computational engineering analysis of the hydraulic-fracturing process

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Computational Engineering Analysis of the

Hydraulic‐Fracturing Process M. Grujicic *1, R. Yavari, S. Ramaswami, J. S. Snipes, R. Galgalikar

Department of Mechanical Engineering

Clemson University, Clemson SC 29634, USA

*[email protected]

Received 14 May 2014; Accepted 25 February 2014; Published May 2014

© 2014 Science and Engineering Publishing Company Abstract

Hydraulic fracturing (including horizontal drilling) is a

technology widely employed to significantly increase the

rate and extent of oil and natural gas production from deep

(ca. 2 km) shale reservoirs. Past advancements in this

technology have been mainly made using experimental,

empirical and trial‐and‐error approaches. Since the use of

modern computational techniques has benefited many

industries (such as transportation, defense, biomedical,

pharmaceutical, etc.), it is hoped that the use of these

techniques can yield similar benefits in oil and natural gas

extraction from deep‐seated reservoirs. The present work

provides a full three‐dimensional finite‐element analysis of

the main stages of hydraulic‐fracturing‐stimulated fuel

extraction from such reservoirs. Challenges associated with

such an analysis due to nonlinear character and coupling

between the mechanical response of saturated porous rock

formations, their fracturing behavior and the fluid flow

through the fracture and the flow‐induced fracturing were

all addressed. In contrast to the prior finite‐element analyses,

the present work addresses the issues related to the

intersection of naturally‐occurring fissures within the fuel‐

bearing rock formations and the profile of sand‐injection into

the hydraulic‐fracturing fluid. The results obtained are used

to judge the potential of hydraulic‐fracturing process

optimization in maximizing the fuel‐extraction yield.

Keywords

Hydraulic Fracturing; Analysis of Porous Media; Finite Element

Modeling

Introduction

The subject of the present work is computational

modeling and simulations of the hydraulic fracturing

(also known as “fracking”) process used for extraction

of oil and/or natural gas from deep shale formations.

Since this process has been mainly developed and

advanced using purely empirical trial‐and‐error

approaches, it is hoped that the use of an engineering

analysis and the employment of multi‐physics

computational methods and tools can make this

process more economical, speed up its further

development and, possibly, help address some of the

concerns raised regarding the potential impact of this

process on the environment. Based on the foregoing,

the concepts most pertinent to the present work are: (a)

the basics of hydraulic fracturing; and (b) prior work

involving the use of advanced computational

techniques in the analysis of the hydraulic fracturing

process. These two aspects are reviewed briefly in the

remainder of this section.

The Basics of Hydraulic Fracturing

Deep shale formations, which were created tens of

millions of years ago, contain substantial deposits of

trapped oil and natural gas. In recent years, new deep‐

shale‐formation fracture‐stimulation technologies

(such as hydraulic fracturing, including horizontal

drilling) have enabled the extraction of these oil and

natural gas deposits in a time‐efficient and cost‐

effective manner.

Since the shale reservoirs in question are located ca. 2

km below the surface, while natural water reservoirs

are located at depths smaller than approximately 400

m, oil/natural gas extraction is claimed by the oil and

gas industry to be safe to the environment. To further

ensure that the impact on the fresh‐water aquifer is

minimal, the holes drilled into the ground (used to

reach the deep shale reservoirs) are lined with steel

pipes (fixed in place using cement), so forming a

barrier (commonly referred to as casing) between the

bore and the surrounding earth. Despite these efforts

by the oil and gas industry, the true impact

(particularly its long‐term component) of hydraulic‐

fracturing/horizontal‐drilling on the environment

remains a subject of debate.

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In order to facilitate the discussion, presented below,

of the basic steps associated with hydraulic‐fracturing,

a simple schematic of this process is depicted in Figure

1. Typically, the hydraulic‐fracturing process involves

the following basic steps: (a) the initial step – within

this step, a vertical hole is drilled while ensuring the

stability of the wellbore and integrity of the drill bit

(via the use of a drill pipe). The drill pipe is a thick‐

walled, flexible, kilometer‐long steel pipe which is

attached to the more‐rigid drill stem and, in turn, to

the drill bit to form the so‐called “drill string.” To

prevent drill‐bit from overheating and help with the

rock‐cutting extraction, drilling is carried out in the

presence of circulating water; (b) the casing‐construction

step – when the wellbore is significantly deeper than

any local aquifer, drilling is temporarily ceased, the

drill pipe removed and the wellbore lined with steel

pipes. Next, wet cement is pumped down the pipe

under sufficiently high pressure to ensure that, once

the cement has reached the bottom of the hole, it can

flow upward and fill the gap between the casing and

bore‐hole wall. Upon hardening of the cement, a low‐

permeability barrier (called surface casing) is created

between the bore‐hole and the deep shale

surroundings. To further minimize the exchange of

fluids between the bore and the surrounding aquifers,

additional casing layers may be employed; (c) the

horizontal‐drilling step – the vertical drilling and casing

construction continue until the desired depth (also

known as the “kick‐off point”) is reached. Beyond this

point, the direction of drilling begins to acquire a

horizontal component, and ultimately becomes

horizontal. One of the perceived advantages of

horizontal drilling is that one vertical hole can be used

to generate multiple horizontal‐bore sections,

minimizing the potential negative effect of drilling to

the surface environment. When the desired length of

the horizontal section(s) of the bore is reached, the

drilling gear is removed and the horizontal casing is

placed and secured; (d) the casing‐perforation step –

since, at this point, the rock formation containing

trapped oil/gas is isolated (in a fluid‐exchange sense)

from the bore due to the presence of the impermeable

cement/steel casing, preventing seepage of the oil/gas

into the wellbore, a local connection between the

reservoir and bore must be established. This is done

by lowering and guiding a specialized shape‐charge

gun to the desired location within the wellbore. The

gun is next fired to create perforations in the casing

and cracks/holes in the adjacent rock formation; (e)

hydraulic‐fracturing step – upon the removal of the

perforating gun from the wellbore, a mixture of water,

FIGURE 1. A SCHEMATIC OF THE HYDRAULIC FRACTURING/FRACKING PROCESS INVOLVING HORIZONTAL DRILLING.

