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1 8. Gas wells There is no qualitative difference between the oil and gas flow. However, since the gas is lighter and less viscous than the oil, effects are negligible for oil may be important for gas flow, and vice versa. 8.1 Physical properties 8.1.1 Molecular model Many gas properties can be related to a concept of gas as molecules flying around in a room. The speed, and thus energy, increases with temperature. The distance between the molecules is very large, so that there is no interaction between them except for random and elastic collisions. The pressure is caused by collisions between the molecules and the walls. The general equation of state: pV nRT can be justified / derived from this model. This gives good prediction for most gases at low pressures and temperatures well above the dew point. Figure 8.1 Molecular gas model When gas is compressed, the volume decreases so that the distance between the molecules becomes smaller. There will then be more interactions between them. Van der Waals included molecular volume and attraction/repulsion. The deviation from the ideal state equation can be expressed by the z-factor linked to reduced pressure and temperature (ratios: pressure/critical-pressure and temperature/critical-temperature). For petroleum gases the z-factor often estimated graphically by the Standings-Katz chart, Standing /1952 /. Yarborough & Halls extended equation of state / 1974 / give very similar results. 8.1.2 Density Density equals the ratio between mass and volume. Gas density depends on molecular weight: M, pressure and temperature. From the state equation we can express the relationship m nM pM V V zRT (8-1) It is worth noting that even if the gas density is less than the oil density, the gas density, at reservoir conditions it can still be large compared what we associate with gas. (As a rule of thumb, the gas density in the reservoir approximately equals the reservoir pressure in bar: thus at pressure 100 bar, the density will be of the order of 100 kg/m3) Prod.karakterist28.06.2017 skr/221/HA 2 Compressibility quantifies how density is changed by pressure. For a given temperature, we can estimate from the general state equation. If the z-factor changes little with pressure, the compressibility corresponds to the inverse of pressure cg 1 d dp T z d p 1 dz 1 1 p p dp z T p dp T p (8-2) The result means that the compressibility decreases rapidly when the pressure increases. We have all experienced that a ball becomes harder when pumped up. The compressibility due to change in temperature at constant pressure can be derived similarly. 8.2 Inflow 8.2.1 Forcheimer’s equation The Forcheimer equation combines adds “turbulence” to viscous friction dp v v 2 dr k (8-3) 8.2.2 Inflow performance Solved for stationary, radial flow of gas, gives the pressure profile pe2 go p oTz 1 1 2 re p oTz g q g p r o ln q g T kh r T o 2 2 h 2 r re 2 with re>>rw, pe2 pw2 Ak qg A qg2 (8-4) where Ak A p oT z g ln( re / rw ) To go p oT z To kh 2 2 2 h rw g : gas viscosity at averaged flow conditions z : z-faktor at averaged flow conditions Figure 8.2 illustrates the flow characteristics of a gas well, with -factor estimated with Tek's correlation (6-7). For the sake of comparison, Figure 8.2 also show flow characteristics calculated for no turbulent pressure loss: = 0, Prod.karakterist28.06.2017 skr/221/HA 3 Figure 8.2 Inflow performance Backpressure equation An alternative expression of inflow performance is by "backpressure equation" 2 q g C ( p R2 p w ) n . introduced by the U.S. Bureau of Mines in 1929: The parameters: n and C are determined by multi-rate production test.They lack connection to reservoir and fluid parameters comparable to (8-4). 8.2.3 Incompressible approximation By using the identity: (pe2–pw2) = (pe–pw) (pe+pw), we can express the flow characteristics of the gas, (8-4), as pe pw g Bg re go Bg 2 ln qg q 2kh rw 2h 2 rw g (8-5) Here B g is the formation factor at average flow conditions. The relations is analogous to the inflow performance of liquid, with turbulence included. By equation (8-5) is assumed: pR2 pw2 pr pw pR pw 2 pR pR pw This implies the error margin: p pw 1 p 1 R 1 w 2 pR 2 pR For example: if the reservoir pressure is 200 bar and the well pressure 190, the error neglecting gas expansion becomes: = 0.5. (1-190/200) = 0.025, ie 2.5% Prod.karakterist28.06.2017 skr/221/HA 4 8.3 Flow in production tubing 8.3.1 Pressure in static gas column By integrating the pressure balance: dp gdh 0 along the pipe, the relationship between static tubing head pressure and bottom pressure becomes pth pwe g x L (8-6) zR T L : length along tubing M : gas mol weight : M 28.97 g g : spesific gravity Linear approximation Series development of exponential gives: e x 1 x x 2 2! x3 3! ... Using the 2 first terms (8-6) can be re-written pth 1 g x L g x L g x L 1 g x L 1 ..... 