 1 9

Using Eq. 19, a table of rw vs. FOI can be developed with the FOI value of the minimum

estimated from economics or greater.

Using Prats dimensionless effective wellbore radius and knowing the optimum Fcd is 1.6363,

rw/xf can be found, as shown inFigure 6.

Figure 6. Prats dimensionless effective wellbore radius (Valko 2005)

Thus, the optimum rw/xf value is 0.2534, which can be used for finding the optimum xfresults

in the maximum FOI value. It is important to note that there is a maximum theoretical value of


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xfthat is equal to the drainage radius (re), which is used as a maximum constraint for the fracture

half-length calculation.

It is also important to mention that the work thus far only assumes the theoretical value and

does not include the practical aspect of hydraulic fracturing. For example, is it possible to

achieve a fracture half-length equal to the drainage radius of the reservoir? Is it possible to have

all of the injected fluid contained in the pay zone, or intended zone? Valko introduced a

parameter called the dimensionless proppant number (Nprop) as shown in Eq. 20 (Valko 2001).

4

. . 2 0 According to Valko, since the proppant cannot be contained in the pay zone and within thedrainage area and for large treatments there is a great uncertainty as to where the proppant goes

in both horizontal and vertical directions, there is a practical limit to the dimensionless proppant

number (Valko 2001). The practical N number is less than or equal to 0.1 for medium and high

permeability formations (50 mD and above), while for low permeability reservoirs a

dimensionless proppant number more than 0.5 is rarely realised. Therefore, another condition

is applied in this work to the calculation performed, which is the use of N of 0.1 or less for

formations with a permeability of 50mD and above, and 0.5 or less for the formations with a

permeability of less than 50mD (Valko 2001).

Also, it is preferable to have the fracture half-length longer than the damage radius to eliminate

the impact of damage on production. If the fracture half-length is increased, the width also has

to be increased to maintain the optimum Fcd value.Thus, the proppant volume required is

increased so that the economic value resulted from increasing production by increasing the

volume of the fracture versus the economic saving made on the proppant cost by keeping the

fracture half-length less than the damaged radius has to be evaluated. Finally, Eq. 8 is used to

calculate the fracture width, since everything else is known.


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In the case of low permeability reservoirs, the fracture width calculated by the method above

can be small. Practically, the fracture width has to be large enough for the proppant to be placed

in the fracture to keep it open. Therefore, another condition to be applied is that the fracture

width has to be at least two to three (2-3) times the mesh proppant grain diameter. In such a

case, starting with a fracture width of 2-3 times the proppant grain diameter, the length can be

calculated using N value of 0.5 or lower (applying the condition that the calculated half-length

fracture is equal to or less than drainage radius). By analysing Eq.8, for low permeability

reservoirs, it can be inferred that a long fracture is required for a certain minimum width

determined to result in optimum Fcd value. Thus, in cases of low permeability low drainage

radius reservoirs, the theoretical limitation on the maximum possible fracture half-length in

addition to the practical limitations previously mentioned may result in F cdvalues more than

1.6363.

An excel workbook is developed to perform all the calculations mentioned in this section.

Further work can be performed by linking this workbook to Eclipse Dynamic Simulator so that

refinements and calculations are performed automatically.

2.1.1 Cases Input for Testing the Procedure Used to Determi ne the Optimum H ydrauli c Fracture

Dimensions for a Well

For the first objective, two cases from literature (Valko and Economides 1995) are used to test

the analysis of this work, perform the calculations for the optimum fracture dimensions and

perform the Eclipse Dynamic Simulation. The results match very well, as shown in the results

and discussion chapter.

The two cases are for a low permeability reservoir and a high permeability reservoir. Each

reservoir has six water injectors and one oil producer. The oil producer is to be hydraulically


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fractured to increase the reservoirs production rate. It is necessary to determine the optimum

hydraulic fracture dimensions, estimate the increase in production rate and develop the new

production forecast.

2.1.1.1

Low Permeability Reservoir Case

Input data for the low permeability reservoir case are shown inTable 1.

