**9. Modeling pressure drop due to perforations in a fractured reservoir**

To confirm these initial predictions, a more fundamental approach was implemented to predict the near-wellbore pressure drop – following Huang and Ayoub, 2007 [56]. The problem of

<sup>4</sup> Flow up a 10-inch (0.254 m) ID pipe will be turbulent at this rate (Re = 3.61 × 106).

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 perforations, into the wellbore.

$$
\Delta p\_{\text{perf}} = \Delta p\_{\text{sc}} + \Delta p\_f + \Delta p\_{\text{se}} \tag{8}
$$

where:

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.

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

The next logical step is to consider turbulent flow. For either completion case, flow up the well‐

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.

**9. Modeling pressure drop due to perforations in a fractured reservoir**

4 Flow up a 10-inch (0.254 m) ID pipe will be turbulent at this rate (Re = 3.61 × 106).

To confirm these initial predictions, a more fundamental approach was implemented to predict the near-wellbore pressure drop – following Huang and Ayoub, 2007 [56]. The problem of

The openhole scenario suggests that a contin‐

Order of magnitude calculations for those effects are presented later.

proposed to be small – calculations demonstrating this will follow.

bore will likely be turbulent above the reservoir.4

370 Effective and Sustainable Hydraulic Fracturing

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

Δp<sup>f</sup> pressure drop due to friction in the perforation tunnel

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

**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

#### **Perforation Orifice Losses**

"Since the width reduction [due to turning] takes place only on a small portion of the fracture, the friction loss, i.e., the incremental pressure above the pressure for a straight fracture, is small unless the perforation angle is off the preferred direction by a large angle." The situation during production is more complicated since the width will be smaller than during fracturing because

Of more interest is twisting. When the fracture half-length becomes large enough, a twisting component can result as the fracture evolves to realign with the far-field stresses. For the

> s

α azimuth angle from the horizontal minimum stress direction (well is projected into a

This local stress increase is superimposed on local stress concentrations and causes a nearwellbore width reduction during injection or production. As can be seen in Figure 7 in Weng's [12] paper, most of this loss can be eliminated by appropriate drilling direction, to keep α' small. Turning then does not appear to be too substantial of a pressure loss mechanism.

The twisting component is more significant. Additional frictional losses may result from the specific connectivity of starter fractures near the wellbore; specifically, how do fractures that initiate at an angle to the wellbore (normal to the smallest local principal stress) propagate, twist and align (or not) to direct injection fluid to or to collect production fluid from a more dominant master fracture. For simplicity, assume one dominant master fracture surviving a short distance radially from the wellbore. Fractures from the perforations reorient and/or link to connect with this main fracture. The multiply fractured region connects the master fracture to the perforations and the wellbore. If the wellbore orientation falls outside of a specified range, these starter fractures do not link up (they grow independently until fracture friction causes only one to survive. Production will be through discrete fractures. An extensive, multiply fractured zone was proposed by Weng [12] as being capable of causing a significant frictional zone. This becomes quite a complicated consideration during production – are multiple, nonlinked fractures at the wellbore an impediment (friction, propensity for width

With twisting and interacting fractures, only approximate calculations are attempted – just to assess the relative order of magnitude of the frictional loss that might be anticipated. Adapting Weng's equation 18 [12] for flow through a multiplicity of near-wellbore fractures for a

 a=+ - (9)

Do Perforated Completions Have Value for Engineered Geothermal Systems

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373

( ) <sup>2</sup> sin ' *c HMIN HMAX HMIN*

twisting component, Weng incorporated a local increase in the closure stress:

 s

of the direction of the pressure gradient.

where:

α' 90° - α

horizontal plane)

ss

reduction) or an advantage (wider pressurized fractures)?

Newtonian fluid (parallel plate flow):

**Figure 2.** Total pressure loss through perforations for 60,000 BWPD (0.11 m3/sec). The legend shows the nominal per‐ foration diameter through the casing and the number of shots per meter connecting with the fracture. The abscissa is the actual contact length of the fracture along the wellbore. For practical lengths and open perforations, this type of loss is inconsequential.

estimated using the formulae for an orifice shown previously. The actual value for the discharge coefficient is uncertain because conventionally some of that is embodied in the entrance and exit losses being calculated separately. To be conservative, it will be calculated in the same fashion as above, shown by Crump and Conway [8] in 1988. For the generic situation chosen, a pressure drop of 3.48 kPa, ~0.5 psi is estimated. Finally, for the sudden expansion into the wellbore, the minor loss, Δpse, is estimated as 2.73 kPa (0.4 psi). The sum is again a remarkably small loss. Figure 2 demonstrates this.

While more complex models may need to be developed to account for tortuosity and packing perforations, this simple model suggests that the pressure drop difference between perforated completions and openhole will be small. For practical lengths and open perforations, this type of loss is inconsequential. However, other losses have to be considered ….

From experience in fracturing, there is evidence for near wellbore flow impedance. Simple calculations suggest that it is not strictly due to the perforations themselves. It seems that the real issue then remains choking skin associated with near-wellbore fracture turning and twisting and interlinking. The best discussion of this is Weng, 1993, [12] who proposed approximations of frictional losses during injection. If it is assumed that reciprocal losses might be approximated during production, some gross approximations are possible. First, fracture turning5 is probably not a significant issue. Weng [12], in discussing fracture turning states

<sup>5</sup> Weng [12]delineated the turning stage as being related to fracture growth with tip rotation in a plane collinear with the wellbore.

"Since the width reduction [due to turning] takes place only on a small portion of the fracture, the friction loss, i.e., the incremental pressure above the pressure for a straight fracture, is small unless the perforation angle is off the preferred direction by a large angle." The situation during production is more complicated since the width will be smaller than during fracturing because of the direction of the pressure gradient.

Of more interest is twisting. When the fracture half-length becomes large enough, a twisting component can result as the fracture evolves to realign with the far-field stresses. For the twisting component, Weng incorporated a local increase in the closure stress:

$$
\sigma\_c = \sigma\_{\rm HMDN} + \left(\sigma\_{\rm HMAX} - \sigma\_{\rm HMDN}\right) \sin^2 \alpha' \tag{9}
$$

where:

α azimuth angle from the horizontal minimum stress direction (well is projected into a horizontal plane)

α' 90° - α

estimated using the formulae for an orifice shown previously. The actual value for the discharge coefficient is uncertain because conventionally some of that is embodied in the entrance and exit losses being calculated separately. To be conservative, it will be calculated in the same fashion as above, shown by Crump and Conway [8] in 1988. For the generic situation chosen, a pressure drop of 3.48 kPa, ~0.5 psi is estimated. Finally, for the sudden expansion into the wellbore, the minor loss, Δpse, is estimated as 2.73 kPa (0.4 psi). The sum is

**Figure 2.** Total pressure loss through perforations for 60,000 BWPD (0.11 m3/sec). The legend shows the nominal per‐ foration diameter through the casing and the number of shots per meter connecting with the fracture. The abscissa is the actual contact length of the fracture along the wellbore. For practical lengths and open perforations, this type of

1 10 100 1000

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

**Perforation Orifice Losses**

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

While more complex models may need to be developed to account for tortuosity and packing perforations, this simple model suggests that the pressure drop difference between perforated completions and openhole will be small. For practical lengths and open perforations, this type

From experience in fracturing, there is evidence for near wellbore flow impedance. Simple calculations suggest that it is not strictly due to the perforations themselves. It seems that the real issue then remains choking skin associated with near-wellbore fracture turning and twisting and interlinking. The best discussion of this is Weng, 1993, [12] who proposed approximations of frictional losses during injection. If it is assumed that reciprocal losses might be approximated during production, some gross approximations are possible. First, fracture

5 Weng [12]delineated the turning stage as being related to fracture growth with tip rotation in a plane collinear with the

is probably not a significant issue. Weng [12], in discussing fracture turning states

again a remarkably small loss. Figure 2 demonstrates this.

0.0001

0.001

0.01

0.1

**Pressure Loss Through Perforation (kPa)**

loss is inconsequential.

1

10

100

1000

372 Effective and Sustainable Hydraulic Fracturing

turning5

wellbore.

of loss is inconsequential. However, other losses have to be considered ….

This local stress increase is superimposed on local stress concentrations and causes a nearwellbore width reduction during injection or production. As can be seen in Figure 7 in Weng's [12] paper, most of this loss can be eliminated by appropriate drilling direction, to keep α' small. Turning then does not appear to be too substantial of a pressure loss mechanism.

The twisting component is more significant. Additional frictional losses may result from the specific connectivity of starter fractures near the wellbore; specifically, how do fractures that initiate at an angle to the wellbore (normal to the smallest local principal stress) propagate, twist and align (or not) to direct injection fluid to or to collect production fluid from a more dominant master fracture. For simplicity, assume one dominant master fracture surviving a short distance radially from the wellbore. Fractures from the perforations reorient and/or link to connect with this main fracture. The multiply fractured region connects the master fracture to the perforations and the wellbore. If the wellbore orientation falls outside of a specified range, these starter fractures do not link up (they grow independently until fracture friction causes only one to survive. Production will be through discrete fractures. An extensive, multiply fractured zone was proposed by Weng [12] as being capable of causing a significant frictional zone. This becomes quite a complicated consideration during production – are multiple, nonlinked fractures at the wellbore an impediment (friction, propensity for width reduction) or an advantage (wider pressurized fractures)?

With twisting and interacting fractures, only approximate calculations are attempted – just to assess the relative order of magnitude of the frictional loss that might be anticipated. Adapting Weng's equation 18 [12] for flow through a multiplicity of near-wellbore fractures for a Newtonian fluid (parallel plate flow):

$$
\Delta p\_{mf} = \frac{12\,\mu q}{h} \frac{L\_m}{\overline{w}^3} \tag{10}
$$

openhole requirements) are estimated. This depends strongly on the number of effective perforations. The number of effective perforations is strictly governed by gun characteristics, especially phasing and density as well as in-situ conditions. It appears that for typical perfo‐ ration diameters the casing hole perforation diameter is a secondary parameter – unless it

Using the same generic reservoir, the power requirements for lifting and to overcome the estimated pressure losses in the perforations and the multiply fractured region are shown in

**Completion Losses**

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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

1 10 100 1000

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

**Figure 3.** Total pressure loss through perforations and near-wellbore region for 60,000 BWPD (0.11 m3/sec) – Comple‐ tion Losses. The legend shows the nominal perforation diameter through the casing and the number of shots per me‐ ter connecting with the fracture. The abscissa is the actual contact length of the fracture along the wellbore. For

This might be considered in terms of incremental cost. Figure 5 shows that with enough connectivity these costs could be manageable. An overall economics evaluation would be

7 Sometimes referred to as a secondary minimum principal stress. This strictly indicates the minimum principal stress at

hreservoir nominal depth to midpoint of producing fracture(s) -- 2,000 m TVD

length fracture communicates with wellbore -- 10, 20, 50, 100, 500 m

becomes extremely small.

hf

Figure 4, using additional assumptions shown below.

η pump efficiency (dimensionless) … 0.50 was used

hset setting depth for pump -- 500 m TVD

0.1

practical lengths and open perforations, this type of loss is inconsequential.

the borehole wall, not necessarily aligned with the far-field minimum principal stress.

1

10

100

**Pressure Loss Through Completion (kPa)**

required.

1000

10000

where:

Δpmf pressure loss for flow through the multiply fractured region

h unit height

m dynamic viscosity

q volumetric flow rate in each fracture per unit height

Lm half-length of multiply fractured zone *<sup>w</sup>*¯average aperture for each connecting fracture

Apply this relationship to estimate the pressure drop. On a meter by meter basis it is possible to assume that the fracture width would be the width of an openhole fracture (assume 2 mm constant along the wellbore length) divided by the number of effective perforations;<sup>6</sup> similarly for the flow rate per fracture. For 60° phasing and 6 spf, one can envision something over six effective perforations per meter, giving an effective width of 0.33 mm (as opposed to 2 mm for an openhole bi-winged fracture). Assuming a velocity of 2.5 m/s through each perforation, the required perforated wellbore length (net perforated length) would be 160 m (see earlier) and the total inflow per meter would need to be 6.9 x 10-4 m3 /s/m. The greater the half-length of the zone of multiple fractures (distance away from wellbore), Lm, the greater the pressure losses. Assume 5 m (Weng [12] found a typical transition at about 15 ft. in some of his calculations), the pressure loss can be estimated as 27 kPa (4 psi). The key variables are of course the number of effective perforations per unit length (phasing and spm) and the length over which linkage would occur. The presumption is that the length required for linkage can be reduced when the stimulation is carried out by breaking down all perforations, drilling at acceptable angles, and initiating at low rate. The troublesome aspect of this logic here is that more perforations give a higher pressure drop – so the ideal situation would be to use fewer, which is counter-intuitive and would increase the perforation tunnel losses. Figure 3 shows order of magnitude pressure losses for first order approximations of losses through the perforation tunnels themselves (see Figure 2) and from the friction estimated in the multiply fractured region.
