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2 Static ehaviour of natural gas and its low in pipes 43 9 X Static ehaviour of natural gas and its flow in pipes Ohirhian, P. U. University of Benin, Petroleum Engineering Department, Benin City, Nigeria. peter@ohirhian.com, okuopet@yahoo.com Astract A general differential equation that governs static and flow ehavior of a compressile fluid in horizontal, uphill and downhill inclined pipes is developed. The equation is developed y the comination of Euler equation for the steady flow of any fluid, the Darcy Weisach formula for lost head during fluid flow in pipes, the equation of continuity and the Colerook friction factor equation. The classical fourth order Runge-Kutta numerical algorithm is used to solve to the new differential equation. The numerical algorithm is first programmed and applied to a prolem of uphill gas flow in a vertical well. The program calculates the flowing ottom hole pressure as 44.8 psia while the Cullender and Smith method otains 44 psia for the 700 ft (aove perforations) deep well Next, the Runge-Kutta solution is transformed to a formula that is suitale for hand calculation of the static or flowing ottom hole pressure of a gas well. The new formula gives close result to that from the computer program, in the case of a flowing gas well. In the static case, the new formula predicts a ottom hole pressure of 640 psia for the 790 ft (including perforations) deep well. Ikoku average temperature and deviation factor method otains 639 psia while the Cullender and Smith method otaines 64 psia for the same well.. The Runge-Kutta algorithm is also used to provide a formula for the direct calculation of the pressure drop during downhill gas flow in a pipe. Comparison of results from the formula with values from a fluid mechanics text ook confirmed its accuracy. The direct computation formulas of this work are faster and less tedious than the current methods. They also permit large temperature gradients just as the Cullender and Smith method. Finally, the direct pressure transverse formulas developed in this work are comined wit the Reynolds numer and the Colerook friction factor equation to provide formulas for the direct calculation of the gas volumetric rate Introduction The main tasks that face Engineers and Scientists that deal with fluid ehavior in pipes can e divided into two road categories the computation of flow rate and prediction of pressure at some section of the pipe. Whether in computation of flow rate, or in pressure transverse, the method employed is to solve the energy equation (Bernoulli equation for

3 436 Natural Gas liquid and Euler equation for compressile fluid), simultaneously with the equation of lost head during fluid flow, the Colerook (938) friction factor equation for fluid flow in pipes and the equation of continuity (conservation of mass / weight). For the case of a gas the equation of state for gases is also included to account for the variation of gas volume with pressure and temperature. In the first part of this work, the Euler equation for the steady flow of any fluid in a pipe/ conduit is comined with the Darcy Weisach equation for the lost head during fluid flow in pipes and the Colerook friction factor equation. The comination yields a general differential equation applicale to any compressile fluid; in a static column, or flowing through a pipe. The pipe may e horizontal, inclined uphill or down hill. The accuracy of the differential equation was ascertained y applying it to a prolem of uphill gas flow in a vertical well. The prolem came from the ook of Ikoku (984), Natural Gas Production Engineering. The classical fourth order Runge-Kutta method was first of all programmed in FORTRAN to solve the differential equation. By use of the average temperature and gas deviation factor method, Ikoku otained the flowing ottom hole pressure (P w f) as 43 psia for the 700 ft well. The Cullender and Smith (96) method that allows wide variation of temperature gave a P w f of 44 psia. The computer program otaines the flowing ottom hole pressure (P w f ) as 44.8 psia. Ouyang and Aziz (996) developed another average temperature and deviation method for the calculation of flow rate and pressure transverse in gas wells. The average temperature and gas deviation formulas cannot e used directly to otain pressure transverse in gas wells. The Cullender and Smith method involves numerical integration and is long and tedious to use. The next thing in this work was to use the Runge-Kutta method to generate formulas suitale for the direct calculation of the pressure transverse in a static gas column, and in uphill and downhill dipping pipes. The accuracy of the formula is tested y application to two prolems from the ook of Ikoku. The first prolem was prediction of static ottom hole pressure (P w s). The new formula gives a P w s of 640 psia for the 790ft deep gas well. Ikoku average pressure and gas deviation factor method gives the P w s as 639 psia, while the Cullender and Smith method gives the P w s as 64 psia. The second prolem involves the calculation of flowing ottom hole pressure (P w f). The new formula gives the P w f as 4 psia while the average temperature and gas deviation factor of Ikoku gives the P w f as 43 psia. The Cullender and Smith method otains a P w f of 44 psia. The downhill formula was first tested y its application to a slight modification of a prolem from the ook of Giles et al.(009). There was a close agreement etween exit pressure calculated y the formula and that from the text ook. The formula is also used to calculate ottom hole pressure in a gas injection well. The direct pressure transverse formulas developed in this work are also comined wit the Reynolds numer and the Colerook friction factor equation to provide formulas for the direct calculation of the gas volumetric rate in uphill and down hill dipping pipes.

4 Static ehaviour of natural gas and its low in pipes 437 A differntial equation for static ehaviour of a compressile fluid and its flow in pipes The Euler equation is generally accepted for the flow of a compressile fluid in a pipe. The equation from Giles et al. (009) is: dp vdv d sin dhl 0 () g In equation (), the plus sign (+) efore d sin corresponds to the upward direction of the positive z coordinate and the minus sign (-) to the downward direction of the positive z coordinate. The generally accepted equation for the loss of head in a pipe transporting a fluid is that of Darcy-Weisach. The equation is: The equation of continuity for compressile flow in a pipe is: H W = L f L v () gd A (3) Taking the first derivation of equation (3) and solving simultaneously with equation () and () we have after some simplifications, dp d f W sin. A dg W d A g dp (4) All equations used to derive equation (4) are generally accepted equations No limiting assumptions were made during the comination of these equations. Thus, equation (4) is a general differential equation that governs static ehavior compressile fluid flow in a pipe. The compressile fluid can e a liquid of constant compressiility, gas or comination of gas and liquid (multiphase flow). By noting that the compressiility of a fluid (C f ) is: Equation (4) can e written as: d C f () dp