Table 1: Low permeability case input

Horizontal Permeability 0.5 mD

Formation Height 105 ft

Fracture Height 35 ft

Fracture Permeability 60000 mD

Proppant Mesh Diameter 70 mesh (420 m)

Drainage radius 2100 ft

Wellbore radius 0.328 ft

Damage skin 0

2.1.1.2

High Permeability Reservoir Case

Input data for the high permeability reservoir case is shown inTable 2.

Table 2: High permeability case input

Horizontal Permeability 500 mD

Formation Height 150 ft

Fracture Height 30 ft

Fracture Permeability 100000 mD

Proppant Mesh Diameter 40 mesh (840 m)

Drainage radius 1000 ft

Wellbore radius 0.328 ft

Damage skin 0


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2.2 I ncorporating Hydraul ic Fractures in Reservoir Dynamic Simu lators

To estimate the impact of hydraulic fracturing on reservoir production and/or injection rates, a

high permeability channel in a refined box model is simulated using Eclipse Reservoir

Simulator. The results of the optimum dimensions for hydraulic fracture (Section2.1)are used

to create a high permeability channel near the wellbore, extending away from it by fracture

half-length in each direction. In the two cases of this work, the model is refined to have the

width of the gridblocks equal to the hydraulic fracture width, as shown in Figure 8 andFigure

10.

In both cases, the models have six water injectors and one hydraulically fractured oil producer,

as shown in Figure 7 andFigure 9.A permeability multiplier, NTG multiplier and porosity

multiplier are used for the gridblocks representing the fracture to mimic the hydraulic fracture

permeability, increase the porosity of the gridblocks to 100% and increase NTG to 100%.Table

3 summarises the Eclipse dimensions used for the two cases.

Table 3: Model dimensions summary

Item Low permeability case High permeability case

Number of gridblocks 11 x 1000 x 3 11 x 10000 x 5

Model dimensions (ft) 410.8 x 0.014 x 35 87.2 x 1.783 x 30


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Figure 7. Low permeability case model

Figure 8. Hydraulic fracture for low permeability case


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Figure 9. High permeability case model

Figure 10. Hydraulic fracture for high permeability case

In other cases, models can be refined so that the gridblock width is less than the fracture width

(i.e. the fracture channel includes more than one gridblock in the width direction). In the case

of large models, local gridblock refinement can be performed instead to mimic the same


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procedure followed in this work. Also, fracture half-length should be taken into consideration

of refinement in case fracture half-length estimated is less than gridblock length.

2.3

An Algorithm for Modell ing I nduced Fractures Created Dur ing I njection under

F ractur ing Conditi ons in Reservoir Dynamic Simulators

The following sections introduce the methodology used in this work to develop an algorithm to

be used to simulate the propagation, recession and dimensions induced fractures created due to

injection under fracturing conditions.

2.3.1

Fracture Height Growth

As mentioned in the Section1.1.4.2,Ahmed and Economides notation (Economides 1992) is

used in this work. The variables to be estimated as shown in the notation (Figure 11)are the

upper (hu) and lower (hd) height growth.

Figure 11. Ahmed and Economides notation for fracture height (Valko and Economides

1995)

The solution can be achieved by solving for two unknowns in two equations (Eqs. 21 and 22).


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,

. { , , 1, , , ,

, , 1, , , , } , 2 . 21

,

. { , , 1, , , , , , 1, , , , } , 3 . . 22

where:

1 2

. . 2 3

1 2 . . 2 4 2 . 2 5 2 . 2 6

1 1 . 2 . 2 2 2 2 .tan 1 1 1 . . 2 7


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1 1 . 2 . 2 2

2 2 .tan 1 1 1 . . . 2 8 It should be noted that the following limits (Eqs. 29 through 32) are required to perform the

calculation:

1,, 42 . 2 9

1,, 42 . 3 0

1,, 4 2 . 3 1 1,, 2 . . 3 2

where: hpis the perforation interval length, huis the upper height growth, hdis the lower height

growth, is the density, 1 is the minimum horizontal stress in the targeted layer, 2 is the

minimum horizontal stress in the upper layer, 3 is the minimum horizontal stress in the lower

layer, K(C,2) is fracture toughness in the upper layer, k0 is a constant, k1 is the slope of net

pressure, k00 is the pressure at the middle of the crack, K(C,3) is fracture toughness in the lower

layer and pcpis the pressure at mid perforation.