PLEASE SEE TEXT FOR DETAILS.


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chemicals and sand are pumped, under high pressure,

into the wellbore and, via the aforementioned

perforations, into the deep underground reservoir

formations. The chemicals used (typical concentrations

of which are in the range of 0.1 to 0.5 vol. %) have a

number of roles, such as: (i) reducing friction

accompanying flow of the hydraulic‐fracturing fluid;

(ii) inhibiting bacteria formation; and (iii) minimizing

the tendency for sand‐particle coalescence. The

application of high pressure causes hydraulic‐

fracturing/fracking of the deep underground reservoir,

while the presence of sand (also referred to as

“proppant”) within the hydraulic‐fracturing fluid

ensures that the cracks remain open upon removal of

the applied pressure. These processes stimulate a

higher rate of extraction of oil and gas from the deep

underground reservoirs; (f) stimulation‐segment

isolation step – next, specially designed plugs are used

to isolate the newly‐created stimulation segments, and

the casing‐perforation and hydraulic‐fracturing steps

repeated to generate additional stimulated segments.

This process is repeated multiple times along the

horizontal section of the well, the section which may

extend several kilometers; (g) production step – once all

the desired stimulation segments are created, the

isolation plugs are drilled out, the pressure is relieved,

and the extraction of the previously trapped fuel

begins. In the initial portion of this step, the extraction

consists mainly of the hydraulic‐fracturing fluid. This

fluid is separated from the fuel and either: (i) recycled

for use in subsequent hydraulic‐fracturing operations;

or (ii) safely disposed of in accordance with

government regulations. Subsequent extractions

consist mainly of the released fuel which first flows

through the horizontal section and then up the vertical

section of the wellbore.

Prior Hydraulic‐Fracturing Computational Work

Modeling of hydraulic fracturing is a fairly

complicated task, since it must account for coupling

between four basic processes: (a) deformation of the

rock formation induced by the injected‐fluid pressure

acting on the fracture/crack faces; (b) viscous‐fluid

flows within the fracture; (c) propagation/extension of

the fracture into the rock formation, induced by the

sustained application of the hydraulic‐pressure/fluid‐

flow; and (d) hydraulic‐fracturing fluid “leakoff” from

the fracture into the adjacent rock‐formation. The

challenges associated with hydraulic‐fracturing

process modeling are further compounded by the

nonlinear characters of the differential/algebraic

equations governing these four processes. These

equations include: (a) a mechanical constitutive

(algebraic‐type) relation linking the pressure within

the fracture to the fracture‐opening and, in turn, to the

deformation of the surrounding rock formation; (b) a

nonlinear fluid‐flow differential equation relating the

rates of flow and fluid accumulation to the fracture

opening and the gradient of the fluid pressure; (c) a

fracture‐propagation law (e.g. a linear‐elastic fracture

mechanics, LEFM, relation which links the fracture‐

length increase to the relative magnitude of the fluid‐

induced stress intensity factor and its critical rock‐

formation toughness‐quantifying counterpart); and (d)

a diffusion‐type differential or algebraic relation

governing fluid leakoff through the fracture faces.

The analytical‐modeling efforts reported in the open

literature [e.g. Jeffrey et al. (2001)] have focused on

simplified (and predetermined) fracture geometries

(e.g. penny‐shaped cracks) and utilized simplified

distributions (e.g. uniform distribution) of the pressure

over the fracture faces. These simplifications could be

considered as being of a major character when

analyzing the full‐scale hydraulic fracturing process

and, consequently, analytical solutions have

demonstrated relatively little utility in these contexts.

This is the reason that the majority of the most recent

non‐experimental efforts have focused on the use of

advanced numerical methods and tools in order to

analyze the hydraulic‐fracturing process.

In the late 1970s, the so‐called “pseudo‐3D” model,

perhaps the first reported hydraulic‐fracturing

numerical model, was developed by Clifton (1989). In

order to address the phenomena such as: (a) complex

three‐dimensional geometry of the hydraulic

fracturing; and (b) complex interactions between the

four coupled phenomena/processes discussed above,

additional models have been proposed by Adachi et al.

(2007), and Zhang et al. (2007). Subsequent efforts used

innovative computational methods and tools, such as

the extended finite element methods (XFEM)

[Lecampion (2009)], capable of handling singularities

such as those associated with the crack tip, within the

LEFM formalism, and the discrete‐element methods

(methods which treat porous material not as a

continuum but rather as an assembly of interacting

and/or bonded particles) [Zhao et al. (2008), Grujicic et

al. (2013ab)]. The most recent numerical‐modeling

efforts have focused on issues such as: (a) potential

contamination of shallow aquifers [Gassiat et al.

(2013)]; (b) optimization of the hydraulic‐fracturing

process [Zhu et al. (2013)]; and (c) identification of the

factors affecting well productivity [Lv et al. (2013)].


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Despite all these recent advances in the numerical

analysis of the hydraulic‐fracturing process, this

engineering and scientific field needs additional

maturing before its predictions can become a bona fide

complement or substitute to the experimental field‐test

data. The work presented in the current manuscript is

an attempt to further advance the numerical approach

to the computational investigation of the hydraulic‐

fracturing process, and is a continuation of our

previous study [Grujicic et al. (2013c)].

Main Objective

The main objective of the present work is to utilize

cohesive‐zone finite elements in order to numerically

investigate the hydraulic‐fracturing process resulting

in the formation of vertical fractures. The cohesive‐

zone finite‐element approach is an alternative to the

finite‐element approach based on the use of the LEFM.

