**Appendix I — Pressure losses during production (Literature survey)**

The most cited work for perforation pressure losses is usually Karakas and Tariq, 1991 [23], Although their skin values were designed for permeable formations, the methodology is useful for thinking about pressure losses that might be incurred. They incorporated additive skin components that accounted for vertical and horizontal convergence and phasing. Inclination mechanical skin can also be considered. Presume that the perforation skin for a vertical well can be represented as:

**Figure 6.** For practical perforated lengths, single phase erosion will not be likely. Even if erosion does occur it should only reduce the velocities and the pressure loss in the perforations.

$$\mathbf{s}\_p = \mathbf{s}\_H + \mathbf{s}\_V + \mathbf{s}\_{wb} \tag{11}$$

kH, are equal and the density is 6 spf (giving the space between the perforations, h, as 0.2 ft. For 60° phasing we have c1 = 3 x 10-4, c2 = 7.509, rwD = 0.333 and the skin as 3.67 x 10-3. Since there is no horizontal permeability to speak of one can chose to ignore the horizontal convergence for a vertical fracture aligning with a vertical wellbore. Alternatively, the vertical convergence concept can be ignored for a transverse fracture intersecting a vertical well and axisymmetric convergent flow is required. Similar simplifications are possible for horizontal wells. For these

( )

*w perf*

*w wD*

aq (13)

379

(14)

*r r*

Do Perforated Completions Have Value for Engineered Geothermal Systems

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æ ö æ ö

1 10 2 1 2

aq*r L*

*H V DD*

*s s hr*

, , log , <sup>2</sup>

*h r a a r a b br b*

= = = += +

*D pD pD pD*

Transverse Fracture 0, ln ln

For the example being considered we find a1 =, a2 =, a =, b1 =, b2 =, b = αθ = 0.813, giving:

Transverse Fracture 0, ln ln 0.89

2 *s p <sup>Q</sup> p s kh* m

p

the pressure drop for a longitudinal fracture is negligible for a highly conductive fracture. Transverse fracturing may even give a negative skin although the simplifications adopted may

Kabir and Salmachi, 2009, [25] described relationships for perforation skin calculation during injection, using well-known relationships; extending concepts from Karakas and Tariq, 1991 [23], representing the skin as a superposed combination of convergence to the perforations (but presuming flow from the matrix, whereas the considerations here are fracture-flow dominated, damage (the analog here could be choking or micro-annular pressure drop) and

crushing (perforation infill and poor fracture connectivity could be relevant.

*H V D pD*

*s s hr*

®= = =

1 10 2 1 2

aq*r L*

®= =

*V H*

*s s*

Longitudinal Fracture 0, 10

0.2, , log , <sup>2</sup>

*D pD pD pD*

*h r a a r a b br b*

= = = += +

*perf*

*r*

*h*

not be appropriate. In either case, skin is small.

*perf*

Longitudinal Fracture 0, 10 0.45

*V H*

*s s*

If the pressure drop to anticipate under steady state conditions is:

*perf perf*

*h r*

*L h*

*perf perf*

1

*ab b*

®= = ç ÷ <sup>=</sup> ç ÷ ç ÷ ç ÷ <sup>+</sup> è ø è ø

( )

*w perf*

*w wD*

aq

D = (15)

*r r*

æ ö æ ö

1

®= = ç ÷ = = - ç ÷ ç ÷ ç ÷ <sup>+</sup> è ø è ø

*ab b*



two cases (vertical well):

where:

sp perforation skin, dimensionless


The wellbore skin accounts for perforation phasing. Karakas and Tariq, 1991, [23] suggested that it is quite small for phasing less than 120°. Its direct application here (for fracture flow only) may partially account for microannular restrictions although this was not the original intent. Consider a dimensionless radius (rwD) and the wellbore skin (swb):

$$r\_{wD} = \frac{r\_{w\nu}}{r\_w + L\_{perf}} \quad s\_{w\nu} = c\_1 e^{c\_2 r\_{wD}} \tag{12}$$