5 438 Natural Gas fw sin dp A dg d W C f A g (6) Equation (6) can e simplified further for a gas. Multiply through equation (6) y, then f W sin dp g dg d W C f A g (7) The equation of state for a non-ideal gas can e written as p zr (8) Sustitution of equation (8) into equation (7) and using the fact that pdp dp, gives d d fw zr p sin dp dg zr d W zrc f g p (9) The cross-sectional area (A) of a pipe is 4 d d 4 6 (0) Then equation (9) ecomes:

6 Static ehaviour of natural gas and its low in pipes 439 fw zr sin.639 d d g zr. d.639w zrc f 4 g d () The denominator of equation () accounts for the effect of the change in kinetic energy during fluid flow in pipes. The kinetic effect is small and can e neglected as pointed out y previous researchers such as Ikoku (984) and Uoyang and Aziz(996). Where the kinetic effect is to e evaluated, the compressiility of the gas (C f ) can e calculated as follows: For an ideal gas such as air, C f. For a non ideal gas, C f = p z p z p. Matter et al. (97) and Ohirhian (008) have proposed equations for the calculation of the compressiility of hydrocaron gases. For a sweet natural gas (natural gas that contains CO as major contaminant), Ohirhian (008) has expressed the compressiility of the real gas (C f ) as: C f p For Nigerian (sweet) natural gas K =.038 when p is in psia The denominator of equation () can then e written as KW zrt, where K = constant. M g d 4 P Then equation () can e written as dy (ABy) d G ( ) y () where.639fw zrt M sin KW zrt y p, A, B, G. 4 gd M zrt gmd The plus (+) sign in numerator of equation () is used for compressile uphill flow and the negative sign (-) is used for the compressile downhill flow. In oth cases the z coordinate is taken positive upward. In equation () the pressure drop is y - y, with y > y and incremental length is l l. Flow occurs from point () to point (). Uphill flow of gas occurs in gas transmission lines and flow from the foot of a gas well to the surface. The pressure at

7 440 Natural Gas the surface is usually known. Downhill flow of gas occurs in gas injection wells and gas transmission lines. We shall illustrate the solution to the compressile flow equation y taking a prolem involving an uphill flow of gas in a vertical gas well. Computation of the variales in the gas differential equation We need to discuss the computation of the variales that occur in the differential equation for gas efore finding a suitale solution to it.. The gas deviation factor (z) can e otained from the chart of Standing and Katz (94). The Standing and Katz chart has een curve fitted y many researchers. The version that was used in this section of the work that of Gopal(977). The dimensionless friction factor in the compressile flow equation is a function of relative roughness ( / d) and the Reynolds numer (RN). The Reynolds numer is defined as: vd Wd RN (3) Ag The Reynolds numer can also e written in terms of the gas volumetric flow rate. Then W = Q Since the specific weight at ase condition is: The Reynolds numer can e written as: p M 8.97Gg p (4) z T R z T R R N G P Q g Rgdg zt () By use of a ase pressure (p ) = 4.7psia, ase temperature (T ) = 0 o R and R = 4 R N = 007Q G g d g (6) Where d is expressed in inches, Q = MMSCF / Day and g is in centipoises. Ohirhian and Au (008) have presented a formula for the calculation of the viscosity of natural gas. The natural gas can contain impurities of CO and H S. The formula is:

8 Static ehaviour of natural gas and its low in pipes 44 Where g xx xx xx xx p xx = z T p (7) In equation (7) g is expressed in centipoises(c p), p in (psia) and Tin ( o R) The generally accepted equation for the calculation of the dimensionless friction factor (f) is that of Colerook (938). The equation is:. log f 3.7d RN f (8) The equation is non-linear and requires iterative solution. Several researchers have proposed equations for the direct calculation of f. The equation used in this work is that proposed y Ohirhian (00). The equation is f log a log a x (9) Where. a,. 3.7d R x =.4 log log RN log RN.0693 d After evaluating the variales in the gas differential equation, a suitale numerical scheme can e used to it. Solution to the gas differential equation for direct calculation of pressure transverse in static and uphill gas flow in pipes. The classical fourth order Range Kutta method that allows large increment in the independent variale when used to solve a differential equation is used in this work. The solution y use of the Runge-Kutta method allows direct calculation of pressure transverse.. The Runge-Kutta approximate solution to the differential equation

9 44 Natural Gas dy f(x, y) at x xn dx given that y y when x x is y y o (k (k k 3 ) k 4 ) 6 where k Hf(x, y ) o o k Hf(x H, yo k ) k3 Hf(xo H, yo k ) k Hf(x H, y k ) 4 o 3 xn xo H n n numer of applications o o (0) () The Runge-Kutta algorithm can otain an accurate solution with a large value of H. The Runge-Kutta Algorithm can solve equation (6) or (). The test prolem used in this work is from the ook of Ikoku (984), Natural Gas Production Engineering. Ikoku has solved this prolem with some of the availale methods in the literature. Example Calculate the sand face pressure (p wf) of a flowing gas well from the following surface measurements. Flow rate (Q) =.3 MMSCF / Day Tuing internal diameter (d) =.996in Gas gravity (G g) = 0.6 Depth = 790ft (ottom of casing) Temperature at foot of tuing (T w f ) = 60 o F Surface temperature (T s f) = 83 o F Tuing head pressure (p t f) = psia Asolute roughness of tuing () = in Length of tuing (l) = 700ft (well is vertical) Solution When length ( ) is zero, p = psia That is (x o, y o ) = (0, ) By use of step Runge-Kutta H = 700ft.