All of the parameters in these two equations are inputs except hdand huare the variables to be

solved for. These two variables are calculated at each time step and the fracture height results

is used in PKN- calculation for determining the fracture half-length and width.

In this work, these two equations were combined and simplified. The final complete version of

these two equations with two unknowns are shown in appendixA.1.


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2.3.2 Fracture Width and Half-L ength

As mentioned in Chapter1,the PKN geometry assumes an elliptical shape in the vertical and

the horizontal directions for the hydraulic fracture. It assumes that the height, hf, is constant and

that the half-length, xf, is considerably greater than the width, wf. The method used in this work

to simulate induced fractures developed due to injection under fracturing conditions is the PKN-

method, which assumes:

i. The power law surface growth, which is represented in Eq. 33.

. . 3 3 where A is the fracture surface area at time t, Ae is the fracture surface area at the end ofpumping, t is time, teis the time at the end of pumping, and is the exponent of fracture length

growth and is constant during the injection period.

ii. Carter equation I for leak-off, which is shown by Eqs. 34 and 35.

. . . . 3 4 which has an integrated form of:

2 . . . . . 3 5 where is leak-off velocity, is the leak-off coefficient and t is the time elapsed since thestart of leak-off.

iii.

is the exponent and is assumed to be known. It is equal to 4/5 fo r the case with no

leak-off and it is reasonable to assume that the exponent remains the same in the

presence of leak-off.


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With the above assumptions, the material balance at any time during injection can be written as

Eq. 36.

__ 2 (3 2 ) 2 . 3 6 where A is the fracture surface area at time t, which equals x f.hf; i is half of the injection rate or

the injection rate for one wing of the fracture, hfis fracture height, __ is the average width ofthe fracture, Sp is the spurt-loss coefficient, CL is the leak-off coefficient, t is time, is the

exponent of fracture length growth and is the Euler Gamma Function, which can be calculatedusing Eq. 37.

t

3 7 Substituting for the fracture surface area to incorporate fracture half-length and fracture height,

the material balance can be written as shown in Eq. 38.

__ 2 (3 2 ) 2 3 8 To solve for fracture half-length, Eq. 35 can be re-arranged as shown in Eq. 39.

__ 2 2 (3 2 ) . . . 3 9

The procedure to be followed for the calculations consists of using the input data in simple

calculations and testing conditions at each time step. In the beginning, at each time step a

comparison of bottom-hole flowing pressure to the fracture initiation pressure is performed and

fracture is only initiated if the former is greater than or equal to the latter. Once the fracture is

initiated, it either closes, remains open, closes then reopens, or propagates. At each time step

following the fracture initiation, condition testing is performed and a decision is made on the


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fracture simulation for the next time step. In case the bottom-hole pressure is greater than or

equal to the fracture propagation pressure, the PKN- calculation method is performed and the

half-length and width of the fracture are estimated. For the PKN- model, the fractures

dimensions can be estimated as shown in Eqs. 40 through 43.

, = . 4 0 where pn,wis the net wellbore pressure, pbhpfis the bottom-hole flowing pressure and minis the

minimum horizontal stress.

Once the net wellbore pressure is calculated, the maximum fracture width at the wellbore can

be found thus:

, = , . . 4 1 where E is the plane strain modulus (which can be calculated from Youngs modulus and

Poissons ratio) and ww,0is the maximum fracture width at the wellbore.

To solve the maximum fracture width at the wellbore, Eq. 41 can be re-arranged as shown in

Eq. 42.

