**3. Modeling leakoff into permeable NF in UFM**

The main assumptions for the current modelling of leakoff from HF into the intercepted NF in UFM are given below.


Hydraulic fractures propagating in the rock are modeled in accordance with existing approach in UFM model. The Schematic of the complex HF interaction with permeable NF is shown in Figure 2 and Figure 4.

UFM. The crossing model, showing good comparison with existing experimental data, did not account for the effect of fluid viscosity and flow rate on the crossing pattern. More recently a new advanced OpenT crossing model, taking into the account the impact of fluid and NF

The modelling approach used in UFM to predict the leakoff into the NFs is presented below.

The main assumptions for the current modelling of leakoff from HF into the intercepted NF

**•** The rock formation contains vertical discrete deformable fractures (NFs), which are initially closed but conductive because of their pre-existing apertures (due to surface roughness, etc).

**•** The propagation direction of the hydraulic fractures is not affected (unless intercepted) by closed and not invaded natural fractures. The intercepted NF could affect HFN propagation

**•** The original fluid inside the natural fractures and in the reservoir (oil, gas, water) is

**•** Fracturing fluid can be incompressible or compressible and its rheology can be Newtonian

**•** When intercepted by the main hydraulic fracture, a natural fracture may remain closed, while still being able to accept fracturing fluid, or may be mechanically opened by fracturing fluid pressure depending on the magnitude of the fracturing fluid pressure, confining

**•** The natural fractures can be opened by fluid pressure that exceeds the normal stress acting

**•** The original natural fracture has width *w <sup>0</sup>* and are filled with reservoir fluid with pressure

**•** Fluid flow invaded into the natural fracture develops along NFs. Invaded fracturing fluid into the NF may reach the end of NF, break the rock, and start to propagate into the rock accordingly to previously implemented propagation rules (only if the NF is opened). Fracture re-initiation from other points along the NF other than its ends (offsets) is not

stresses applied on natural fracture, and frictional properties of natural fracture.

properties, have been developed [2] and integrated in UFM [18].

**3. Modeling leakoff into permeable NF in UFM**

even from closed parts (when shear slippage takes place)

**•** The rock material is assumed to be permeable and elastic.

on them and/or experience Coulomb type frictional slip.

**•** The natural fractures are assumed to contain pore space and are permeable.

in UFM are given below.

292 Effective and Sustainable Hydraulic Fracturing

compressible and Newtonian.

**•** The flow inside NFs is assumed to be 1D.

equal to pore pressure *p <sup>0</sup> =p res*.

modeled at this time.

or power law.

**Figure 2.** Hydraulic fracture intercepting natural fracture and possible situation to model

Fracturing fluid invasion into the two wings of the NF needs to be considered separately. Four possible regions can co-exist in each wing of the NF encountered by HF (Figure 4):


When a natural fracture is intercepted by the hydraulic fracture, the fluid pressure in the hydraulic fracture transmits into the natural fracture. If the fluid pressure is less than the normal effective stress on the natural fracture, the natural fracture remains closed. Even closed natural fractures may have hydraulic conductivities much larger than the surrounding rock matrix, and in this case fracturing fluid will invade the natural fractures more than leakoff into the surrounding matrix. If the portion of injected fluid is lost into closed natural fractures from the main HF, the HF growth could be affected.

For a closed fracture, the equivalent fluid conductivity is expected to change with the fluid pressure since contact deformation is a function of effective normal stress. This pressureinduced dilatancy and the associated increase in conductivity are important in increasing leakoff. Also, any reduction in effective contact stress may result in fracture sliding, which can lead to local stress variations and slip induced fracture dilation, which can in turn change the overall conductivity of fracture networks.

( ) 2 0,

Here *s* is coordinate along NF, and total leakoff coefficient from the walls of the natural fracture

2

*m f tot f L L*

*q h hC q q s t t s*

+ += =

r

*rock rock v c tot vc rock rock rock*

*C C C C*

= =

*rock rr rr*

m

j

j

pm

( ) 2

*v*

*V*

*L*

*C t*

*rock r rT*

2

*v*

=

*μr* - reservoir fluid viscosity in the porous media

per unit area that previously leaked off into the reservoir.

*C*

*cT* - total compressibility of reservoir

alent term (see (5c)) [42] Where *<sup>C</sup>*¯

*kr* - permeability of rock matrix



*pr* - reservoir pressure

r v

¶ ¶

is equal to combined leakoff coefficient [41]

reservoir zone as shown in equations (5a - 5b)

*Ctot rock*

with

*μf*

*ρf*

*ϕr* - reservoir porosity

( )

(4)

295

(5)

t

¶ ¶ - (3)

2 2

4

*rock rock*

*vv c*

+ +

*CC C*

Where leakoff coefficient for the filtration zone in the rock and leakoff coefficient for the

, a 2 2

In the case of multiple fluids, the invaded zone can be described by replacing *Cν* with equiv‐

permeability of all the filtrate fluids leaked off up to the current time, and *V <sup>L</sup>* is the fluid volume

*v f s f f*

 j

> m

*kpkp <sup>C</sup> pp p*

D D = = D= -

*c f r r*

= D D= -

*k c <sup>C</sup> p pp p*

, b

c

*<sup>v</sup>* is calculated using the average viscosity and relative

*rock*

Hydraulic Fracturing in Formations with Permeable Natural Fractures

http://dx.doi.org/10.5772/56446

The governing processes in first three regions listed above should be modeled, and the modeling approaches in different regions (also referred to as zones in the following context) are different due to different flow behaviors and rock/fluid properties.