10 Static ehaviour of natural gas and its low in pipes 443 The Runge-Kutta algorithm is programmed in Fortran 77 and used to solve this prolem. The program is also used to study the size of depth(length ) increment needed to otain an accurate solution y use of the Runge-Kutta method. The first output shows result for onestep Runge-Kutta (Depth increment = 700ft). The program otaines psia as the flowing ottom hole pressure (P w f ). TUBING HEAD PRESSURE = PSIA SURFACE TEMPERATURE = DEGREE RANKINE TEMPERATURE AT TOTAL DEPTH = DEGREE RANKINE GAS GRAVITY = E-00 GAS FLOW RATE = MMSCFD DEPTH AT SURFACE = FT TOTAL DEPTH = FT INTERNAL TUBING DIAMETER = INCHES ROUGHNESS OF TUBING = E-004 INCHES INCREMENTAL DEPTH = FT PRESSURE PSIA DEPTH FT To check the accuracy of the Runge-Kutta algorithm for the depth increment of 700 ft another run is made with a smaller length increment of 000 ft. The output gives a p wf of psia. as it is with a depth increment of 700 ft. This confirmes that the Runge- Kutta solution can e accurate for a length increment of 700 ft. TUBING HEAD PRESSURE = PSIA SURFACE TEMPERATURE = DEGREE RANKINE TEMPERATURE AT TOTAL DEPTH = DEGREE RANKINE GAS GRAVITY = E-00 GAS FLOW RATE = MMSCFD DEPTH AT SURFACE = FT TOTAL DEPTH = FT INTERNAL TUBING DIAMETER = INCHES ROUGHNESS OF TUBING = E-004 INCHES INCREMENTAL DEPTH = FT PRESSURE PSIA DEPTH FT

11 444 Natural Gas In order to determine the maximum length of pipe (depth) for which the computed P w f can e considered as accurate, the depth of the test well is aritrarily increased to 0,000ft and the program run with one step (length increment = 0,000ft). The program produces the P w f as psia.. TUBING HEAD PRESSURE = PSIA SURFACE TEMPERATURE = DEGREE RANKINE TEMPERATURE AT TOTAL DEPTH = DEGREE RANKINE GAS GRAVITY = E-00 GAS FLOW RATE = MMSCFD DEPTH AT SURFACE = FT TOTAL DEPTH = FT INTERNAL TUBING DIAMETER = INCHES ROUGHNESS OF TUBING = E-004 INCHES INCREMENTAL DEPTH = FT PRESSURE PSIA DEPTH FT Next the total depth of 0000ft is sudivided into ten steps (length increment =,000ft). The program gives the P w f as psia for the length increment of 000ft. TUBING HEAD PRESSURE = PSIA SURFACE TEMPERATURE = DEGREE RANKINE TEMPERATURE AT TOTAL DEPTH = DEGREE RANKINE GAS GRAVITY = E-00 GAS FLOW RATE = MMSCFD DEPTH AT SURFACE = FT TOTAL DEPTH = FT INTERNAL TUBING DIAMETER = INCHES ROUGHNESS OF TUBING = E-004 INCHES INCREMENTAL DEPTH = FT PRESSURE PSIA DEPTH FT

12 Static ehaviour of natural gas and its low in pipes 44 The computed values of P w f for the depth increment of 0,000ft and 000ft differ only in the third decimal place. This suggests that the depth increment for the Range - Kutta solution to the differential equation generated in this work could e a large as 0,000ft. By neglecting the denominator of equation (6) that accounts for the kinetic effect, the result can e compared with Ikoku s average temperature and gas deviation method that uses an average value of the gas deviation factor (z) and negligile kinetic effects. In the program z is allowed to vary with pressure and temperature. The temperature in the program also varies with depth (length of tuing) as s T = GTG current length + T s f, where, (T wf GTG T f ) Total Depth The program otains the P w f as psia when the kinetic effect is ignored. The output is as follows: TUBING HEAD PRESSURE = PSIA SURFACE TEMPERATURE = DEGREE RANKINE TEMPERATURE AT TOTAL DEPTH = DEGREE RANKINE GAS GRAVITY = E-00 GAS FLOW RATE = MMSCFD DEPTH AT SURFACE = FT TOTAL DEPTH = FT INTERNAL TUBING DIAMETER = INCHES ROUGHNESS OF TUBING = E-004 INCHES INCREMENTAL DEPTH = FT PRESSURE PSIA DEPTH FT Comparing the P w f of psia with the P w f of psia when the kinetic effect is considered, the kinetic contriution to the pressure drop is psia psia = psia.the kinetic effect during calculation of pressure transverse in uphill dipping pipes is small and can e neglected as pointed out y previous researchers such as Ikoku (984) and Uoyang and Aziz(996) Ikoku otained 43 psia y use of the the average temperature and gas deviation method. The average temperature and gas deviation method goes through trial and error calculations in order to otain an accurate solution. Ikoku also used the Cullendar and Smith method to solve the prolem under consideration. The Cullendar and Smith method does not consider the kinetic effect ut allows a wide variation of the temperature. The Cullendar and Smith method involves the use of Simpson rule to carry out an integration of a cumersome function. The solution to the given prolem y the Cullendar and Smith method is p w f = 44 psia. If we neglect the denominator of equation (), then the differential equation for pressure transverse in a flowing gas well ecomes