As pointed out earlier, the use of the LEFM and the

zero‐opening crack‐tips led to stress singularities at

the crack tip. Such singularities posed a severe

numerical challenge to the finite element method,

which could be resolved only through the use of crack‐

tip‐tracking adaptive‐meshing schemes and through

the use of special hybrid elements [e.g. Grujicic and

Cao (2002)]. These remedial schemes are

computationally quite costly, rendering a full three‐

dimensional analysis of the hydraulic‐fracturing

process impractical or impossible. In addition to stress

singularities, zero‐opening crack tips gave rise to

singularities in the degenerate partial differential

equation governing viscous flow of the hydraulic‐

fracturing fluid through the fracture [Peirce and

Detournay (2008)]. Remedy of this problem was

associated with yet further increases in the

computational cost.

In sharp contrast to the LEFM, the cohesive‐zone

finite‐element approach treats rock‐formation

fractures as having finite opening even in the un‐

cracked state. Consequently, stress singularities and

the associated singularities in the degenerate partial

differential equation are avoided. This leads to a

considerably reduced computational cost. Additional

savings in the computational cost through the use of

cohesive‐zone elements results in close tracking of the

crack‐tip position not being required (the position of

the crack tip is a normal outcome of the finite element

method). In sharp contrast, in the case of LEFM finite‐

element analysis, fracture evolution entails

(computationally expensive) tracking of the current

location of the crack‐tip. In addition, the cohesive

zone‐based finite element approach offers additional

capabilities of interest for modeling the hydraulic‐

fracturing process, such as: (a) initiation of new cracks

within the rock formation, as well as coalescence of the

existing cracks and fragment formation; and (b)

initiation of the fracture within the borehole casing.

The advantages of the cohesive‐zone finite element

approach identified above enable this method to be

employed in the analysis of a large‐scale hydraulic‐

fracturing process.

The finite element method which utilizes the cohesive‐

zone approach has already been applied with success

to the analysis of fracture in a variety of

materials/systems including metals, ceramics,

polymers, and hybrids [Grujicic et al. (2009b, 2012)].

This approach was recently used by Zhang et al. (2012)

and Zhang et al. (2010) to investigate the hydraulic‐

fracturing process. The present work advances the

approach and the analysis reported by Zhang et al.

(2012) and Zhang et al. (2010) in the direction of (i)

revealing the role natural fissures intersected by

hydraulic fractures play in the degree of fuel‐reservoir

stimulation; and (ii) clarifying the effect of sand

concentration and its injection profile (i.e. variation of

the sand concentration with time) into the hydraulic‐

fracturing fluid on the success of the hydraulic‐

fracturing process.

Porous-Medium/Fluid-Flow/Fracture Coupled Analysis

Modeling and simulation of hydraulic fracturing is a

complex endeavor and involves mathematical and

numerical treatment of two interacting/coupled

phenomena/processes, each of which is itself fairly

complex [Dassault Systèmes (2011)]. The two

processes involved include: (a) flow of the hydraulic‐

fracturing fluid through the wellbore, perforations and

fractures, and the accompanying additional fracturing;

and (b) fluid flow within the surrounding porous rock

formation and its accompanying deformation. The two

phenomena are coupled through: (a) the fluid leakoff

through the wellbore/fracture surfaces; and (b) the

fluid pressure acting as traction on the fracture

surfaces. Due to space limitations, only a qualitative

synopsis of the main concepts and functional relations

associated with the modeling and simulation of

hydraulic fracturing is provided in this section.

Continuum Analysis of Porous Media

Pore fluid diffusion/stress‐coupling types of problems

involve single‐phase, partially‐ or fully‐saturated fluid


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flow through porous media. Such problems can be

analyzed under a variety of conditions, such as: (a)

including/excluding the pore fluid weight; (b)

including/excluding heat transfer due to conduction in

the soil skeleton and the pore fluid, and convection

due to the flow of the pore fluid; (c) time‐

dependent/transient or time‐invariant/steady‐state

scenarios; (d) including/neglecting nonlinear

geometrical and/or material effects; and (e) including

potential contacts between the model components.

1) Porous Medium Effective‐Stress Principle

It is a common practice to assume that the porous

medium consists of a solid skeleton and

(connected/isolated) pores filled with up to two

distinct fluids: (a) a nearly‐incompressible “wetting

liquid”; and (b) a compressible gas. A dry medium

contains only the gas; a partially‐saturated medium

contains both fluids; and a fully‐saturated medium

contains only the wetting liquid. The wetting liquid

can be present either as a free‐flowing or a trapped

liquid (in the case of isolated pores and/or presence

of a material which absorbs the liquid and forms a

gel). It should be noted that the present work is

concerned with a saturated porous medium which

does not contain trapped fluid, the conditions

which are commonly encountered in the hydraulic‐

fracturing process.

When analyzing the total (true Cauchy) stress at a

material point within the porous medium, σ , this

quantity is commonly assumed to be composed of

three parts: (a) the so called “effective stress,” σ ,

associated with the solid skeleton; (b) the pressure

associated with the wetting liquid, wp ; and (c) the

pressure associated with the gas/air, ap .

In the analysis of porous media, two parameters

are often encountered/used in order to quantify the

fraction of the porous‐medium volume occupied

by the fluid: (a) porosity, n – the ratio of the volume

of voids to the total volume; and (b) void ratio, e –

the ratio of the volume of voids to the sum of the

volumes occupied by the solid skeleton and the

trapped fluid. The two quantities are related by the

equation nne 1 .

It should be noted that porosity, n, or void ratio, e,

(and, in the case of a partially saturated porous

medium, saturation, s) are state variables of the

porous medium which define morphological (and

saturation) state of the porous medium and evolve

with deformation/loading of this medium.