#### **4. Basic governing equations**

#### **4.1. Continuity of fluid volume (mass)**

The equation for the continuity of incompressible fluid volume has the form

$$\frac{\partial q\_{\rm NF}}{\partial \mathbf{s}} + \frac{\partial A}{\partial t} + q\_{\rm L} = 0, \qquad q\_{\rm L} = \frac{2hC\_{\rm tot}^{\rm rock}}{\sqrt{t - \tau(\mathbf{s})}}, \quad A = \sigma \hbar \mathbf{h} \tag{1}$$

where

*qNF* (t) - volumetric flow rate through a cross section of area *A* of natural fracture [m3 /s]

*A* - cross sectional area of the natural fracture

*qL* - the volume rate of leakoff per unit length

*ϖ* - average hydraulic fracture width (different from *w,* fracture opening by fluid pressure exceeding normal stress)

*h* - fracture height

*Ctot* - total leakoff coefficient from the wall of natural fracture

More generally in the case of compressible fluid the equation (1) should account for fluid density *ρ <sup>f</sup>* and mass flux *q <sup>m</sup> .* Considering the rate of change of fluid mass per unit length in a fracture *m*˙ , continuity of fluid mass in the fracture is governed by equation

$$\frac{\partial q\_m}{\partial \mathbf{s}} + \dot{m} + \rho\_f q\_L = 0 \tag{2}$$

or along the fracture of constant length

Hydraulic Fracturing in Formations with Permeable Natural Fractures http://dx.doi.org/10.5772/56446 295

$$\frac{\partial q\_m}{\partial \mathbf{s}} + \frac{\partial (\rho\_f \varpi \hbar)}{\partial t} + \rho\_f q\_L = 0, \qquad q\_L = \frac{2\hbar C\_{tot}^{\text{rock}}}{\sqrt{t - \tau(\mathbf{s})}} \tag{3}$$

Here *s* is coordinate along NF, and total leakoff coefficient from the walls of the natural fracture *Ctot rock* is equal to combined leakoff coefficient [41]

$$\mathbf{C}\_{tot}^{\rm rock} = \mathbf{C}\_{vc}^{\rm rock} = \frac{2\mathbf{C}\_{v}^{\rm rock}\mathbf{C}\_{c}^{\rm rock}}{\mathbf{C}\_{v}^{\rm rock} + \sqrt{\mathbf{C}\_{v}^{\rm rock2} + 4\mathbf{C}\_{c}^{\rm rock2}}} \tag{4}$$

Where leakoff coefficient for the filtration zone in the rock and leakoff coefficient for the reservoir zone as shown in equations (5a - 5b)

$$\begin{aligned} \mathbf{C}\_{v}^{\text{rock}} &= \sqrt{\frac{k\_{r}\rho\_{r}\Delta p}{2\mu\_{f}}} = \sqrt{\frac{k\_{r}\rho\_{r}\Delta p}{2\mu\_{f}}}, \Delta p = p\_{f} - p\_{s} & \mathbf{a} \\ \mathbf{C}\_{c}^{\text{rock}} &= \sqrt{\frac{k\_{r}\rho\_{r}\mathbf{C}\_{T}}{\pi\mu\_{r}}} \Delta p\_{r} & \Delta p = p\_{f} - p\_{r} & \mathbf{b} \\ \overline{\mathbf{C}}\_{v} &= \frac{2\left(\mathbf{C}\_{v}\right)^{2}\sqrt{t}}{V} & \mathbf{c} \end{aligned} \tag{5}$$

with

For a closed fracture, the equivalent fluid conductivity is expected to change with the fluid pressure since contact deformation is a function of effective normal stress. This pressureinduced dilatancy and the associated increase in conductivity are important in increasing leakoff. Also, any reduction in effective contact stress may result in fracture sliding, which can lead to local stress variations and slip induced fracture dilation, which can in turn change the

The governing processes in first three regions listed above should be modeled, and the modeling approaches in different regions (also referred to as zones in the following context)

> 2 0, , ( )

+ += = =

*qNF* (t) - volumetric flow rate through a cross section of area *A* of natural fracture [m3

*ϖ* - average hydraulic fracture width (different from *w,* fracture opening by fluid pressure

More generally in the case of compressible fluid the equation (1) should account for fluid density *ρ <sup>f</sup>* and mass flux *q <sup>m</sup> .* Considering the rate of change of fluid mass per unit length in

> 0 *<sup>m</sup> f L*

++ =

r

*<sup>q</sup> m q*

a fracture *m*˙ , continuity of fluid mass in the fracture is governed by equation

*s*

¶

*rock*

¶ ¶ - (1)

t

v

¶ & (2)

/s]

are different due to different flow behaviors and rock/fluid properties.

The equation for the continuity of incompressible fluid volume has the form

*NF tot L L <sup>q</sup> <sup>A</sup> hC q q Ah s t t s*

overall conductivity of fracture networks.

294 Effective and Sustainable Hydraulic Fracturing

**4. Basic governing equations**

where

**4.1. Continuity of fluid volume (mass)**

¶ ¶

*A* - cross sectional area of the natural fracture

*qL* - the volume rate of leakoff per unit length

or along the fracture of constant length

*Ctot* - total leakoff coefficient from the wall of natural fracture

exceeding normal stress)

*h* - fracture height

 $\phi\_r$  - reservoir porosity


*L*

*V*


In the case of multiple fluids, the invaded zone can be described by replacing *Cν* with equiv‐ alent term (see (5c)) [42] Where *<sup>C</sup>*¯ *<sup>v</sup>* is calculated using the average viscosity and relative permeability of all the filtrate fluids leaked off up to the current time, and *V <sup>L</sup>* is the fluid volume per unit area that previously leaked off into the reservoir.

#### **4.2. Pressure drop along closed NF**

The pressure drop along closed NF can be expressed from Darcy's law

$$\sigma\_{\rm NF} = -\frac{k\_{\rm NF}A}{\mu\_f} \frac{\Delta p}{\mathcal{L}(t)} , A = \varpi h \tag{6}$$

Where constants *C* and *σ \**

stress *σ<sup>n</sup>*

**4.4. The width of closed invaded NF**

pressure-dependent permeability as [10]

shear displacement *us* and dilation angle *ϕdil*

*eff n f*

the enhanced 2D DDM approach [40,43,44].

= -

 s

s

s

v

v

**4.5. Shear failure**

s

s

t

( )

*i i ij ij j ij ij j n nn n ns s j j*

å å

accounting for the mechanical opening in HFN from Eq. (12).


