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 Note : Time q reservoir scale is dn dt added 1
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 gz d d otential Flid d g K gz d g K gz g d K , atm z gz d gdz d atm atm z z 4.6, b b b z z z z g d b Eq4. & Eq4.3 here
5 ??? gz 5 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. gz d d
6 Exeriments erformed ith a variety of different liqids revealed that the la can be generalized as d 4.8 here = ermeability μ = viscosity ρ = density of flid /μ = K/g -- > /μ = K/g K = /μ g K d Note : 4.4 g d d gz d 6
7 4.3 Sign convention Φ 1 Φ Linear flo l 1 l d 4.9 Radial flo d 4.10 Φ 1 r r Φ 7
8 4.4 Units: Units Conversion 8 d 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 gz d d
9 9 In Darcy nits nits hybrid atm d and c hen nit Darcy Darcy d / 1 /, 1 sec / 1 1
10 10
11 Darcy eqation In absolte nits cgs nits 11 sec sec / sec / /, sec / / sec / A q gm oise gm gm dyne gm dyne z l g dyne dz g d d dz θ
12 Darcy eqation 1 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 13 dz d A q sin d A q, / ft A c ft z l md g g si D bbl q ater 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 14
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 P=K=Coefficient of Permeability f 3 g g c 1 md m s 15
16 4.5 Real Gas otential The flid otential fnction d gz4.6 b For an incomressible flid consant gz4.7 M zrt d b gz f, T For a comressible flid or or b zrt M d gz RT M b zd RT z d d gdz M 4.0 d d gdz gz
17 4.6 Datm Pressre From eq4.0 d d gdz d From eq4.1 q A d A d 4. Note: Ψ otential is freqently refered to as the datm ressre 17
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 f q A d 4.3 dr For q=const. across any radial area A=π rh rh d f q h d dr r r q ln h 1 dr r r r d q rh dr 4.4 at r r e e f q r ln h r e
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 4.6 h S sin factor, dim ensionless nmber 19
20 Eq4.5 and Eq4.6 P e P f q ln h r r e S 4.7 In field nit, Eq4.7 is P e PI P f P e qbo re 141. ln S 4.8 h r q P f oil rate ressre STB / D dradon h re Bo ln S r PI = rodctivity index of a ell a direct measre of the ell erformance 0
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 Tstimlation 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 sin 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 3
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 4
25 Alternative manner of normalizing the effective ermeability o S S 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 5
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 6
27 -- Water dislaces oil Ideal or iston-lie dislacement Interface ater front Injecting ell Prodcing ell Water zone r r S 1 S or ' r Water zone Oil zone r ro ' ro s o s o c The favorable tye of dislacement ' r ' ro o ' ro M 1 and ' r iston lie dislacement here M = end oint mobility ratio endo int relative ermeabilities const. Oil recovered = ater injected in a linear reservoir bloc 7
28 Non-ideal dislacement Injecting ell Prodcing ell Non ideal Oil Common recov ered M dislacement or 1 6 ater injected 8
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 9
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 ~ 30 heating on oil the of ansive crde the of on distillati control mobility effects side tertiary some ith rocesses ery re Secondary method Thermal ex 3 1 cov
31 Tertiary flooding recov ering the oil remaining in the reservoir after a conventional secondary recov ery roject. After secondary recov ery 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 31
32 3 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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