Figure 6.1 Model for radial flow of fluids to the wellbore

Transcription

1 In Chater 3 e develoed the oundations o luid lo through orous media and introduced Darcy s La; the undamental lo euation to deribe this behavior. In this chater e ill exand Darcy s La and investigate various orms and alications useul or engineers. We ill begin ith various geometries or single hase, incomressible lo, migrate to comressible lo euations, and end ith multihase lo euations, by alying multihase lo concets rom Chater 5. The second section o this chater resents the mathematical develoment o governing euations and associated conditions or deribing luid lo. This ill lay the rameor to solutions in receding chaters. 6. Alications o Darcy s La 6.. Radial Flo U to no the euations develoed or luid lo assume linear lo. Hoever, lo into a ellbore rom a cylindrical drainage region is radial as shon in Figure 6.. Figure 6. Model or radial lo o luids to the ellbore The to conditions are given by an external boundary ressure, e, located at boundary radius, r e and the internal bottomhole loing ressure, at the ellbore radius, r. In terms o Darcy velocity, v A rh d dr (6.) 6.

2 Note since the radius direction is measured ositive in the direction oosite to lo, the otential gradient is ositive. E. (6.) can be searated and integrated ith resect to the to boundary conditions, resulting in, h( e r B o ln e r ) (6.) Euation (6.) is exressed in Darcy units. Conversion to ield units yields the olloing idely used exression or radial lo o a single-hase luid. Examle 6. h( e r 4.B o ln ) e r (6.3) A ell is roducing in a radial reservoir at a bottom hole ressure o 5,500 si. The reservoir ressure is 6,000 si. Oil viosity is 0.5 c and the ormation volume actor is.5 bbl/stb. I the ermeability o the reservoir is 0 md, the thicness is 30 t and the drainage radius is,000 t., at hat rate ill the ell roduce? The ell bore radius is 6". I, by alying artiicial lit method, the bottom hole ressure is reduced to 3,000 si, at hat rate ill the ell roduce? Solution Using E. 6.3, (0)(30)(6,000 5,500) 4.(0.5)(.5) ln, stbd I the bottomhole ressure is reduced to 3,000 sia, e ill get, (0)(30)(6,000 3,000) 4.(0.5)(.5) ln, ,47stbd 6.

3 a substantial increase in roduction by loering the bottom hole loing ressure through artiicial lit. Examle 6. A ell is roducing under reservoir ressure maintenance roject. The reservoir ressure is 8,000 si and the minimum ossible ellbore ressure ithout using artiicial lit is 7,500 si. Well test analysis shos a sin actor o +4. Determine:. Well lo rate under these conditions (steady-state radial lo). Well lo rate ater an acid job that imroves the sin actor to S = 0. Also, given: Bo o h r r e Solution =.5 Res. bbl/stb = 0.5 c = 00 md = 30 t = 6 inches =,000 t. The radial lo euation can readily be modiied to include steady state sin, h( e r 4.B o ln e r Subseuently, the lo rate or the damaged case is, (00)(30)(8,000 7,500) 4.(0.5)(.5) ln ,0 stbd Ater the acid treatment the rate increases to, (00)(30)(8,000 7,500), ln (0.5)(.5) 0 863stbd Permeability o Combination Layers ) S (6.4) Most orous rocs have satial variations o ermeability and may be comrised o distinct layers, blocs, or concentric rings o constant ermeability. To determine the average ermeability o such a system, consider the olloing cases. 6.3

4 Case I: Layered reservoirs ithout crosslo Reservoir rocs are interbedded ith imermeable shales or silts such that no crosslo exists beteen sand beds (Figure 6.). In this case, t = l and t = = Figure 6. Linear lo, arallel combination o beds Substituting Darcy s euation in or lo rates, results in, h t h h h L hich reduces to, or h t Examle 6.3 L L h h h L L n i h i i n h i i L (6.5) What is the euivalent linear ermeability o our arallel beds having eual idths and lengths under the olloing conditions? Bed thicness, h, t. ermeability,, md

