which net pressure draws fluid into the capillary?

Fracture Pressure level Analysis and Perforation Pattern

Hoss Belyadi , ... Fatemeh Belyadi , in Hydraulic Fracturing in Unconventional Reservoirs, 2022

Net Pressure level

Internet force per unit area is one of the virtually important pressures to consider in hydraulic fracturing. Cyberspace pressure is the energy required for propagating fractures and creating width during the frac chore and refers to the excess pressure over the frac pressure required to extend the fractures. Net pressure level is essentially the difference between the fracturing fluid pressure and the closure pressure and is the driving machinery behind fracture growth. The more pressure inside a fracture, the more potential there is for growth. The term net pressure is simply used when the fracture is open. If the fracture is closed, cyberspace pressure is equal to 0. Cyberspace pressure depends on various parameters such every bit Immature's modulus, fracture meridian, fluid viscosity, fluid rate, full fracture length, and tip pressure level. Net pressure is too referred to as procedure zone stress and tin exist calculated using Eq. (9.fifteen) or Eq. (9.sixteen).

Equation nine.15. Net force per unit area, equation 1.

P _net=Net pressure level, psi

BHTP=Bottom-hole treating pressure, psi

P _c=Closure force per unit area (approximately minimum horizontal stress), psi

BH ISIP=Bottom-pigsty ISIP, psi, BH ISIP=ISIP+P _h.

Equation 9.16. Net pressure.

E=Immature's modulus, psi

h=Fracture summit, ft

Q=Rate, bpm

L=Total fracture length, ft

P _tip=Fracture tip force per unit area, psi.

As tin can be seen from Eq. (9.16), Young's modulus is raised to the power of 3/4 while the fluid rate, viscosity, and total fracture length are only raised to the power of ane/4. This shows that Immature'southward modulus has more impact on net force per unit area compared to viscosity, rate, and length. As a upshot, the Young's modulus measurement of a formation is a key parameter in fracture propagation. Fracture-tip pressure is a quantity that is non piece of cake to find, notwithstanding, dissimilar numerical simulations depending on a wide range of assumptions (e.g. fracture tip with or without fluid lag) volition provide estimates of the fracture tip pressure, Bao et al. (2016). In hydraulic fracturing, a dynamic gap zone betwixt fracture tip and fracturing fluid following the tip exists which tin can affect the fracture tip pressure level.

When net pressure (y-axis) versus time (x-axis) is plotted on the log–log plot during a live frac stage treatment, a net force per unit area nautical chart can be constructed. A cyberspace force per unit area nautical chart is as well referred to as a Nolty chart, and is used during the hydraulic frac handling to follow diverse pressure trends throughout the stage. Net pressure level charts are used to estimate various fracture propagation behavior at different points in time. Every bit previously discussed, since cyberspace force per unit area is the driving mechanism behind the fracture growth, information technology tin be used to predict the fracture dimension. Company representatives in the field rely heavily on the Nolty nautical chart during the treatment since it is very accurate in conventional reservoirs. In unconventional reservoirs, this nautical chart is nevertheless a useful tool to determine the fracture propagation, just it is non as accurate equally it is in the conventional reservoirs. Fig. ix.5 shows the concept of net force per unit area during the treatment, which tin be used to brand critical decisions.

Effigy 9.v. Net force per unit area interpretation.

•

If the pressure response in the Nolty chart is similar to trend #1, it is an indication of independent height and unrestricted length extension during the treatment (slightly positive gradient).

•

If the slope of the net pressure line is zip (trend #ii), it represents independent superlative and mayhap openning upward more than fractures with fluid loss. Information technology indicates a less-efficient length extension.

•

Trend #3 pressure response during treatment is bad news because the formation is giving upwardly and at that place is a loftier possibility of a tip screening-out (sanding-off) if sand is not cutting on time.

•

Trend #4 is basically a full screen-out and the pump room needs to exist prepare to come offline equally presently equally the force per unit area starts rising dramatically to avoid exceeding the pressure level limitations on the casing and equipment.

•

Tendency #v illustrates uncontrolled fracture height growth.

Cyberspace pressure typically ranges between 100 and 1400   psi. In some instances, internet pressure could be higher. If net pressure is much higher than 1400   psi, this could be due to almost-wellbore brake or large tip plasticity.

Instance

Estimate net pressure if closure pressure is obtained from a step-rate examination to be 6500   psi and ISIP is 4700   psi. The well has a TVD of 6800' and used an eight.8-ppg frac fluid to pump the job.

$\mathrm{BH}\mathrm{ISIP}=\mathrm{ISIP}+P_{\mathrm{h}}=4700+(0.052\times \mathrm{eight.8}\times 6800)=7811\mathrm{psi}$

$P_{\mathrm{net}}=\mathrm{BH}\mathrm{ISIP}–P_{\mathrm{c}}=7811-6500=1312\mathrm{psi}$

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Fracture pressure assay and perforation blueprint

Hoss Belyadi , ... Fatemeh Belyadi , in Hydraulic Fracturing in Unconventional Reservoirs (Second Edition), 2022

Net pressure level

Net pressure level is i of the most important pressures to consider in hydraulic fracturing. Net pressure is the energy required for propagating fractures and creating width during the frac job and refers to the backlog force per unit area over the frac pressure required to extend the fractures. Cyberspace pressure is essentially the deviation between the fracturing fluid pressure level and the closure pressure level and is the driving machinery behind fracture growth. The more pressure inside a fracture, the more potential there is for growth. The term net pressure is only used when the fracture is open. If the fracture is airtight, net pressure is equal to 0. Net force per unit area depends on various parameters such as Young's modulus, fracture tiptop, fluid viscosity, fluid rate, total fracture length, and tip pressure. Net pressure is as well referred to as procedure zone stress and can exist calculated using Eq. (9.15) or Eq. (9.xvi).

