Department of Petroleum Engneerng and Appled Geophyscs SOLUTION - Examnaton paper for TPG4160 Reservor Smulaton Academc contact durng examnaton: Jon Kleppe Phone: 91897300/73594925 Examnaton date: May 22, 2014 Examnaton tme (from-to): 1500-1900 Permtted examnaton support materal: D/No prnted or hand-wrtten support materal s allowed. A specfc basc calculator s allowed. Other nformaton: Language: Englsh Number of pages: 14 Number of pages enclosed: 0 Checked by: Date Sgnature
Fnal Exam page 2 of 15 Queston 1 (26x0,5 ponts) Explan brefly the followng terms as appled to reservor smulaton (short sentence and/or a formula for each): a) Control volume b) Mass balance c) Taylor seres d) Numercal dsperson e) Explct f) Implct g) Stablty h) Upstream weghtng ) Varable bubble pont j) Harmonc average k) Transmssblty l) Storage coeffcent m) Coeffcent matrx n) IMPES o) Fully mplct p) Cross secton q) Conng r) PI s) Stone s relatve permeablty models t) Dscretzaton u) Hstory matchng v) Predcton w) Black Ol x) Compostonal y) Dual porosty z) Dual permeablty Soluton a) Control volume small volume used n dervaton of contnuty equaton b) Mass balance prncple appled to control volume n dervaton of contnuty equaton c) Taylor seres expanson formula used for dervaton of dfference approxmatons (or formula: f (x + h) = f (x) + h f (x) + h 2 f (x) + h 3 f (x) +...) 1! 2! 3! d) Numercal dsperson error term assocated wth fnte dfference approxmatons derved by use of Taylor seres e) Explct as appled to dscretzaton of dffusvty equaton: tme level used n Taylor seres approxmaton s t f) Implct as appled to dscretzaton of dffusvty equaton: tme level used n Taylor seres approxmaton s t+δt g) Stablty as appled to mplct and explct dscretzaton of dffusvty equaton: explct form s condtonal stable for Δt 1 φµc 2 k (Δx)2, whle mplct form s uncondtonally stable
Fnal Exam page 3 of 15 h) Upstream weghtng descrptve term for the choce of moblty terms n transmssbltes ) Varable bubble pont term that ndcates that the dscretzaton og undersaturated flow equaton ncludes the possblty for bubble pont to change, such as for the case of gas njecton n undersaturated ol j) Harmonc average averagng method used for permeabltes when flow s n seres k) Transmssblty flow coeffcent n dscrete equatons that when mulpled wth pressure dfference between grd blocks yelds flow rate. l) Storage coeffcent flow coeffcent n dscrete equatons that when mulpled wth pressure change or saturaton change n a tme step yelds mass change n grd block m) Coeffcent matrx the matrx of coeffcent n the set of lnear equatons n) IMPES an approxmate soluton method for two or three phase equatons where all coeffcents and capllary pressures are computed at tme level of prevous tme step when generatng the coeffcent matrx o) Fully mplct an soluton method for two or three phase equatons where all coeffcents and capllary pressures are computed at the current tme level generatng the coeffcent matrx. Thus, teratons are requred on the soluton. p) Cross secton an x-z secton of a reservor q) Conng the tendency of gas and water to form a cone shaped flow channel nto the well due to pressure drawdown n the close negborhood. r) PI the productvty ndex of a well s) Stone s relatve permeablty models methods for generatng 3-phase relatve permeabltes for ol based on 2-phase data t) Dscretzaton convertng of a contneous PDE to dscrete form u) Hstory matchng n smulaton the adjustment of reservor parameters so that the computed results match observed data. v) Predcton computng future performance of reservor, normally followng a hstory matchng. w) Black Ol smplfed hydrocarbon descrpton model whch ncludes two phases (ol, gas) and only two components (ol, gas), wth mass transfer between the components through the soluton gas-ol rato parameter. x) Compostonal detaled hydrocarbon descrpton model whch ncludes two phases but N components (methane, ethane, propane,...). y) Dual porosty denotes a reservor wth two porosty systems, normally a fractured reservor z) Dual permeablty denotes a reservor wth two permeabltes (block-to-block contact) n addton to two porostes, normally a fractured reservor Queston 2 (1+2+2+2+4+4 ponts) Answer the followng questons related to the dervaton of reservor flud flow equatons: a) Wrte the mass balance equaton (one-dmensonal, one-phase) b) Lst 3 commonly used expressons for relatng flud densty to pressure c) Wrte the most common relatonshp between velocty and pressure, and wrte an alternatve relatonshp used for hgh flud veloctes. d) Wrte the expresson for the relatonshp between porosty and pressure. e) Derve the followng partal dfferental equaton (show all steps):
