**8. Relative order of magnitude calculations — Pressure losses**

A comparative analysis is done to assess the magnitude of perforation friction losses. Tor‐ tuosity associated with shear fractures intersecting the wellbore can be included using methods proposed by Weng, 1993, [12] and Haney et al., 1995, [57] with the assumption that they work for production in the same fashion that they do for hydraulic fracturing. The simplest possible representation of near-wellbore pressure loss, ignoring tortuosity, is that the pressure drop is strictly due to orifice losses using calculations that can be readily inferred from Bernoulli's equation along with a discharge coefficient, Cd. Consider a hypothetical case:

TD to reservoir top 2000 m Reservoir net thickness 500 m Net to gross 1 Perforated reservoir thickness up to 500 m Inclination 0° Average reservoir temperature 200°C required mass flow rate per well 100 kg/sec Required volumetric flow rate 62,000± BWPD (sand face) Average bottomhole fluid density1 , 877.6 kg/m3 Average viscosity 0.14 cP

Perforation casing diameter.0127,.0254 m

Perforation phasing 60°2

Formation pressure gradient 10.18 kPa/m or 0.45 psi/ft

Wellbore inclination vertical

Hydraulic fracture width 2 mm

Drilled hole diameter in reservoir 0.254 m, 10 inch

$$\begin{aligned} \Delta p\_{\text{perfavour}} &= \frac{1}{2} \frac{16}{\pi^2} \frac{\text{Q}^2 \rho}{N\_p^2 d^4 \text{C}\_d^2} \quad \text{C}\_d = \left( 1 - e^{-\frac{2.2D}{\mu^{0.1}}} \right)^{0.4} \\\Delta p\_{\text{perfours}} &= \frac{8}{\pi^2} \frac{\left( \pi \frac{d^2}{4} \right)^2 v^2 \rho}{N\_p^2 d^4 \text{C}\_d^2} = \frac{v^2 \rho}{2 N\_p^2 \text{C}\_d^2} \end{aligned} \tag{4}$$

To determine conditions for laminar flow, Re ≤ 2,000,for an openhole longitudinal fracture, hydraulic radius was used to approximate the fracture as a slot-like wellbore intersection:

*HR V hw HR*

The maximum velocity of fluid through the fracture will be based on one half of the total flow rate presuming a symmetrical, bi-winged fracture, and the cross sectional area of the fracture

To maintain laminar flow, the fracture contact length, h, along the wellbore can be estimated as:

4

The fracture height would need to be about 346 m (with connection along the wellbore) to avoid going into turbulent flow for an openhole completion. While not strictly speaking impossible in a 500 m reservoir, it may prove difficult to accommodate this much longitudinal wellbore contact openhole. Additionally, this gives us a maximum velocity under 0.08 m/s for

To assess the flow regime for production of a geothermal reservoir through a cased, cemented and perforated completion, first consider the concept of effective perforations. Assume that only perforations within less than 30° of the preferred fracture plane are expected to signifi‐ cantly contribute to fracturing/flow (Behrmann and Elbel, 1991 [13]). It may be assumed that these same perforations will also contribute the vast majority of production flow, due to their connectivity to the fracture(s). For simplicity, the model developed here assumes that effective

*kg m <sup>Q</sup> <sup>m</sup> <sup>s</sup> h w m m*

3

*Pa s*

( ) Re *<sup>Q</sup> h w* r

m

3

Re 2000 1.4 10


877.6 0.1104

 = -= - = ×

r

m

<sup>4</sup> Re

w nominal and vertically constant fracture width at wellbore

on that side. After some rudimentary manipulation it is found that:

h fracture height intersecting the vertical wellbore

V fluid velocity in the fracture at the wellbore

r

m

where:

HR hydraulic radius

ρ fluid density

laminar flow.

μ dynamic viscosity

( )

*h w*

= = <sup>+</sup> (5)

Do Perforated Completions Have Value for Engineered Geothermal Systems

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

369

<sup>=</sup> <sup>+</sup> (6)

(7)

0.002 346

2

For economic viability, it has been suggested that a minimum of 100 kg/s of water needs to be generated (at this reservoir temperature).3 This is 0.114 m3/s (~60,000 BWPD) downhole. Friction in the tubulars is considered to be second order at this point – and similar for cased and openhole completions.

It has been argued that the casing, cement and perforating a geothermal well would result in poor production, due to restrictive flow. There is also concern that the pressure drop associated with the perforations would be much too large or severe. To investigate the nature of orifice losses near the wellbore, a simple theoretical model was developed to compare pressure drop between an openhole and cased/perforated wellbores. To start the assessment, the flow regime was assessed to confirm that both completion types are in turbulent flow.