These coefficients were tabulated by Karakas and Tariq [23] from their numerical work. As an example, suppose, the wellbore radius is rw = 0.5 ft., the perforation length, Lperf = 1.0 ft., the perforation radius, rperf = 0.5 inches and that the vertical and horizontal permeability, kV and kH, are equal and the density is 6 spf (giving the space between the perforations, h, as 0.2 ft. For 60° phasing we have c1 = 3 x 10-4, c2 = 7.509, rwD = 0.333 and the skin as 3.67 x 10-3. Since there is no horizontal permeability to speak of one can chose to ignore the horizontal convergence for a vertical fracture aligning with a vertical wellbore. Alternatively, the vertical convergence concept can be ignored for a transverse fracture intersecting a vertical well and axisymmetric convergent flow is required. Similar simplifications are possible for horizontal wells. For these two cases (vertical well):

$$\begin{aligned} h\_D &= \frac{h\_{perf}}{L\_{perf}}, r\_{pD} = \frac{r\_{perf}}{2h\_{perf}}, a = a\_1 \log\_{10} r\_{pD} + a\_2, b = b\_1 r\_{pD} + b\_2\\ \text{Longitudinal Feature} &\to s\_H = 0, s\_V = 10^4 h\_D^{b-1} r\_D^b\\ \text{Transverse Feature} &\to s\_V = 0, s\_H = \ln \left(\frac{r\_w}{\alpha\_\theta (r\_w + L\_{perf})}\right) = \ln \left(\frac{r\_{wD}}{\alpha\_\theta}\right) \end{aligned} \tag{13}$$

For the example being considered we find a1 =, a2 =, a =, b1 =, b2 =, b = αθ = 0.813, giving:

$$\begin{aligned} \text{In}\_{D} &= 0.2, r\_{pD} = \frac{r\_{pref}}{2h\_{pref}}, a = a\_1 \log\_{10} r\_{pD} + a\_2, b = b\_1 r\_{pD} + b\_2 \\ \text{Longitudinal Factor} & \to s\_H = 0, s\_V = 10^a h\_D^{b-1} r\_{pD}^b = 0.45 \\ \text{Transverse Feature} & \to s\_V = 0, s\_H = \ln \left(\frac{r\_w}{a\_\theta \left(r\_w + L\_{pref}\right)}\right) = \ln \left(\frac{r\_{wD}}{a\_\theta}\right) = -0.89 \end{aligned} \tag{14}$$

If the pressure drop to anticipate under steady state conditions is:

*p H V wb ss ss* =++ (11)

25.4 mm, 13.12 Effective spm 12.7 mm, 13.12 Effective spm 25.4 mm, 6.56 Effective spm 12.7 mm, 6.56 Effective spm

where:

sp perforation skin, dimensionless

0.01

0.1

1

**Perforation Velcoity (m/s)**

10

100

378 Effective and Sustainable Hydraulic Fracturing

swb wellbore skin, dimensionless

sH skin due to horizontal convergence, dimensionless

only reduce the velocities and the pressure loss in the perforations.

sV skin due to vertical flow convergence, dimensionless

The wellbore skin accounts for perforation phasing. Karakas and Tariq, 1991, [23] suggested that it is quite small for phasing less than 120°. Its direct application here (for fracture flow only) may partially account for microannular restrictions although this was not the original

1 10 100 1000

**Fracture Contact Length Along Wellbore (m)**

**Figure 6.** For practical perforated lengths, single phase erosion will not be likely. Even if erosion does occur it should

**Perforation Velocity**

<sup>1</sup> *wD <sup>w</sup> c r*

These coefficients were tabulated by Karakas and Tariq [23] from their numerical work. As an example, suppose, the wellbore radius is rw = 0.5 ft., the perforation length, Lperf = 1.0 ft., the perforation radius, rperf = 0.5 inches and that the vertical and horizontal permeability, kV and

2

*r L* = = <sup>+</sup> (12)

intent. Consider a dimensionless radius (rwD) and the wellbore skin (swb):

*wD wb w perf r r s ce*

$$
\Delta p\_s = \frac{\mathbb{Q}\mu}{2\pi kh} \mathbf{s}\_p \tag{15}
$$

the pressure drop for a longitudinal fracture is negligible for a highly conductive fracture. Transverse fracturing may even give a negative skin although the simplifications adopted may not be appropriate. In either case, skin is small.