1 pw zRT zRT 2 zRT 2 3 zRT 2 3 When the parameter group in parentheses is much smaller than 1, the higher order terms may be neglected and the pressure approximated as pˆ th pw w g x L (8-7) -----8.3.2 Producing gas well Under normal flow conditions in pipes of constant diameter, we can neglect acceleration. By expressing the gas density at the general state equation, the flow equation becomes 8 fm 2 zRT 1 g x dx 0 dp pdx 2 5 zRT d p (8-8) Usually, parameter groups in the parentheses "( )" vary so little along the production pipe that we can consider them as constants. Integration of (8-18) then gives 2 pth2 g x L 8 f z RT e zRT pw2 2 5 g d x 2 g x L zR T 1 e 2 2 m The relationship above express a simple relationship between pressure and flow rate: 2 2 pth qg Ag A f (8-9) pw pw Prod.karakterist28.06.2017 skr/221/HA 5 Mg x L Ag e zRT 2 2 2 8f 8 f z p0 T o 2 zRT Af 2 g 1 Ag 2 g d 5 T 0 1 Ag g xd 5 M x Along a production pipe, the parameter groups Ag and Af can usually considered constants. This means that we can estimate these from production data, by plotting: pth / pw 2 versus: qg / pw 2 The plot should define a straight line. The intersection with the y-axis provides parameter: Ag, while the slope provides parameters and Af. Figure 8.6 compares shut-in pressure along the production pipe. Production is set relatively high: 20 Sm3 / s, which here gives the flow speed between 12 and 14 m / s. Since the speed increases towards the top, the pressure drops profile curves slightly. Figure 8.3 Pressure profiles in gas well 8.3.3 Horizontal pipes For a horizontal pipe, the static pressure contribution is equal to zero: gx= 0. By letting gx og towards zero, it can be shown (with l’Hopital’s rule) that the friction parameter goes asymptotically towards Prod.karakterist28.06.2017 skr/221/HA 6 Af g x 0 fM T zL 4 p o Rd 5 T o 2 Equation (8-9) then simplifies to the Weymouth equation for pressure loss in horizontal pipes: 2 2 5 0 T p pi Rd q 0 u 4 p f zM T L 2 g 0.5 (8-10) 8.2 Temperature change Temperature change affects the physical properties of the fluid and may cause precipitation of liquid, or solid, or evaporation. For gas, pressure drop can lead to considerable temperature changes. Energy balances are used to keep account of temperature, heat transfer and work. 8.2.1 Energy balance The first law of thermodynamics claims that energy can change form, but not disappear. We can then put up energy accounting. The steam engine (or the Sterling Machine) is often used as concept. Energy change can then be formulated and illustrated as dE dq dw (8-11) dq: heat supplied | J | dw: work performed | J | Figure 8.4: The energy balance For gases the internal energy (molecular velocity) is mainly related to temperature, little impacted by pressure (distance between molecules). The relationship is often reasonably linear as illustrated below, where: cv denotes internal energy pr. mole, corresponding to the heat capacity at constant volume 1st Law of Thermodynamics is also relevant when fluid flows as illustrated below. Energy changes by flow through pipe, pumps and compressors can be estimated from this Prod.karakterist28.06.2017 skr/221/HA 7 Figure 8.5 Energy accounting for an open system Energy forms Molecular energy can be understood as speed and vibration of the molecules and depens on temperature. The relationship is illustrated below. Slope: cv represents internal energy pr. mol and corresponds heat capacity at constant volume. As long as the distance between molecules is large, interactions between molecules can be neglected. The molecular energy will then be unaffected by pressure. Within defined temperature range, change in the molecular energy may be approximated: dEi ncv dT (8-12) Figure 8.6 Internal energy as function of temperature Mechanical energy associated with the flow can be expressed by Bernoulli's equation. In energy units dEm Vdp mvdv mgdh (8-13) Energy