13 446 Natural Gas dy A By dl where.639fw zrt A g d M M sin 8.79Gg sin B zrt RTz () (3) (4) The equation is valid in any consistent set of units. If we assume that the pressure and temperature in the tuing are held constant from the mid section of the pipe to the foot of the tuing, the Runge-Kutta method can e used to otain the pressure transverse in the tuing as follows. dy d KzGgQ p T 4 grd T y GgQ fzrt 9.940Gg Sin y gd zrt The weight flow rate (W) in equation () is related to Q (the volumetric rate measurement at a ase pressure (P ) and a ase temperature (T )) in equation () y: () W = Q (6) Equation () is a general differential equation that governs pressure transverse in a gas pipe that conveys gas uphill. When the angle of inclination ( ) is zero, sin is zero and the differential equation reduces to that of a static gas column. The differential equation () is valid in any consistent set of units. The constant K =.038 for Nigerian Natural Gas when the unit of pressure is psia. The classical 4 th order Runge Kutta alogarithm can e used to provide a formula that serves as a general solution to the differential equation (). To achieve this, the temperature and gas deviation factors are held constant at some average value, starting from the mid section of the pipe to the inlet end of the pipe. The solution to equation () y the Runge Katta algorithm can e written as: p p y. (7) Where

14 Static ehaviour of natural gas and its low in pipes p 3 u aa y x 0.x 0.36x 4.96x.48x 0.7x x 0.7x G Q f z RT 7.94G sin aa gd zzr u x g g av g av g avav G Q f z L gd G sin L z T R When Q = 0, equation (7) reduces to the formula for pressure transverse in a static gas column. L In equation (7), the component k 4 in the Runge Kutta method given y k 4 = H f(x o + H, y + k 3) was given some weighting to compensate for the fact that the temperature and gas deviation factor vary etween the mid section and the inlet end of the pipe. Equation (7) can e converted to oil field units. In oil field units in which L is in feet, R = 4, temperature is in o R, g = 3. ft/sec, diameter (d) is in inches, pressure (p) is in pound per square inch (psia), flow rate (Q ) is in MMSCF / Day, P = 4.7 psia and T = 0 o R.,the variales aa, u and x that occur in equation () can e written as: u.3090g g Q f z av T av L d x G g L sin z av Tav The following steps are taken in order to use equation (7) to solve a prolem.. Evaluate the gas deviation factor at a given pressure and temperature. When equation (7) is used to calculate pressure transverse in a gas well, the given pressure and temperature are the surface temperature and gas exit pressure (tuing head pressure).. Evaluate the viscosity of the gas at surface condition. This step is only necessary when calculating pressure transverse in a flowing gas well. It is omitted when static pressure transverse is calculated. 3. Evaluate the Reynolds numer and dimensionless friction factor y use of surface properties. This step is also omitted when considering a static gas column. 4. Evaluate the coefficient aa in the formula. This coefficient depends only on surface properties.

15 448 Natural Gas. Evaluate the average pressure (p a v) and average temperature (T a v). 6. Evaluate the average gas deviation factor.(z a v ) 7. Evaluate the coefficients x and u in the formula. Note that u = 0 when Q = Evaluate y in the formula. 9. Evaluate the pressure p. In a flowing gas well, p is the flowing ottom hole pressure. In a static column, it is the static ottom hole pressure. Equation (7) is tested y using it to solve two prolems from the ook of Ikoku(984), Natural Gas Production Engineering. The first prolem involves calculation of the static ottom hole in a gas well. The second involves the calculation of the flowing ottom hole pressure of a gas well. Example Calculate the static ottom hole pressure of a gas well having a depth of 790 ft. The gas gravity is 0.6 and the pressure at the well head is 300 psia. The surface temperature is 83 o F and the average flowing temperature is 7 o F. Solution Following the steps that were listed for the solution to a prolem y use of equation (7) we have:. Evaluation of z factor. The standing equation for P c and T c are: P c (psia) = G g 37. Gg T c ( o R) = G g. Gg Sustitution of G g = 0.6 gives, P c = 67. psia and T c = 38. o R. Then P r = 300/67. = 3.4 and T r = 43/38. =. The Standing and Katz chart gives z = Steps and 3 omitted in the static case..309 Gg Q f z T Gg p sin 4. aa L d zt Here, G g = 0.6, Q = 0.0, z = 0.78, d =.996 inches, p = 300 psia,. p a v= T = 43 o R and L = 700 ft. Well is vertical, =90 o, sin =. Sustitution of the given values gives: aa = / ( ) = psia Reduced p a v = 470. / 67. = 3.68 T a v = 7 o F = 77 o R

16 Static ehaviour of natural gas and its low in pipes 449 Reduced T a v = 77/38. =.6 From the standing and Katz chart, z a v = In the static case u = 0, so we only evaluate x Sin90 x aa 8. 3 p y x 0.x 0.36x 4.96x.48x 0.7x Sustitution of a = 66696, x = and P = 300 gives y p p y psia 640 psia o Ikoku used 3 methods to work this prolem. His answers of the static ottom hole pressure are: Average temperature and deviation factor = 639 psia Sukkar and Cornell method = 634 psia Cullender and Smith method = 64 psia The direct calculation formula of this work is faster. Example 3 Use equation (7) to solve the prolem of example that was previously solved y computer programming. Solution. Otain the gas deviation factor at the surface. From example, the pseudocritical properties for a 0.6 gravity gas are, P c = 67. psia. and T c = 38., then Pr = / 67. = 3.6 T r = 43 / 38. =. From the Standing and Katz chart, z =0.78. Otain, the viscosity of the gas at surface condition. By use of Ohirhian and Au p equation, xx z p Then g cp Evaluation of the Reynolds numer and dimensionless friction factor 007QGg RN.34 0 gd

17 40 Natural Gas The dimensionless friction factor y Ohirhian formula is f log a log a x Where a /3.7d,. /R N x.4 log log RN log RN.0693 d Sustitute of , d.996, R.340 gives f Evaluate the coefficient aa in the formula. This coefficient depends only on surface properties..309 G Q f z T G p sin g g aa L d z T Here, G g = 0.6, Q =.3 MMSCF/Day, f = 0.07, z = 0.78, d =.996 inches, p = psia, T = 43 o R, z = 700 ft Sustitution of the given values gives; aa = ( ) 700 = Evaluate P a v p p 0.aa psia av 6. Evaluation of average gas deviation factor. Reduced average pressure = p a v / p c = 37.6 / 67. = 3.46 av L / Where is the geothermal gradient. L T a v at the mid section of the pipe is 80 ft. Then, T a v = = 8. o R Reduced T a v = 8. / 38. =.6 Standing and Katz chart gives z a v = Evaluation of the coefficients x and u G gl x zavav G gqf zavav L u d N

