Chapter 5 Darcy s Law and Applications
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1 Chater 5 Darcy s La and Alications 4.1 Introdction Darcy' s la q reservoir Note : Time scale is added dn dt
2 4. Darcy s La; Flid Potential K h 1 l h K h l Different Sand Pac Different K
3 The ressre at any oint in the flo ath in fig 4. gh gz g h z Darcy s exerimental la as roved to be indeendent of the direction of flo in the earth s gravitational field. a + g h z hg + gz 4. From Eq.4.1 K dh here K is a const. 4.3
4 + gz Φ d g K gz d g K gz g d K Φ , 1 atm z gz d atm + Φ 4.6, b b b z z z z g d b + Φ Eq4. & Eq4.3 here
5 + gz Φ Φ Φ A b A B B b P P P P B A P P b B b A B A z z g d z z g d z z g d The reason for this Eq4.6 is that flid flo beteen to oints A and B is governed by the difference in otential beteen the oints, not the absolte otentials, i.e.
6 Exeriments erformed ith a variety of different liqids revealed that the la can be generalized as dφ 4.8 here ermeability μ viscosity density of flid K dφ Note : 4.4 g /μ K/g -- > /μ K/g K /μ g
7 4.3 Sign convention Φ 1 Φ Linear flo l 1 l dφ 4.9 Radial flo dφ 4.10 Φ 1 r r Φ
8 4.4 Units: Units Conversion d Φ 3 / / L L L T LT M L M T L d 1 / 1 L L L T L M LT M T L Darcy s eqation or In cgs nits In SI nits m gz + Φ
9 In Darcy nits nits hybrid atm d and c hen nit Darcy Darcy d / 1 /, 1 sec / 1 1
10
11 Darcy eqation In absolte nits cgs nits sec sec / sec / /, sec / / sec / A q gm oise gm gm dyne gm dyne z l g dyne dz g d d Φ + Φ dz θ
12 Darcy eqation In Darcy nits dz g d + sec sec /, sec / sec / 3 3 A q gm c z l Darcy g atm dz θ
13 Darcy eqation + dz d A q γ θ γ sin d A q, / ft A c ft z l md g g si D bbl q ater γ In field nits or here dz θ
14 Note: In this text 1 Darcy or 1 Darcy 10-8 or 1 Darcy 10-1 m 1 m In other boo 1 Darcy Darcy m 1 Darcy 1 m
15 Conversion of Permeability md and Coefficient of Permeability m/s g 3 8 ft / s c 1 md ft si / ft H 6.4 lb / ft O m c H O 1 lb m lb c ft / s PKCoefficient of Permeability f 3 g g c 1 md m s
16 4.5 Real Gas otential The flid otential fnction Φ d b + gz 4.6 For an incomressible flid consant Φ + gz 4.7 Φ or or M zrt d b + gz zrt RT d + gz b M M b zd f, T For a comressible flid RT z dφ d + gdz M 4.0 d dφ + gdz + gz 4.19
17 4.6 Datm Pressre From eq4.0 dφ d + gdz dψ From eq4.1 q A dφ A dψ 4. Note: Ψ otential is freqently refered to as the datm ressre
18 4.7 Radial Steady State Flid; Well Stimlation Assme 1Steady state Darcy s la for the radial flo of single hase oil 0 e and ressre rofile const. t Reservoir is homogeneos 3Well is erforated across the entire formation thicness. ith time q q A d 4.3 dr For qconst. across any radial area Aπrh π rh d dr d q π rh q r 1 d dr f πh r r q r f ln 4.4 πh r dr at r r e e f q r ln πh r e 4.5
19 In drilling a ell P ellbore >P formation md flo from ellbore to formation md article ssended in the md lg the ore saces Redcing Damage zone r < r < ra sin q S πh 4.6 S sin factor, dim ensionless nmber
20 Eq4.5 and Eq4.6 q re Pe Pf ln + πh r S 4.7 In field nit, Eq4.7 is P P q Bo re 141. ln S 4.8 h r e f + 3 q oil rate STB / D h PI P P ressre dradon r e f e Bo ln + S r PI rodctivity index of a ell a direct measre of the ell erformance
21 The varios methods of stimlation a Removal of sin S - Determination of sins - ressre bild test - If S>0, acid treatment - Hydrochloric acid if formation limestone - Md acid if formation is sandstone b Increasing the effective ermeability a hydralic fractring - sandstone b fractre-acidising - carbonate reservoir c Viscosity redction μ o high PI lo The T stimlation o Steam Soaing fig
22 d Redction of the oil formation volme e factor Bo Bo RB/STB can be minimized by choosing the correct srface searator,or combination of searator. e Redction in the ratio re/r - it is seldom considered as a means of ell stimlation f Increasing the ell enetration h
23 4.8 To-Phase Flo:Effective and Relative Permeabilities Darcy eqation In Single hase flo d g d 6 dz in Darcy γ sinθ in nits field nits To-hase flo---oil and ater or oil and gas, or gas and ater --- > Effective ermeability in Darcy eqation Effective ermeability f flid satration effevtive < absolte o, : effective ermeability
24 S S or connate residal or oil irredcible satration ater satration and ro S r S o S S Normalized crve Shae of crves Wettability of roc srface
25 Alternative manner of normalizing the effective ermeability S S o c o K K ro r S S o o o S S S S S S c c 4.3 In the dislacement of one immiscible flid by another r ro K K r ro Note : r f s, h h caillary & gravity forces
26 4.9 The Mechanics of Slementary Recovery --Slementary recovery by dislacing the hydrocarbons toards the rodcing ell ith some injected flid. injected flid --ater becase - - availability - - lo cost - - high secific gravity --Mobility of any flid λ λ * r /μ Relative mobility r /μ --Water dislaces oil
27 -- Water dislaces oil Ideal or iston-lie dislacement Interface ater front Injecting ell Prodcing ell Water zone Water zone Oil zone λ r r S 1 S or ' r The favorable tye of dislacement ' r ' ro o ' ro M 1 and ' r iston lie dislacement here M end oint mobility ratio endoint relative ermeabilities const. Oil recovered ater injected in a linear reservoir bloc
28 Non-ideal dislacement Injecting ell Prodcing ell Non ideal Oil Common recovered M dislacement > or 1 6 ater injected
29 Mobility Control Mobility ratio M Mobility Mobility of of the the dislacing dislaced flid flid ' rd ' ro d o Where d in rd is dislacing flid Mobility control to redce M olymer olymers, flooding sch as increase olysaccharide, d are dissolved in the injection ater
30 combstion sit in injection steam injection ater hot temeratre the g increa by redced drastically be can of order crdes viscos very For decrease method Thermal o o d o sin 1000 ~ heating on oil the of ansive crde the of distillation control mobility effects side tertiary some ith rocesses ery re Secondary method Thermal ex 3 1 cov
31 Tertiary flooding recovering the oil remaining in the reservoir after a conventional secondary recovery roject. After secondary recovery traing of oil drolets becase of srface tension ro After secondary recovery To dislace the oil ith a flid hich is solble in it So A Sor S So. B A Sor To flood ith a flid hich is miscible or artially miscible ith the oil Eliminating srface tension
32 Gas Petrolem Liqid LPG flooding LPG Miscible oil from H C light extracting control mobility C t A t oil CO flooding dioxide Carbone o ro.. 3..,,. 1 tan flooding solton Micellar flooding t Srfac
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