balance for open systems For an open system, the energy balance is expressed by putting the energy forms above into the first law (8-11) Prod.karakterist28.06.2017 skr/221/HA 8 ncv dT Vdp mvdv mgdh dq dw (8-14) Work Work performed by expansion can be expressed: dw pdV (8-15) Heat transfer Supplied heat will depend on the temperature difference between the system and surroundings: T -Ta and the contact surface: A. Fourier law says proportionality, so that the change in thermal energy may be expressed dq UA T Ta dt (8-16) - Where: U is the heat transfer coefficient | w/(m2K) |, constant for heat conduction For flow pipe flow at moderate speed, the temperature change is often dominated by heat conduction, so that the energy balance (8-14) simplifies to: ncv dT dq . With eq. (8-16), this gives a change in temperature with time: T t Ta T to Ta e UA t to ncv or with distance, if we follow the fluid volume flowing through the pipe. Heat can also be supplied by the heating element, radiation and the like. Flow equation Changes in pressure and speed can be quantified by the flow equation. For pipe flow: (1-4) given below in energy units, consistent with (8-14) f 2 Vdp mgdh mvdv m v dx 0 2d 8.4.2 Adiabatic model Adiabatic approximation implies that work, heat transfer and friction are neglected: dw = 0, dpf = dq = 0. Energy change by expansion: pdV remain in the system. Inserted in (8-14), this gives the energy balance for adiabatic flow: ncv dT pdV Vdp mvdv mgdh 0 (8-17) The last 3 members of (8-17) correspond to frictionless flow (1-4). If the fluid is at rest, the sum of these is zero and (8-17) simplifies: ncv dT pdV 0 With the general equation of state, we can eliminate the temperature: nRdT pdV Vdp . Combination gives: cvVdp cv R pdV 0 . Dividing by pV and integrating from some selected initial state: pi,Vi , provides the adiabatic process equation ( "adiabatic equation of state") pV k piVi k Where the "adiabatic exponent" is defined as: k cv R cv c p cv Prod.karakterist28.06.2017 skr/221/HA (8-18) 9 When energy remains in the system, there will only be one free thermodynamic variable. This can be selected and all others expressed as a function thereof. For example, by expressing volume by the general equation of state and insert into (8-18), cooling during adiabatic expansion is estimated p T Ti pi k 1 k (8-19) 8.4.3 Stability of stationary gas We previously estimated how the pressure in the stagnant gas change with altitude, assuming dynamically stable gas column. Figure 8.7 illustrates a small amount of gas that rises slightly. Dynamic stability will imply that it then sinks back. This will happen if the density in the ascending gas is greater than what it becomes surrounded by. For a down sinking gas quantity, the opposite applies. Figure 8.7 Perturbation of stagnant gas When a gas volume rises, it expands so that the pressure is equal to the gas that surrounds it. If the ascending gas preserves temperature, the density be greater if it was colder: Ti Ti 1 . It will thus be dynamically stable if the temperature rises with height Expansion will affect the temperature. If the ascending gas not exchanging heat, the energy balance (8-17) becomes: ncv dT nRdT mgdh 0 . This provides temperature gradient dT gM g dh cv R c p M=m/n : mol weight |kg/Kmol| Prod.karakterist28.06.2017 skr/221/HA (8-20) 10 c p : specific heat capacity cp=cv+R |J/(kg K)| The air heat capacity is about 1005 J / (kg K). This gives the gradient aka “adiabatic lapse rate”: -dT / dh = 0.01K / m. When the sun warms the soil so that the gradient becomes larger, air will circulate upward so that the temperature gradient approaches 1 C/ 100 meters. Heat radiation from the soil may make the air colder near the surface and gradient positive. Such "inversion" makes air pollution accumulate. This is common in northern cities during the winter. ---- 8.5 Flow through valves and restrictions 8.5.1 Compressible flow When acceleration forces dominate, the flow equation becomes: dp vdv 0 . We neglect heat transfer and express density by the adiabatic equation (8-18) and the definition: m / V . Inserted in the flow equation and integrated from upstream pressure: pi and velocity: vi~0, to outlet pressure and velocity: pc ,vc , this gives k 1 2k pi pc k vc 1 k 1 i pi (8-21) Expressed as mass flow : m c vc Ac , this provides “Thornhill-Craver’s equation” m Ac pi 2k M k 1 zi RTi 2 k 1 pc k pc k 