18 Static ehaviour of natural gas and its low in pipes 4 8. Evaluate y 3 p 3 u aa y x 0.x 0.36x 4.96x.48x 0.7x x Where u = 666, x = 0.684, p = psia and aa = Then, y psia 9. Evaluate p (the flowing ottom hole pressure) p p y psia 4 psia The computer program otains, the flowing ottom hole pressure as psia. For comparison with other methods of solution, the flowing ottom hole pressure y: Average Temperature and Deviation Factor, P = 43 psia Cullender and Smith, P = 44 The direct calculating formula of this work is faster. The Cullendar and Smith method is even more cumersome than that of Ikoku.t involves the use of special tales and charts (Ikoku, 984) page The differential equation for static gas ehaviour and its downhill flow in pipes The prolem of calculating pressure transverse during downhill gas flow in pipes is encountered in the transportation of gas to the market and in gas injection operations. In the literature, models for pressure prediction during downhill gas flow are rare and in many instances the same equations for uphill flow are used for downhill flow. In this section, we present the use of the Runge-Kutta solution to the downhill gas flow differential equation. During downhill gas flow in pipes, the negative sign in the numerator of differential equation () is used..the differential equation also reaks down to a simple differential equation for pressure transverse in static columns when the flow rate is zero. The equation to e solved is: Where y p, dy d (A By) G ( ) y (8).639 f W zrt M sin KW zrt A, B, G 4 gd M zrt gmd Also, the molecular weight (M) of a gas, can e expressed as M = 8.97Gg.

19 4 Natural Gas Then, the differential equation (8) can e written as: G s in f z R W g g d G z R d g d z R W g d G g (9) The differential equation (9) is valid in any consistent set of units. The relationship etween weight flow rate (W) and the volumetric flow rate measured at a ase condition of pressure and temperature (Q ) is; The specific weight at ase condition is: W = Q (30) p M 8.97Ggp (3) z T R z T R Sustitution of equations (30) and (3) into differential equation (9) gives: dp dl f zg g Q 9.940G sin g gd z zr R Gg Q 4 grd (3) The differential equation (3) is also valid in any consistent set of units. Solution to the differential equation for downhill flow In order to find a solution to the differential equation for downhill flow (as presented in equation (9) and (3) we need equations or charts that can provide values of the variales z and f. The widely accepted chart for the values of the gas deviation factor (z) is that of Standing and Katz (94). The chart has een curve fitted y some researchers. The version used in this section is that of Ohirhian (993). The Ohirhian set of equations are ale to read the chart within 0.777% error. The Standing and Katz charts require reduced pressure (Pr) and reduced temperature (T r). The Pr is defined as Pr = P/Pc and the T r is defined as T r = T / T c; where Pc and T c are pseudo critical pressure and pseudo critical temperature, respectively. Standing (977) has presented equations for Pc and T c as functions of gas gravity (G g). The equations are: Pc = G g 37.Gg (33) T c = G g. Gg (34)

20 Static ehaviour of natural gas and its low in pipes 43 The differential equation for the downhill gas flow can also e solved y the classical fourth order Runge-Kutta method. The downhill flow differential equation was tested y reversing the direction of flow in the prolem solved in example 3. Example 4 Calculate the sand face pressure (p w f) of an injection gas well from the following surface measurements. Flow rate (Q) =.3 MMSCF / Day Tuing internal diameter (d) =.996 in Gas gravity (G g) = 0.6 Depth = 790ft (ottom of casing) Temperature at foot of tuing (T w f) = 60 o F Surface temperature (T s f) = 83 o F Tuing head pressure (P s f ) = 4 psia Asolute roughness of tuing () = in Length of tuing (L) = 700ft (well is vertical) Solution Here, (x o, y o ) = (0, 4) By use of I step Runge-Kutta H = 700 The Runge-Kutta algorithm is programmed in Fortran 77 to solve this prolem. The output is as follows. TUBING HEAD PRESSURE = PSIA SURFACE TEMPERATURE = DEGREE RANKINE TEMPERATURE AT TOTAL DEPTH = DEGREE RANKINE GAS GRAVITY = E-00 GAS FLOW RATE = MMSCFD DEPTH AT SURFACE = FT TOTAL DEPTH = FT INTERNAL TUBING DIAMETER = INCHES ROUGHNESS OF TUBING = E-004 INCHES INCREMENTAL DEPTH = FT PRESSURE PSIA DEPTH FT The other outputs from the program (not shown here) indicates that the contriution of kinetic effect to pressure transverse during down hill flow is also negligile. The program also shows that an incremental length as large as 700 ft can yield accurate result in pressure transverse calculations.