(nine.fifteen) $\mathrm{Cyberspace}\text{pressure level},\mathrm{Eq}.\left(9.1\right)\begin{array}{c}{P}_{\mathrm{cyberspace}}=\text{BHTP}-P_{\mathrm{c}}\\ \text{or}\\ P_{\mathrm{net}}=\mathrm{BH}\text{ISIP}-P_{\mathrm{c}}\end{array}$

P _net is the cyberspace pressure (psi), BHTP is the bottom-hole treating pressure (psi), P _c is the closure force per unit area which is approximately minimum horizontal stress (psi), and BH ISIP is the bottom-hole ISIP (psi), BH ISIP = ISIP + P _h.

(9.16) $P_{\mathrm{net}}=\frac{E^{¾}}{h}{\left(\mu \times Q\times L\right)}^{¼}+P_{\mathrm{tip}}$

E is the Young's modulus, psi; h is the fracture height, ft; Q is the rate, bpm; 50 is the total fracture length, ft; and P _tip is the Fracture tip pressure, psi.

Every bit can exist seen from Eq. (ix.16), Young'southward modulus is raised to the ability of 3/4 while the fluid rate, viscosity, and total fracture length are only raised to the power of 1/iv. This shows that Immature'south modulus has more impact on internet pressure compared to viscosity, charge per unit, and length. Every bit a result, the Young'southward modulus measurement of a formation is a key parameter in fracture propagation. Fracture-tip pressure is a quantity that is not piece of cake to find, however, different numerical simulations depending on a wide range of assumptions (e.grand., fracture tip with or without fluid lag) volition provide estimates of the fracture tip pressure, Bao et al. (2016). In hydraulic fracturing, a dynamic gap zone betwixt fracture tip and fracturing fluid following the tip exists which tin can bear upon the fracture tip pressure level.

When internet pressure (y-axis) versus fourth dimension (x-axis) is plotted on the log-log plot during a alive frac stage treatment, a net pressure chart can be constructed. A net force per unit area chart is also referred to as a Nolty chart, and is used during the hydraulic frac treatment to follow diverse force per unit area trends throughout the phase. Net pressure level charts are used to estimate various fracture propagation beliefs at different points in time. As previously discussed, since internet pressure is the driving mechanism behind the fracture growth, it can exist used to predict the fracture dimension. Company representatives in the field rely heavily on the Nolty nautical chart during the treatment since it is very accurate in conventional reservoirs. In anarchistic reservoirs, this chart is however a useful tool to make up one's mind the fracture propagation, but it is non every bit accurate equally information technology is in the conventional reservoirs. Fig. nine.5 shows the concept of net force per unit area during the treatment, which can be used to make critical decisions.

Fig. 9.5. Net pressure level interpretation.

•

If the pressure response in the Nolty chart is similar to trend #1, it is an indication of contained height and unrestricted length extension during the treatment (slightly positive gradient).

•

If the slope of the net pressure level line is zero (trend #ii), information technology represents independent superlative and possibly opening up more fractures with fluid loss. It indicates a less-efficient length extension.

•

Trend #3 pressure response during handling is bad news because the formation is giving up and there is a high possibility of a tip screening-out (sanding-off) if sand is not cut on time.

•

Tendency #four is basically a total screen-out and the pump room needs to be set up to come offline as soon as the pressure starts ascent dramatically to avoid exceeding the pressure limitations on the casing and equipment.

•

Tendency #5 illustrates uncontrolled fracture elevation growth.

Internet pressure typically ranges between 100 and 1400   psi. In some instances, net pressure could be college. If internet pressure is much higher than 1400   psi, this could be due to well-nigh-wellbore restriction or large tip plasticity.

Example

Approximate net pressure level if closure pressure is obtained from a step-charge per unit test to be 6500   psi and ISIP is 4700   psi. The well has a TVD of 6800′ and used an viii.8-ppg frac fluid to pump the job.

$\mathrm{BH}\text{ISIP}=\text{ISIP}+P_{\mathrm{h}}=4700+\left(0.052\times 8.8\times 6800\right)=7811\mathrm{psi}$

$P_{\mathrm{net}}=\mathrm{BH}\text{ISIP}–P_{\mathrm{c}}=7811-6500=1312\mathrm{psi}$

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Hydraulic Fracturing

Boyun Guo PhD , ... Xuehao Tan PhD , in Petroleum Production Engineering (2d Edition), 2022

14.11.1.1 Analyzing the cyberspace pressure from diagnostic injection tests

Matching the net force per unit area of a main fracture treatment without diagnostic injection data offers niggling value. The analysis of diagnostic injection data tin can provide the closure stress, fluid efficiency, leakoff coefficient, and the organization permeability resulting from fluid leakoff which could exist larger than the reservoir permeability. Cyberspace pressure friction match of the diagnostic injection data should too be conducted. This match should be reviewed before proceeding with the analysis of the main handling itself. Consistency between the parameters obtained from both matches should be maintained and deviation recognized. The first estimate of efficiency and leakoff is obtained from the diagnostic injection or calibration treatment refuse analysis. The calibration treatment provides a straight measurement of the efficiency using the graphical G-plot analysis. Then calibration with a model that estimates the geometry of the fracture provides the corresponding leakoff coefficient ( Meyer and Jacot, 2000). This leakoff coefficient determination is model dependent.