, 2 ( ) . 4 2 The average fracture width is related to the maximum fracture width at the wellbore, as shown

in Eq. 43.

__

0.628319 , . . 4 3 where __ is the average fracture width and 0.628319 is the shape factor (/5). The shape factorcontains /4 because the vertical shape is an ellipse. Also, it contains another factor (4/5) which

accounts for the lateral variation of the width for the PKN model (Yang 2011). Once the average

fracture width has been estimated, the fracture half-length is related to it as shown in Eq. 39.


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To determine the fracture half-length and width, fracture height is required as an input. From

fracture height calculation Section2.3.2,the result is used as an input to the PKN- calculation

to estimate the maximum fracture width near the wellbore which is then used to carry the of the

calculations to estimate the fracture average width and half-length.

To incorporate the fracture in the dynamic model, local grid refinement is performed based on

the new fracture height, width and half-length calculated after each time step. Also,

permeability, NTG and porosity multipliers should be applied to the gridblocks representing

the fracture. The logic behind that should be performing NTG and porosity multiplying to result

in gridblocks NTG of 1 and porosity of 100%. As for the permeability multipliers, analogues

can be used to relate the fracture width to certain fracture permeability value. Then,

permeability multiplier should be used to increase the gridblocks permeability to the fracture

permeability. The algorithm for the procedure described is shown inFigure 12.


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Figure 12. Algorithm for estimating fracture dimensions created due to injection under

fracturing conditions


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| Chapter 3Results and Discussion

3 Results and Discussion

This chapter introduces the results of the investigation to determine the optimum hydraulic

fracture dimensions for low and high permeability reservoir cases. Furthermore, it discusses the

results of modelling hydraulic fractures and induced fractures created due to injection under

fracturing conditions in reservoir dynamic simulators. For each of these topics, the results are

discussed and further improvements are suggested.

3.1 Determin ing the Optimum Hydrauli c F racture Dimensions for a Well

The optimum hydraulic fracture width and half-length are estimated using the Excel workbook

developed, based on the derived optimum dimensionless fracture conductivity, the practical

dimensionless proppant number constraint and the minimum possible hydraulic fracture width

related to proppant grain diameter.

3.1.1 Low Permeabil ity and H igh Permeabil i ty Case Resul ts

The results of the low permeability case and the high permeability case are tabulated and

discussed below. The same procedure can be followed to determine the optimum hydraulic

fracture dimensions for other cases, to which the issues discussed also apply.


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3.1.1.1 Low Permeability Reservoir Case

The optimum hydraulic fracture dimensions estimated for the low permeability case are shown

inTable 4.

Table 4: Low permeability case results

Half-length (xf) 1027 (ft)

Fracture width (wf) 0.014 (ft)

Fracture volume (vf) 1010 (ft3)

FOI 4.2

Skin due to fracture (sf) -6.676

N 0.5

Fcd 1.643

Although the results shown in Table 4 represent the optimum fracture dimensions for the

maximum folds of increase, it can be inferred fromFigure 13 andFigure 14 that reducing the

fracture volume to half will result in a reduction of only 0.5 in FOI. For a fracture volume of

about 1000 ft3, FOI of about 4.2 is calculated, while FOI of 3.7 is calculated for a fracture

volume of 570 ft3. Therefore, the cost of fracturing material and the expected increase in

production due to the larger FOI value should be taken in consideration when determining the

optimum hydraulic fracture dimensions for a well.


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Figure 13. Net present bbls increase in production and fracture volume vs. folds of

increase for low permeability case

For economics to be calculated there are multiple factors that need to be taken into

consideration. These include:

i.

Capacity of the production system and facilities. If additional equipment and/or larger

equipment sizes are required, there is an additional cost encountered for each barrel of

oil produced and the additional folds of increase also incur more cost. Thus, the

comparison should include this factor when estimating the saving versus the cost.

ii.

Other small projects that require investment (e.g. artificial lift, increasing the injection

rate and/or deploying EOR methods). Ranking of the additional fracturing volume cost

and gain versus the cost and gain of other methods should be performed to facilitate the

selection of the best project for investment (the one with the highest net present value

index, the highest net present value and/or the highest internal rate of return).

iii. Additional investment in production facilities due to earlier water and/or gas

breakthrough. Since created hydraulic fractures are high permeability channels, they can

result in faster breakthrough of water and/or gas from aquifers, injectors and/or gas cap


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to producers. The dimensions of the hydraulic fracture should be selected carefully to

delay breakthrough since this requires larger equipment sizes and/or additional

equipment for separation and treatment.

iv. Depth of damage due to drilling and/or completion phases. The depth of the damaged

zone near the wellbore might be longer than the optimum fracture half-length in a few

special cases. In such a case, the optimum fracture half-length can be increased (while

also increasing the width to result in the same optimum fracture conductivity) to bypass

formation damage.

All of these factors should be taken into account when selecting the optimum dimensions of the

hydraulic fracture.

3.1.1.2

High Permeability Reservoir Case

The optimum hydraulic fracture dimensions estimated for the high permeability case are shown

inTable 5 andFigure 14.

Table 5: High permeability case results

Half-length (xf) 218 (ft)

Fracture width (wf) 1.783 (ft)

Fracture volume (vf) 23322 (ft3)

FOI 2.77

Skin due to fracture (sf) -5.13

N 0.1

Fcd 1.6363


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Figure 14. Net present bbls increase in production and fracture volume vs. folds of

increase for high permeability case

Similar to the low permeability case, it can be observed that for FOI of 2.5, a fracture volume

of 12500 ft3 is required. Increasing the fracture volume to 23300 ft3 (approximately double),

will result in only approximately 0.25 increase in FOI value. Therefore, the cost of the fracturing

material and the additional gain in oil production should be taken into consideration before

deciding the optimum fracture dimensions for a well, as has been shown in the two cases tested

above.


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3.2 I ncorporating Hydraul ic Fractures in Reservoir Dynamic Simulators

The increases in production due to hydraulic fracture for the low permeability case and the high

permeability case are shown inFigure 15 andFigure 16.

Figure 15. Increase in cumulative oil production for low permeability case

Figure 16. Increase in cumulative oil production for high permeability case


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It can be observed that the production increase for the low permeability case is higher than the

increase for the high permeability case. Creating a 600,000 mD channel inside a 0.5 mD

reservoir provides a high permeability pathway for the fluid to flow through, which significantly

impacts the production. On the other hand, for the high permeability case, the reservoir is able

to produce at a good rate and thus the impact of the high permeability channel is not as great as

for the low permeability case. Also, it can be observed that the average FOI is different from

the estimated FOI from the optimum hydraulic fracture calculations. The first reason for this is

that the calculations were based on the assumption of pseudo-steady state flow conditions, while

in a dynamic simulation the well flows under transient flow conditions followed by pseudo-

steady state conditions. The second reason is that water breakthrough can be expected to occur

earlier for the hydraulic fracture case, impacting the well-lifting ability and thus the oil

production rate. Once the water has created a path to the well, water production will increase

rapidly, since the fracture will act as a short circuit for water conducting. Therefore, it is

expected that for wells with hydraulic fractures, there is faster water and/or gas breakthrough

and a subsequent reduction in oil production.


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3.3 Algori thm for M odell ing I nduced F ractures Created Dur ing I njection under

F ractur ing Conditi ons in Reservoir Dynamic Simulators

The impact of implementing the algorithm shown in Figure 12 in the reservoir simulator is

examined using the injectors of the two cases from Section 3.1. The procedure followed

involved manually stopping the simulator after each time step to check the pressure and then

manually performing the calculations, refinement and properties multiplying. This resulted in

an average increase of 42% in run time for a simulation period of 200 days with an average

time step length of 10 days as shown inTable 6.

Table 6: Run time and storage capacity requirement for base cases and cases with

induced fracture modelling

It can be inferred that the run time and storage capacity for the low permeability case are more

than those for the high permeability case. The reason is the fracture dimensions of the low

permeability case is smaller than the fracture dimensions of the high permeability case and

therefore more gridblocks are required due to more refinement.