21 44 Natural Gas Neglecting the kinetic effect, the Runge-Kutta algorithm can e used to provide a solution to the differential equation (3) as follows Here p p y (3) 3 p 3 u aa y x 0.x 0.3x.x.x 0.6x..x 0.6x f z T G g Q G g sin p aa L gd z z T R T R u f z T GgP Q av av L Gg sin L, x gd z T R zavtav R f = Moody friction factor evaluated at inlet end pipe T = Temperature at inlet end of pipe T a v = Temperature at mid section of pipe = 0.(T + T ) p = Pressure at inlet end of pipe z = Gas deviation factor evaluated with p and T z = Gas deviation factor calculated with temperature at mid section (T a v) and pressure av at the mid section of pipe (p a v ) given y pav p 0. aa p = Pressure at exit end of pipe, psia p = Pressure at inlet end of pipe Note that p > p and flows occurs from point () to point () Equation (3) is valid in any consistent set of units. Equation (3) can e converted to oil field units. In oil field units in which L is in feet, R = 4, temperature(t) is in o R, g = 3. ft/sec, diameter (d) is in inches, pressure (p) is in pound per square inche (psia), flow rate (Q ) is in MMSCF / Day and P = 4.7 psia, T = 0 o R. The variales aa, u and x that occur in equation (3) can e written as: G f z T Q G sin p g g aa L d zt G f z T Q L G Sin L g av av g u =, x d zavtav

22 Static ehaviour of natural gas and its low in pipes 4 Example Use equation (3) to solve the prolem of example 4 Solution Step : otain the gas deviation factor at the inlet end T = 83 o F = 43 o R P = 4 psia Gg = 0.6 By use of equation (33) and (34) Pc (psia) = x x 0.6 = 67. psia Tc ( o R) = x0.6. x 0.6 = 38. o R Then, P r = 4/67. = 3.784, T r = 43/38. =. The required Ohirhian equation is z z.390 r r log r Fc Where z r r Fc Tr( Tr(37.9 Tr( Tr))) Sustitution of values of Pr = and Tr =. gives z = Step Evaluate the viscosity of the gas at inlet condition. By use of Ohirhian and Au formula (equation 7) xx g cp Step 3 Evaluation of Reynolds numer (R N ) and dimensionless friction factor (f). From eqn. (6) R The dimensionless friction factor can e explicitly evaluated y use of Ohirhian formula (equation 9) d E 4 a E E E 6 x.4log E log log Sustitution of values of a, and x into f = f log a log a h gives

23 46 Natural Gas Step 4 Evaluate the coefficient aa in the formula. This coefficient depends only on surface (inlet) properties. Note that the pipe is vertical = 90 o and sin 90 o = aa Step Evaluate the average pressure (p a v) at the mid section of the pipe given y av p p 0. aa Step 6: Evaluate the average gas deviation factor (za v). Reduced average pressure(p a v r ) = 40./67. = 3.7. av L where geothermal gradient given y : T T L o T a v at mid section of pipe (80 ft) then, is: av R Reduced T av = 8./38. =.6 Sustitution into the Ohirhian equation used in step, gives z = 0.80 Step 7: Evaluate the coefficients x and u x u Step 8: Evaluate y p 3 aa 3 u y..x 0.6x.x.x 0.6x x 0.x 0.3x Sustitution of u = ,p = 4 psia, aa = and x = gives y Step 9: Evaluate p, the pressure at the exit end of the pipe psia 38 psia Pressure drop across 700 ft of tuing is 4 psia 38 psia = 7 psia This pressure drop may e compared with the pressure drop across the 700 ft of tuing when gas flows uphill against the force of gravity. From example 3, tuing pressure at the surface = psia when the ottom hole pressure (inlet pressure) = 4 psia. Then pressure drop = 4 psia psia = 43 psia. The pressure drop during down hill flow is less than that during up hill flow. The general solution (valid in any system of units) to the differential equation for downhill flow was tested with slight modification of a prolem from the ook of Giles et al (009). In the original prolem the pipe was horizontal. In the modification used in this work, the pipe

24 Static ehaviour of natural gas and its low in pipes 47 was made to incline at 0 degrees from the horizontal in the downhill direction. Other data remained as they were in the ook of Giles et al.. The data are as follows: Example 6 Given the following data, Length of pipe (L) = 800ft z = z = z a v = (air is flowing fluid) p = 49. psia = psf = 78 psf W = 0.7 l/sec Q = ft 3 /sec P = 4.7 psia = 6.8 psf T = 60 o F = 0 o R T = T a v = 90 o F = 0 o R G g =.0 (air) R = 44 = 390 x 0-9 l sec/ft d = 4 inch = ft Asolute Roughness () = ft calculate the exit pressure (p ) at 800 ft of pipe. Solution Step : Otain the gas elevation factor at inlet end, z =.0, air in flowing fluid Step : Otain the viscosity of the gas at inlet condition. Viscosity of gas is l sec / ft (given) Step 3: Evaluate the Reynolds numer and friction factor Gg Q R g R d z T 9 o 3 Here, Gg.0(air), 6.8 psf, Q ft /sec,g 3. ft /sec, d ft, R 44, 3900, z.0 (air), T 0 R Then R d From Moody chart, f = Step 4: Evaluate the coefficient aa in the formula f ztgg Q 7.940GgSin aa - gd z T R z T R L

25 48 Natural Gas Then aa = ( ) 800 = Step Evaluate the average pressure (p a v) at the mid section of the pipe p av p psf Step 6: Evaluate average gas derivation factor (z a v ) for air, z a v =.0 Step 7: Evaluate the coefficients x and u 7.94GgSinL x zavtav R 0 44 u av av f z T Gg Q L gd z T R Step 8: Evaluate y p 3 aa 3 u y..x 0.6x.x.x 0.6x x 0.x 0.3x Where x = 0.036, (..x + 0.6x ) =.336, (-.x +.x -0.6x 3 ) = ( x + 0.x -0.3x 3 ) = = = Step 9: Evaluate p, the pressure at the exit end of the pipe Then, y psia p psf 46. psia 44 Pressure drop = 49. psia 46. psia = 3.3 psia When the pipe is horizontal, p (from Fluid Mechanics and Hydraulics) is 4.7 psia. Then, pressure drop = 49. psia 4.7 psia = 3.8 psia