*p AC D AC D*

*sn n ss s*

*AC D AC D A*

=+ = -

*ij i ij ij j ij ij j ij*

*j j ij*

*p s*

 vv

(reference stress state) are determined from field data, *k <sup>o</sup>* is the initial

= *k* (10)

Hydraulic Fracturing in Formations with Permeable Natural Fractures

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297

*eff eff dil*

( )

+

*d h*

å å (12)

*d*

2.3 2.3/2 2 2

 f  f

> s

> > (11)

 s

NF permeability (reservoir permeability under in-situ conditions), *σ <sup>n</sup>* is the normal stress on

The width *ϖ* of closed NF invaded by the treatment fluid (hydraulic aperture) is related to the

12 *NF*

The hydraulic width *ϖ* can be evaluated from Barton-Bandis model following approach [11,36,38], i.e. directly from Eq. (11) for given effective normal stress *σ eff*, reference effective

*ref* , initial hydraulic fracture aperture *ϖo* (related to the roughness of fracture surface),

 f

v

<sup>0</sup> , | |tan( ),

= ++ = =

 v

1 9 1 9

+ +

*s res s s dil dil eff eff ref ref n n*

The second term in Eq. (11) represents the shear-induced dilation which contributes to the NF permeability (Eq.9). Shear induced dilation is related to the frictional slip which occurs when the shear stress reaches the frictional shear strength of the natural fractures *τ<sup>s</sup>* =*λ*(*σ<sup>n</sup>* − *p*) . In the present study the NF propagation due to the shear induced slip is not considered, but the contribution of shear slippage zone to the NF enhanced permeability is evaluated based on

, 1

The fracture surface slip (shear displacement) *us* can be found as shear displacement discon‐ tinuity *Ds* calculated for the closed sliding elements following Coulomb frictional law *τ* ≥*τ<sup>s</sup>* =*λ*(*σ* − *p* ) from elasticity equations in the stress shadow calculation approach with

*u*

the NF, *p* is the pressure in the NF, and *u <sup>s</sup>* is shear-induced displacement (slippage).

Or for mass flux

$$q\_m = -\rho\_f \frac{k\_{\rm NF}}{\mu\_f} A \frac{\partial p}{\partial s} \tag{7}$$

Then the pressure can be calculated from

$$\frac{\partial p}{\partial \mathbf{s}} = -\frac{\mu\_f}{\rho\_f k\_{\rm NF} A} q\_m = -\frac{\mu\_f}{\rho\_f k\_{\rm NF} \varpi \hbar} q\_{m'} \quad \text{at the inlet} \colon p = p\_{\rm in}(t) \tag{8}$$

Here


#### **4.3. Change of NF permeability due to stress and pressure changes**

The leakoff into the natural fracture, or permeability of the natural fracture, is highly pressure dependent when pressure of invading fluid exceeds reservoir pressure but still is below the closure pressure. In general, permeability of natural fracture is a function of normal stress on NF, shear stress (or shear displacement due to shear slippage), and fluid pressure, and can be represented as a combination of permeability due to normal stress and permeability due to shear slippage [26]:

$$\begin{aligned} k\_{\rm NF} &= f(k\_{\rm NF}^{\rm n}, k\_{\rm NF}^{s})\\ k\_{\rm NF}^{\rm n} &= f\_1(k\_{o^\*}, \sigma\_{n'} \, p \, ), \quad k\_{\rm NF}^{s} = f\_2(u\_{s'}, \phi\_{\rm diff})\\ k\_{\rm NF}^{\rm n} &= k\_o \left\{ \mathrm{C} \ln \left[ \frac{\sigma^\*}{\sigma\_n - p} \right] \right\}^3 \end{aligned} \tag{9}$$

Where constants *C* and *σ \** (reference stress state) are determined from field data, *k <sup>o</sup>* is the initial NF permeability (reservoir permeability under in-situ conditions), *σ <sup>n</sup>* is the normal stress on the NF, *p* is the pressure in the NF, and *u <sup>s</sup>* is shear-induced displacement (slippage).

#### **4.4. The width of closed invaded NF**

**4.2. Pressure drop along closed NF**

296 Effective and Sustainable Hydraulic Fracturing

Then the pressure can be calculated from

r

*kNF* - permeability of natural fracture

*A* - cross sectional area of closed NF



shear slippage [26]:

*pin* - fluid pressure at the inlet

m

Or for mass flux

Here

*μf*

*ρf*

The pressure drop along closed NF can be expressed from Darcy's law

*NF*

*f NF f NF*

 rv

**4.3. Change of NF permeability due to stress and pressure changes**

*NF*

m

*m f*

 m

r

*f k A <sup>p</sup> <sup>q</sup> A h L t*

, ( )

*NF*

*f <sup>k</sup> <sup>p</sup> q A*

, at the inlet : ( ) *f f m m in*

The leakoff into the natural fracture, or permeability of the natural fracture, is highly pressure dependent when pressure of invading fluid exceeds reservoir pressure but still is below the closure pressure. In general, permeability of natural fracture is a function of normal stress on NF, shear stress (or shear displacement due to shear slippage), and fluid pressure, and can be represented as a combination of permeability due to normal stress and permeability due to

1 2

s

*NF o n NF s dil*

*p*

\* ln

ì ü ï ï é ù <sup>=</sup> í ý ê ú ï ï - î þ ë û

s

*k fk p k fu*

= =

*n*

(,)

s

*n s NF NF NF n s*

*k fk k*

=

*k kC*

*n NF o* 3

 f

(9)

( , , ), ( , )

¶ =- =- <sup>=</sup> ¶ (8)

m

*<sup>p</sup> <sup>q</sup> <sup>q</sup> ppt s kA k h*

v

*s*

<sup>D</sup> <sup>=</sup> - = (6)

¶ = - ¶ (7)

The width *ϖ* of closed NF invaded by the treatment fluid (hydraulic aperture) is related to the pressure-dependent permeability as [10]

$$
\sigma = \sqrt{12k\_{\rm NF}} \tag{10}
$$

The hydraulic width *ϖ* can be evaluated from Barton-Bandis model following approach [11,36,38], i.e. directly from Eq. (11) for given effective normal stress *σ eff*, reference effective stress *σ<sup>n</sup> ref* , initial hydraulic fracture aperture *ϖo* (related to the roughness of fracture surface), shear displacement *us* and dilation angle *ϕdil*

$$\begin{aligned} \boldsymbol{\sigma} &= \frac{\boldsymbol{\sigma}\_{0}}{1 + 9 \frac{\boldsymbol{\sigma}\_{\text{eff}}}{\boldsymbol{\sigma}\_{\text{n}}^{\text{ref}}}} + \boldsymbol{\sigma}\_{s} + \boldsymbol{\sigma}\_{\text{res}}, \quad \boldsymbol{\sigma}\_{s} = \boldsymbol{u}\_{s} \, | \, \text{tan}(\boldsymbol{\phi}\_{\text{d}\text{i}}^{\text{eff}}), \quad \boldsymbol{\phi}\_{\text{d}\text{i}}^{\text{eff}} = \frac{\boldsymbol{\Phi}\_{\text{d}\text{i}}}{1 + 9 \frac{\boldsymbol{\sigma}\_{\text{eff}}}{\boldsymbol{\sigma}\_{\text{n}}^{\text{ref}}}} \\ \boldsymbol{\sigma}\_{\text{eff}} &= \boldsymbol{\sigma}\_{n} - \boldsymbol{p}\_{f} \text{(s)} \end{aligned} \tag{11}$$

#### **4.5. Shear failure**

The second term in Eq. (11) represents the shear-induced dilation which contributes to the NF permeability (Eq.9). Shear induced dilation is related to the frictional slip which occurs when the shear stress reaches the frictional shear strength of the natural fractures *τ<sup>s</sup>* =*λ*(*σ<sup>n</sup>* − *p*) . In the present study the NF propagation due to the shear induced slip is not considered, but the contribution of shear slippage zone to the NF enhanced permeability is evaluated based on the enhanced 2D DDM approach [40,43,44].

$$\begin{aligned} \sigma\_n^i - p^i &= \sum\_j A^{ij} \mathbb{C}\_{nn}^{ij} D\_n^j + \sum\_j A^{ij} \mathbb{C}\_{ns}^{ij} D\_s^j \\ \pi^i &= \sum\_j A^{ij} \mathbb{C}\_{sn}^{ij} D\_n^j + \sum\_j A^{ij} \mathbb{C}\_{ss}^{ij} D\_s^j \\ &\quad \pi^i \end{aligned} \qquad A^{\dagger j} = \mathbf{1} - \frac{d\_{ij}^{2,3}}{\left(d\_{ij}^2 + h^2\right)^{2.3/2}} \tag{12}$$

The fracture surface slip (shear displacement) *us* can be found as shear displacement discon‐ tinuity *Ds* calculated for the closed sliding elements following Coulomb frictional law *τ* ≥*τ<sup>s</sup>* =*λ*(*σ* − *p* ) from elasticity equations in the stress shadow calculation approach with accounting for the mechanical opening in HFN from Eq. (12).

*f*

0 0

<sup>1</sup> *<sup>n</sup> <sup>n</sup> fl fl*

Where *<sup>w</sup>*¯ is average fracture opening, and *n'* and *K'* are fluid power law exponent and

*fl fl fl f*

With Reynolds number (*N* Re) for the power law fluid between parallel plates and *Fanning*

1- (T ) <sup>1</sup> *<sup>T</sup> p p*

1

*B*

' 1

¶ = - ¶ (16)

1/2 <sup>3</sup>

(17)

(18)

r

10

e

, ;


= - - - (15)

>*σ <sup>n</sup>*) will be handled as fluid flow in HFN and have been

(14)

http://dx.doi.org/10.5772/56446

299

Hydraulic Fracturing in Formations with Permeable Natural Fractures

r

where *B* is bulk modulus (fluid elasticity) in Pa and *c <sup>f</sup> =*1*/B* is fluid compressibility in Pa-1. More generally, change in fluid density due to changes in pressure and temperature for the low compressibility fluids through the known values of density *ρ <sup>0</sup>* and temperature T0 at pressure

*<sup>d</sup> dp dp <sup>c</sup> dt B dt dt*

0

0 2'1

( *<sup>w</sup>*(*z*) *<sup>w</sup>*¯ ) <sup>2</sup>*n*'+1 *<sup>n</sup>*' *dz*

+

a

*<sup>H</sup> fl <sup>∫</sup> H fl*

<sup>3</sup> , *<sup>f</sup>*

r

1 '2 ' 2 ' '

r

*n n nn*

2'1 '

*V w N f <sup>n</sup> <sup>K</sup>*


*<sup>p</sup> <sup>f</sup> q q <sup>w</sup> dp q H s HH <sup>w</sup> f ds*

¶ æ ö <sup>=</sup> - = ç ÷ - ¶ ç ÷ è ø

Re ' 2

16 log 3 ' 7.4

*n w*

æö é ù <sup>+</sup> æ ö ç ÷ ê ú ç ÷ è ø ê ú è ø ë û

3 2 1

*n*

*p q q s HH w*

r

b

r r= =

*p* 0 has form (*β* here is volumetric expansion coefficient)

described before [1] depending on the flow regime:

; *<sup>φ</sup>*(*<sup>n</sup>* ')= <sup>1</sup>

**4.7. Fluid flow in opened NF**

( <sup>4</sup>*<sup>n</sup>* ' <sup>+</sup> <sup>2</sup> *<sup>n</sup>* ' ) *n*'

Turbulent fluid flow (NRe>4000):

friction factor (*f*) defined as

*<sup>α</sup>*<sup>0</sup> <sup>=</sup> <sup>2</sup>*<sup>K</sup>* ' *φ*(*n* ')*n*'

consistency index.

The fluid flow in the opened NF (*p <sup>f</sup>*

Laminar fluid flow: Poiseuille Law [45]

r

**Figure 3.** Stress Shadow Effect from opened (blue) HFN and closed parts (grey) of intercepted NFs

For any element *j* in the opened part of HFN (including opened part of intercepted NFs) the input normal displacement discontinuity *Dn j* in Equation (12) is given by known fracture aperture (width) *Dn j* =*w <sup>j</sup>* ≠0 , and the shear stress is zero *τ <sup>j</sup>* =0 . Along the closed part of intercepted NFs the mechanical the opening is zero, *Dn j* =0 , and the pressure and normal stress should be tracked to detect (find) elements sliding in shear. If *τ <sup>i</sup>* ≥*τ<sup>s</sup> i* then element *i* is sliding and dilating in shear, and the shear stress *τ <sup>i</sup>* =*τ<sup>s</sup> i* for this element is used to find fracture surface slip *us i* =*Ds i* from the second equation (12). So, equation (12) could be solved for shear dis‐ placements *Ds j* along opened and sliding in shear parts of total HFN and intercepted NF as schematically shown in Fig.3. In Eq.(12) *C ij* are 2D, plane strain elastic influence coefficients [43] defining interactions between the elements *i* and *j*, and *A ij* are 3D correction factors [44] accounting for the 3D effect due to fracture height *h* depending on the distance between elements *d ij*.

#### **4.6. Fluid density as function of pressure and temperature**

Density of gas as a function of pressure and temperature has form

$$
\rho\_{\text{gas}} = \frac{p \, m\_m}{\text{ZRT}} \tag{13}
$$

Where *p* is pressure, *T* is temperature, *Z* is compressibility, *R* is gas constant, and *m <sup>m</sup>* is molar mass. The changes in pressure and temperature with time produce changes in gas density. Density of a compressible fluid as function of pressure

Hydraulic Fracturing in Formations with Permeable Natural Fractures http://dx.doi.org/10.5772/56446 299

$$\frac{d\rho}{dt} = \frac{\rho}{B} \frac{dp}{dt} = \rho c\_f \frac{dp}{dt} \tag{14}$$

where *B* is bulk modulus (fluid elasticity) in Pa and *c <sup>f</sup> =*1*/B* is fluid compressibility in Pa-1. More generally, change in fluid density due to changes in pressure and temperature for the low compressibility fluids through the known values of density *ρ <sup>0</sup>* and temperature T0 at pressure *p* 0 has form (*β* here is volumetric expansion coefficient)

$$\rho = \frac{\rho\_0}{1 \cdot \beta (\Gamma - T\_0)} \times \frac{1}{1 - \frac{p - p\_0}{B}} \tag{15}$$

#### **4.7. Fluid flow in opened NF**

**Figure 3.** Stress Shadow Effect from opened (blue) HFN and closed parts (grey) of intercepted NFs

input normal displacement discontinuity *Dn*

intercepted NFs the mechanical the opening is zero, *Dn*

should be tracked to detect (find) elements sliding in shear. If *τ <sup>i</sup>*

**4.6. Fluid density as function of pressure and temperature**

Density of a compressible fluid as function of pressure

Density of gas as a function of pressure and temperature has form

*j* =*w <sup>j</sup>*

298 Effective and Sustainable Hydraulic Fracturing

and dilating in shear, and the shear stress *τ <sup>i</sup>*

aperture (width) *Dn*

slip *us i* =*Ds i*

placements *Ds*

elements *d ij*.

*j*

For any element *j* in the opened part of HFN (including opened part of intercepted NFs) the

*j*

from the second equation (12). So, equation (12) could be solved for shear dis‐

along opened and sliding in shear parts of total HFN and intercepted NF as

in Equation (12) is given by known fracture

for this element is used to find fracture surface

≥*τ<sup>s</sup> i*

= (13)

=0 . Along the closed part of

then element *i* is sliding

=0 , and the pressure and normal stress

*j*

≠0 , and the shear stress is zero *τ <sup>j</sup>*

=*τ<sup>s</sup> i*

schematically shown in Fig.3. In Eq.(12) *C ij* are 2D, plane strain elastic influence coefficients [43] defining interactions between the elements *i* and *j*, and *A ij* are 3D correction factors [44] accounting for the 3D effect due to fracture height *h* depending on the distance between

*m*

Where *p* is pressure, *T* is temperature, *Z* is compressibility, *R* is gas constant, and *m <sup>m</sup>* is molar mass. The changes in pressure and temperature with time produce changes in gas density.

*pm ZRT*

*gas*

r

The fluid flow in the opened NF (*p <sup>f</sup>* >*σ <sup>n</sup>*) will be handled as fluid flow in HFN and have been described before [1] depending on the flow regime:

Laminar fluid flow: Poiseuille Law [45]

$$\frac{\partial p}{\partial \mathbf{s}} = -\alpha\_0 \frac{1}{\overline{w}^{2n^\*+1}} \frac{q}{H\_{\boldsymbol{\beta}}} \left| \frac{q}{H\_{\boldsymbol{\beta}}} \right|^{n^\*-1} \tag{16}$$

$$\alpha\_0 = \frac{2K}{\varrho(n')^{n'}} \left(\frac{4n'+2}{n'}\right)^{n'}; \quad \varrho(n') = \frac{1}{H\_{\frac{\mathcal{H}}{\mathfrak{H}\_{\beta}}}} \int\_{H\_{\beta}} \left(\frac{w(z)}{\overline{w}}\right)^{\frac{2n'+1}{n'}} dz$$

Where *<sup>w</sup>*¯ is average fracture opening, and *n'* and *K'* are fluid power law exponent and consistency index.

Turbulent fluid flow (NRe>4000):

$$\frac{\partial p}{\partial s} = -\frac{f \, \rho\_f}{\overline{w}^3} \left. \frac{q}{H\_{\beta}} \right| \frac{q}{H\_{\beta}} \bigg|\_{} , \eta = H\_{\beta} \left( -\frac{\overline{w}^3}{f \, \rho\_f} \frac{dp}{ds} \right)^{1/2} \tag{17}$$

With Reynolds number (*N* Re) for the power law fluid between parallel plates and *Fanning* friction factor (*f*) defined as

$$N\_{\rm Re} = \frac{3^{1-n^\*} 2^{2-n^\*} \rho V^{2-n^\*} \overline{w}^{n^\*}}{K \left(\frac{2n^\*+1}{3n^\*}\right)^{n^\*}}, f \approx \frac{1}{16 \left[\log\_{10}\left(\frac{\varepsilon}{7.4\overline{w}}\right)\right]^2};\tag{18}$$

Where *V* is fluid velocity and *ε* is surface roughness height.

**Darcy fluid flow** through proppant pack of height *h*

$$\frac{\partial p}{\partial s} = -\frac{q\mu\_{\hat{\mu}}}{kh\overline{w}}\tag{19}$$

will take place if the height of the fluid in fracture element become smaller than minimum fluid height. The minimum fluid height is calculated from the condition that pressure drop is equal to pressure drop due to the Darcy flow. The minimum height for turbulent and laminar flow is defined as

$$\begin{aligned} \text{Laminar flow:} \qquad &h\_{\boldsymbol{\beta}}^{\text{min}} = \left(\frac{k\alpha\_0(\boldsymbol{\nu}^\circ)H\_{\boldsymbol{\beta}}}{\overline{\boldsymbol{w}}^{2\boldsymbol{n}^\circ}\boldsymbol{\mu}\_{\boldsymbol{\beta}}}\right)^{1/\boldsymbol{n}^\circ} \frac{1}{q^{\overline{\boldsymbol{n}}^{-1}}}\\ \text{Turbulent flow:} \quad &h\_{\boldsymbol{\beta}}^{\text{min}} = \left(\frac{kH\_{\boldsymbol{\beta}}f\,\rho\boldsymbol{q}}{\overline{\boldsymbol{w}}^{2}\boldsymbol{\mu}\_{\boldsymbol{\beta}}}\right)^{1/2} \end{aligned} \tag{20}$$

Where *μfl* is fluid dynamic viscosity, and *k* is proppant pack permeability. The boundary conditions at the inlet and tip of opened fracture

$$p = p\_f(t), \quad p\_{tip} = \sigma\_n^{tip}(t) \tag{21}$$

**Figure 4.** Details on different possible zones in intercepted permeable NF

stress, but can still be higher than reservoir (pore) pressure

If the NF is opened at intersection element *i* with HF, then fluid pressure at intersection exceed

The tip of opened part of NF or the intersection element with HF (if NF is closed) becomes the

Mention that if NF is completely opened, then it is a part of total hydraulic fracture network (HFN) and handled based on the approach described previously in [1], see also equations (16)-

If NF is closed at intersection element *i* with HF, then fluid pressure is below the local normal

() () *NF of n p pi i* < £ s

(22)

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(23)

*filtr* <sup>=</sup> *ptip*

*open* <sup>=</sup>*σ<sup>n</sup>*

*NF* .

() () *NF f n pi i* > s

inlet (injection point) into the closed invaded part of NF (filtration zone) *pin*

**5.1. Opened part of NF**

(21).

the local normal stress on NF:

where *p <sup>f</sup>* is known fluid pressure at the intersection with natural fracture.

#### **5. Combined fluid flow into the opened and closed parts of invaded NF**

As I mentioned before, natural fracture can be closed, closed but invaded with fracturing fluid, closed and filled with pressurized reservoir fluid, or opened (Figure 4). The partially opened NF can contain opened, invaded, pressurized and closed parts which are dynamically changing with time. When fronts (positions of the boundaries between co-existing parts in invaded NF) change or propagate, the velocity of each propagating front can be considered as velocity of the corresponding fluid front in the relevant part of the invaded NF.

To properly model invasion of fracturing fluid into the NF, the propagation of each front should be modelled, and in different parts of the invaded NF different governing equations should be satisfied.

**Figure 4.** Details on different possible zones in intercepted permeable NF

#### **5.1. Opened part of NF**

Where *V* is fluid velocity and *ε* is surface roughness height.

*fl p q s khw*

m

will take place if the height of the fluid in fracture element become smaller than minimum fluid height. The minimum fluid height is calculated from the condition that pressure drop is equal to pressure drop due to the Darcy flow. The minimum height for turbulent and laminar

min 0

*fl n*

*kH f q <sup>h</sup> w*

( ), ( ) *tip f tip n p pt p t* = =

**5. Combined fluid flow into the opened and closed parts of invaded NF**

As I mentioned before, natural fracture can be closed, closed but invaded with fracturing fluid, closed and filled with pressurized reservoir fluid, or opened (Figure 4). The partially opened NF can contain opened, invaded, pressurized and closed parts which are dynamically changing with time. When fronts (positions of the boundaries between co-existing parts in invaded NF) change or propagate, the velocity of each propagating front can be considered as

To properly model invasion of fracturing fluid into the NF, the propagation of each front should be modelled, and in different parts of the invaded NF different governing equations

is known fluid pressure at the intersection with natural fracture.

velocity of the corresponding fluid front in the relevant part of the invaded NF.

s

2

*fl*

is fluid dynamic viscosity, and *k* is proppant pack permeability. The boundary

m

r

m

*fl*

æ ö = ç ÷ ç ÷ è ø

*w*

a

*k nH*

æ ö = ç ÷ ç ÷ è ø

min

*fl*

( ' Laminar flow:

*h*

Turbulent flow:

conditions at the inlet and tip of opened fracture

= - ¶ (19)

1/ '

*n fl*

1 )

2 ' <sup>1</sup> <sup>1</sup> '

*fl n*

*q*


(21)

(20)

1/2

¶

**Darcy fluid flow** through proppant pack of height *h*

300 Effective and Sustainable Hydraulic Fracturing

flow is defined as

Where *μfl*

where *p <sup>f</sup>*

should be satisfied.

If the NF is opened at intersection element *i* with HF, then fluid pressure at intersection exceed the local normal stress on NF:

$$p\_f(i) > \sigma\_n^{\rm NF}(i) \tag{22}$$

The tip of opened part of NF or the intersection element with HF (if NF is closed) becomes the inlet (injection point) into the closed invaded part of NF (filtration zone) *pin filtr* <sup>=</sup> *ptip open* <sup>=</sup>*σ<sup>n</sup> NF* . Mention that if NF is completely opened, then it is a part of total hydraulic fracture network (HFN) and handled based on the approach described previously in [1], see also equations (16)- (21).

If NF is closed at intersection element *i* with HF, then fluid pressure is below the local normal stress, but can still be higher than reservoir (pore) pressure

$$p\_o < p\_f\text{(i)} \le \sigma\_n^{NF}\text{(i)}\tag{23}$$

#### **5.2. Filtration zone (closed part of NF invaded by fracturing fluid)**

The fluid pressure, width and flow rate along the filtration zone can be calculated from given pressure *pin filtr* <sup>=</sup> *pf* (*i*) which satisfies condition (23). The flow rate along filtration zone can be iteratively solved from the system of equations (24) with flow rate from the inlet to filtration zone *qin filtr* which is a part of total solution

For the general case of compressible reservoir fluid and non-zero total leakoff from the walls of pressurizes NF into the rock, the leakoff to the rock from the NF part filled with pressurized

> *c NF r r*

The governing equations to calculate fluid pressure, width, and flow rate along the pressurized zone from known influx *q filtr/pres* and pressure *p filtr/pres* are similar to Equations (24) for filtration zone with replacing fracturing fluid with reservoir fluid and using *ϖpres* as hydraulic width of

Notice that the pressure at the front between opened and filtration zones along invaded NF (or at HF/closed NF intersection point) and the time step are the inputs for the new pressure, width, and flow rate calculation in closed invaded part of intercepted NF. Initial influx *q* in can be prescribed based on the pressure in NF from the previous time step, and then the solution scheme will be applied from the intersection towards the end of NF with tracking the incre‐ mental mass balance (fluid injected to NF at current time step) to define the end of filtration front, and tracking pressure in the rest of NF (not invaded part, filled with reservoir fluid) to track the end of pressurized zone (fluid pressure equal to reservoir pressure). The flow rate and pressure are tracked and corresponding front positions are to be updated iteratively until in the pressurized zone of NF the following condition is satisfied (which indicates the position

> ( ) 0 ( ) *open filtr pres*

*pL L L p*

*qL L L*

*open filtr pres r*

At the end of pressurized zone pressure is equal to the reservoir pressure and flow rate is zero. If the end of NF is reached and pressure is above the reservoir pressure, then Equation (27) is replaced with condition *q(L NF )=0*. During pumping the pressurized zone extends until the end of NF, and then gradually shrinks, while the lengths of invaded zone and opened zones increase. Eventually whole NF will be filled with fracturing fluid, so the NF will contain only filtration and/or opened zones. When NF is completely opened, it becomes a part of total HFN. The elements in closed part of invaded NF can slip under some conditions (for example, due to the stress field change from stress shadow), influencing the calculations of total NF perme‐

Each element in the closed part of intercepted NF is checked for shear slip possibility. The fracture surface slip (the shear displacement *us* ) can be found as shear displacement discon‐ tinuity calculated for the closed sliding elements satisfying Coulomb frictional law *τ* ≥*τ<sup>s</sup>* =*λ*(*σ* − *pf* ) , from elasticity equations (12) accounting for the mechanical opening in HFN.

++ =

= D D= - (26)

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++ = (27)

*k c <sup>C</sup> p pp p*

reservoir fluid is defined by the compressibility controlled leakoff coefficient

, *rock r rT*

j

pm

pressurized zone of invaded NF, and *c <sup>T</sup>* as reservoir fluid compressibility.

of the end of the pressurized zone)

ability *k NF*.

$$\begin{cases} \frac{\partial p}{\partial s} = -\frac{\mu\_f}{k\_{NF}\sigma\_{filtr}} q\_f & \text{at the inlet} : p = p\_{in}^{filtr}(t) \\\\ \frac{\partial q}{\partial s} + \frac{\partial (\sigma\_{filtr}h)}{\partial t} + q\_L = 0, q\_L = \frac{2hC\_{tot}}{\sqrt{t - r\_o(s)}} \\\\ k\_{NF} = \frac{\sigma\_{filtr}}{12} \\\\ \sigma\_{filtr}(p) = \frac{\sigma\_0}{1 + 9\frac{\sigma\_{eff}}{\sigma\_{eff}^4}} + \sigma\_s + \sigma\_{res}, \quad \sigma\_s \doteq u\_s \mid \tan(\phi\_{diff}^{eff}), \quad \sigma\_{eff} = \sigma\_n - p(s), \end{cases} \tag{24}$$

The length of filtration zone can be calculated by the tracking the volume of fracturing fluid leaked into the NF by marching from the inlet along the NF. At the end of filtration zone mass balance should be satisfied and fluid pressure will be higher or equal to the reservoir pressure

$$\begin{aligned} \, \, q\_{in}^{\text{filr}} \, \text{dt} &= dVol\_{\text{frac}}^{\text{filr}} \{ \text{dt} \} + dVol\_{\text{leak}}^{\text{filr}} \{ \text{dt} \} \\ \, \, p\_r &< p^{\text{end} \text{ fillr zone}} < \sigma\_n^{\text{NF}} \end{aligned} \tag{25}$$

The flowrate (filtration/pressurized front velocity) at the last element of filtration zone is used for calculations in the pressurized zone. The position of the front between the filtration zone and pressurized zone should be tracked, giving velocity of filtration front and pressure *pin pres* as input for solution in pressurized zone.

If NF is partially opened, then in the solution scheme for the filtration zone the intersection (inlet) element is replaced by the tip element *i* of the opened part of NF with *p*(*i*)=*σ<sup>n</sup> NF* (*i*) .

#### **5.3. Closed pressurized part of NF (filled with reservoir fluid).**

Mention that if the leakoff coefficient for filtration zone *C<sup>ν</sup> rock* =0 , then *pr* <sup>=</sup> *pf end filtr zone* <sup>&</sup>lt;*σ<sup>n</sup> NF* , and there will not be pressurized zone in NF ( *pr* = *pf end filtr zone* <sup>&</sup>lt;*σ<sup>n</sup> NF* , *<sup>L</sup> pressurized* =0 ).

For the general case of compressible reservoir fluid and non-zero total leakoff from the walls of pressurizes NF into the rock, the leakoff to the rock from the NF part filled with pressurized reservoir fluid is defined by the compressibility controlled leakoff coefficient

**5.2. Filtration zone (closed part of NF invaded by fracturing fluid)**

which is a part of total solution

( ) 2 0,

*q h hC q q s t t s*

<sup>ï</sup> ¶ ¶ <sup>ï</sup> + += = <sup>ï</sup> ¶ ¶ - ïï

ï = - =

*filtr tot L L*

vv

*<sup>p</sup> <sup>q</sup> pp t sk h*

2

*NF filtr*

v

m

12

*filtr*

v

302 Effective and Sustainable Hydraulic Fracturing

*<sup>p</sup> <sup>k</sup>*

ï +

v

*NF*

v

ïî

ì¶

ï ¶

í <sup>ï</sup> <sup>=</sup> <sup>ï</sup> ï

( )

1 9

as input for solution in pressurized zone.