Consequently, the appropriate evolution equations

for these quantities must be defined.

2) Equilibrium Equation for a Porous Medium

The first porous‐medium governing equation

involves an equilibrium equation which, within the

current configuration, is generally expressed using

the principle of virtual work.

3) Constitutive Response of Porous‐Medium

Components

The porous medium is generally considered as a

mixture of the solid‐skeleton phase, entrapped

fluid (assumed to be integrated within the solid

matter), and voids (filled with wetting and non‐

wetting fluids). Thus, to completely define the

constitutive behavior of a porous medium, one

must specify: (a) volumetric responses of the

wetting, non‐wetting and trapped fluids. As

mentioned earlier, the contribution of the non‐

wetting fluid and that of the trapped fluid

(typically lumped with the solid material response)

are ignored in the present work; (b) volumetric

response of the solid skeleton; and (c) deviatoric

response of the solid skeleton.

Volumetric Constitutive Response of the Wetting Liquid:

Under isothermal conditions at the reference

temperature, at which the contribution of the

thermal strains can be neglected, the volumetric

constitutive response of the wetting fluid is

typically expressed by a pressure vs. density

relation.

Volumetric Constitutive Response of the Solid Skeleton:

As far as the volumetric constitutive response of

the solid skeleton is concerned, it is also defined by

a solid‐material density g vs. pressure relation.

Deviatoric Response of the Solid Skeleton: The

deviatoric constitutive response of the solid

skeleton is assumed to be governed by the

extended linearized Drucker‐Prager model

[Grujicic et al. (2009a)]. Before this model can be

applied to compute the effective stress within the

solid skeleton, one must determine the fraction of

the porous‐medium strain which is allotted to the

solid skeleton, i.e. the effective strain.

The effective strain tensor can be defined as the

difference between the overall strain tensor

experienced locally by the porous medium and the


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so‐called “moisture‐swelling strain tensor,” defined

as the sum of: (a) a volumetric strain tensor,

resulting from the wetting‐liquid pressure acting

on the solid matter; and (b) the volumetric strain

tensor produced by the entrapped liquid and gel

formation (ignored in the present work).

4) Wetting‐Liquid Continuity Equation in Porous

Media

The second governing equation for the porous

medium is the wetting‐liquid continuity equation

which relates the rate of change of the fluid mass at

a point to the net flux of the fluid at the same point.

5) Porosity Evolution Equation

This equation must be specified in order to make

the system of governing equations determinate.

Material Models

Within the present work, three types of material

constitutive models were used in order to define: (a)

the effective mechanical response of the solid skeleton;

(b) the fracture of the porous medium and the

associated fluid exchange between the rock formation

and the wellbore; and (c) the mechanical response of

the casing.

1) Extended Drucker‐Prager Solid‐Material Model

The rock‐shale‐formation layers were modeled

using the extended Drucker‐Prager material model.

This type of material model is generally used to

represent the constitutive behavior of the (frictional)

granular and geological (e.g. rock‐type) materials

which display pressure‐dependent yield behavior

(or, more specifically, which display a higher

resistance toward inelastic deformation under

higher pressures).

2) Cohesive‐Zone Porous Material Model

Hydraulic fracturing was modeled by placing

cohesive‐zone materials between the

solid/continuum portion of the model. A schematic

of three adjoining cohesive elements in the crack‐

tip region is given in Figure 2. In contrast to the

bulk‐continuum materials for which the

constitutive behavior was described in terms of

stresses, strains, strain rates, etc., the constitutive

behavior of (interfacial) cohesive materials was

more conveniently and physically more

appropriately (due to negligible thickness of the

interface) defined in terms of the (normal and

tangential) traction versus separation functional

relations.

FIGURE 2. A SCHEMATIC OF THREE ADJOINING COHESIVE

ELEMENTS IN THE CRACK‐TIP REGION, WITH THE

DIRECTIONS OF TANGENTIAL AND NORMAL/LEAKOFF

FLOW INDICATED.

3) Casing Material Model

The casing was composed of steel pipes and

cement‐based bonding layers. For simplicity, the

material of the casing was homogenized and

treated as an isotropic linear‐elastic material, with

its elastic properties scaling weighted by the

volume fractions of the steel and cement materials

in the casing.

Problem Formulation and Analysis

As mentioned earlier, the hydraulic‐fracturing process

is generally used in order to increase the output from

and lifetime of deep‐shale reservoirs of oil and natural

gas. This is accomplished by: (i) increasing the surface

area of the fuel‐bearing rock formation; and (ii)

providing a low‐resistance flow path for the fuel being

extracted. Hydraulic fracturing accomplishes these

goals through the use of high‐pressure fluids which

can overcome high compressive stresses within the

rocks and cause crack formation and growth. The

efficiency of hydraulic fracturing is mainly affected by:

(a) the extent of induced fracture; (b) the connectivity

between the fracture and the well bore; and (c) the

extent to which hydraulically‐induced fracture

intersects with naturally‐occurring rock fissures.

As discussed in greater detail in the Introduction

section of this manuscript, hydraulic fracturing is a

complex process, which involves a number of well‐

defined steps. Since these steps were described in

great detail in the Introduction section, a similar

discussion will not be provided here. However, as will

be shown later, the main hydraulic‐fracturing steps,

i.e.: (i) the geostatic equilibration step following initial

drilling/casing‐construction; (ii) hydraulic‐fracturing

fluid‐pumping step; (iii) proppant‐injection step; and

(iv) production step, are analyzed in the present work.


7

The main problem analyzed in the present work

involves a finite‐element analysis of the hydraulic

fracturing enhancement in the rate of extraction of the

fuel from the deep‐shale reservoir. Two specific

aspects of the problem are the focus of the work: (i) the

role of the natural fissures intersected by the hydraulic

fractures in enhancing the rate of fuel recovery from

the reservoir; and (ii) the effect of the sand

concentration in the hydraulic‐fracturing liquid and its

injection profile on the success of the hydraulic‐

fracturing process.