26 Static ehaviour of natural gas and its low in pipes 49 Direct calculation of the gas volumetric rate The rate of gas flow through a pipe can e calculated if the pipe properties, the gas properties, the inlet and outlet pressures are known. The gas volumetric rate is otained y solving an equation of pressure transverse simultaneously with the Reynolds numer and the Colerook friction factor equation. Direct calculation of the gas rate in uphill pipes Ohirhian(00) comined the Weymouth equation with the Reynolds numer and the Colerook friction factor equation to arrive at an equation for the direct calculation of the gas volumetric rate during uphill gas flow. In this section, the formula type solution to the differential equation for horizontal and uphill gas flow is comined with the Reynolds numer and the Colerook equation to arrive at another equation for calculating the gas volumetric rate during uphill gas flow. Comination of the pressure transverse formula for uphill gas flow (equation (7) in oil field units, with the Reynolds numer, equation (6 ) and equation (8 ) which is the Colerook friction factor equation, leads to: Q. - log f 3.7 d.. (36). R - log N 3.7 d (37) - d R - d N log. 007 G 3.7 d 007 G (38) 007 G g g x p - p - S Lx 6 6 a d B z T x a z av T av x c g Where, S x sin L z av T av G g 4.90 G g L B d G g sin p z T z av is computed with T av = 0.(T + T ) and p av = 0.(p + p ). is viscosity of gas at p and T. The suscript refers to surface condition.

27 460 Natural Gas x x 0.x 0.36x 3 a x 4.96x.48x 0.7x 3 x x 0.7x c The aove equations are in oil field units in which d is expressed in inches, L in feet, in centipoises, T in degrees rankine and pressure in pounds per square inches. In this system of units, p = 4.7 psia, T = 0 R, z =.0. The suscript refers to surface condition in the gas well. Example 7 A gas well has the following data: L = 700ft, G g = 0.6, 90, z = 0.78, z av = 0.8, f = 0.076, T = 43 R, T av = 8. R, d =.996 in, asolute roughness of pipe= in.,p = 4 psia, p = psia,, viscosity of gas at p and T. Calculate the flow rate of cp the well ( Q ) in MM SCF / Day, Solution Sustituting given values, x B S x a x 0.x 0.36x x =.378 x 4.96x.48x 0.7x x =.4096 x x 0.7x c = x x0.680 =

28 Static ehaviour of natural gas and its low in pipes 46 From example 3, actual Reynolds numer is.34e06 and f = Then, R N f = 898. Q - d log log G 3.7 d g =.4 MMSCF / Day Direct calculation of the gas volumetric rate in downhill flow Comination of equation (3) in oil field units, with the Reynolds numer, equation (6 ) and equation (7 ) which is the Colerook friction factor equation, leads to:. - log. (39) f 3.7 d R - log N 3.7 d.. (40) Q - d R - av d av N log 007 G g 007 G 3.7 d g. (40) 007 G g d B z T x d z av T av x f x e 0. J ( S Lx ) 6 d 0. J p - p, if B S 6 x e 4) (4a) J p - - p, if B S 6 (4) x x 0.x 0.3x 3 d x -.x.x - 0.6x 3 e x. -.x 0.6x f B 4.90 G g L d

29 46 Natural Gas S x G g sin p z T sin L z av T av G g z av and av are evaluated with T av = 0.(T + T ) and p av = (p p ) / (p + p ) The suscript refers to surface condition and to exit condition in the gas injection well. The aove equations are in oil field units in which d is expressed in inches, L in feet, in centipoises, T in degrees rankine and pressure in pounds per square inches. In this system of units, p = 4.7 psia, T = 0 R, z =.0. The suscript refers to surface condition in the gas well. Example 8 A gas injection well has the following data: L = 700ft, G g = 0.6, θ = 90 o, sin90 o =, z = , z av = 0.8, T = 43 o R, T av = 43 o R, d =.996 in, p = 4 psia, p = 37.9 psia, asolute roughness of tuing() = in. Calculate the gas injection rate in MMSCF / Day. Take the viscosity of the at surface condition( ) as cp and the average viscosity of the gas( av ) as 0.04 cp. Solution d x B S x x 0.x 0.3x 3 = = x -.x.x - 0.6x 3 e = x. -.x 0.6x = f

30 Static ehaviour of natural gas and its low in pipes 463 Since B > S, J p - p is used J ( S Lx d ) 6 = ( ) z T x d z av T av x f G g x e = J ( S Lx ) 6 d d B z T x d z av T av x f From example, actual Reynolds numer is and f = Then, RN f = d Q av log log G 3.7 d g =.7 MMSCF / Day By use of an average viscosity of cp, the calculated Q =.3 MMSCF / Day. Taking cp as the accurate value of the average gas viscosity, the asolute error in estimated average viscosity is (0.000 / 0.04 =.43 %). The equation for the gas volumetric rate is very sensitive to values of the average gas viscosity. Accurate values of the average gas viscosity should e used in the direct calculation of the gas volumetric rate. Conclusions. A general differential equation that governs static ehavior of any fluid and its flow in horizontal, uphill and downhill pipes has een developed.. classical fourth order Runge-Kutta numerical method is programmed in Fortran 77, to test the equation and results are accurate. The program shows that a length ncrement as large as 0,000 ft can e used in the Runge-Kutta method of solution to differential equation during uphill gas flow and up to 700ft for downhill gas flow 3. The Runge-Kutta method was used to generate a formulas suitale to the direct calculation of pressure transverse in static gas pipes and pipes that transport gas uphill or downhill. The formulas yield very close results to other tedius methods availale in the literature.