1 pi pi (8-22) The figure below shows the flow as function of downstream pressure. The red graph shows incompressible flow, almost coinciding with (8-22) for relative pressure drop <5%.For outlet pressure equal to inlet pressure: pc = pi, zero flow. When the outlet pressure is zero, zero flow is also predicted by (8-22); obviously unphysical. Prod.karakterist28.06.2017 skr/221/HA 11 Figur 8.8 Mass flow through orifice, calculated by different approximations When the pressure drop across a valve or restriction is less than 5% (pc / pi> 0.95), corresponding to Mach: Ma <0.25, we can neglect compressibility and calculate as for incompressible fluid 8.5.2 Critical flow Figure 8.8 shows maximum rate at approximately at 50% pressure drop. Measured rate matches the prediction up to this, but further reduction of downstream pressure does not reduce the rate. From (8-22), the maximum estimate by: dm / dpc 0 This is fulfilled when k p* 2 k 1 pi k 1 (8-23) Maximum throughput reached for the pressure ratio (8-23), governed by the adiabatic exponent. For air at 20 C : k 1.4 . This gives critical pressure ratio: p * / pc = 0.53. Setting (8-23) into (8-22) provided maximum mass flow reached by reducing the downstream pressure k 1 m* Ac pi kM zi RTi 2 k 1 k 1 (8-24) Further reduction of the downstream pressure will therefore not change neither the pressure in the nozzle, nor mass flow. The speed in nozzle can be estimated by setting (8-23) into (8-21) Prod.karakterist28.06.2017 skr/221/HA 12 vc* kRTc* 2k RTi k 1 M M (8-25) This is recognized as sound velocity, ie the velocity of pressure waves. Sound velocity is approximately constant, but the outlet pressure and density depends on the upstream pressure. Figure 8.9 illustrates the relationship upstream pressure and rate, utreknet from (8-22) and (8-24) Figure 8.9 Choke characteristics, for constant downstream pressure: 20 bar 8.5.3 Supercritical flow When the downstream pressure falls below the critical ratio: ps p * the flow out expand, thus accelerating further and briefly exceeding the sound velocity. This causes pressure shocks and severe turbulence as illustrated below Figure 8.10: Critical outflow To maintain the flow speed above sound velocity, the outlet must be designed so that the flow can accelerate evenly. This can be sought outlet geometry as illustrated below Prod.karakterist28.06.2017 skr/221/HA 13 Figur 8.11: Supercritical outflow, Laval-choke 8.5 Future production We will ignore here possible influx of water and assuming that the reservoir produces with gas expansion. We can formally quantify the expansion with compressibility equation c 1 dV VR dp c VR TR Dp dV (8-26) TR : compressibility : gas volume of the reservoir (constant) : reservoir temperature (constant) : pressure reduction : volume change Compressibility derived (8-1) gives the relationship is expressed as p z dV VR d z p If the pore volume in the reservoir is constant, the change is: dV, equal to the volume produced. It is convenient to express the production volume at standard conditions (we could possibly use mass produced, or moles). We can express the production rate of volume change over time, divided by the formation factor 2 1 dV To p d z To d p VR o qg VR o Bg dt p TR z dt p p TR dt z (8-27) Combining (8-27) with the previously derived system relations, we can predict production profiles. The combination can be done analytically, but often done numerically. This predicts how various improvements of the production system will affect production rates and earnings. t N p q g dt VR ti T o pi p p oTR zi z The reservoir volume must correspond to gas initially in the place: VR Bgi N Combined, this the classical mass balance for the gas reservoirs Prod.karakterist28.06.2017 skr/221/HA (8-28) p oTR zi N piT o 14 p pi z zi N p pi 1 pi N p 1 N zi N zi (8-29) This predicts that the ratio pressure/z-factor: p/z, should decline proportionally to the cumulative production. (We have previously shown that if a closed reservoir containing small compressible oil, the pressure will fall proportionally with cumulative production.) If the measured p/z plots as a straight line against cumulative production, this indicates that production is driven by the expansion. We can then predict the gas content in the reservoir by extrapolating it to zero pressure. Figure 8.12: Production history for the Frigg field Prod.karakterist28.06.2017 skr/221/HA