Numerical investigation of such a problem typically

involves the following steps: (a) specifying the

geometrical model; (b) specifying the meshed model;

(c) defining the material constitutive models; (d)

specifying initial conditions; (e) specifying boundary

conditions and loading; (f) specifying computational

algorithm and tool; and (g) estimating computational

accuracy, stability and cost.

Geometrical Model: The geometrical model analyzed in

the present work involves a three‐layer hollow

circular‐disk computational domain with the

following overall dimensions: outer radius = 200 m,

inner radius = 0.1 m, top layer thickness = 10 m,

middle layer thickness = 20 m, bottom layer thickness

= 20 m. The top surface of the computational domain is

assumed to be located at the depth of 2100 m. The

three layers analyzed include the middle layer, which

is the primary target of fuel extraction, and the top and

bottom shale layers. The inner hole in the geometrical

model was used to represent the vertical wellbore. A

schematic of the geometrical model used is displayed

in Figure 3(a).

Meshed Model: Due to the inherent symmetry of the

geometrical model about the x=0 plane, only one half

of the geometrical model described in the previous

FIGURE 3. (A) GEOMETRICAL AND (B) MESHED MODELS USED IN THE PRESENT FINITE ELEMENT ANALYSIS OF THE

HYDRAULIC‐FRACTURING PROCESS.


8

section was meshed and analyzed. The meshed model

consisted of three distinct sections: (a) a bulk section

used to represent the three rock‐formation layers. This

region is discretized using 9600 eight‐node continuum

degrees of freedom. Each of these elements also

contains an additional set of four nodes located at the

crack mid‐surface. These nodes were used to place the

additional degrees of freedom associated with the

cohesive‐zone pore‐pressure elements, i.e. the

tangential mass flow rate; and (c) a membrane region

covering the central hole of the computational domain,

and representing the wellbore casing. This section was

meshed using 480 four‐node membrane elements. A

typical finite‐element mesh used in the present work is

displayed in Figure 3(b).

Material Models: The material models used in the

present work were overviewed in Section III. It should

be recalled that: (a) an extended linearized Drucker‐

Prager model was used for the bulk materials; (b) a

linear‐elastic/linear‐damage traction vs. separation

material constitutive law with a quadratic traction‐

interaction damage‐initiation criterion and an energy‐

based mixed‐mode damage‐evolution law were

employed to model hydraulically‐induced fracturing

within the rock‐formation. In addition, a constitutive

model was used to describe the tangential (i.e. parallel

to the crack faces) flow of the hydraulic‐fracturing

fluid through the cohesive elements and their leakoff

through the crack faces; and (c) the wellbore casing

was modeled using a linear‐elastic isotropic material.

Initial Conditions: To define the initial (equilibrated)

state of the materials within the computational domain,

the following quantities and their spatial distribution

(i.e. depth‐dependence) were specified: (a) void ratio;

(b) pore pressure; and (c) gravity‐induced, geostatic,

orthotropic (compressive) stress field with the

maximum (i.e. the least negative) principal stress

being aligned in a direction orthogonal to the

cohesive‐element fracture faces (i.e. in the y‐direction).

The depth‐dependence of these quantities in the initial

configuration of the computational domain is shown

in Figures 4(a)–(c), respectively.

It should be recalled that, within the present finite‐

element model, the cohesive‐zone mid‐plane is

orthogonal to the y‐direction. Examination of the

results displayed in Figure 4(c) reveals that the normal

stress within the x‐y plane is the lowest in the y‐

direction. This is consistent with the fact that fracture

extends within a plane of least resistance, that is,

within a plane which is perpendicular to the direction

of the minimum/least‐negative principal in situ (crack‐

closing) compressive stress.

Boundary Conditions and Loading: As mentioned earlier,

the analysis carried out involved four distinct loading

steps. The following boundary/loading conditions

were applied to all four steps: (i) the symmetry

boundary conditions were applied to the x=0 plane; (ii)

the zero‐displacements in the direction of the local‐

surface normalwere applied to the bottom and the

circumferential faces of the model; (iii) the top face of

the model was subjected to the uniform (overburden)

normal surface traction; and (iv) a distributed

gravitational load was applied to all portions of the

model in the negative z‐direction. The

boundary/loading conditions unique to the four

phases were applied as follows: (a) geostatic step – in

this step, the so‐called shut‐in pressure was first

applied, in the form of surface tractions, to the

wellbore in order to ensure zero‐stress conditions

along the surface of the wellbore. Then, the overall

mechanical equilibrium was ensured through the

application of the geostatic computational analysis; (b)

hydraulic‐fracturing step – in this 30‐minute‐long

transient step, an initial 8 m long vertical, centrally‐

located perforationwas first created within the casing

of the middle layer by assigning an initial crack

opening to the associated cohesive elements. Then, a

2.5 m3/min (or 15 barrels/min) volumetric flow rate of

the hydraulic‐fracturing fluid, ramped from zero over

the first 200 s, was assigned to the perforated/pre‐

cracked cohesive elements of the casing. During this

step, increases in pressure caused: (i) crack‐opening

enlargement within the pre‐cracked and water‐filled

cohesive elements; (ii) the tangential flow of the

hydraulic‐fracturing fluid into the adjacent cohesive

elements and their fracturing; and (iii) leakoff of the

hydraulic fracturing fluid into the surrounding bulk

material; (c) proppant‐injection/retention step – in this

120‐minute‐long transient step, the sand present

within the previously‐injected fluid was retained

within the rock fractures in order to maintain these

fractures open (for providing a low‐resistance flow

path for the fuel to be extracted). This was

accomplished by: (i) terminating fluid injection into

the well, while allowing the increased pore pressure

within the fracture to bleed off into the formation; and

(ii) fixing the cohesive‐element nodes at their positions

attained at the completion of the hydraulic‐fracturing

step; and (d) production step – in this 240‐hour‐long

transient step, a (lower) drawdown pressure of 20 kPa

was applied to the wellbore nodes of the fracture


9

cohesive elements, which promoted reverse leakoff of

the fuel into the cohesive elements and, in turn, into

the wellbore.