The generic data above were used, assuming a vertical wellbore with a longitudinal fracture 2 mm wide for both completion methods (openhole or cased, cemented, and perforated). Later the consequences of multiple interconnected fractures associated with fracturing through perforations are considered. Although it could be argued that fracture width close to the wellbore would likely be significantly greater for a cased hole, the analysis is too sensitive to width to modify it arbitrarily.

<sup>1</sup> The generic geothermal fluid properties were based on the composition of water from Roosevelt Hot Springs, Utah (Capuano and Cole, 1981), which is a 1% NaCl solution.Ershaghi et al., 1983, [24] measured viscosity for a 1% NaCl brine at 200° C to be 0.139 cP, which is almost exactly the same value given by the NIST thermodynamic tables for pure water under downhole pressure.In addition, the density of water is 878.31 kg/m3 at this pressure.

<sup>2</sup> For 60° phasing and 6 shots per foot, there would be 2 perforations at 180° most likely to communicate – other shots may have a complicated interconnectivity, but this assumption is conservative.

<sup>3</sup> J. Moore, personal communication, 2012.

To determine conditions for laminar flow, Re ≤ 2,000,for an openhole longitudinal fracture, hydraulic radius was used to approximate the fracture as a slot-like wellbore intersection:

$$\text{Re} = \frac{4HR\rho V}{\mu} \quad HR = \frac{hw}{2\left(h + w\right)}\tag{5}$$

where:

Perforation casing diameter.0127,.0254 m

Formation pressure gradient 10.18 kPa/m or 0.45 psi/ft

Drilled hole diameter in reservoir 0.254 m, 10 inch

*perforations*

generated (at this reservoir temperature).3

and openhole completions.

width to modify it arbitrarily.

3 J. Moore, personal communication, 2012.

0.1

m

æ ö -

è ø

*D*

This is 0.114 m3/s (~60,000 BWPD) downhole.

(4)

0.4 2.2 <sup>2</sup>

2

r

*v*

2

*p d pd*

2 24 2

*<sup>Q</sup> <sup>p</sup> C e NdC*

*d*

æ ö ç ÷ ç ÷

8 4

was assessed to confirm that both completion types are in turbulent flow.

under downhole pressure.In addition, the density of water is 878.31 kg/m3 at this pressure.

may have a complicated interconnectivity, but this assumption is conservative.

p

*<sup>p</sup> NdC NC*

2

p

è ø D= =

*perforations d*

p

<sup>2</sup> <sup>2</sup> 2

*p d*

ç ÷ D = =- ç ÷

1 16 <sup>1</sup>

r

2 24 2 2 2

For economic viability, it has been suggested that a minimum of 100 kg/s of water needs to be

Friction in the tubulars is considered to be second order at this point – and similar for cased

It has been argued that the casing, cement and perforating a geothermal well would result in poor production, due to restrictive flow. There is also concern that the pressure drop associated with the perforations would be much too large or severe. To investigate the nature of orifice losses near the wellbore, a simple theoretical model was developed to compare pressure drop between an openhole and cased/perforated wellbores. To start the assessment, the flow regime

The generic data above were used, assuming a vertical wellbore with a longitudinal fracture 2 mm wide for both completion methods (openhole or cased, cemented, and perforated). Later the consequences of multiple interconnected fractures associated with fracturing through perforations are considered. Although it could be argued that fracture width close to the wellbore would likely be significantly greater for a cased hole, the analysis is too sensitive to

1 The generic geothermal fluid properties were based on the composition of water from Roosevelt Hot Springs, Utah (Capuano and Cole, 1981), which is a 1% NaCl solution.Ershaghi et al., 1983, [24] measured viscosity for a 1% NaCl brine at 200° C to be 0.139 cP, which is almost exactly the same value given by the NIST thermodynamic tables for pure water

2 For 60° phasing and 6 shots per foot, there would be 2 perforations at 180° most likely to communicate – other shots

 r

*v*

Perforation phasing 60°2

Wellbore inclination vertical

368 Effective and Sustainable Hydraulic Fracturing

Hydraulic fracture width 2 mm

HR hydraulic radius

h fracture height intersecting the vertical wellbore

w nominal and vertically constant fracture width at wellbore

ρ fluid density

V fluid velocity in the fracture at the wellbore

μ dynamic viscosity

The maximum velocity of fluid through the fracture will be based on one half of the total flow rate presuming a symmetrical, bi-winged fracture, and the cross sectional area of the fracture on that side. After some rudimentary manipulation it is found that:

$$\text{Re} = \frac{\rho Q}{\mu \left(h + w\right)}\tag{6}$$

To maintain laminar flow, the fracture contact length, h, along the wellbore can be estimated as:

$$h = \frac{\rho Q}{\mu \text{Re}} - w = \frac{877.6 \frac{\text{kg}}{\text{m}^3} \times 0.1104 \frac{\text{m}^3}{\text{s}}}{2000 \times 1.4 \times 10^{-4} \text{ Pa} \cdot \text{s}} - 0.002 \text{ m} = 346 \text{ m} \tag{7}$$

The fracture height would need to be about 346 m (with connection along the wellbore) to avoid going into turbulent flow for an openhole completion. While not strictly speaking impossible in a 500 m reservoir, it may prove difficult to accommodate this much longitudinal wellbore contact openhole. Additionally, this gives us a maximum velocity under 0.08 m/s for laminar flow.