5 Solution Alying E. (6.5) here n = 4, (50)(5) (00)(8) (30)(5) (80)(0) 33md ( ) Similar to linear lo, e can develo average ermeability euations or radial lo systems. I the beds are arranged in arallel as shon in Figure 6.3, it can be shon that E. (6.5) is still valid. Figure 6.3 Radial lo, arallel combination o beds Case II: Comosite Reservoirs A comosite reservoir is deribed by variations in roerties occurring aay rom the ellbore. These variations could be induced by drilling and comletion ractices (invasion o luids into the reservoir), by injection o ater during aterlooding oerations or could be natural to the reservoir. The system is simliied to a set o dierent blocs arranged in series as shon in Figure 6.4. Figure 6.4 Linear lo, combination o beds in series 6.5

6 6.6 In this case, t = l = = 3 and t = + + 3, Substituting Darcy s euation in or the ressure dro, results in, 3 3 L L L h L h hich can be ritten in general orm as: n i i i L L (6.6) A similar exression can be develoed or radial lo o multile beds (Figure 6.5) Figure 6.5 Radial lo, combination o beds in series h r e r h r r h r e r ln ln ln Simliying to general terms, n i i i r i r r e r ln ln (6.7)

7 Examle 6.4 Consider a radial system comrised o three zones ith the olloing roerties. Layer Inner Radius Outer Radius Permeability Calculate the average ermeability. Solution Using E. (6.7), Examle 6.5 ln 0 3.4md ln ln 50 ln For a coniguration shon belo, calculate the horizontal and the vertical average ermeabilities. 50 Solution a. I lo is in the horizontal direction, then the beds are arranged in a arallel seuence. Thereore E. (6.5) results in the average horizontal ermeability. (7)(40) (5)(0) (4)(70) h 60md b. I the lo is vertical, the beds are in series and alying E. (6.6) rovides the average vertical ermeability. v md

8 Darcy s La and Filtration The API static iltration test is indicative o the rate at hich ermeable ormations are sealed by the deosition o a mudcae ater being enetrated by the bit. The lo o mud iltrate through a mudcae is deribed by Darcy's la. Thus, the rate o iltration is given by, dv dt here the variables are deined belo and on Figure 6.6. dv /dt = the iltration rate, cc/s, A m h mc = the ermeability o the mudcae, darcies, = the area o the ilter aer, cm, = the ressure dro across the mudcae, atm, = the viosity o the mud iltrate, c, and = the thicness o the ilter (mud) cae, cm. mc A (6.8) h m mc(t) P in A Filter aer mudcae h mc P out Figure 6.6 Schematic o iltration rocess At any time, t, during the iltration rocess, the volume o solids in the mud that has been iltered is eual to the volume o solids deosited in the ilter cae. Deining sm, as the volume raction o solids in the mud and, is the volume raction o solids in the cae, a volume balance euation can be ritten. thereore, sm V m h mc A (6.9) V h mc (6.0) A sm 6.8

9 Inserting this exression or h mc and integrating, results in [Bourgoyne, 99], V A * t (6.) sm Notice the iltrate volume is roortional to the suare root o time; thereore tyical iltrate loss data is lotted as shon belo in Figure 6.7. V s is the surt loss volume hich V V s t occurs rior to the buildu o mudcae. Examle 6.6 Figure 6.7 Examle o iltrate volume loss during iltration test A standard iltration test is oerated at 00 sig (6.8 atm) through a cross-sectional area o 45 cm. Consider 5 ml o iltrate ith a viosity o.0 c is collected in 30 minutes. The cell is disassembled and the mudcae thicness is measured to be 3/3. ermeability o the mudcae. Solution Rearranging E. (6.0) to solve or the volume raction ratio, e obtain, sm V 5 cc Ah mc cm 45cm *3/ 3in *.54 in Substituting this value into E. (6.) and solving or ermeability, 5 { cc} { d} * 45 { cm c 0.065md } *0.4666*6.8atm *30min Determine the 6.9

10 6..3 Comressible Flo Euations develoed earlier or liuid lo considered the luids to be incomressible, ith a constant density in the lo system, hich is not a bad assumtion or many liuid lo roblems. Hoever, gas is so highly comressible that euations or liuid lo cannot be used. Derivation o gas lo rate or a linear horizontal system ollos. Darcy s euation in dierential orm can be ritten as, 3 3 A dp t.7x0 *5.65 dl bbl 3 A dp 6.38x0 dl here is the lo rate at reservoir conditions, rc/d. (6.) Naturally, it is most desirable to determine gas lo rate at surace, or standard conditions in units o standard cubic eet er day (/d). For steady-state conditions, the lo rate in d ( ) is constant everyhere in the system at any ressure. Beginning ith Boyle's la or a real gas under isothermal conditions, here: P V Z T res T = absolute ressure, sia; = volume, cu t, = comressibility actor; PV zt = reservoir temerature, R; res PV zt = temerature at standard conditions, R. (6.3) For volumetric lo rates Euation 6.3 can be ritten as, hich can be rearranged to, P zt res P zt (6.4) P T z res res res (6.5) P T res z 6.0