0

s

s

v

*eff ref n*

*p p*

**5.3. Closed pressurized part of NF (filled with reservoir fluid).**

Mention that if the leakoff coefficient for filtration zone *C<sup>ν</sup>*

and there will not be pressurized zone in NF ( *pr* = *pf*

pressure *pin*

zone *qin filtr*

The fluid pressure, width and flow rate along the filtration zone can be calculated from given

iteratively solved from the system of equations (24) with flow rate from the inlet to filtration

( )

*rock*

*o*

t

*in*

( ) , | |tan( ), ( ),

The length of filtration zone can be calculated by the tracking the volume of fracturing fluid leaked into the NF by marching from the inlet along the NF. At the end of filtration zone mass balance should be satisfied and fluid pressure will be higher or equal to the reservoir pressure

> () () *filtr filtr filtr in frac leak end filtr zone NF r n*

s

The flowrate (filtration/pressurized front velocity) at the last element of filtration zone is used for calculations in the pressurized zone. The position of the front between the filtration zone and pressurized zone should be tracked, giving velocity of filtration front and pressure *pin*

If NF is partially opened, then in the solution scheme for the filtration zone the intersection

(inlet) element is replaced by the tip element *i* of the opened part of NF with *p*(*i*)=*σ<sup>n</sup>*

*q dt dVol dt dVol dt*

= +

< <

*p u p s*

*filtr s res s s dil eff n*

<sup>ï</sup> = ++ = = - <sup>ï</sup>

 v *eff*

 s  s

*rock* =0 , then *pr* <sup>=</sup> *pf*

*NF* , *<sup>L</sup> pressurized* =0 ).

*end filtr zone* <sup>&</sup>lt;*σ<sup>n</sup>*

(24)

(25)

*pres*

*NF* ,

*NF* (*i*) .

*end filtr zone* <sup>&</sup>lt;*σ<sup>n</sup>*

 f

, at the inlet : ( )

*f filtr*

*filtr* <sup>=</sup> *pf* (*i*) which satisfies condition (23). The flow rate along filtration zone can be

$$\mathbf{C}\_{c}^{\text{rock}} = \sqrt{\frac{k\_{r}\rho\_{r}c\_{T}}{\pi\mu\_{r}}}\Delta p\_{r} \quad \Delta p = p\_{\text{NF}} - p\_{r} \tag{26}$$

The governing equations to calculate fluid pressure, width, and flow rate along the pressurized zone from known influx *q filtr/pres* and pressure *p filtr/pres* are similar to Equations (24) for filtration zone with replacing fracturing fluid with reservoir fluid and using *ϖpres* as hydraulic width of pressurized zone of invaded NF, and *c <sup>T</sup>* as reservoir fluid compressibility.

Notice that the pressure at the front between opened and filtration zones along invaded NF (or at HF/closed NF intersection point) and the time step are the inputs for the new pressure, width, and flow rate calculation in closed invaded part of intercepted NF. Initial influx *q* in can be prescribed based on the pressure in NF from the previous time step, and then the solution scheme will be applied from the intersection towards the end of NF with tracking the incre‐ mental mass balance (fluid injected to NF at current time step) to define the end of filtration front, and tracking pressure in the rest of NF (not invaded part, filled with reservoir fluid) to track the end of pressurized zone (fluid pressure equal to reservoir pressure). The flow rate and pressure are tracked and corresponding front positions are to be updated iteratively until in the pressurized zone of NF the following condition is satisfied (which indicates the position of the end of the pressurized zone)

$$\begin{aligned} \, \, q(L\_{open} + L\_{film} + L\_{pres}) &= 0\\ \, \, p(L\_{open} + L\_{film} + L\_{pres}) &= p\_r \end{aligned} \tag{27}$$

At the end of pressurized zone pressure is equal to the reservoir pressure and flow rate is zero. If the end of NF is reached and pressure is above the reservoir pressure, then Equation (27) is replaced with condition *q(L NF )=0*. During pumping the pressurized zone extends until the end of NF, and then gradually shrinks, while the lengths of invaded zone and opened zones increase. Eventually whole NF will be filled with fracturing fluid, so the NF will contain only filtration and/or opened zones. When NF is completely opened, it becomes a part of total HFN. The elements in closed part of invaded NF can slip under some conditions (for example, due to the stress field change from stress shadow), influencing the calculations of total NF perme‐ ability *k NF*.

Each element in the closed part of intercepted NF is checked for shear slip possibility. The fracture surface slip (the shear displacement *us* ) can be found as shear displacement discon‐ tinuity calculated for the closed sliding elements satisfying Coulomb frictional law *τ* ≥*τ<sup>s</sup>* =*λ*(*σ* − *pf* ) , from elasticity equations (12) accounting for the mechanical opening in HFN. If some elements in closed not disturbed part of NF are sliding then the corresponding shear stress is involved in the stress shadow calculations and thus influences simulations results.

**•** Save elements information (volume, pressure, flow rates) for the next time step

e International Conference for Effective and Sustainable Hydraulic Fracturing, 20-22 May 2013, Brisbane, Australia

**•** Make the NF elements a part of total network (HFN) and include pressure calculations along different zones of NF into the whole iterative scheme to calculate pressure, time step, and

**Treatment of closed permeable NFs**

After pressure, opening, flow rate and time step calculations in opened part of HFN

Create elements along intercepted NF when HF encounters it (for new intersection): HFN = HFNopened+HFNclosed . Update open zone in NF if necessary Evaluate possibility of shear sliding at each element from Mohr-Coulomb criterion and update hydraulic width and NF conductivity from Equations (10), (11) and (12)

Based on the pressure calculations in opened HFN and known time step d*t* evaluate initial guess *in q* for flow rate of fracturing fluid into closed NF

Solve for pressure, flow rate and width by marching from inlet towards the end of NF. Equations (24) Follow volume (mass) balance to indicate the end of filtration zone and to switch to reservoir fluid in pressurized zone. Equation (25)

End of NF (i=N)

**Yes No**

*<sup>r</sup> q* 0, *p p*

*q p pr* 0,

*r r q p p q p p or* 0, 0,

*r r q p p q p p or* 0, 0,

End of pressurized zone Increase *in q*

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Reduce *in q*

Figure 5 The proposed algorithm to account for leakoff from HF into the permeable NFs (*i=N* indicates the end of NF)

**Figure 5.** The proposed algorithm to account for leakoff from HF into the permeable NFs (*i=N* indicates the end of NF)

Solution along closed NF found, proceed to HFN new time step calcs

*New time step calculations in Opened part of HFN*

*<sup>r</sup> q* 0, *p p*

*<sup>q</sup>* <sup>0</sup> *or <sup>r</sup> q* 0, *p p*

*<sup>q</sup>* <sup>0</sup> Increase *qin*

Reduce *in q*

12

**6.2. Fully coupled numerical approach**

front positions.

**•** Discretize the NF when HF intercepts it