To assess the flow regime for production of a geothermal reservoir through a cased, cemented and perforated completion, first consider the concept of effective perforations. Assume that only perforations within less than 30° of the preferred fracture plane are expected to signifi‐ cantly contribute to fracturing/flow (Behrmann and Elbel, 1991 [13]). It may be assumed that these same perforations will also contribute the vast majority of production flow, due to their connectivity to the fracture(s). For simplicity, the model developed here assumes that effective perforations are fully open for flow, and are perfectly aligned with the preferred fracture plane. The model therefore fails to account for tortuosity or proppant packing in order to compare the best case scenario for using perforations with that of an openhole fractured completion. Order of magnitude calculations for those effects are presented later.

pressure drop can be broken up into three parts: pressure drop from the contraction between the fracture and the perforation, pressure drop from friction within the perforation itself and pressure drop from the expansion between the perforation to the wellbore. These are shown schematically in Figure 1, representing flow regimes going from a fracture, through the

Δpsc pressure drop from sudden contraction/expansion between fracture and perforation

Δpse pressure drop due to sudden expansion between perforation tunnel and wellbore

pressure drop due to friction in the perforation tunnel

**sc**

**Figure 1.** Proposed flow regime near the wellbore through effective perforations.

For 12 spf (39.37 spm) at 60° phasing, parallel effective perforations are spaced 15.24 cm apart. For this system, transition from the fracture "cells" to the perforation is equivalent to going from a 15.24 cm x 0.20 cm slit to a 1.27 cm pipe. One could therefore account for the first phase of pressure loss as the result of sudden contraction. Using conventional fluid mechanics principles, the minor loss in a sudden contraction similar to this are again estimated to be negligible (0.92 kPa, 0.13 psi). The friction loss through each perforation tunnel, Δp<sup>f</sup>

, is

**f**

**se**

*perf sc f se* D =D +D +D *p p pp* (8)

Do Perforated Completions Have Value for Engineered Geothermal Systems

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

371

perforations, into the wellbore.

where:

tunnel

Δp<sup>f</sup>

Maximum laminar velocity was determined through perforations with 60° phasing, 12 spf (39.37 spm) and 1.27 cm casing-hole diameters. This phasing guarantees that 1/3 of the perforations will be within 30° or less of the preferred fracture plane. While a more detailed study would need to take place to determine the pressure drop due to misalignment of the perforations (reorientation and/or microannulus flow), for now it is assumed that the best oriented perforations are perfectly aligned with the fractures. As discussed earlier, it is also assumed that the aligned perforations accept virtually all flow. The maximum velocity for laminar conditions would be approximately 0.0251 m/s. The number of perforations for laminar flow with this velocity is too large to be feasible. Perforation flow would theoretically be turbulent (based on the Reynolds' number). The length over which this occurs is very small however and the losses through the perforation tunnel itself – if it is clear of debris – are proposed to be small – calculations demonstrating this will follow.

The next logical step is to consider turbulent flow. For either completion case, flow up the well‐ bore will likely be turbulent above the reservoir.4 The openhole scenario suggests that a contin‐ uous fracture height with laminar flow could fit vertically within this reservoir. While possible, it isn't probable that this will be the case (over 346 m). Alternatively the perforations will almost certainly see nonlaminar effects. Depending on the fracture height, turbulent flow will also ex‐ ist within the fracture, but may not fully develop in the short interval of the perforations.

A correlation often adopted in chemical engineering piping design (Towler and Sinnott, 2012 [55]) was used as a measuring stick for the maximum allowable fluid velocity (to avoid erosion and to carry any entrained solids; the latter is likely not relevant for geothermal applications). A common rule of thumb for sizing pipes for liquid flow is to impose a velocity of 3 m/s. Realistically, this may be extremely conservative if the flow is single phase and solids-free. Nevertheless, for purposes of illustration, this limit is adopted - the maximum velocity through the perforations was set at 2.5 m/s (this conservatively low velocity can account for situations where two-phase flow may unexpectedly occur). Compared to perforation requirements for laminar flow, using this design velocity vastly reduces the number of perforations required to make up the total flow rate (62,000 BWPD). Assuming 60° phasing, and 6 spf, the required perforated height restrict velocity to 2.5 m/s would be ~160 m. This is still a substantial number of perforations and relaxing the critical velocity restriction would seem to make sense.