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Dynamic characteristics of rolling piston machines

Alison Subiantoro , Kim Tiow Ooi , in Positive Displacement Machines, 2022

Forces and torques

During operation, the interaction of the driving forces past the motor and the changes in the pressure of the working fluid results in a complex interaction of all the moving components (rotor, vane, vane spring, eccentric and shaft) and the stationary components (cylinder, trounce and bearings). A detailed forcefulness analysis is required to understand the dynamic characteristics of a rolling piston compressor. Such studies take been carried out since the early years (Pandeya & Soedel, 1978; Yanagisawa & Shimizu, 1985c) until now and is the centre of any dynamic analysis of a rolling piston motorcar. Force analysis is also needed in the design procedure to ensure that the parts are potent enough to withstand all the loads. Some of the major force notations, particularly at the vane, are shown in Fig. half dozen.

Fig. 6. Principal force notations.

The net force per unit area forces across the vane in the ten-direction and along the vane in the y-direction can exist computed with Eqs (10), (11), respectively. The inertial force of the vane and the bound strength are governed by Eqs (12), (13), respectively. Finally, the vane tip contact force can be obtained by solving Eq. (14). Subscripts vs and vt bespeak the vane side and the vane tip, respectively. More discussions on these equations tin be found in Pandeya and Soedel (1978) or Yanagisawa and Shimizu (1985c), among others.

(x) $F_{p,x}=\left(p_{\mathrm{ane}}-p_{\mathrm{two}}\right)y_{v,\mathit{ext}}{l}_{\mathrm{five}}$

(eleven) $F_{p,y}=\left(p_{\mathit{out}}{t}_v-p_1\left(0.5t_5+R_{v}sin\alpha \right)-p_2\left(0.5t_v-R_{v}sin\alpha \right)\right)l_v$

(12) $F_i=-m_v\frac{d^2{y}_{v,\mathit{ext}}}{d{t}^2}$

(13) $F_{\mathrm{south}}={\mathrm{chiliad}}_s\left(y_{5,\mathit{ext},\mathit{max}}-y_{5,\mathit{ext}}\right)$

(14) $F_{\mathit{vt},n}=\frac{{\eta }_{f,\mathit{vs}}{F}_{p,x}\left(l_v+t_v{\eta }_{f,\mathit{vs}}\right)+\left(F_{p,y}+F_{\mathrm{southward}}+F_i\right)\left(y_{v,\mathit{ext}}-y_{v,\mathit{gap}}\right)}{\left\{\begin{array}{c}\left(cos\alpha +{\eta }_{f,\mathit{vt}}sin\alpha \right)\left(y_{\mathrm{five},\mathit{ext}}-y_{v,\mathit{gap}}-\mathrm{two}{\eta }_f{R}_{5}sin\alpha \right)\\ +{\eta }_{f,\mathit{vs}}\left(sin\alpha -{\eta }_{f,\mathit{vt}}cos\alpha \right)\left(t_v{\eta }_{f,\mathit{vs}}+l_v+y_{v,\mathit{ext}}-2R_v\left(\mathrm{i}-cos\alpha \right)\right)\end{array}\right\}}$

where

α is the vane tip contact angle

η _f is friction coefficient.

F is force.

one thousand _s is the leap stiffness

l _v is the length of the vane, which is slightly shorter than the sleeping accommodation length, 50 _c , due to the clearance gaps

thou _v is the mass of the vane

p _ane is the pressure in the compression/belch chamber.

p ₂ is the force per unit area in the suction chamber.

p _out is the pressure in the shell.

y _{5, ext} is the length of the portion of the vane that is exposed to the pressures.

y _{v, gap} is the distance between the vane tip and the reaction betoken

The friction forces at the vane tip and the vane sides are governed by Eqs (15), (16), respectively. When the vane, the vane slot and the rotor outer wall are made of metal and the surfaces are lubricated, a friction coefficient of 0.15 is typically assumed.

(15) $F_{\mathit{vt},f}={\eta }_{f,\mathit{vt}}{f}_{\mathit{vt},\mathrm{due\; north}}$

(16) $F_{\mathit{vs},f}={\eta }_{f,\mathit{vs}}{f}_{\mathit{vs},n}$

The variations of vane side pressure and vane tip contact forces with rotor's angular position are shown in Fig. 7. The force per unit area force increases steadily at the beginning every bit the exposed vane length increases and the pressure in the pinch chamber rises simultaneously. It then reaches its peak at a rotor angle of approximately 200° so drops steadily as the vane exposed length decreases. The vane tip contact strength is a combination of three factors, i.e., vane inertia, leap and pressure, but it is virtually significantly influenced by the pressure variations in the chambers.

Fig. vii. Variations of vane side pressure and vane tip contact forces of a rolling piston compressor with rotor'southward angular position.