FIGURE 4. INITIAL CONDITIONS USED IN THE FINITE

ELEMENT ANALYSIS TO PRESCRIBE DEPTH‐ AND LAYER‐

DEPENDENCES OF: (A) VOID RATIO; (B) PORE PRESSURE; AND

(C) NEGATIVES OF THE THREE PRINCIPAL STRESSES.

Computational Algorithm and Tool: All the calculations

carried out in the present work involved the use of a

transient, porous‐solid/viscous‐fluid coupled implicit

finite‐element algorithm. The analysis was carried out

under isothermal conditions, i.e. no thermal effects

associated with the viscous‐fluid flow or the porous

medium in elastic deformation/fracture were

considered. All the calculations were performed using

ABAQUS/Standard, a general‐purpose finite element

solver [Dassault Systemes (2011)]. To take advantage

of the more advanced features of this tool, many

aspects of the model associated with the spatially‐

varying initial conditions and time‐varying boundary

and loading conditions were handled through the use

of the appropriate user subroutines.

Computational Accuracy, Stability and Cost: A standard

mesh sensitivity analysis was carried out (the results

not shown for brevity) in order to ensure that the

results obtained were accurate, i.e. insensitive to

further reductions in the size of the elements used.

Due to the use of the implicit numerical‐solution

algorithm, the analysis carried out was

unconditionally stable. A typical analysis involving

the aforementioned durations of the four hydraulic‐

fracturing steps, followed by a detailed post‐

processing data reduction analysis, required on

average 90 minutes of (wall‐clock) time on a 12‐core,

3.0 GHz machine with 16 GB of memory.

TABLE 1. POROUS, COHESIVE AND CASING MATERIAL MODEL

PARAMETERS AND THE HYDRAULIC‐FRACTURING PROCESS PARAMETERS

USED IN THE PRESENT WORK

Parameter Symbol Units Value

Poisson’s Ratio shale/target

N/A 0.2

casing 0.3

Young’s Modulus

Eshale

GPa

8.0

Etarget 12.0

Ecasing 2.0

Friction Angle shale

deg 29

target 36

Strength Ratio Kshale

N/A 1.0

Ktarget 0.95

Dilation Angle shale

deg 29

target 36

Compressive Strength shale

MPa 30

target 38

Interfacial Stiffness Knn, Kss, Ktt GPa/m 85

Damage Initiation Tractionstshale

kPa 100

ttarget 320

Fracture Energy Gn, Gs, Gt kJ/m2 28

Fracture Exponent N/A 2.284

Fluid Viscosity kPa s 1.0

Leakoff Coefficient Kleak‐off kg/m4/s 5.88E‐7

Draw‐down Pressure Pdraw‐down MPa 20

Depth, m

Void

Ratio

,NU

2100 2110 2120 2130 21400.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

(a)

Top Layer

MiddleTarget Layer

Bottom Layer

Depth, m

Pore

Pre

ssure

,MP

a

2100 2110 2120 2130 214024.1

24.2

24.3

24.4

24.5

24.6

24.7

(b)

Top Layer

Middle Target Layer

Bottom Layer

Depth, m

Negativ

eofth

eP

rinci

palS

tress

es,

MP

a

2100 2110 2120 2130 2140

11

12

13

14

15

16

17

18

Boundary 1

Boundary 2

11

22

33

(c)

Top Layer

Middle Target Layer

Bottom Layer


10

Results and Discussion

As mentioned earlier, the present work dealt with a

finite‐element analysis of the hydraulic‐fracturing

enhancement of the rate of extraction of the fuel from

the deep‐shale reservoir, and focused on revealing: (i)

the role of the natural fissures intersected by the

hydraulic fractures in enhancing the rate of fuel

recovery from the reservoir; and (ii) the effect of sand

concentration in the hydraulic‐fracturing liquid and its

injection profile on the success of the hydraulic‐

fracturing process. A summary of the material‐model

parameters and of the hydraulic‐fracturing process

parameters (not specified in the sections dealing with

the initial and the boundary/loading conditions) is

provided in Table 1.

Prototypical Results

Before the results revealing the effect of natural

fissures and proppant concentration/injection‐profile

are presented, a few prototypical results (for the case

of target rock formation without natural fissures) are

shown and discussed in this section.

Locations of the hydraulic‐fracture front at four (300 s,

600 s, 900 s, and 1200 s) times during the

pumping/hydraulic‐fracturing step are depicted, as a

contour plot, in Figure 5. The results show that the

fracture primarily extends, within the 20 m‐thick rock‐

formation target layer, in the radial (i.e. x‐) direction,

and that the fracture growth in the vertical (i.e. z‐)

direction is restricted within the target 20 m‐thick

middle layer, despite the presence of the surrounding

(more compliant) shale layers.

FIGURE 5. A CONTOUR PLOT SHOWING A PORTION OF THE

X–Z “FRACTURE” PLANE AND THE LOCATIONS OF THE

HYDRAULIC‐FRACTURE FRONT AT FOUR (300 S, 600 S, 900 S,

AND 1200 S) DIFFERENT TIMES DURING THE

PUMPING/HYDRAULIC‐FRACTURING STEP.