11 Substituting E (6.5) or reservoir lo rate in E. (6.), and searating variables results in, L AT dx P T d ( ) z ( ) (6.6) Note that viosity and z-actor are strong unctions o ressure; thereore to solve E. (6.6) to aroximations ere develoed. Figure 6.8 illustrates the variation in viosity and z- actor ith ressure. In the lo-ressure region ( < 000 si), z = constant and can be evaluated at average ressure ( + )/ z, c g z = constant constant g z ressure,sia Figure 6.8 viosity-z-actor variation ith ressure Subseuently, E. (6.6) can be integrated, resulting in here, {d} {md} A {t } T {R} P {sia} L {t} {c} AT ( ) (6.7) P T( z) i L 6.

12 In terms o lab use e change to darcy units. The linear lo euation or a comressible gas becomes, AT ( in out ) (6.8) 000P T( z) i L here, {c/sec} {md} A {cm } T {R} P {atm} L {cm} {c} A second aroximation occurs at high ressures ( > 4000 sia). Insection o Figure 6.8, shos that at high ressures the sloe is constant; thereore z z (6.9) Substituting E. (6.9) in or the viosity/z-actor roduct o E. (6.6) and integrating, results in AT ( ) (6.0) P T( z) L In laboratory or, the linear lo o comressible gas is in Darcy units and is given by, AT ( in out ) (6.) 000P T( z) L Shortcomings exist or both aroximations hen ut to ractical use. The derivations o both ere based on the assumtion that and z ere constants and ere removed rom the integral. In act, and z both vary substantially ith ressure. Also, both euations reuire and z to be evaluated at some "average ressure," hich is diicult to estimate. 6.

13 These diiculties involving true average ressure and the assumtion that and z are constants can be avoided by using the real-gas seudoressure, or real-gas otential m(), resented by Al-Hussainy et al in 966. m( ) d (6.) z b Relacing the integral term in E. 6.6 ith m() and solving results in, AT [( m( ) m( )] (6.3) P TL The advantages o E. (6.3) are it s alicability or the entire ressure range and the avoidance o averaging ressure and assuming z is constant. The disadvantage is the calculations or m(). Examle 6.7 Assume steady state, horizontal lo o a comressible gas through a core as shon in the diagram belo. Find the ermeability o the samle. g P T Tout = 0.0 c = atm = 60F = 70F P in =.45atm P out =atm A=cm L=cm V=000cc t=500sec The outlet lo rate o = 000/500 = cc/sec. The lo rate in standard conditions rom E. (6.5) is, 50.96cc / sec

14 Solving E. (6.8) or ermeability, 000()(.0)()(.96) md 50 (.45 ) Similar euations can be derived or radial lo. For the sae o comleteness the three orms are given belo. Lo-ressure aroximation: ht P T ( ) e r ( z) ln e i r (6.4) High-ressure aroximation: Real Gas otential: ht P T ( e r ( z)ln e r ) (6.5) ht [ m( ) m( )] e (6.6) r P T ln e r 6.4

15 6..4 High-velocity lo Darcy's exeriments and the resulting euations used by etroleum engineers ere based on strictly laminar lo; that is,. At increased lo rates, this relationshi is no longer valid (see Figure 6.9). At high velocity lo, the use o Darcy's La in the orm resented so ar ould lead to calculated ermeabilities that are less than the true ermeability o the roc. Turbulent lo Q/A Laminar lo / /L Figure 6.9 Eect o high-velocity lo on Darcys Euation It should be evident that or a given ressure dierential, the velocity o gas is greater than the velocity o liuid due to the viosity dierence. Thereore, this high velocity lo eect has a signiicant imact on gas ell testing such as isochronal and deliverability tests. In the laboratory, accurate results reuire the correction or Klinenberg Eect at lo ressures and the non-darcy lo a high ressures. To resolve this issue o non-darcy lo, Forchheimer in 90 roosed a uadratic lo euation to deribe both laminar and turbulent conditions. Units o the arameters are: dp dl {atm} {darcy} L {cm} {gm/cc} v v (6.7) 6.5

16 {c} v {cm/sec} {atm-sec /gm} hereis the non-darcy lo coeicient. dimensions o /length; thereore the conversion actor is given by, atm s 7 *3.0889x0 t gm Dimensional analysis reveals that has A ey to alying the Forchheimer euation is to estimate a value or. Methods develoed to calculate are based on exerimental or, correlations, and rom the Forchheimer euation. To determine exerimentally, the rocedure is to irst measure the absolute ermeability o each o the core samles and then to aly a series o increasing ressure dierentials across each samle by loing air through the core lugs at ever increasing rates. Knoing the lo rates and ressure dierentials across the lugs, the coeicient o inertial resistance can be directly calculated using a linear version o the Forchheimer euation (6.7). The results are shon in Figure 6.0 in hich is lotted as a unction o the absolute ermeability over the range o core samles tested. Figure 6.0 Laboratory determined or non-darcy lo [Dae, 978] 6.6

17 A relationshi is usually derived o the orm, cons tant (6.8) a in hich the exonent, a is a constant. For the exerimental results shon in Figure 6.0, the seciic relationshi is, 0.73x0 { t } (6.9).045 { md} This examle is or single-hase gas lo ith a limited range o orosity variation (neglecting the eect o orosity on non-darcy lo). Many correlations exist or in the literature, ith most using absolute ermeability as the correlating arameter. See Li, et al., 00 or a comrehensive literature revie o correlations. Figure 6. is idely oular correlation develoed by Firoozabadi and Katz in 979. Figure 6. Correlation o ith absolute ermeability [Firoozabadi & Katz, 979] 6.7

18 The emirical relationshi develoed as, 0.33x0 { t } (6.30).0 { md} Li (00) attemted to uniy the various correlations into a single general euation. The results o his or lead to, c(, ) (6.3).5.5c.5c3 here c, c and c 3 are constants or a given orous media, ith c + c 3 =. It as determined or arallel caillary tubes that c = 0 and c 3 =, hile or caillary tubes in series, c = and c 3 = 0. These are uer and loer limits, ith actual orous media residing beteen the to endoints. The last alternative to estimate is to rearrange the Forchheimer euation in the orm a b (6.3) A cartesian lot o / vs. ill result in a sloe o b and a y-intercet o a (Figure 6.). / Sloe = b Intercet = a Figure 6. Cartesian lot o Forchheimer euation In laboratory alications, a and b are derived or linear, steady state lo in Darcy units. P L T a z (6.33) g A T 6.8

19 5 ML P Tz b 8.76x0 (6.34) T A Temerature is exressed in R, molecular eight (M) in gm/gm-mole and is in cm -. The gas ermeability is solved rom the y-intercet (a), and the non-darcy coeicient is solved rom the sloe (b); resectively. In reservoir conditions the lo is radial and the units are reuently in ield units. Thereore, 4zT r a ln e h r g (6.35) 3.6x0 zt g b h r (6.36) here is in t - and is md. Examle 6.8 The olloing isochronal test data are available rom a gas ell. Additional inormation available are: S g = 48% = 3% = 5.5 md r e = 980 t. h = 54 t. g = c r = 0.76 t z = r = 798 sia T = 64 F g = 0.65 Find the non-darcy coeicient rom the a) exerimental method, b) emirical correlation and c) rom the Forchheimer euation.,mmd, sia

20 Solution a. Alying the exerimental relationshi o E. (6.9), results in = 4.5 x 0 9 t -. Recall, this relationshi as derived or a seciic set o samles and thereore is not a general euation to use. b. Alying the emirical euation (6.30) gives = 3.0 x 0 9 t -. c. Plotting the isochronal test data, /, sia /mmd y = x R = , mmd Results in a sloe (b) = sia /mmd and an intercet (a) = sia /mmd. Substituting this sloe into E. (6.36) results in, x0 (54) (.76) x0 t 3.6x0 (.8743)(64)(0.65) and the intercet into (E. 6.35) gives. 4(.055)(.8743)(64) 980 g ln 5. 4md (38.33)(54).76 The last value o is considered the most accurate because actual data or the reservoir as used to generate the anser. 6.0

21 6..5 Fracture Flo In Chater e diussed a simle dual orosity model comosed o matrix and racture orosity. The majority o the storage o luids is in the matrix. Hoever, in terms o luid lo and lo caacity the ractures dominate and the matrix tyically contributes a minor amount. Consider a matrix bloc o negligible orosity intersected by a racture o idth, and height, h as shon in Figure 6.. The lo rate is arallel to the racture length, hich coincides to the rincial direction. The volumetric lo through a slot can be Figure 6.. Unit model or racture ermeability deribed by Bucingham s euation or lo through slots o ine clearance, ( n A) h 3 here n is the number o ractures er unit area. L (6.37) 6.

22 Tortuosity can be included in the lo euation be multilying and dividing the right-hand side by the unit length, L. Euation (6.37) becomes, ( n A) h 3 L (6.38) Furthermore, e can substitute (E..) into E. (6.38) to include orosity in the lo euation. A L (6.39) Euation (6.39) is a modiied Bucingham euation or lo through multile ractures o tortuous ath. I e comare E. (6.39) ith Darcy s La or lo through orous media, A L e can obtain an exression or ermeability. (6.40) (6.4) This exression is in similar orm to the caillary tube models here the caillary radius, r and constant 8 are relaced by the racture idth, and constant. units the resulting relationshis are: In terms o Darcy 5 { darcy} 84.4x0 { cm }* / 6 { darcy} 54.5x0 { in }* / (6.4) Note that the eect o multile ractures on ermeability is accounted or in the orosity term. Examle 6.9 A cubic bloc ( t 3 ) o a carbonate roc has a single horizontal racture o idth cm. What is the racture ermeability? I the number o ractures as doubled; i.e., ractures/t hat is the increase in racture ermeability? I lo direction is arallel to the racture and matrix system, hat is the average ermeability o the bloc i the matrix ermeability is md.? Water as injected ith a viosity o c and a ressure dro o 0 sia as recorded. What is the matrix and racture lo rates or these conditions? 6.

23 Solution a. From Chater, the racture orosity, =.00635/(*.54) = 0.0%. The racture ermeability is calculated rom E. (6.4), x0 ( ) *(0.000) 7md Recall that the racture orosity is deined as the racture ore volume ith resect to the bul volume o the roc. Thereore this ermeability is the racture ermeability ith resect to that same volume. This is dierent rom isolating the racture only. The ermeability ould be ininite in the racture in this case. A urther case ould be i e deine the racture orosity as unity ith lo resect to the bloc area, then ermeability ould be 340 darcys. This is the average ermeability or the racture layer. b. The number o racture or the bloc is n = ractures/t * t =. Subseuently, the ermeability ould double to 4 md. c. I lo is arallel to the alternating layers o matrix and ractures, e can use E. (6.5) to determine the average ermeability. m h m h h t (md)(.00008){ t} (340000md)( t) 7md t or m 7 7md d. The racture lo rate is, 3 (7md)( t )(0 sia).7x bd (c)( t) and a similar euation or the matrix results in m = 0.0 bd. Notice the roduction rate is 70 times greater in the racture than the matrix, illustrating the imortance o ractures on roduction rate. 6.3

24 Using the deinition o S v and alying to ractures, e obtain, S v S v A s V h ( n A) * ( ( n A) * h h ) L L (6.43) Since the racture idth is much smaller than the racture height, then / >> /h, and E. (6.43) simliies to, S v (6.44) Substituting this relationshi or seciic surace areas into the exression or ermeability (E. 6.4), results in a Carmen Kozeny tye euation or ractures. (6.45) 3S v We can generalize this relationshi urther by relacing the constant ith a racture shae actor, s. Subseuently, the eective zoning actor, T can be deined as, * (6.46) T s Substituting into E. (6.45) results in a generalized Carmen-Kozeny euation or ractures. (6.47) S T v This exression can be alied to identiy and characterize hydraulic lo units; subseuently, e can deine H c as the hydraulic unit characterization actor or naturally ractured systems as, H S (6.48) c T v This actor can be uantiied ith the aid o core analysis and etrograhic image analysis. 6.4

25 Eect o Fracture Shae The constant in E. (6.45) is only valid or a racture shae o a rectangular aralleloied. To investigate this behavior, consider a racture ith an ellisoidal shae as shon in Figure 6.3. Figure 6.3. Eect o racture shae on the ermeability orosity relationshi Assuming the lo only occurs in the racture netor and that no matrix orosity exists, then the seciic surace area er unit ore volume is deined only by the ellisoidal cylinder. A s V h L (6.49) 4 3 L * h h (6.50) 4 Combining Es. (6.49) and (6.50) and simliying, S v 3 h (6.5) h Based on the remise that racture idth is much smaller than the racture height, / >> /h, and also assuming the suare root o h >> (less than 5% error), then the exression or seciic surace area is reduced to, 6.5

26 S v 3 (6.5) Thus clearly demonstrating the eect o racture shae and the reliance o racture geometry in the hydraulic unit characterization actor. Hydraulic radius o a racture An eective hydraulic radius or the racture can be derived by euating the Hagen Poiseuille euations or a caillary tube and racture. The resulting relationshi is, solving or hydraulic radius, r h, 4 r h 8 r h 3 h / 4 3 h 3 (6.53) (6.54) We can relate the hydraulic radius to ermeability through the Kozeny euation or caillary tubes, r 3 h (6.55) I e consider the racture as a slit, then S v = / can be substituted into E. (6.55), h (6.56) 3 3 S v The imortance o Euations (6.55) and (6.56) is related to the eect o racture height on the ermeability orosity relationshi. For examle, racture height can be determined i the other arameters are estimated by core, log or other tye o analysis. 6.6

27 6..5 Multihase Flo Darcy s La can be extended to multihase lo using concets develoed in Chater 5 on relative ermeability. In terms o suericial velocity or a D roblem, ri i ui (6.57) i x here i = o, g and deending on the hase and ermeability is the eective ermeability o the hase. The otential or each hase is given by i d i i g( z z d ) (6.58) id i here z is the elevation above a horizontal datum, z d is the elevation o the reerence datum and id is the hase ressure at the reerence datum. The hase ressures are related through caillary ressure curves. The eective ermeability o a hase are unctions o the saturation, and vary according to the relative ermeability curves. An examle o alying relative ermeability curves to lo rates is given in Section or revie. 6. Dierential Euations or Fluid Flo The objective o this section is to understand and develo undamental euations or luid lo through orous media. Direct alications occur in ell testing, simulation, and roduction decline analysis, and ill orm the basis or later develoment o secondary recovery rocesses. Detailed derivations can be ound in Collins (96), Lee (98), Matthes and Russell (967), and Musat (98). A mathematical derition o luid lo in a orous medium can be obtained by combining the olloing hysical rinciles: () the la o conservation o mass, () a momentum euation, e.g., Darcy's la and (3) an euation o state. In lo o any tye the conservation rincile is simly a statement that some hysical uantity is conserved, i.e., neither created nor destroyed. In luid lo in a orous medium, the most signiicant uantity conserved is mass. 6.7

28 . La o conservation o mass mass into mass out o net amount o rate o the element the element mass introduced accumulation Figure 6.4. Volume element in Cartesian coordinates Begin by alying the rincile o continuity to an arbitrary volume element ithin the lo region. As an examle, consider the three-dimensional case shon in Figure 6.4. The volumetric comonents o lo into the element in the x, y and z directions are denoted by v x, v y, and v z, resectively. These are volumetric lo rates er unit o cross-sectional area. Thus, the mass lo rate into the element in the x-direction is u yz x The mass lo rate in the x-direction out o the element is u (u ) yz x x here (u x ) is the change in mass lux that occurs ithin the element. The net lo rate in the x-direction (amount-in less the amount-out) is ( u ) yz x Similar exressions can be ritten or the y and z directions. Assuming no mass is generated or lost in the element, the amount o net mass change in the element in a time element t is, 6.8

29 t u u u V x y z y x z z x Dividing by the volume(v)-t roduct and taing the limit as x, y, z, and t aroach zero, results in the continuity euation or lo o luid in a orous media, and can be ritten as, ( v) G ( ) (6.59) t here G reresents mass introduced (source) or mass removed (sin) ithin the system.. Momentum Euation Darcy's la exresses the act that the volumetric rate o lo er unit cross-sectional area (v) at any oint in a uniorm orous medium is roortional to the gradient in otential ()in the direction o lo at that oint. The la is valid or laminar lo and can be ritten as: v (6.60) In the above euation, note that lo occurs in the direction o decreasing otential. Hubbert has studied Darcy's la and its imlications uite extensively and shoed that y tt t d gz (6.6) o here z is the height above an arbitrary datum lane. Subseuently, Darcy s La becomes v giˆ z (6.6) Also, note in Es. (6.60 and 6.6) that ermeability is a tensor roerty. Including all cross terms, ermeability can be exanded to: xx xy xz yx yy yz (6.63) zx zy zz I the ermeability is aligned ith the rincial directions, then only the three diagonal terms are resent. I the media is considered isotroic, then the three diagonal terms reduce to a single valued ermeability. Combining Darcy s La ith the continuity euation results in, 6.9

30 i ( ) G ( ) (6.64) t 3. Euation o state Various euations o state are used in deriving the lo euations. An euation o state seciies the deendence o luid density on the luid ressure and temerature T. Thus, deending on the actual luid(s) resent, an aroriate euation o state must be chosen. The simliest examle is to consider the luid incomressible, d 0 (6.65) dp Thereore the density is constant and can easily be eliminated rom E. (6.64). An imortant class o lo euations is the isothermal lo o luids o small and constant comressibility; i.e., an ideal liuid. c (6.66) P The resulting governing dierential euation becomes, c P P (6.67) t The assumtions to obtain E. (6.67) are: a. comressibility is small and constant b. ermeability is isotroic and constant c. orosity is constant d. gradient suared terms are small e.? Alternative solutions can be obtained given constraints or assumtions are relaxed. Several o the ossibilities are:. Pressure deendent orosity and ermeability. incomressible luid 3. anisotroy (aligned ith rincial directions) 4. comressible luid 5. multihase lo 6.30

31 A inal imortant class o lo euations deribes the lo o a real gas through a orous media. The euation o state or a comressible luid; i.e., real gas is given by, The resulting governing euation becomes, PM (6.68) z (, T) RT c z t z (6.69) To account or the non-linearity in the gas governing euation, three aroaches have been taen to handle the ressure deendent luid roerties.. Lo ressure aroximation. High ressure aroximation 3. Real gas otential or seudoressure unction (Al-Hussainy, et al.,966) m( ) d (6.70) z b Figure 6.5 Diagram illustrating variations o z ith ressure. Beore develoing solutions to the dierential euations or lo through orous media, e should oint out that a dierential euation deribes only the hysical la or las hich aly to a situation. To obtain a solution to a seciic lo roblem, one must have not only the dierential euation, but also the boundary and initial conditions that characterize the articular situation o interest. Common conditions used in luid lo are: 6.3

32 Initial and Boundary Conditions Initial condition P(r,0) P i The cases o interest are (a) an ininite acting reservoir the case here a ell is assumed to be situated in a orous medium o ininite extent, (b) bounded reservoir the case in hich the ell is assumed to be located in the center o reservoir ith no lo across the external boundary, and (c) constant otential outer boundary - the case in hich the ell is assumed to be located in the center o an area ith constant ressure along the external boundary. Listed belo and shon in Figure 6.6 is a derition o these boundary conditions. a. ininite reservoir ψ(, t) ψ i b. Constant otential boundary ψ( r e, t) ψ e c. Bounded cylindrical reservoir 0 l n here l n is the distance measured arallel to the unit vector n. d. Constant lux boundary v n ρ ψ μ l n e. Diontinuity in orous media 6.3

33 Figure 6.6 Schematic o the boundary conditions or radial lo, constant lo rate (ater Matthes and Russell) Freuently lo roblems must be considered in hich a diontinuity in the ermeability o the medium exists. The roer boundary conditions at such a boundary are arrived at rom hysical considerations. Suose, or examle, that the domain o lo is comosed o to regions, and, having dierent ermeabilities, and. Since ressure at any given oint must be single-valued, the irst reuirement is =. Also since hat enters the boundary rom one side must come out on the other side, the velocities normal to the boundary must be eual on the to sides. l n l n here l n is distance measured normal to the boundary. The last set o boundary conditions deribes the hysical conditions that occur at the inner condition (at the ellbore). I roduction rate is constant then rom Darcy s La, A I otential is constant then ( r, t). 6.33