31 464 Natural Gas 4. The direct pressure transverse formulas developed are suitale for wells and pipelines with large temperature gradients.. Contriution of kinetic effect to pressure transverse in pipes that transport gas is small and ca n e neglected.. 6. The pressure transverse formulas developed in this work are comined with the Reynolds numer and Colerook friction factor equation to provide accurate formulas for the direct calculation of the gas volumetric rate. The direct calculating formulas are applicale to gas flow in uphill and downhill pipes. Nomenclature p = Pressure = Specific weight of flowing fluid v = Average fluid velocity g = Acceleration due to gravity in a consistent set of umits. d = Change in length of pipe = Angel of pipe inclination with the horizontal, degrees dh l = Incremental pressure head loss f = Dimensionless friction factor L = Length of pipe d = Internal diameter of pipe W = Weight flow rate of fluid C f = Compressiility of a fluid C g = Compressiility of a gas K = Constant for expressing the compressiility of a gas M = Molecular weight of gas T= Temperature R N = Reynolds numer = Mass density of a fluid =Asolute viscosity of a fluid z = Gas deviation factor R = Universal gas constant in a consistent set of umits. Q = Gas volumetric flow rate referred to P and T, = specific weight of the gas at p and T p = Base pressure, asolute unit T = Base temperature, asolute unit z = Gas deviation at p and T usually taken as G g = Specific gravity of gas (air = ) at standard condition g = Asolute viscosity of a gas = Asolute roughness of tuing GTG = Geothermal gradient f = Moody friction factor evaluated at outlet end of pipe.

32 Static ehaviour of natural gas and its low in pipes 46 z = Gas deviation factor calculated with exit pressure and temperature of gas z = Gas deviation factor calculated with exit pressure and temperature of gas p = Pressure at exit end of pipe. p = Pressure at inlet end of pipe p > p T = Temperature at exit end of pipe T = Temperature at inlet end of pipe T a v = 0.( T + T ) z av Gas deviation factor evaluated with T a v and average pressure (p a v ) given y p p a v 0.aa in uphill flow, and in downhill flow SI Metric Conversion Factors F - 3 / 8 C ft *E-0 m m.40 * E 00 cm lf E 00 N lm E - 0 kg psi E 03 Pa l sec / ft E 0 Pa.s cp.0 E - 3 Pas pav p 0. aa foot 3 / second (ft 3 / sec) x.8368 E 0 = metre 3 / sec ( m 3 / s ) foot 3 / second x 8.64 *E - 0 = MMSCF / Day MMSCF / Day x.7407 E + 0 = foot 3 / second (ft 3 / sec) MMSCF / Day x E + 0 = metre 3 / sec ( m 3 / s ) MMSCF / Day = E + 04 ft 3 /hr ft 3 /hr = E 06m 3 /s * Conversion factor is exact. References. Colerook, C.F.J. (938), Inst. Civil Engineers,, p 33. Cullender, M.H. and R.V. Smith (96). Practical solution of gas flow equations for wells and pipelines with large temperature gradients. Transactions, AIME 07, pp Giles, R.V., Cheng, L. and Evert, J (009).. Schaum s Outline Series of Fluid Mechanics and Hydraulics, McGraw Hill Book Company, New York. 4. Gopal, V.N. (977), Gas Z-factor Equations Developed for Computer, Oil and Gas Journal, pp Ikoku, C.U. (984), Natural Gas Production Engineering, John Wiley & Sons, New York, pp

33 466 Natural Gas 6. Matter, L.G.S. Brar, and K. Aziz (97), Compressiility of Natural Gases, Journal of Canadian Petroleum Technology, pp Ohirhian, P.U.(993), A set of Equations for Calculating the Gas Compressiility Factor Paper SPE 74, Richardson, Texas, U.S.A. 8. Ohirhian, P.U.: Direct Calculation of the Gas Volumeric Rates, PetEng. Calculators, Chemical Engineering Dept. Stanford University, California, U.S.A, Ohirhian, P.U. and I.N. Au (008), A new Correlation for the Viscosity of Natural Gas Paper SPE 0639 USMS, Richardson, Texas, U.S.A. 0. Ohirhian, P.U. (00) Explicit Presentation of Colerook s friction factor equation. Journal of the Nigerian Association of Mathematical physics, Vol. 9, pp Ohirhian, P.U. (008). Equations for the z-factor and compressiility of Nigerian Natural gas, Advances in Materials and Systems Technologies, Trans Tech Pulications Ltd, Lauisrtistr. 4, Stafa Zurich, Switzerland... Ouyang, I and K. Aziz (996) Steady state Gas flow in Pipes, Journal of Petroleum Science and Engineering, No. 4, pp Standing, M.B., and D.L. Katz (94) Density of Natural Gases, Trans AIME 46, pp Standing, M.B. (970), Volumetric and Phase Behavior of Oil Field Hydrocaron Systems, La Hara, California.

34 Natural Gas Edited y Primoà ¾ Potoà  nik ISBN Hard cover, 606 pages Pulisher Sciyo Pulished online 8, August, 00 Pulished in print edition August, 00 The contriutions in this ook present an overview of cutting edge research on natural gas which is a vital component of world's supply of energy. Natural gas is a comustile mixture of hydrocaron gases, primarily methane ut also heavier gaseous hydrocarons such as ethane, propane and utane. Unlike other fossil fuels, natural gas is clean urning and emits lower levels of potentially harmful y-products into the air. Therefore, it is considered as one of the cleanest, safest, and most useful of all energy sources applied in variety of residential, commercial and industrial fields. The ook is organized in chapters that cover various aspects of natural gas research: technology, applications, forecasting, numerical simulations, transport and risk assessment. How to reference In order to correctly reference this scholarly work, feel free to copy and paste the following: Peter Ohirhian (00). Static Behaviour of Natural Gas and its Flow in Pipes, Natural Gas, Primoà ¾ Potoà  nik (Ed.), ISBN: , InTech, Availale from: InTech Europe University Campus STeP Ri Slavka Krautzeka 83/A 000 Rijeka, Croatia Phone: +38 () Fax: +38 () InTech China Unit 40, Office Block, Hotel Equatorial Shanghai No.6, Yan An Road (West), Shanghai, 00040, China Phone: Fax: