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by: Ron Huel


Ron Huel
GPA 3.77

S. Kam

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S. Kam
Class Notes
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This 31 page Class Notes was uploaded by Ron Huel on Tuesday October 13, 2015. The Class Notes belongs to PETE 2031 at Louisiana State University taught by S. Kam in Fall. Since its upload, it has received 34 views. For similar materials see /class/222992/pete-2031-louisiana-state-university in Petroleum Engineering at Louisiana State University.

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Date Created: 10/13/15
Permeability Permeability Permeability defines porous medium s ability to transmit fluids ie permit fluid flow Absolute permeability Permeability is defined through Darcy s law Darcy s law in a simple form q kA Ap u Ax where q volumetric flow rate A cross sectional area it viscosity of fluid dx P pressure difference k absolute permeability Permeability concept Permeability H Darcy 1856 French Cix i1 Engineer v lmwmg of Hemy p on ow 01 13451 1 me Hubbarl Eermeabilitl Darcy s law For steadystate linear ow FEM Pz y L gt For steadystate radial ow note A2mh 1114 Vw P1 n W 7 P w 2717 dri 1ampde r1 P dr 7k d 27th 39wr IPW p permeability Darcy s law In general Darcy s law can be expressed with the fluid potential CD q pipgz u kA dd kA dp dz q pg u ds ds ds Hydrostatic head Importance of this term depends on the application and inclination angle s the direction of flow 2 coordinate system a inclination angle permeability Darcy s law Example Derive a simple Darcy s law expression for the example iven below Answer q d5 dz dP dP h 52 1dPdP0 sz 7pi 1 dipepighpg d5 15 amp d5 d5 dz L is is dz L 3 1 0 1 st Pghso fist PgO FPgUHXJrL zrz2 39zrz2 07L 39zrz2 07L CA CA h kA h q pg q pg 1 F i y L y L permeability 39 5 Units of Darcy 5 law In calculations one may use any consistent unit system such as cgs or SI units People sometimes use a unit system called Darcy units NOT a consistent unit system BUT easy to use in the field For inclined flow Darcy s law in a differential form is M L j u where q volumetric flow rate cm3ls A cross sectional area cm s distance in the direction of flow and is always positive cm 2 vertical coordinate positive in downwards cm u viscosity of fluid 52 dplds fluid pressure gradient atmlcm k absolute permeability of the porous media my p density of fluid g cm g acceleration of gravity 980665 cml s2 Note that 1 atm 147 psia 10133 x 106 dynescm2 101325 Pa And 1 poise 1gcm s 01 kgm s 01 Pa s 1 cp0001 Pa s 1 mPa s permeability Oil field units In some cases calculations in field units are preferred For horizontal linear flow Darcy s law in oilfield units q 1 ki 71 7 72 887 2 u L For horizontal radial flow Darcy s law in oilfield units q 1 kh p 7 pw 1412 u 1nre rW where q production rate bbllday A cross sectional area ft2 k absolute permeability md h thickness ft L length ft u viscosity of fluid cp p pressure psi r distance in radial direction ft Note that In a natural logarithm Qermeability nutty unit A rxw sue quotaquot Lump vmusur How 0519 n259 05 m Mm xsa P v mlg msg nmmuuy k nu ma mug of mum 1 Congressible um Qermeability Unit of Permeability From Darcy s law A y L L A A1 7 L2 L2 k l M l LT 2 Permeability has a unit of length sguared mean square pore diameterquot closely related to porethroat size 1 darcy 1000 md M 9869 x 103 em2 9869 x 1039 mm2 W 1062 x1039 ft2 Therefore 1 darcy 1 x 103912 m2 1pm2 germeabili Range of Permeability Values 103 Permeability affected by Eermeabiliy Averaging Permeability For horizontal linear flow the average permeability is n beds in series q AR AIHAPZ A we 15411127 4414239 qn me 131333 germeabili Averaging Permeability For horizontal radial flow the average permeability is n beds in series k 7 1 n beds in parallel Eermeabiliy Averaging Permeability Linear Beds in Series Notice that the pressure drops are additive 171 1 4P1 P2P2 p3 p3 1174 q e Lu4l u Writing the equalent expressions from Darcy s equation for flow linear beds39 41 quII 41 quhu Since the flow rates cross sections and Viscosities assuming that modest change in pressure does not alter viscosity are equal in each bed then L kw h Lilqr VL 7L t L ii L r 1 A The average permeability Kavg is the value which can be used assuming a single bed and yield the same flow rate as for the series of beds Qerm eability Averaging Permeability Linear Beds in Parallel Often ow in the reservoir is through parallel strata having differing permebilities The total flow is the sum of the individual ows in each zone qi1112 43 The equivalent expression from Darcy s equation is km gAJPl 7P1 klAlltPl 7P1 klAlPt 1 31th 11 l L L L Simplifying Z kl Ii it x If the average permeability of the equation is used with the total area in the Darc equation the calculated flow rate will be equal to the total ow rate from the individual beds The equation assumes that no cross flow occurs between the individual beds If this is not true then some error will result Qerm eability Averaging Permeability Radial Beds in Series Depositicnal it is hard to imagine radial beds in series occurring in an actual reservoir Ho ever quot quot quot 39 r quotr 39L lte ation ui re er uii piupenie maican occur in the vicinity of wellbores both producers and injectors during drilling production and stimulation operations Noting that the pressure drops are in series then P2 7 Pw Fe 7 P11Pn 7 PW and from Darcy s equation for radial flow 111110 rw qJlulrc ru 1 111139 rH 708mgll 708le 708kah Solvmg for K then avg39 KW fake 1111 0 73 7 in 1117 ra 2 1114 in The equation below has had wide usage in determining the impact of wellbore damage or stimulation due to acidizing and fracturing and productivity increases due to healing near the ellbore Qermeability Averaging Permeability Radial Beds in Parallel Mostsedimentaryresen strataui 39 39 h 1 H39 the r quot 39 quot 0 p quot 111quot tiiiLmie 39 additive assuming no crossriow 1 41 42 513 Cn Therefore 08kmgi Pg 7 12 7 70311151 713w hosr2150 713 111110 1h 11 1111 2 rn 11111012 rw 0r 7 1171 K3113 rubW 7 11 K mg kh is also called the llow capacityquot of a bed in 7 darcy feet of millidarcyieet The summation of the capacities of the beds is usually termed the total ca acity ofthe producing formation Where n s the total capacity of a produ we formatio 1 cal ul e or etermined bet 1 t e l modem stimulation methods will often allow economic producing rates to be attaine h gt Eermeability Averaging Permeability Example Determine the average permeability for the following dia linear hypothetical layers of porous me Parrallel Series zone seg length perm zone Seg Length perm Li ft md ki Li ft mo ki 1 50 40 1 40 2 100 10 2 100 10 3 25 500 3 25 500 Parrallel Answer zone seg length perm kiLi average k 155001175 Li ft md ki 3306313132 1 50 40 2000 md 2 100 10 1000 3 25 500 12500 15500 Series linear zone Seg Length perm Li ft m kl 1 50 4o 2 100 10 1o 3 25 m I 005 1548672566 permeability Averaging Permeability Example Determine the average permeability for the following radial hypothetical layers of porous media Assume wellbore radius 6 inches Parrallel Series rw6quot zone seg thickness perm zone radial diatance perm hi ft md ki delri md ki 1 40 1 40 2 100 10 2 100 10 3 25 500 3 25 500 Parrallel Answer zone seg thickness perm kihi average k 15500l175 39 md ki 8806818182 1 40 2000 2 100 10 1000 3 25 500 12500 15500 Series rw6quot lnrelrw 5860786223 zone radial diatance perm 39 md ki lnrilri1 1 40 4615120517 2 100 10 1091989748 3 25 500 0153675959 averane k lmdl 260613355 permeability Anisotropic Permeability In reality permeability is a function of direction or orientation anisotropy Anisotropic property should be represented by a tensor in principle not by a scalar k q VPpg u Principal values of permeability calculated by eigenvalue and eigenvector problem mathematically det 21 0 where 2 eigenvalue In other words with given principle directions k1 0 0 E 0 k2 0 More in linear algebra or 0 k 0 solid mechanics 3 permeability Anisotropic Permeability k1 o o k 0 k2 o k k21 kn k2 0 0 k3 k31 k32 k33 cf normal vs shear stress strain permeability Anisotropic Permeability Not in principal Ire I n If in principal direction F kx39 ku39 7 kn1 kzy k 0 k 0 kz permeability Anisotropic Permeability kuu 0 kuv 0 k Let u and vthe principal directions and n H kx z x and 2 In any other coordlnate system Ie k k Zr 22 kwkw k 7k kn Mcos 20 2 2 k km kw 7 kuu kw COS 20 one can rewrite this equation for 2 k k2 km and kW km kzx 7 W W sinsz permeability Anisotropic Permeability Example Given the 2dimensional permeability tensor in a principal direction compute the permeability tensor for a coordinate system rotated 20 clockwise relative to the principal coordinate system 15 0 NFL 10 Answer kxx 151o2 151o2 cos40 1083265 151o2 151o2 cos4D 1416735 k X k zx 151o2 sin4D 1862783 10833 1863 16062 1863 14167 One should be able to calculate kuv and o from kxz Frequency of appearance permeability Permeability Distribution I 09 normal distribution permeability 10g210gk g quotklkzkyukn Geometric mean 1 Frequency of appearance ln permeability permeability Deviation from Darcy s Law 1 NonDarcy effect Darcy s law applies only to laminar flow cf for laminar flow in pipe NRe p v dl p lt 2000 where Nre Reynolds number q flow rate p density porosity 8 average sand grain size diameter 4 liquid density A crosssectional area Similar to the case in pipes transition from laminar to turbulent flow occurs at around 2000 Frictional factor 2 is defined as NR6 zooo MM Dimensionless AP 4 2 where L length of porous medium A 6 AP pressure difference qu permeability Deviation from Darcy s Law Flow in pipe laminar vs turbulent laminar flow occurs when NRe p v dl p lt 2000 turbulent flow occurs when NRe p v dl p gt 2000 gt QM Stream lines do not intersect in a laminar flow regime because viscous force is dominant Pressure pfnq1 112 pfnq1 flow rate permeability Deviation from Darcy s Law 1 NonDarcy effect AP L Viscous force dominant Inertia force dominant L 43A 43A A 3m transition Fanning friction factor 8w 101 102 103 104 105 10B Reynolds number NRe permeability NonDarcy effect Example Given the following input data what is the maximum injection velocity below which Darcy s law is applicable ie nondarcy effect negligible u 10p 03 0 lgcc 5 100um grain diameter Answer 7 M M NR3 2000 M or 200015 ConSIstency In unlts 3 2000xlcp10 Pa39sxos 16p 39Zmx 7 3 76 2000x1gccxlkingx100mxlo 39 1000g 1m 1 Note that u is Darcy velocity superficial velocity volumetric flux and one may calculate the velocity in terms of interstitial velocity ul Why dependence on grain size permeability Deviation from Darcy s Law 2 Klinkenberg Effect Labmeasured permeability is required in fluid flow calculation Core permeability is usually measured using dry gas inert Klinkenberg found that at low pressures measured gas permeability deviates from liquid permeability due to gas slippage at the wall of the pores where kgpm permeability measured by gas b kL true permeability ie measured by liquid kg Pm MEN b empirical constant m dependent on rock and gas pm mean core pressure pinlet puma2 For a compressible phase Darcy s law can be described as PM Pq P P 13 and q s E P 7P 2 u L permeability Deviation from Darcy s Law 2 Klinkenberg Effect true permeability measured by 50 liquid solution that we are k L 1 i E E looking for kkL g pm Cquot 40 a o 39 E a 3quot 2 3 4 5 39 deViated Permeabilitf Reciprocal of mean core pressure 1latm measured by gas at high p77 permeability Klinken berg effect Example Determine the permeability of a sandpack after Klinkenberg correc 39 Permeability measurement using air Upstream P ow rate sc Core diameter cm 25 atm ccmin Core Ie cm 10 2 downstream P atm 1 3 Air viscosity cp 00175 4 64 8 Answer SE k7 ucp Lcm 2 P q 5 E z 7 as as A 5 4 A cm Pm PM atm Pm PM E 5 2 g 5 E 4 a y 214m 4 2803 Lsc k darcy k md Pm g 4 E ccsec atm 1alm a 4 4 0 24 0005707006 5707006369 15 01566666667 42 06 0005350318 5350318471 05 1 08 0005136306 5136305732 25 4 D D 5 k 007133753 qisclPinquot27Poulquot2 reciprncal nfmean panprEssurE k L moi 4 28 1Pm 1latm 39 permeability Deviation from Darcy s Law 3 Compressible flow a Slightly compressible liquid Darcy s law is still valid with mass flux incorporated dx Assuming the density varies as slightly compressible p pa expcP Chain rule gives LP LPE paoaxWH E 1dP dx dP dx dx dx cp dx Therefore mass rate m is given by m E pl 7 p2 uc L permeability Deviation from Darcy s Law 3 Compressible flow b Highly compressible gas ideal gas Boyles law implies P V Using Darcy s law and mass balance ie kAp dP pq 7 and pq pbqb constant u dx P 1 390 3905 7 Jul11be and poc b any base state Combining these relationships ii P PA gt A u Pb dx A u dx Integration gives kA P2 7P2 P P qb 1 D and choosmg Pb P M 2 P 2 Finally mean 4 q gill Pm permeability Deviation from Darcy s Law 4 Nonisothermal radial gas flow 703x10quot khRZ ePWZ In Tyzln r E rw whe re q production rate 1000 scfday T temperature R k absolute permeability md h thickness ft L length ft p pressure psi P P y and z are Viscosity and compressibility factor evaluated at 2 W permeability Recapping Darcy s Law and Its Implications Darcy s equation for real gas ow k4 d d 7 Pga Darcy s equation to radial ow of single phase uid replacing A Zirr 2m kh dp 73 q If pE is the outer boundary pressure and is assumed constant because the oils withdrawn could be considered to be replenished by uids crossing in from the other boundary re and puf owing bottom hole pressure we can Write the Darcy s equation in terms of pressures and radius of well 1 In 1 2rk7 Pg ow germeabili Radial Pressure Pro les for a Well with Skin ve S Ve S and zero S 9 r 7 7 13 s A In 2W rw w ri damaned radius by mud drilling etc Relative Permeability Relative permeability Effective Permeability Multiphase flow in porous media Darcy s law still applies Effective permeability permeability of one fluid in the presence of other fluids in porous media k J g dx where kj is the effective permeability of phase j Absolute permeability is no longer valid Note that k gt k always Relative permeability Relative Permeability It is more convenient to use relative permeability krj kjlk the relative permeability of phase kkA dx kkmA W w W kk AdP 7 rg g qg gdx Therefore relative permeability represents fraction to the absolute permeability Relative permeability Saturation Pore space in a reservoir rock occupied by multiple phases Saturation Si Volume of each phase in the pore space Relative permeability is a function of saturation Does not include fluids interaction at the interface ie saturation does not tell the distribution of fluid in pm see interfacial phenomena capillary pressure ie SW volume of water pore volume le Vp For a reservoir above bubble point pressure BPP SW S0 1 For a reservoir below BPP SW So Sg 1 Relative permeability Key features krcurves are nonlinear and monotonic 1 functions of Swconvex to SW k 9 Flow of a phase always hindered by 9 the presence of other phases The sum of kW and kr0 always less than k one The same for three phase flow Two endpoint permeabilities correspond to Swi and 130 rw Nonwetting phase endpoint perm kme is larger than wetting phase end point perm kme nonwetting phase in the bigger pores 3 o 0 s 15quot 1 SW Typical oilwater relative permeability curve Relative permeability Key features Mobile Saturation Relative permeability g g at a as W a 13 01 Invasion of nonwetting phase into Invasion of wetting phase into non we ingphase saturated medium we ingphase saturated medium Drainage lmbibition Relative permeability Rel Penn Hysteresis due to C hange in Direction of Saturation Change Relative permeability Wettability concepts Wetting phase coats the surface of rock grains Wetting phase fills smaller pores first larger pores occupied by nonwetting phase In pores wetting phase occupies corners and crevices nonwetting phase occupies centers Therefore wetting phase prefers pore throat than pore body 2 m WIIIIIIMUIIIIIIM 1111111101 WWWWMW WWWW WWWm WWWWWMWIMIIIIIJWIIIA BOT Bundleof Tubes odel Relative permeability Effect of Wettability Porelevel description Residual Wetting Phase Distribution Residual Nonwetting Phase Distribution wettin Wetting Saturation Satura n lmmobile Wettin Saturation usually water usually oil Flow Direction gt Relative permeability Effect of Wettability 1 1 593 krwe kme kro kr kr kro e km km 0 swi 15quot 0 M 1Sr SW 1 1 Waterwet medium Oilwet medium Change in SWi and SM Change in Endpoint mobility Relative permeability Effect of OilWet Weathered Core Chemicals in Mud Wat erWet at Relative permeability Normalizing relative perm Relative permeability curve depends on Swi and SM Need normalization to get rid of these effects k A swesw M 1eswrsa laswrsm SW 7 SW Defining S1 S S then kMASm and km B17S Parameters A B m and n are the intercepts and slopes of relative permeability vs normalized saturation S plot Coreytype relative permeability Relative permeability Normalizing relative perm U E WW KVEI Relative perm w e 0 an rw kro 0 m Relative perm Kr o 1 0 N o D Em Nomalized Water Saturation S nunm Wale Salmalmn SW Relative permeability Normalizing relative perm log Relative perm log Kr y 1 9996X 02964 25 log Nomalized Water Sat log 8 log Relative perm log Kr y 29881X 00512 log 1 S kW AS39 therefore 10gkw 10g A m log S km 307 Squot therefore 10gkg10gB n10g17 S Topics in Advanced Level Pala un nnrmnahilih THREEPHASE RELATIVE PERMEABILITIES NaarWygal Henderson method Stone s methods Model and ll based twophase data NaarWygaIHenderson method uses only the irreducible water saturation Swir as the parameter This method is recommended only when no other data is available or when they have been conclustvely proved to be valid to match known twophase data Even in the later case it is advisable to test the method With few data points form laboratow core tests 7 Si 1 33 5 ZSilir 13 Iquot S 9 4 l 3 2 sgu I 739s Relative nermeahility km 5 km HS km 51 5 S For the oil phase km sum bu where bw Km 13 ng lsg S 2 So39Som S 2 Sw39swir o 1SWiISOm 139SwirSom from 2 phase data bg Stone bw and 139 are from 2phase data Relative permeability so minimum ROS which lies between 025 Sm and 05 SW so is adjusted to match 3phase data such that Sam is less than the residual oil saturations in OW flow and in 06 flow Fayers and Mathews 1984 correlation to estimate Sam so a Sm 1a s where saturation in the 06 system an a 39 SW residual oil saturation in OW system SmEl residual oil 139Swir Sorg Sg 1 39 Swil 39 Sorg Stone H Km W km 0 krg km 39 krg Km km Sm 3w Krw km SW Krg krg Sn SW 0T kg krg 8g Relative permeability Implicitly assumes kw k ch in OW system and SgU in 06 system mg Stone H is successful in most cases but in some cases it produces a negative km which is then interpreted as zero Dietrich and Bondor 1976 eqn Avoids ve km using kru the maximum relative 03 permeability to the oil and is assumed km krw krog i 1rg m 39 km 19 Relative permeability 1 u 7 390 n a u a n 5 mg k u 6 04 krg a A n 2 u 2 DD 02 04 J as us 10 D 0 0 2 3 qumd 1 n n a k smne g DE 2 i 77 NaanygaLHendersun S a D 4 SwiFD Zn 2 f Expenmemi data Figure 3 24mm Relamel eimeabdm Dara and the Svnmesized mm on Relamel emieabilm39 Relative nermeahilitu Mobility and Mobility Ratio Mobility of a uid is de ned as the ratio of its permeability to its Viscosity It is denoted by Greek letter L It is a measure of its mobility within the porous medium A For oil and water we can write Mobility and Mobility Ratio Relative permeability Mobility Ratio is the ratio of the mobility of the displacing uid say water t0 the M inability of the displaced uid say oil It is denoted by letter kalixphzcirig A1 7 AmpIacmg Udisplnr171g Adapt ng kdupiared Udmplnceri Fur water displacing oil M is Mwhm in icy1w Ili5011 ofthe key parameters that needs to he considered in a particularly in an enhance recoveiy pro ect If displacement DCClll39S in a lioi success or failure of the hsplaceinent process displacement process 39izontal direction inability ratio Will essentially dictate the


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