The angular velocity of the rotor, ω _r , can exist plant from Eq. (17). Even when the shaft turns at a constant speed, the rotor'south angular velocity may fluctuate because of the dynamic interaction of the forces. The rotor's angular acceleration tin exist computed from the torque balance equation of the rotor, equally expressed in Eq. (18). More discussions on these equations can exist found in Pandeya and Soedel (1978), among others.

(17) ${\omega }_r=\frac{{\delta }_{\mathit{rot},\mathit{cyl}}\left(\mathrm{ii}\pi {\mu }_{\mathit{oil}}{\omega }_{\mathrm{southward}}{\mathrm{fifty}}_c{R_{\mathrm{east}}}^{\mathrm{three}}-R_r{F}_{\mathit{vt},\mathrm{due\; north}}{\delta }_{\mathit{ecc},\mathit{rot}}\right)}{2\pi {\mu }_{\mathit{oil}}\left(l_c{R_e}^3{\delta }_{\mathit{rot},\mathit{cyl}}+\left({R_r}^4-{R_{\mathrm{due\; east}}}^{\mathrm{four}}\right){\delta }_{\mathit{ecc},\mathit{rot}}\right)}$

(18) $I_r\frac{d{\omega }_r}{\mathit{dt}}=\frac{2\pi {\mu }_{\mathit{oil}}{l}_c{R_e}^3\left({\omega }_s-{\omega }_r\right)}{{\delta }_{\mathit{ecc},\mathit{rot}}}-F_{\mathit{vt}}{R}_v-\frac{2\pi {\mu }_{\mathit{oil}}\left({R_r}^4-{R_e}^4\right){\omega }_{\mathrm{r}}}{{\delta }_{\mathit{rot},\mathit{cyl}}}$

where

δ _{ecc, rot} is the clearance gap between the eccentric and the inner surface of the rotor

δ _{rot, cyl} is the clearance gap between the endfaces of the rotor and the cylinder.

μ _oil is the viscosity of the lubricating oil

ω _s is the athwart velocity of the shaft.

The variations of rotor'southward angular velocity and vane-slot linear velocity with rotor's angular position are shown in Fig. viii. The shaft rotates at a constant speed of 2800 rev/min (or 293   rad/s). Equally can be seen, the rotor'south angular velocity is non constant although the shaft'south speed is constant. Moreover, the rotor's velocity of between 26 and 36   rad/due south is only effectually 10% of the shaft's speed of 293   rad/s. This loftier relative velocity volition issue in the high friction loss between the roller and the eccentric to be discussed later. The vane-slot linear velocity fluctuates in a sinusoidal manner as the vane extends and retracts during operation.

Fig. 8. Variations of the rotor angular velocity and the vane-slot linear velocity with rotor'due south angular position.

Frictions

One of the of import factors affecting the performance of a rolling piston compressor is the frictional losses betwixt the rubbing parts. Half-dozen main friction locations in a rolling piston machine and the models to calculate each of the frictional power losses are shown in Eqs (19)–(25). More than discussions on these equations tin can be found in Pandeya and Soedel (1978) or Yanagisawa and Shimizu (1985c), amid others.

•

Rubbing between the eccentric and the inner surface of the rotor can exist computed according to the shear of the lubricating oil in the gap due to relative tangential velocity of the two surfaces as expressed in Eq. (xix).

(19) $P_{f,\mathit{ecc},\mathit{rot}}=\frac{2\pi {\mu }_{\mathit{oil}}{\left({\omega }_s-{\omega }_r\right)}^{\mathrm{ii}}{l}_c{R_e}^3}{{\delta }_{\mathit{ecc},\mathit{rot}}}$

•

Rubbing between the rotor's and the cylinder'southward endfaces can be computed according to the shear of the lubricating oil in the gap due to the relative rotational velocity between the two surfaces as expressed in Eq. (20).

(20) $P_{f,\mathit{rot},\mathit{cyl}}=\frac{2\pi {\omega }_{\mathrm{s}}{\omega }_r{\mu }_{\mathit{oil}}\left({R_r}^{\mathrm{four}}-{R_e}^{\mathrm{iv}}\right)}{{\delta }_{\mathit{rot},\mathit{cyl}}}$

•

Rubbing betwixt the eccentric'south and the cylinder's endfaces can exist computed according to the shear of the lubricating oil in the gap due to the relative rotational velocity between the 2 surfaces, of which the cylinder is stationary, as expressed in Eq. (21).

(21) $P_{f,\mathit{ecc},\mathit{cyl}}=\frac{\mathrm{ii}\pi {{\omega }_{\mathrm{south}}}^2{\mu }_{\mathit{oil}}\left(2{R_e}^4-{R_r}^4\right)}{{\delta }_{\mathit{ecc},\mathit{cyl}}}$

•

Rubbing between the vane tip and the rotor wall tin be computed past multiplying the vane tip friction force and the sliding velocity of the vane tip as expressed in Eq. (22).

(22) $P_{f,\mathit{vt}}=F_{\mathit{vt},f}\left(R_r{\omega }_r+\left(R_c-R_r\right){\omega }_s\frac{cos\theta }{cos\alpha }\right)$

•

Rubbing between the vane sides and the slot wall can be computed by multiplying the friction forces and the vane's linear velocity as expressed in Eq. (23).

(23) $P_{f,\mathit{vs}}=\left(F_{\mathit{vs},f\mathrm{ane}}+F_{\mathit{vs},f2}\right)\frac{d{y}_{v,\mathit{ext}}}{\mathit{dt}}$

•

Rubbing betwixt the shaft and the bearing tin be modelled as the shear of the lubricating oil in the gap as expressed in Eq. (24). For a more comprehensive model, Reynolds lubrication equation can be modelled.

(24) $P_{f,\mathit{bearing}}=\frac{2\pi {\mu }_{\mathit{oil}}{{\omega }_s}^{\mathrm{ii}}{l}_{\mathit{bearing}}{R_{\mathit{shaft}}}^3}{{\delta }_{\mathit{bearing}}}$

The total frictional loss of a rolling piston compressor is obtained past summing up all the frictional loss components. Mechanical efficiency of the compressor can then be calculated co-ordinate to Eq. (25). If in that location is no friction loss, all the compressor power is used to compress the refrigerant.

(25) ${\varepsilon }_{\mathit{mech}}=\frac{P_{\mathit{refr}}}{P_{\mathit{refr}}+P_{f,\mathit{total}}}=\frac{{\dot{\mathrm{yard}}}_{\mathit{refr}}\left(h_{\mathit{disc},\mathit{real}}-h_{\mathit{suct}}\right)}{{\dot{m}}_{\mathit{refr}}\left(h_{\mathit{disc},\mathit{real}}-h_{\mathit{suct}}\right)+P_{f,\mathit{total}}}$

The variations of the major frictional losses with rotor's angular position of the benchmarked rolling piston compressor model are shown in Fig. 9. The clearance gaps used in this case are:

Fig. ix. Variations of various friction forces of a rolling piston compressor with rotor'south angular position.

•

25   μm between the eccentric and the inner surface of the rotor,

•

18   μm betwixt the rotor's and the cylinder's endfaces, and

•

347   μm Between the eccentric'south and the cylinder's endfaces

Fig. 9 shows that the friction between the roller and the eccentric is the near dominant. It accounts for more than than 40% of the total loss. Frictions at the vane side and at the vane tip follow, while the others are less pregnant. It is useful to discover that based on the friction equations above, frictional losses are dependent on geometry, operating conditions and textile properties. Therefore, for a given set of operating conditions and material properties, the compressor'due south geometrical configuration can be optimized to reduce frictional losses. The vane tip friction can be completely removed by attaching the vane rigidly to the rotor, which results in a modified rolling piston machinery chosen the swing piston design. In such pattern, the vane slot is replaced past a rocker to allow for the modified vane swiveling movement.

Vibration

As a compressor operates, the complex interactions between the motor, the moving parts and the stationary parts may create imbalance forces and torques that will produce vibration. Suspension systems are needed to minimize this event. The suspension systems used for rolling piston compressors are different from that for reciprocating compressors. There is typically no inner intermission arrangement due to the small vibration excited by the pinch mechanism, hence, the compressor is supported only past the outer interruption system.

The vibration characteristics of a rolling piston compressor can be divided into three different stages: starting, steady-state performance and stopping (Yanagisawa, Mori, Shimizu, & Ogi, 1984). These tin be modelled from the torque balances of the rotating and the stationary parts. The master torque components are the motor output torque, the refrigerant pressure, the vane contact forces and the mechanical losses. The intermission organisation provides the supporting and the damping moments to counter-balance the torques. The suspension system is typically modelled as a spring with a set of stiffness and damping characteristics.

At the starting stage, pressures in both chambers are initially equal and the motor torque rises steadily following the acceleration of the rotating parts. The motor torque is the master moment acting on the stationary part. Thus, the maximum torsion of the stationary function occurs according to the acceleration finish of the rotating role and then damped vibration follows. Vibration is not significantly afflicted by the inertia, just it decreases with a higher spring stiffness.

During the steady-state functioning stage, pressure in the pinch chamber varies according to the rotor's angular position. The motor responds according to these changes. The rotating role rotates at an average speed with sinusoidal fluctuations, while the stationary part vibrates sinusoidally with a phase lag. The vibration aamplitude of the moving parts is by and large influenced by the refrigerant pressure level and the rotational inertia, while the vibration of the stationary part decreases with a higher moment of inertia but is not sensitive to the spring constant, different that at the starting stage.

Finally, at the stopping phase, the motor torque is zero. The maximum torsion of the stationary office occurs just after the compressor power-off. After that, damped vibration follows. Similar that at the starting stage, vibration decreases with a stiffer bound and the inertia effect is non significant.

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Integrated Hydraulic Fracture Design and Well Performance Analysis

Mei Yang , ... Guan Qin , in Unconventional Oil and Gas Resource Handbook, 2022

12.1.four.1 Nolte–Smith Analysis

The Nolte–Smith test uses a log–log plot of net pressure versus pump fourth dimension to predict fracture geometry. A negative slope usually indicates unconfined height growth, resulting in "penny" shape fracture geometry. A shallow, positive slope indicates fair to adept fracture height confinement, and unrestricted fracture extension. A steep, positive slope indicates that something has restricted fracture half-length growth, essentially a tip screen-out, which does not necessarily occur at the fracture tip, but can exist virtually the wellbore. Operationally, screen-out tin lead to a condition where continued injection of fluid inside the fracture requires pressures in backlog of the safety limitations of the wellbore or wellhead equipment and results in wellbore cleaning before resumption of operations. Finally, a flat slope indicates that fracture extension has been drastically slowed, which could be the start of extensive height growth, or an acceleration of fluid loss due to the opening of natural fractures.

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Anarchistic Natural Gas

V. Kuuskraa , in Advances in Energy Systems and Applied science, Book 3, 1982

3 Calculation of Gas In Place

With the bachelor production data and a definition of certain of the reservoir backdrop (e.m., pressure, internet pay thickness), one tin can utilize reservoir simulation to determine the gas volume of the natural fracture arrangement (ϕh) and thus the amount of complimentary gas in identify. Such an arroyo was taken in the study by Lewin and Assembly using the reservoir data in Table XXIII. The findings of the simulation study were applied to the analytic areas to determine the technically recoverable gas and the gas in place for the Appalachian Basin (Table XXIV).

TABLE XXIII. Results of Computer Simulation of Devonian Shale Gas Production

Reservoir data	Areas
	I, Two	Five	Seven	VIII
Original reservoir pressure (psi)	500	632	197	421
Flowing lesser hole pressure (psi)	100	100	50	100
Reservoir temperature (°F)	100	100	100	100
Internet formation thickness (ft)	500	500	200	500
Gas gravity	0.half-dozen	0.vi	0.6	0.6
Gas volume (ϕh)	1-four	two.4	2.0	two.5
Flow potential (kh)	10–12	7.5	xxx.0	87.5

Technically recoverable gas (MMcf)	Areas
	I	II	V	VII	VIII
With shooting	410	350	340	90	340
With small-scale fractures	500	430	430	100	360

	Areas
	I	Two	5	VII	VIII
Gas in place (MMcf)	800	670	660	170	450

TABLE XXIV. Footing of Devonian Technically Recoverable and Gas in Identify Estimates

Area	Undrilled area (square miles)	% of areal unit	30-year recovery/well w/shooting (MMcf)	Technically recoverable gas from area (Bcf)	Assumed recovery efficiency (%)	Estimated gas in place (Bcf)
I	1050	50	411	921	47	1959
		25	328	367		782
		25	205	230		489
Two	1609	33	349	791	50	1581
		33	244	553		1106
		33	175	396		793
3	1098	50	348	815	50	1630
		25	278	326		651
		25	175	205		410
IV	848	33	376	449	48	935
		33	263	314		654
		33	158	189		393
V	1690	33	338	804	51	1577
		33	236	562		1101
		33	169	402		789
VI	2032	33	264	755	60	1259
		33	185	529		882
		33	132	378		629
VII	15,261	33	96	2063	60	3438
		33	67	1440		2399
		33	48	1031		1719
VIII	2955	33	362	1506	76	1982
		33	253	1053		1385
		33	181	753		991
Ix and 10	17,776	33	338	8460	51	16,588
		33	237	5932		11,631
		33	169	4230		8294
Xi	12,332	33	267	4636	60	7727
		33	187	3247		5412
		33	134	2327		3878
Full	57,000			45,664		83,064

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Proceedings of the ninth International Briefing on Foundations of Calculator-Aided Process Design

Haoshui Yu , ... Truls Gundersen , in Computer Aided Chemic Engineering, 2022

Two-stage Reverse Osmosis Desalination Model

For each stage of the RO, the permeate flux across the membrane is proportional to the membrane permeability A _i and the cyberspace pressure level driving force P _i ^net every bit shown in Eq. (11), where ΔP _f is the transmembrane force per unit area departure, /2 is the boilerplate pressure drop between the feed and the retentate streams, and $\overline{P_{\mathit{os},i}}$ is the average osmotic force per unit area departure.

(11) $J_i^p=A_i{P}_i^{\mathit{internet}}=A_i\left(\Delta P_f-\Delta P_i^d/2-\overline{P_{\mathit{bone},i}}\right)i=1,\mathrm{ii}$

For each stage of the RO, $\overline{P_{\mathit{os},i}}$ is a role of the concentration polarization factor (f _pol,i ), the concentration of the feed and retentate streams (c _i ^f and c _i ^r ), and the osmotic pressure of the feed stream (P _os,i ). The average osmotic pressure can be calculated by the Eq. (12). It is obvious that with the increment of recovery ratio (Y _i ), the concentration polarization factor increases as shown in Eq. (13). Besides, only NaCl is the target contaminant, which is considered to be removed in this study, and the osmotic pressure is approximated by Van't Hoff equation equally shown in Eq. (fourteen).

(12) $\overline{P_{\mathit{os},i}}=f_{\mathit{pol},i}\times \frac{c_i^f}{c_i^r}\times P_{\mathit{os},i}i=1,2$

(13) $f_{\mathit{pol},i}=e^{\frac{0.7Y_i}{\mathit{Nepv}}}i=1,2$

(14) $P_{\mathit{os},i}=2\times \frac{\rho {\mathrm{Five}}_H{\mathit{RT}}_{\mathit{in},i}}{M_{\mathit{NaCl}}}\times \left(c_i^r-c_i^p\right)i=\mathrm{i},2$

Here, N _epv stands for the number of elements in each force per unit area vessel, M _NaCl , R and ρ are the molar mass of NaCl, universal gas constant, and mass density of the stream, respectively.

The concentration of permeate and retentate flows can exist calculated by Eqs. (15) and (16).

(15) $c_i^p=\left(\mathrm{i}-\mathit{rejection}\right)\times c_i^{f}i=\mathrm{one},\mathrm{two}$

(16) $c_i^r=\frac{\mathit{rejection}\times c_i^f}{\mathrm{i}-Y_i}i=\mathrm{ane},\mathrm{two}$

The total permeate flow rate of the system can be calculated past Eq. (17).

(17) $Q_p={\mathrm{South}}_e{\mathit{NPV}}_i\sum _{i=1,2}{J}_i^{p}i=\mathrm{one},2$

Here, S _east and NPV _i are the active surface surface area for a membrane element and the corresponding number of membrane elements of a RO block.

The energy required past the circulating pump can exist calculated by Eq. (xviii). Here, P _cp ^f is the discharge pressure of the circulating pump. The intermediate pressure pump and high-pressure pump free energy consumption can be calculated in a similar way.

(18) ${\mathrm{East}}_{\mathit{cp}}=\frac{Q_{\mathit{ww}}}{{\eta }_{\mathit{cp}}}\times \left(P_{\mathit{cp}}^f-P_{\mathit{atm}}\right)$

So the overall energy consumption of the RO arrangement tin can exist calculated by Eq. (19).

(19) $E_{\mathit{net}}={\mathrm{Due\; east}}_{\mathit{cp}}+{\mathrm{East}}_{\mathit{hpp}}+{\mathrm{Eastward}}_{\mathit{mpp}}+E_{\mathit{rec}}$

The Total Annualized Cost (TAC) is the summation of annualized capital letter cost and annual operating cost equally shown in Eq. (xx).

(20) $\mathit{TAC}={\mathit{CC}}_{\mathit{tot}}+{\mathit{OC}}_{\mathit{tot}}$

The majuscule toll consists of all the equipment toll in the organization and the membrane cost in the RO process. However, the cost of estrus exchangers in the chemical plant is excluded since the Duran-Grossmann model just provides the energy target and not the detailed rut exchanger network configuration. In essence, if the Heat Recovery Approach Temperature (HRAT) is called properly, the cost of the estrus exchanger network exerts little effect on the whole system. The operating cost includes the utility cost in the heat exchanger network, profit from ORC ability generation and the consumption of ability in the RO process. It should be noted that negative TAC indicates that the profit from ability generation of the ORC outweighs the capital cost and operating cost of the organization and vice versa.

In this study, the unit price of desalted h2o product is chosen every bit the objective function shown in Eq. (21).

(21) $\text{Specific Cost}=\frac{\mathit{TAC}}{365\times 24\times Q^p}$

So the overall model of the integrated arrangement for simultaneous waste matter heat utilization and wastewater treatment can be written every bit follows:

$\begin{array}{l}\text{Minimizing}\text{Specific Cost}\\ \mathrm{southward}.\mathrm{t}.\text{Duran}‐\text{Grossmann model}\\ \text{ORC model}\\ 2‐\text{stage RO model}\\ \text{Economic correlations}\end{array}$

Notwithstanding, it should exist noticed that only the nearly important equations are presented in this newspaper. For a detailed clarification of the models, the above-mentioned references (Yu et al., 2022, Karuppiah et al., 2022) provide all relevant data.

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Architectural drinking glass to resist wind-borne debris impacts

D.B. Hattis , in Architectural Glass to Resist Seismic and Extreme Climatic Events, 2009

Abstract:

The phenomenon of internal pressurization of partially enclosed buildings has been long recognized and understood past engineers and represents more than a threefold increment in internal pressure level, which significantly increases the net pressures for which the building envelope must be designed. Unintended internal pressurization of partially enclosed buildings occurs when the building envelope is breached. One type of alienation, drinking glass breakage, is most likely to occur at or very close to the fourth dimension that the edifice is exposed to its design wind load: breakage during a hurricane or other windstorm as the result of impact from air current-borne debris. In this case the building changes from being enclosed to beingness partially enclosed and may immediately be exposed to higher internal pressures that may exceed its designed capacity. The first office of this chapter traces the history of standards evolution and regulation in the United states of america to address this situation. The 2d part provides a survey of current design solutions for fenestration to comply with these windborne debris touch on standards.

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Vibrations of composite beat out-type structures

Eelco Jansen , in Stability and Vibrations of Thin Walled Composite Structures, 2022

8.1.3.1 Pressure loading

In the static state , i.eastward., the time-independent example, the radial pressure load is conservative. The pressure loading is causeless to exist constant over the shell surface and is therefore axisymmetric. The net force per unit area (positive inward) is the difference between the external and internal pressures and will exist denoted by p _eastward ,

(8.13) $\tilde{p}=p_e$

If the net pressure is directed outward (i.eastward., is negative) so p _{due east}   =   −p _i , where p _i is the net internal force per unit area,

The nonstationary load in the case of nonlinear vibrations is given past

(viii.fourteen) $\hat{p}=q=\hat{q}\left(x,y\right)\cos \omega t$

where q is the specified radial loading (with spatial distribution of the vibration mode).

In the case of dynamic buckling and parametric excitation the nonstationary pressure (constant over the shell surface) is given by

(8.15) $\hat{p}={\hat{p}}_{0}f\left(t\right)$

where ${\hat{p}}_0$ is a constant and f(t) a specified function of time. For footstep loading (the dynamic buckling example), f(t)   = u(t), where u(t) is the unit of measurement pace role, and for a pulsating load (the parametric excitation instance), f(t)   =   cos Ω _e t, where Ω _eastward is the excitation frequency.

The possibilities in the dissimilar cases for the radial loading p are summarized in Table eight.i. It is possible to extend the models used in the present chapter to include the flutter beliefs in an external supersonic menstruum. In this flutter analysis case, the outer surface of the beat out is exposed to a high Mach number supersonic flow directed parallel to its axis, as is illustrated in Ref. [5]. In Ref. [5], the aerodynamic pressure level is obtained from linear piston theory, which is oft applied and the simplest theory available for modeling this problem. For completeness, the corresponding aerodynamic loads are also included in Table 8.one.

Table 8.1. Summary of loads [u(t)   =   unit step function]

Case	N 0	T 0	p
Static state	${\tilde{N}}_0$	${\tilde{T}}_0$	p e
Preflutter			${\tilde{p}}_{ae}+p_e$
Dynamic buckling	${\hat{N}}_{0}u\left(t\right)$	${\hat{T}}_{0}u\left(t\right)$	${\hat{p}}_{0}u\left(t\right)$
Parametric excitation	${\widehat{\mathrm{Due\; north}}}_0\cos {\mathrm{\Omega }}_{e}t$	${\hat{T}}_0\cos {\mathrm{\Omega }}_{e}t$	${\hat{p}}_0\cos {\mathrm{\Omega }}_{\mathrm{east}}t$
Nonlinear vibration			$\hat{q}\left(x,y\right)\cos \omega t$
Flutter			${\hat{p}}_{ae}$

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Hydraulic fracturing

James Speight , in Shale Oil and Gas Production Processes, 2022

5.5.2 Proppant ship and placement

To create a hydraulic fracture, fluid is injected at loftier rate and pressure into a wellbore and into a germination that is open up to the wellbore. Viscous fluid flow within the fracture as well as other effects creates the net force per unit area required to generate the created width profile and the created fracture pinnacle. The volume of fluid pumped will bear upon the created fracture length – nevertheless, without pumping a propping agent into the fracture, the created fracture will close once the pumping operation ceases. The flow of crude oil and natural gas from the formation into the fracture is dependent on the propped fracture dimensions. The really important characteristics of a fracture are the propped width, height, and length distributions; therefore, proppant transport considerations are very of import in designing a hydraulic fracture treatment.

Proppant placement in the fractures is governed past a serial of mechanisms involving the interaction between the fracturing fluid and proppant. For example, proppant density and size take a determining affect on proppant settling, which, in plough, impacts placement of the proppant in the fracture. The settling rate of the proppant is directly proportional to the divergence in density between the fracturing fluid and proppant, and inversely proportional to the fluid viscosity. This condition makes settling an important consideration when pumping low-viscosity Newtonian fluids, as are typically used in horizontal multi-fracture treatments conducted in some shale formations. However, while much attention is typically given to density, diameter can actually be of greater importance in a fracturing treatment since settling velocity is proportional to particle diameter squared (Stoke'southward police force, which is an expression for the frictional force – also called drag force – exerted on spherical objects with pocket-sized Reynolds numbers i.eastward., very small particles) in a mucilaginous fluid, thus having an exponentially larger effect on settling rate than fluid viscosity. This may not fully draw settling under dynamic atmospheric condition in a slurry state of affairs but does illustrate that smaller and lighter proppant are easier to place.

The first fluid pumped into a well during a fracture treatment (the prepad) is used (i) to fill the casing and tubing, (two) examination the system for pressure, and (iii) intermission downwards the formation after which the pad fluid (the gummy fracturing fluid simply without the propping amanuensis) used during the handling, is pumped into the well. The purpose of the pad is to create a tall, wide fracture that volition accept the propping amanuensis. Post-obit the pad, the fluid containing propping agent (the slurry) is pumped and moves into the fracture. The propping agent particles movement up, out, and down the fracture with the slurry and may settle in the fracture as a result of gravitational forces. The settling velocity increases as the diameter and density of the propping amanuensis increase and as the density and viscosity of the fracturing fluid decrease. To minimize proppant settling, propping agents that are smaller in bore and/or less dense, every bit well as a more pasty fluid, can exist used.

There are other factors that must be included when trying to compute the propped fracture dimensions. For case, the type of fracture fluid affects transport of the proppant – a linear-structured fracture fluid will not transport proppants likewise as fluids with construction, such every bit a cross-linked fracture fluid or a viscoelastic surfactant fluid. Geologic realities likewise must be considered; for example, no fracture is exactly vertical, and the walls of a fracture are rarely (if at all) smooth. If there are turns and ledges along the fracture walls, these geologic features tend to reduce proppant settling when compared with the theoretical equations for transport in shine-wall, parallel-plate systems. Other issues such as (i) fracture height growth during and later on pumping operations, (ii) fluid loss in layered formations, and (iii) slurry viscosity also affect the propped fracture dimensions (Gidley et al., 1989; Smith et al., 1997).

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