Additionally, the algorithm is not fully automated in the Eclipse Dynamic Simulator, but it can

be easily observed that such an increase in run time is not very significant. Still, there are other

Low Permeability Case High Permeability Case

Run Time for Case with No

Fracture (Minutes)3.3 3.1

Run Time for Case with

Induced Fracture (Minutes)4.9 4.1

Storage Required for Case with

No Fracture (Megabytes)359 342

Storage Required for Case with

Induced Fracture (Megabytes)573 512


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factors that can increase the expected run time, such as performing the calculations within the

dynamic simulator and having a case of smaller fracture dimensions which results in increasing

the number of gridblocks due to more refinement.

The numerical three-dimensional fracture models are based on moving coordinate system

governed bypartial differential equations with five unknowns that need to be solved for either

explicitly or implicitly (Xiang 2011). Such calculations are time consuming since for each time

step fiveunknowns are required to be solved for in addition mass and pressure. In contrast to

the numerical three-dimensional fracture models, the proposed algorithm has only two sets of

equations to be solved for the fracture three dimensions. Therefore, a reduction in both run time

and storage capacity is achieved by implementing the proposed algorithm compared to using

three-dimensional models. Still, due to the assumptions and simplifications in both PKN-

model and Ahmed and Economides notation after Simonson analysis mentioned in

Section1.1.4,the accuracy of the results of using this algorithm is expected to be lower than

those of the numerical three-dimensional fracture models. Therefore, the algorithm presented

provides a good approximation for modelling induced fractures growth with reduced simulation

run time and storage capacity compared to three dimensional fracture models. Also, it provides

more accurate results compared to the simple two-dimensional models that assumes fixed

fracture height.


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| Chapter 4Conclusions and Recommendations

4 Conclusions and Recommendations

The work presented is of significant economic and environmental importance for oil and gas

companies. In this work, optimum dimensionless hydraulic fracture conductivity is

mathematically determined for pseudo-steady state and steady state flow conditions. The

optimum Fcdis found to be 1.6363 and this value, in addition to the practical limitations, is then

used in an algorithm to determine and simulate the optimum hydraulic fracture dimensions.

Refinement and gridblock properties multiplying are then performed to incorporate the effect

of hydraulic fractures on production and injection forecasts from reservoir dynamic models.

Additionally, in this work an algorithm to simulate induced fractures created due to injection

under fracturing conditions is developed. The advantage of the algorithm is that it incorporates

the pressure data and field constraints from the dynamic simulator to simulate the fracture

progression, recession and dimensions for the full life of the field. Also, the algorithm acts as

an alerting tool for fracture growth into unintended zones. All in all, the results of this work

represent a thorough tool for determining the optimum hydraulic fracture dimensions and

simulating the impact of induced fractures on the fields production and injection forecasts

using reservoir dynamic simulators.

Future work can include incorporating the algorithms in reservoir dynamic simulators so that

the calculations and induced fracture modelling can be performed automatically in the

background.


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| References

References

Anon. a 2013. Fracture mechanics. PetroWiki (16 September 2013 revision),

http://petrowiki.org/Fracture_mechanics(accessed 3 August 2014).

Anon. b 2013. Fracture. Schlumberger Oilfield Glossary, 14 November 2013,

http://www.glossary.oilfield.slb.com/en/Terms/f/fracture.aspx(accessed 20 July 2014).

Anon. c 2014. What is Fracking?. Energy From Shale, 27 June 2014,

http://www.energyfromshale.org/hydraulic-fracturing/what-is-fracking (accessed 14 July

2014).

Anon. d 2014. Natural Gas Extraction - Hydraulic Fracturing. EPA, 16 July 2014,

http://www2.epa.gov/hydraulicfracturing(accessed 4 August 2014).