Kabir and Salmachi, 2009, [25] described relationships for perforation skin calculation during injection, using well-known relationships; extending concepts from Karakas and Tariq, 1991 [23], representing the skin as a superposed combination of convergence to the perforations (but presuming flow from the matrix, whereas the considerations here are fracture-flow dominated, damage (the analog here could be choking or micro-annular pressure drop) and crushing (perforation infill and poor fracture connectivity could be relevant.

Saleh and Stewart, 1996, [26] elaborated on the Karakas and Tariq [23] considerations for pressure loss and added an additional complexity that is relevant for geothermal as well as shale gas/oil production – a second phase. The conventional production pressure drop allocations are represented by van Everdingen and Hurst's skin and Hawkins's representation:

$$s = \frac{2\pi\Delta p\_s k h}{Q\mu} = \left(\frac{k\_s}{k} - 1\right) \ln\left(\frac{r\_s}{r\_w}\right) \tag{16}$$

flow efficiency – normalized with respect to ideal (cased but no mechanical losses; or undam‐

Lian et al., 2000, [31] describe numerical modeling of perforated completions for fractures. Yildiz, 2006, [32] described methods of approximating a composite skin, as did Furui et al., 2008 [33]. Ehlig-Economides et al., 2008, [34] provided a rationale analysis, presuming flow through perforations directly connected to the hydraulic fracture (this may be impacted by wellbore and perforation deviation from principal stress directions). A halo effect was also considered wherein angularly offset perforations would still be connected along this length. This was acceptable in the high permeability formations that those authors were considering

Zhang et al., 2009, [35] expanded on the work of Ehlig-Economides et al., 2008, [34]. They introduced a model hypothesizing that only perforations between the far-field hydraulic fracture plane and the wellbore actually connect flow through the fracture and the well, for fracpacks. For deviated wells the number of perforations can drop substantially unless multiple injections are carried out on isolated zones. The problem may be more severe in openhole – with the fracture quickly deviating from where it discretely intersects the wellbore and possibly minimizing contact length along the well. Considering only the connected perforations (fracture(s) physically in contact with the wellbore), the pressure drop can be

,

D = (17)

Do Perforated Completions Have Value for Engineered Geothermal Systems

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381

888 *<sup>g</sup> perf f total ppc*

*p q kAN*

*L B*

m

This relationship came from Welling, 1998, [36] simply calculating the pressure drop for liquid

aged openhole...) situations.

considered as:

where:

μ dynamic viscosity, cP

but seems unrealistic in most EGS scenarios.

Δpperf pressure drop through the perforations, psi

Lg length of the propped perforation tunnel, ft

B water formation volume factor, res bbl/STbbl

kp absolute permeability of packed perforation, md

flow through a sand-packed perforation, as follows:

Ap cross-sectional area (nominal) of individual perforation, ft2

qf,total bottomhole total flow rate, BLPD

Nc number of connected perforations

where:

Δps incremental pressure drop occurring in the

wellbore region due to a changed permeability – this could be considered to be reduced aperture or reduced relative permeability or twisting of the fracture, etc.

k virgin permeability


Before low permeability (shale gas and shale liquids) was popular, Tariq et al., 1989, [27] considered production from naturally fractured reservoirs in low matrix permeability environments. "The sharp discontinuities in porosity and permeability created by fractures have a significant impact on the overall fluid flow in the reservoir. Fractures allow rapid conduction of fluids with very little pressure drop because their resistance to fluid flow is much lower than that of the matrix rock. Very high flow rates (30,000 to 50,000 B/D … have been obtained from fractured reservoir wells under a limited pressure drop." They further stated: "In the past, many naturally fractured reservoirs were completed openhole (barefoot). Perforated completions have now become more popular for naturally fractured reservoirs as a result of improvements in drilling technology and in fracture detection techniques. The concern in the perforated completion, however, is the small area open to flow. The productivity of a perforated completion in naturally fractured reservoirs is totally dependent on the hydraulic communication between the perforations and the fracture network. This commu‐ nication, in turn, is dependent on such factors as fracture interval (or fracture density), fracture orientation, number of joint sets, shot density, and perforation length."