The variation of fracture opening over the x‐z plane at

the aforementioned four times during the pumping

step is depicted in Figures 6(a)–(d). A comparison of

the results displayed in these figures shows that, at

earlier times, a fracture is accompanied by an increase

in the extent of its opening over the entire fracture

surface. On the other hand, at the later stages of

fracture, increase in the fracture opening occurs

mainly in the region near the fracture front, where

new fracture surfaces are created. In other words,

previously‐fractured regions appear to have reached

the state of nearly‐constant fracture opening. In other

words, previously‐fractured regions appear to have

reached the state of nearly‐constant fracture opening.

FIGURE 6. VARIATION OF FRACTURE OPENING (IN MM) OVER

THE X–Z PLANE AT FOUR: (A) 300 S; (B) 600 S; (C) 900 S; AND (D)

1200 S TIMES DURING THE

PUMPING/HYDRAULICFRACTURING STEP.

The variation of the pore pressure over the x‐z plane at

the same four times during the pumping step is

depicted in Figures 7(a)–(d). Examination of the results

displayed in these figures reveals that: (a) at the earlier

times, pore pressure at the wellbore site acquires very

high values, which rapidly decline with distance from

the wellbore, undershooting the pressure level present

in the rock formation before fracture (ca. 24.7 MPa).

The pressure then gradually recovers, with an increase


11

in distance from the wellbore, towards its pre‐

pumping value. This behavior is consistent with the

fact that hydraulic‐induced fracturing is controlled by

the creation of new fracture surfaces at the fracture

front, the process which requires a substantial increase

in the hydraulic pressure within the existing fracture.

However, once new fracture surfaces are created, the

high pressure causes their rapid separation and the

high pressure causes their rapid separation and the

associated large drop in the local pore pressure within

the fracture (in the region adjacent to the fracture

front); and (b) at later simulation times, the pressure at

the wellbore site acquires smaller values which are

retained over a large portion of the fracture surface.

However, as in the case of the shorter times within the

pumping stage, pressure still experiences a drop

below, and subsequent recovery towards, its pre‐

pumping level at the fracture‐front region.

FIGURE 7. VARIATION OF PORE PRESSURE (IN MPa) OVER THE

FRACTURED PORTION OF THE X‐Z PLANE AT FOUR: (A) 300 S;

(B) 600 S; (C) 900 S; AND (D) 1200 S TIMES DURING THE

PUMPING/HYDRAULIC‐FRACTURING STEP.

One of the ways of judging the effectiveness of the

hydraulic‐fracturing process in stimulating fuel

extraction from the deep‐seated reservoirs is to

compare the volumetric rates (commonly referred to

as “yield”) of the extracted fuel for the two identical

wellbore and reservoir scenarios except that hydraulic

fracturing is carried out in one case but not in the

other case. Such a comparison was carried out in

Figure 8, in which time‐dependence of the yield was

plotted for the two cases. Examination of the results

displayed in Figure 8 shows that, as a result of

applying hydraulic fracturing, the yield increased by

more than two orders of magnitude. This finding

suggests that, for the combination of the rock‐ and

shale‐formation properties and the set of hydraulic‐

fracturing process conditions considered in the present

finite element analyses, hydraulic fracture yielded

considerable benefits relative to the enhancement in

the fuel‐extraction efficiency.

FIGURE 8. THE EFFECT OF HYDRAULIC FRACTURING ON THE

TEMPORAL EVOLUTION OF THE VOLUMETRIC EXTRACTION

RATES OF THE FUEL.

It should be noted that the results presented in this

section, as well as in the subsequent section, were

obtained using a prototypical sand‐injection profile, as

depicted in Figure 9. On the other hand, in Section

IV.3, the sand‐injection profile was varied and the

effect of this variation was examined. It should be

further noted that sand was not treated explicitly

within the present finite‐element analysis. Rather, its

presence within the hydraulic‐fracturing fluid was

accounted for by quantifying its effect on the effective

viscosity of this fluid. This was done by utilizing the

following fluid‐viscosity, 　, vs. sand‐concentration, c,

functional relation [Adachi et al. (2007); Grujicic et al.

(2010)]:

7.165.011.0 c (1)


12

In addition, the propping effect of sand was accounted

for by fixing the nodes at the fracture surface, within

the sand‐retention and production steps, at their

positions acquired at the end of the pumping step.

FIGURE 9. PROTOTYPICAL SAND‐INJECTION PROFILE IN THE

HYDRAULIC‐FRACTURING FLUID DURING

THE PUMPING STEP.

The Role of Natural Fissures

Rock formations are treated in the present work as a

porous medium consisting of a solid skeleton and fuel‐

saturated fine‐scale interconnected pores. Due to the

fine‐scale nature of the pores, the hydraulic

conductivity (a measure of the ease with which fuel

flows through the porous medium) is not very high.

This negatively affects the rate of fuel extraction from

the deep‐seated reservoirs. However, the rock

formation also contains natural flaws such as cracks,

fissures, crevices, etc. The presence of these flaws

(henceforth referred to as fissures), if intersected by

the hydraulic fractures, may improve the efficiency of

fuel extraction. This could be the result of an increase

in the “effective” fracture surface (the surface through

which the fuel trapped within the target‐rock

formation is entering the fracture) and/or a result of

the introduction of additional larger‐size flow

channels with larger hydraulic conductivity. Due to a

larger difference in the size of the fissures (mm to cm

long), and the hydraulic fractures (tens of meters long),

the fissures could not be modeled explicitly (i.e. as

discrete entities within the continuum rock‐formation).