Anon. e 2014. Hydraulic fracturing. Schlumberger Oilfield Glossary, 11 January 2014,

http://www.glossary.oilfield.slb.com/en/Terms/h/hydraulic_fracturing.aspx (accessed 21

July 2014).

Anon. f 2013. Thermal and Hydraulic Fracturing. PETEX, 9 July 2013,

http://www.petex.com/products/?id=53(accessed 5 August 2014).

Anon.g2014. Mini-Frac (or DFIT) & Caprock Integrity.Big Guns Energy Services, 18 July 2014,

http://www.bges.ca/cat/engineering/mini_frac.php(accessed 13 August 2014).

Anon. h2010. Hydraulic Fracturing: The Process. Fracfocus Chemical Disclosure Registry,20

July 2010, http://fracfocus.ca/hydraulic-fracturing-how-it-works/hydraulic-fracturing-

process(accessed 19 July 2014).

Brons, F., Marting, V. 1961. The Effect of Restricted Fluid Entry on Well Productivity.Journal of

Petroleum Technology, 13(2): 172-173.

Carlson, M.R. 2006. Practical Reservoir Simulation - Using, Assessing, and Developing Results.Oklahoma, USA: PennWell Corporation.

Crain, E. 2013.Fracture Pressure Basics. Crains Petrophysical Handbook, 21 December 2013,

http://www.spec2000.net/10-closurestress.htm(accessed 8 August 2014).
http://petrowiki.org/Fracture_mechanicshttp://petrowiki.org/Fracture_mechanicshttp://www.glossary.oilfield.slb.com/en/Terms/f/fracture.aspxhttp://www.glossary.oilfield.slb.com/en/Terms/f/fracture.aspxhttp://www.energyfromshale.org/hydraulic-fracturing/what-is-frackinghttp://www.energyfromshale.org/hydraulic-fracturing/what-is-frackinghttp://www2.epa.gov/hydraulicfracturinghttp://www2.epa.gov/hydraulicfracturinghttp://www.glossary.oilfield.slb.com/en/Terms/h/hydraulic_fracturing.aspxhttp://www.glossary.oilfield.slb.com/en/Terms/h/hydraulic_fracturing.aspxhttp://www.petex.com/products/?id=53http://www.petex.com/products/?id=53http://www.bges.ca/cat/engineering/mini_frac.phphttp://www.bges.ca/cat/engineering/mini_frac.phphttp://fracfocus.ca/hydraulic-fracturing-how-it-works/hydraulic-fracturing-processhttp://fracfocus.ca/hydraulic-fracturing-how-it-works/hydraulic-fracturing-processhttp://fracfocus.ca/hydraulic-fracturing-how-it-works/hydraulic-fracturing-processhttp://www.spec2000.net/10-closurestress.htmhttp://www.spec2000.net/10-closurestress.htmhttp://www.spec2000.net/10-closurestress.htmhttp://fracfocus.ca/hydraulic-fracturing-how-it-works/hydraulic-fracturing-processhttp://fracfocus.ca/hydraulic-fracturing-how-it-works/hydraulic-fracturing-processhttp://www.bges.ca/cat/engineering/mini_frac.phphttp://www.petex.com/products/?id=53http://www.glossary.oilfield.slb.com/en/Terms/h/hydraulic_fracturing.aspxhttp://www2.epa.gov/hydraulicfracturinghttp://www.energyfromshale.org/hydraulic-fracturing/what-is-frackinghttp://www.glossary.oilfield.slb.com/en/Terms/f/fracture.aspxhttp://petrowiki.org/Fracture_mechanics


55/58

P a g e 44

| References

Dubey, S., Gudmundsson, A. 2012. Field Study and Numerical Modeling of Fracture Networks: Application

to Petroleum Reservoirs. M&S: Schlumberger J. of Modeling, Design, and Simulation 3

(1): 5-9.

Economides, M.J. 1992.A Practical Companion to Reservoir Stimulation. Amsterdam: Elsevier.

Economides, M.J. and Nolte, K.G. 2000. Reservoir Stimulation, third edition. New York: John

Wiley and Sons.