Part of the concern near the wellbore is the choking type of skin [3, 28, 29, 30], where losses due to damage and convergence are isolated in the plane (or within the confines) of a fracture. The effects can be further normalized by looking at various forms of the productivity ratio or flow efficiency – normalized with respect to ideal (cased but no mechanical losses; or undam‐ aged openhole...) situations.

Lian et al., 2000, [31] describe numerical modeling of perforated completions for fractures. Yildiz, 2006, [32] described methods of approximating a composite skin, as did Furui et al., 2008 [33]. Ehlig-Economides et al., 2008, [34] provided a rationale analysis, presuming flow through perforations directly connected to the hydraulic fracture (this may be impacted by wellbore and perforation deviation from principal stress directions). A halo effect was also considered wherein angularly offset perforations would still be connected along this length. This was acceptable in the high permeability formations that those authors were considering but seems unrealistic in most EGS scenarios.

Zhang et al., 2009, [35] expanded on the work of Ehlig-Economides et al., 2008, [34]. They introduced a model hypothesizing that only perforations between the far-field hydraulic fracture plane and the wellbore actually connect flow through the fracture and the well, for fracpacks. For deviated wells the number of perforations can drop substantially unless multiple injections are carried out on isolated zones. The problem may be more severe in openhole – with the fracture quickly deviating from where it discretely intersects the wellbore and possibly minimizing contact length along the well. Considering only the connected perforations (fracture(s) physically in contact with the wellbore), the pressure drop can be considered as:

$$
\Delta p\_{\rm perf} = \frac{888 L\_g \mu B}{k\_p A\_p N\_c} q\_{f, \text{total}} \tag{17}
$$

where:

Saleh and Stewart, 1996, [26] elaborated on the Karakas and Tariq [23] considerations for pressure loss and added an additional complexity that is relevant for geothermal as well as shale gas/oil production – a second phase. The conventional production pressure drop allocations are represented by van Everdingen and Hurst's skin and Hawkins's representation:

<sup>2</sup> 1 ln *ss s*

<sup>D</sup> æ ö æ ö = = - ç ÷ ç ÷

*p kh k r <sup>s</sup> Qk r*

p

Δps incremental pressure drop occurring in the

where:

k virgin permeability

h reservoir thickness Q volumetric flow rate

rw wellbore radius rs damaged radius

ks damaged permeability

380 Effective and Sustainable Hydraulic Fracturing

μ dynamic viscosity of flowing fluid

m

aperture or reduced relative permeability or twisting of the fracture, etc.

orientation, number of joint sets, shot density, and perforation length."

*w*

(16)

ç ÷ è ø è ø

wellbore region due to a changed permeability – this could be considered to be reduced

Before low permeability (shale gas and shale liquids) was popular, Tariq et al., 1989, [27] considered production from naturally fractured reservoirs in low matrix permeability environments. "The sharp discontinuities in porosity and permeability created by fractures have a significant impact on the overall fluid flow in the reservoir. Fractures allow rapid conduction of fluids with very little pressure drop because their resistance to fluid flow is much lower than that of the matrix rock. Very high flow rates (30,000 to 50,000 B/D … have been obtained from fractured reservoir wells under a limited pressure drop." They further stated: "In the past, many naturally fractured reservoirs were completed openhole (barefoot). Perforated completions have now become more popular for naturally fractured reservoirs as a result of improvements in drilling technology and in fracture detection techniques. The concern in the perforated completion, however, is the small area open to flow. The productivity of a perforated completion in naturally fractured reservoirs is totally dependent on the hydraulic communication between the perforations and the fracture network. This commu‐ nication, in turn, is dependent on such factors as fracture interval (or fracture density), fracture

Part of the concern near the wellbore is the choking type of skin [3, 28, 29, 30], where losses due to damage and convergence are isolated in the plane (or within the confines) of a fracture. The effects can be further normalized by looking at various forms of the productivity ratio or Δpperf pressure drop through the perforations, psi

Lg length of the propped perforation tunnel, ft

μ dynamic viscosity, cP

B water formation volume factor, res bbl/STbbl

qf,total bottomhole total flow rate, BLPD

kp absolute permeability of packed perforation, md

Ap cross-sectional area (nominal) of individual perforation, ft2

Nc number of connected perforations

This relationship came from Welling, 1998, [36] simply calculating the pressure drop for liquid flow through a sand-packed perforation, as follows:

$$\begin{aligned} P\_{w\text{\#s}} - P\_{w\text{\#}} &= cQ + dQ^2\\ c = 888 \frac{L\_{gr} \mu B\_w}{k\_{gr} AN} \text{ and } d = 9.08 \times 10^{-13} \frac{\beta\_{gr} L\_{gr} \rho B\_w^2}{A^2 N^2} \text{ and } \beta\_{gr \text{-}oil} &= \frac{6.5 \times 10^4}{k^{0.996}} \end{aligned} \tag{18}$$

perforations. Jackson and Rai [41] suggested, where the subscripts 1 and 2 indicate some fine

(20)

383

http://dx.doi.org/10.5772/56211

*r r*

"The near-perforation damage influences the positive y-intercept significantly more than poor fracture conductivity for the entire length of the fracture." This finding implies how important

Recently, there have been numerous publications to determine near-wellbore skin from productive fractures. Many of these have been diagnostic methods. That in itself is valuable by providing a method for discerning how large these pressure drops in the near-wellbore region can be. Nobakht and Mattar, 2012, described a method for correcting for the apparent skin effect that has been attributed to flow convergence in a horizontal well and/or finite conductivity of the fractures – as well as a number of other mechanisms such as two phase flow. Inappropriate consideration of skin can cause linear flow with skin to appear as transient radial flow with boundaries. This can be overcome by using a square root of time plot (the

'


Do Perforated Completions Have Value for Engineered Geothermal Systems


*<sup>f</sup> kp p*

ln

m


2

p

*q*

time can be a superposed square root) and taking a linear relationship as:

*q*

*m*

drainage from the matrix). They incorporated a convergence skin, sc.

referred to as a modified normalized pressure):

flow, non-Darcy flow and fracture face skin.

*i wf p p mt b*

*i wf*

*p p p b mt q*

This can also be expressed for a gas by using pseudopressure. It can be rewritten as (pm is

'

Most reservoir simulations don't discriminate the near-wellbore specifics in any sort of detail (Xie et al., 2012 [43]). Clarkson et al., 2012, [44] do describe dynamic skin effects. They use a time-dependent intercept, b'(t), to give a time-dependent s'(t). This dynamic skin was associ‐ ated with depletion- and fluid-damage-related fracture conductivity changes, convergent

Bello and Wattenbarger, 2010, [45] pointed out that in many multiply-hydraulically-fractured horizontal wells, skin is observed in a characteristic fourth production region (transient

near wellbore distances:

the near wellbore regime is.

It is not really relevant because this analysis assumes that the fracture perforations are not infilled with any material. However, propped fractures may require consideration of this additional skin. It is known that near-wellbore fracture geometry is likely more complicated and that more than orifice frictional losses or packed perforation losses are involved. Fracture width reduction near the wellbore can result from twisting and turning with associated shear or partial departures from the direction of perforating, through a microannulus and then in the direction of the maximum normal stress. Cherny et al., 2009, [37] considered (in twodimensions) the consequences of micro-annular losses. A relevant question is how they are represented during production. It might be anticipated that pressure drops are even larger because of the different sign of the pressure gradient from the wellbore into the fracture during injection – as opposed to during production. Fallahzadeh and Rasouli, 2012, [38] considered some aspects of stress conditions around cased and cemented wellbores impacting perforation performance. Other references relevant to pressure losses during hydraulic fracturing include Ceccarelli et., 2010, [39] as well as Fallahzadeh et al., 2010 [40].

Jackson and Rai, 2012, [41] have come closest to proposing methodologies for discriminating various types of apparent skin in shale gas plays – including low conductivity fractures (manifested by a ¼ slope), poor connection to the wellbore (choke skin and near fracture face damage), relative permeability effects, fracture skin and casing connections; using the apparent skin intercept concept. They reiterated concepts for poorly connected factors strictly using standard choking analogs [see for example Cinco-Ley and Samaniego, 1981 [29]. This certainly has an effect but also needs to consider tortuosity and perforation damage. The choking skin, sch, has been considered to be:

$$s\_{ch} = \frac{\pi \chi\_s k}{w\_s k\_{fs}}\tag{19}$$

where:

xs damage length, ft

ws width of the fracture over the damaged length, ft

k fracture permeability, md

#### kfs damaged permeability in the fracture, md

The convergent skin macroscopically exists for a fracture that is transverse to the wellbore. It is further increased by convergent flow of some additional complexity into individual perforations. Jackson and Rai [41] suggested, where the subscripts 1 and 2 indicate some fine near wellbore distances:

2

Ceccarelli et., 2010, [39] as well as Fallahzadeh et al., 2010 [40].

*wfs wf*

382 Effective and Sustainable Hydraulic Fracturing

sch, has been considered to be:

where:

xs damage length, ft

k fracture permeability, md

*gr*

*P P cQ dQ*


m

13

*gr w gr gr w*

*L B L B c and d and*

6.5 10 <sup>888</sup> 9.08 10

b

It is not really relevant because this analysis assumes that the fracture perforations are not infilled with any material. However, propped fractures may require consideration of this additional skin. It is known that near-wellbore fracture geometry is likely more complicated and that more than orifice frictional losses or packed perforation losses are involved. Fracture width reduction near the wellbore can result from twisting and turning with associated shear or partial departures from the direction of perforating, through a microannulus and then in the direction of the maximum normal stress. Cherny et al., 2009, [37] considered (in twodimensions) the consequences of micro-annular losses. A relevant question is how they are represented during production. It might be anticipated that pressure drops are even larger because of the different sign of the pressure gradient from the wellbore into the fracture during injection – as opposed to during production. Fallahzadeh and Rasouli, 2012, [38] considered some aspects of stress conditions around cased and cemented wellbores impacting perforation performance. Other references relevant to pressure losses during hydraulic fracturing include

Jackson and Rai, 2012, [41] have come closest to proposing methodologies for discriminating various types of apparent skin in shale gas plays – including low conductivity fractures (manifested by a ¼ slope), poor connection to the wellbore (choke skin and near fracture face damage), relative permeability effects, fracture skin and casing connections; using the apparent skin intercept concept. They reiterated concepts for poorly connected factors strictly using standard choking analogs [see for example Cinco-Ley and Samaniego, 1981 [29]. This certainly has an effect but also needs to consider tortuosity and perforation damage. The choking skin,

*s*

*s fs x k*

The convergent skin macroscopically exists for a fracture that is transverse to the wellbore. It is further increased by convergent flow of some additional complexity into individual

*w k* p

*ch*

*s*

ws width of the fracture over the damaged length, ft

kfs damaged permeability in the fracture, md

 r

<sup>=</sup> <sup>=</sup> = (18)


*k AN A N k*

2 4

= (19)

2 2 0.996

b

*gr oil*


$$q = -\frac{2\pi k\_f \left(p\_2 - p\_1\right)}{\mu \ln\left(\frac{r\_2}{r\_1}\right)}\tag{20}$$

"The near-perforation damage influences the positive y-intercept significantly more than poor fracture conductivity for the entire length of the fracture." This finding implies how important the near wellbore regime is.

Recently, there have been numerous publications to determine near-wellbore skin from productive fractures. Many of these have been diagnostic methods. That in itself is valuable by providing a method for discerning how large these pressure drops in the near-wellbore region can be. Nobakht and Mattar, 2012, described a method for correcting for the apparent skin effect that has been attributed to flow convergence in a horizontal well and/or finite conductivity of the fractures – as well as a number of other mechanisms such as two phase flow. Inappropriate consideration of skin can cause linear flow with skin to appear as transient radial flow with boundaries. This can be overcome by using a square root of time plot (the time can be a superposed square root) and taking a linear relationship as:

$$\frac{p\_i - p\_{wf}}{q} = m\sqrt{t} + b^\circ \tag{21}$$

This can also be expressed for a gas by using pseudopressure. It can be rewritten as (pm is referred to as a modified normalized pressure):

$$p\_m = \frac{p\_i - p\_{w\circ}}{q} - b' = m\sqrt{t} \tag{22}$$

Most reservoir simulations don't discriminate the near-wellbore specifics in any sort of detail (Xie et al., 2012 [43]). Clarkson et al., 2012, [44] do describe dynamic skin effects. They use a time-dependent intercept, b'(t), to give a time-dependent s'(t). This dynamic skin was associ‐ ated with depletion- and fluid-damage-related fracture conductivity changes, convergent flow, non-Darcy flow and fracture face skin.

Bello and Wattenbarger, 2010, [45] pointed out that in many multiply-hydraulically-fractured horizontal wells, skin is observed in a characteristic fourth production region (transient drainage from the matrix). They incorporated a convergence skin, sc.

$$s\_c = \frac{k\_f x\_e \left[ m(p\_i) - m(p\_{w\uparrow}) \right]}{1422 q\_g T} = -\ln\left[ \frac{\pi r\_w}{h} \left( 1 + \sqrt{\frac{k\_V}{k\_H}} \right) \sin\left(\frac{\pi d\_z}{h}\right) \right] \tag{23}$$

t time, days

where:

T reservoir temperature, °R

k absolute permeability, md

s' apparent skin, dimensionless

b intercept of square root time plot, psi2

, John McLennan1

h net reservoir thickness, feet

[52] and Li et al., 2012 [53].

**Author details**

Walter Glauser1

**References**

(2012). , 28-29.

Int4 intercept of field data on [m(pi-m(pwf)/qg vs. √t, psi<sup>2</sup>

/cP/MscfD

Do Perforated Completions Have Value for Engineered Geothermal Systems

http://dx.doi.org/10.5772/56211

385

*<sup>T</sup>* <sup>=</sup> (25)

The message is that there are near-wellbore skins that have been diagnosed using pressure transient analyses on production data from tight formations. Near-wellbore pressure losses could be a dominant mechanism. Anderson et al., 2010, [50] recognized a significant skin effect from pressure loss due to finite conductivity in the fracture system, even if there is no me‐ chanical skin damage at the wellbore. If a square root of time plot is used, the apparent skin

(gas) can be inferred from a y-intercept, b, that represents a constant pressure loss.

1417 *kh s b*

/cP/MscfD

and Ian Walton2

1 Department of Chemical Engineering, U. of Utah, Salt Lake City, USA

2 Energy & Geoscience Institute, U. of Utah, Salt Lake City, USA

Other similar references include Bahrmai et al., 2011 [50], Sun et al., 2011 [51], Byrne et al., 2011

[1] Pritchett, J. The role of reservoir stimulation in the future of the geothermal power industry. In: Reservoir stimulation current understanding and practice, and the path forward, GRC 2012 Annual Pre-Meeting Workshop, September 2012, Reno, Nevada;

'

where:

kf bulk fracture permeability for dual porosity model md

kV vertical permeability

kH horizontal permeability

h net reservoir thickness, feet

m(p) pseudopressure – gas, psi2 /cP

pi initial reservoir pressure, psi

pwf wellbore flowing pressure, psi

qg gas rate, MscfD

rw wellbore radius, ft

dz well position in reservoir

Rationalizing the permeability ratio for fracture flow in a geothermal well makes applying this difficult. The same problem exists for using the basic Karakas and Tariq [23] relationships.

Al-Ahmadi, et al. 2010, [46] observed that while transient linear flow is common in tight gas reservoirs, in shale gas wells, it is accompanied by a significant skin effect – not commonly seen in tight gas wells. They accounted for this with a modified linear flow relationship. For early time, Bello, 2009, [47] and Bello and Wattenbarger, 2009 [48], 2010 [45, 49] treated this as a constant skin effect. For a shale gas reservoir, they indicated:

$$\frac{\left|m\left(p\_i\right) - m\left(p\_{w\circ}\right)\right|}{q\_{\mathcal{g}}} = \tilde{m}\_4 \sqrt{t} + \frac{Int\_4}{1 + \frac{0.45\tilde{m}\_4\sqrt{t}}{Int\_4}}\tag{24}$$

where:

m(p) pseudopressure – gas, psi2 /cP

pi initial reservoir pressure, psi

pwf wellbore flowing pressure, psi

qg gas rate, MscfD

*<sup>m</sup>*˜4slope of line matching linear flow data and passing through origin on √time plot

t time, days

( ) ( ) ln 1 sin

*g H*

Rationalizing the permeability ratio for fracture flow in a geothermal well makes applying this difficult. The same problem exists for using the basic Karakas and Tariq [23] relationships.

Al-Ahmadi, et al. 2010, [46] observed that while transient linear flow is common in tight gas reservoirs, in shale gas wells, it is accompanied by a significant skin effect – not commonly seen in tight gas wells. They accounted for this with a modified linear flow relationship. For early time, Bello, 2009, [47] and Bello and Wattenbarger, 2009 [48], 2010 [45, 49] treated this as

> 4 4

% % (24)

*Int*

0.45 <sup>1</sup>

+

( ) ( ) <sup>4</sup> 4

*<sup>m</sup>*˜4slope of line matching linear flow data and passing through origin on √time plot

= +

*mp mp Int m t q m t*

*q T hk h* p

 p

(23)

*f e i wf wV z*

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

*kx mp mp rk d*

1422

bulk fracture permeability for dual porosity model md

/cP

a constant skin effect. For a shale gas reservoir, they indicated:


*i wf g*

/cP

*c*

384 Effective and Sustainable Hydraulic Fracturing

*s*

kV vertical permeability

qg gas rate, MscfD

rw wellbore radius, ft

dz well position in reservoir

m(p) pseudopressure – gas, psi2

pi initial reservoir pressure, psi

qg gas rate, MscfD

pwf wellbore flowing pressure, psi

kH horizontal permeability

h net reservoir thickness, feet

m(p) pseudopressure – gas, psi2

initial reservoir pressure, psi

pwf wellbore flowing pressure, psi

where:

kf

pi

where:

Int4 intercept of field data on [m(pi-m(pwf)/qg vs. √t, psi<sup>2</sup> /cP/MscfD

The message is that there are near-wellbore skins that have been diagnosed using pressure transient analyses on production data from tight formations. Near-wellbore pressure losses could be a dominant mechanism. Anderson et al., 2010, [50] recognized a significant skin effect from pressure loss due to finite conductivity in the fracture system, even if there is no me‐ chanical skin damage at the wellbore. If a square root of time plot is used, the apparent skin (gas) can be inferred from a y-intercept, b, that represents a constant pressure loss.

$$s' = \frac{kh}{1417T}b\tag{25}$$

where:

T reservoir temperature, °R

k absolute permeability, md

s' apparent skin, dimensionless

h net reservoir thickness, feet

b intercept of square root time plot, psi2 /cP/MscfD

Other similar references include Bahrmai et al., 2011 [50], Sun et al., 2011 [51], Byrne et al., 2011 [52] and Li et al., 2012 [53].