Rather, their effect is modeled through the proper

adjustment of some of the bulk porous‐material model

parameters. Specific parameters adjusted to account

for the effect of the natural fissures include: (i) void

ratio of the porous medium elements bordering the

hydraulic fracture; and (ii) the leakoff coefficient. The

extent of these corrections was treated as a function of

the number density, Nf , and the average size, Vf , of

the fissures. The product of these two quantities

defines the additional porosity associated with the

presence of the fissures, which is next used in the

relations described in Section II.1.1 to adjust the void

ratio of the porous‐medium elements mentioned

above. As far as the effect of fracture‐intersected

fissures on the leakoff coefficient is concerned, an

approach is used within which a functional

relationship is postulated between the hydraulic

conductivity and the soil‐grain size. By treating the

fissures as being equal‐sized, constant aspect ratio A

(>1), oblate spheroidals, the following simplified

functional form for the leakoff coefficient leakoffK was

obtained:

20 log1 AVNKK ffleakoffleakoff (2)

where 0leakoffK is the reference value of this coefficient.

More details regarding the derivation of Eq. (2) will be

provided in a future communication.

The effect of the NfVf product and the fissures’ aspect

ratio on the percent increase in the extraction yield (at

extraction times long enough to ensure a fairly

constant/steady value of the yield) relative to the case

of hydraulic fracturing of “fissure‐free” rock formations

(the reference case), is depicted in Figure 10.

Examination of the results displayed in this figure

reveals that: (a) both an increase in the NfVf product

and the fissures’ aspect ratio result in an increase in

the fuel‐extraction yield. The effect of the NfVf product

can be attributed to an increase in the porous‐medium

permeability due to the increase in the effective void

ratio within this medium. The effect of the fissures’

aspect ratio, on the other hand, is manifested through

an increase in the effective fracture‐surface area; and

(b) for the range of the NfVf product and the fissures’

aspect ratio considered, increases in the fuel‐extraction

yield as high as ca. 5% relative to the reference case

can be obtained.

The Effect of Sand Concentration and Injection Profile

In this section, hydraulic fracturing of fissure‐free rock

formations was again considered. In addition, all the

model/process parameters, except for the sand

concentration and injection profile within the

hydraulic‐fracturing fluid, were set equal to their

values used in the reference case, Section IV.1.

Examination of Figure 9 shows that a prototypical


13

sand‐injection profile involves a linear increase in the

sand concentration with time, beginning with a sand‐

injection start‐time, to a target final value, attained at

the sand‐injection end‐time. Thus, the sand‐injection

profile contains three parameters: the injection start‐

time, the sand‐target final concentration, and the sand‐

injection end‐time (or the sand‐injection duration). In

the present work, the sand‐injection start‐time is fixed

at seven min of the pumping time and the remaining

two parameters are varied in the following range: (a)

the sand concentration ‐ 400 to 800 kg/m3 ; and (b) the

injection duration ‐ 2 to 12 min. The effect of the sand

concentration and the injection duration on the

percent change in the extraction yield relative to the

reference case (characterized by the maximum sand

concentration of 600 kg/m3 and injection duration of 7

min) analyzed in Section IV.1, is depicted in Figure 11.

Examination of the results displayed in this figure

reveals that: (a) there is an optimal combination of the

sand concentration and the injection duration which

maximizes the fluid‐extraction yield; and (b) the

optimal combination of the sand concentration and

injection duration which maximizes the fluid‐

extraction yield is associated with the intermediate

values of the sand concentration and the longest

values of the injection duration.

FIGURE 10. THE EFFECT OF THE NFVF PRODUCT AND THE

FISSURES’ ASPECT RATIO ON THE RELATIVE CHANGE (IN

PERCENT) IN THE EXTRACTION YIELD WITH RESPECT TO

THE CASE OF HYDRAULIC FRACTURING OF “FISSURE‐FREE”

ROCK FORMATIONS (THE REFERENCE CASE).

Careful examination of the hydraulically‐induced

fractures and their extensions over the fracture plane

and fracture opening profile, the results not shown for

brevity, provided rationale for the results displayed in

Figure 11. That is: (a) as the sand concentration

increases, the increased hydraulic‐fracture fluid

viscosity gives rise to an increase in the hydraulic

pressure within the fracture. This condition leads to a

desired enhancement in the extent of hydraulic

fracturing; (b) a further increase in the sand

concentration/fluid viscosity makes the flow of the

hydraulic‐fracturing fluid, within the fracture, a rate‐

controlling process. That is, the associated increase in

the fluid viscosity and hydraulic pressure increases the

opening of the already formed fractures, but does not

significantly contribute to the extension of the fracture

along its length; and (c) the undesirable effects

associated with excessive sand concentrations can be

alleviated if the injection of the sand is done in a more

gradual manner (i.e. if the sand‐injection concentration

is increased slowly).

FIGURE 11. THE EFFECT OF THE SAND CONCENTRATION

AND THE INJECTION DURATION ON THE RELATIVE CHANGE

(IN PERCENT) IN THE EXTRACTION YIELD RELATIVE TO THE

REFERENCE CASE.

Summary and Conclusions

Based on the work presented and discussed in the

present manuscript, the following main summary

remarks and conclusions can be made:

1. Finite‐element analysis of the hydraulic‐fracturing

process used to stimulate fuel‐extraction from deep‐

seated reservoirs can provide a highly beneficial

insight into and quantification of the associated

phenomena and processes. Direct experimental

observation and quantification of these phenomena

could be either quite challenging or impossible.

2. The present work suggests that the reliability and

accuracy of the finite element analysis of the


14

hydraulic‐fracturing processare greatly affected by the

knowledge of the lithography, including fissure/flow

content and the constitutive response of the rock

formation targeted for fuel extraction.

3. The work also shows that the finite element analysis

could be used to optimize the hydraulic fracturing

process by providing a valuable insight into the

optimal start time and duration of different hydraulic‐

fracturing steps, as well as of the optimal process

conditions (e.g. sand concentration and its injection

time into the hydraulic‐fracturing fluid).

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computational engineering analysis of the hydraulic-fracturing process

Documents

shale reservoirs

fracturing behavior

present workprovides

deepseated reservoirs

nonlinear character

horizontal drilling

fluid flowthrough

tens ofmillions of years