Golf-Racht, T. 1982.Fundamentals of fractured reservoir engineering, First edition. Amsterdam:

Elsevier.

Howard G.C. and Fast, c.R.: Optimum Fluid Characteristics for Fracture Extension, Drilling and

Production Prac., API, 261-270, 1957 (Appendix by E.D. Carter).

Hudson, J., and Harrison, J. 2005. Engineering rock mechanics, First edition. Tarrytown, NY:

Pergamon.

Moreno, J., Ligero, E., and Schiozer, D. 2005. Effects of Water Injection under Fracturing

Conditions on the Development of Petroleum Reservoirs. International Congress of

Mechanical Engineering, Ouro Preto, MG, November 6-11.

Nolte, K.G.: Determination of Proppant and Fluid Schedules from Fracturing Pressure Decline,

SPEPE, (July), 225-265, 1986 (originally paper SPE 8341, 1979).

Owen, D. R. J. and Fawkes, A. J. 1983.Engineering Fracture Mechanics: Numerical Methods and

Applications. Swansea: Pineridge Press Ltd.

Richardson, M. 2000. A new and practical method for fracture design and optimisation. SPE 59736,

SPE/CERI Gas Technology Symposium, Calgary, Alberta, Canada, April 35.

Simonson, E., Abou-Sayed, AS., and Clifton, RJ. 1978. Containment of Massive Hydraulic

Fractures, SPEJ, (Feb.), 27-32, 1978.

Shukman, D. 2013. What is fracking and why is it controversial?.BBC NEWS, 27 June 2013,

http://www.bbc.com/news/uk-14432401(accessed 2 August 2014).
http://www.bbc.com/news/uk-14432401http://www.bbc.com/news/uk-14432401http://www.bbc.com/news/uk-14432401


56/58

P a g e 45

| References

Taleghani, A., Ahmadi, M., and Olson, J. 2013. Secondary Fractures and Their Potential Impacts

on Hydraulic Fractures Efficiency. InEffective and Sustainable Hydraulic Fracturing, first

edition, Andrew P. Bunger, John McLennan and Rob Jeffrey, Chap. 38, 785-789. Brisbane:

INTECH.

Valko, P. (2001). HF2D Frac Design Spreadsheet. Lecture conducted from Texas A&M

University, Houston, TX.

Valko, P. (2005). Hydraulic Fracturing Short Course. Lecture conducted from Texas A&M

University, Houston, TX.

Valko, P., and Economides, M. J. 1995.Hydraulic Fracture Mechanics, first edition. Chichester,

England: John Wiley & Sons.

Warpinski, N. R., and Smith, M. B. 1989. Rock mechanics and fracture geometry. In: RecentAdvances in Hydraulic Fracturing, Monograph Vol. 12, Gidley, J. L. et al. (Eds.).

Richardson, TX: SPE.

Xiang, J. 2011. A PKN Hydraulic Fracture Model Study and Formation Permeability

Determination. MS thesis, Texas A&M University, Houston, Texas (December 2011).

Yang, M. 2011. Hydraulic Fracture Optimization with a Pseudo-3d Model in Multi-Layered

Lithology. MS thesis, Texas A&M University, Houston, Texas (August 2011).


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| AppendicesFracture Height Equations

Appendices

A.1Fracture Height Equations

Using Texas Instrument (TI-89) calculator, the two equations with the two unknowns to be solved

for are shown below:

, {.(++).

[()..() ]

. (++)}(++)

{.(++). [(). .() ]+(

+) (++)}.(++)

0.399.( ) . [(+).

. ]

0.399.( ) . (+)..(+) .(+). . +.(++)..+(..)

(++) . . 1 1 = ,2


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II

, {.(++).

[(). . ]

+(+) (++)}(++)

{.(++).[()..() ]

(++)}(++)

0.399.( ) . (+). .

0.399.( ) . (+)..(+) .(+). . .(++)..+(..)

(++) . . 1 2 =

, 3

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