Fnal Exam page 4 of 15 k P = φ c r x µb x B + d(1/ B) dp P f) Reduce the equaton n e) to the smple dffusvty equaton: Soluton 2 P x 2 = (ϕµc k ) P a) For constant cross sectonal area, the contnuty equaton smplfes to: ( x ρu ) = ( φρ ) b) Compressblty defnton: c f = ( 1 V )( V P ). T Real gas law: PV= nzrt. The gas densty may be expressed as: P Z ρ g = ρ S gs Z P S Black Ol descrpton: ρ o = ρ os+ ρ gs R so. c) Darcy's equaton, whch for one dmensonal flow s: u = k P ρg snα µ x. An alternatve equatons s the Forchhemer equaton, for hgh velocty flow (horzontal): P x = u µ k + βun where n s proposed by Muscat to be 2 d) Rock compressblty: c r = ( 1 φ )( φ P ) T e) Substtuton for Darcy s eq.:
Fnal Exam page 5 of 15 x ρ k P = µ x φρ ( ) Flud densty: ρ o = ρ os + ρ gs R so Rght sde ( φρ) = constant B = constant φ (constant /B) + φ P or φρ Left sde (horzontal) x ρ k P = µ x x ( ) = constant φ c r Thus, the flow equaton becomes: B + d(1/b) P dp constant B k P = φ c r x µb x B + d(1/b) dp P f) Startng wth the equaton: k P = φ c r x µb x B + d(1/ B) dp P 1 dφ = constant B dp + φ d(1/b) P dp = constant φ c r B + φ d(1/b) dp k P = constant k P µ x x Bµ x Assume that the group k/µb s approxmately constant: Snce k P x µb x k 2 P µb x 2 d(1/ B) dp c r B = c f B, + d(1/ B) dp = 1 B c + c r f Thus, the equaton becomes 2 P x 2 = (ϕµc k ) P ( ) = c B P
Fnal Exam page 6 of 15 Queston 3 (10 ponts) Use Taylor seres and show all steps n the dscretzaton of the followng equaton: Soluton Rght sde: Thus, k P = φ c r x µb x B + d(1/b) P dp P(x,t) = P(x,t + Δt) + Δt P (x,t + Δt) + ( Δt)2 1! 2! Solvng for the tme dervatve, we get: ( P ) t +Δt = P t +Δt t P Δt Left sde: +1/ 2 1/ 2 P (x,t + Δt) + ( Δt)3 3! + O(Δt). φ c r B + d(1/b) P dp φ c r B + d(1/b) t P +Δt t P dp Δt = + = + Δx /2 1! Δx /2 1! x + x + (Δx /2)2 2! 2 x 2 ( Δx /2)2 2! P (x,t + Δt) +... +... 2 x 2 +... combnaton yelds x µb ) P = +1/ 2 Δx 1/ 2 + O(Δx 2 ). Usng smlar central dfference approxmatons for the two pressure gradents: and P x P x +1/ 2 1/ 2 = P +1 P Δx = P P 1 Δx + O(Δx) + O(Δx). the expresson becomes: x µb ) P µb ) P +1 P Δx µb ) P P 1 +1/ 2 Δx Δx or x µb ) P +1 P +1/ 2 Δx 2 µb ) P P 1 1/ 2 Δx 2 1/ 2
Fnal Exam page 7 of 15 Thus, the dfference equaton becomes: µb ) P +1 P +1/ 2 Δx 2 µb ) P P 1 1/ 2 φ c r Δx 2 B + d(1/b) t +Δt t P +Δt t P dp Δt The terms µb ) +1/ 2 and µb ) 1/ 2 are then computed usng harmonc averages of propertes of blocks -1, and +1, respectvely. Queston 4 (3+5+5 ponts) a) Show all steps n the dervaton of the smple, one dmensonal, radal, horzontal, onephase dffusvty equaton: 1 r r P (r r ) = (φµc k ) P b) Derve the numercal approxmaton for ths equaton usng the transformaton: u = ln(r) c) Explan why the radal grd dmensons n cylndrcal coordnates often are selected accordng to the formula: Soluton a) r +1/ 2 r 1/ 2 = ( r e r w ) 1/ N r In a radal system, the flow area s a functon of radus, and for a full cylnder (360 degrees) the area s: A = 2πrh. Thus, the contnuty equaton may be wrtten (dervaton s not requred): 1 ( r r uρr ) = ( ρφ). Substtutng for Darcy s eqn,: u = k P µ r
Fnal Exam page 8 of 15 we get 1 r r ρr k P = ( ρφ ) µ r Usng defntons of compressbltes (at constant temperature) c r = 1 dφ φ dp c f = 1 dρ ρ dp we rewrte the rght sde as: ( ρφ) = ρ φ + φ ρ dφ = ρ( dp + φ dρ dp ) P = ρφ(c + c ) P r f The left sde may be rewrtten as: 1 P ρr r r r = k 1 µ r ρ r r P r + r P dρ P r dp r = k µ ρ 1 r r r P 2 r + rc P f r It may be shown that: r r P 2 r >> rc P f r Thus, the left sde may be approxmated by: 1 P ρr k r r r µ ρ 1 r r r P r The smple form of the radal equaton then becomes: 1 P (r r r r ) = (φµc k ) P b) For the radal flow equaton, we wll frst make the followng transformaton of the r- coordnate nto a u-coordnate: u = ln(r). Thus, du dr = 1 r and r = e u. The PDE may then be wrtten: P du e u u eu du u dr dr = φµc P k, or 2 P e 2u u = φµc P. 2 k Usng the dfference approxmatons above, we may wrte the numercal for of the left sde as: 2 P e 2u u 2 e 2u P +1 2P + P 1 2 Δu After back-substtuton of r, we wrte the left sde as: P e 2u +1 2P + P 1 = 1 2 2 Δu r P +1 2P + P 1 [ ln(r +1/ 2 /r 1/ 2 )] 2
Fnal Exam page 9 of 15 The rght sde s approxmated as for the lnear equaton. Thus, the complete dfference equaton becomes (no superscrpt means t+δt): 1 P +1 2P + P 1 2 r [ ln(r +1/ 2 /r 1/ 2 )] = P P t, =1,...,N 2 Δt The formula apples to the radal grd block system shown below: The poston of the grd block centers, relatve to the block boundares, may be computed usng the mdpont between the u-coordnate boundares: u = (u +1/ 2 + u 1/ 2 ) /2, or, n terms of radus: r = r +1/ 2 r 1/ 2. Ths s the geometrc average of the block boundary rad. c) Frequently n smulaton of flow n the radal drecton, the grd blocks szes are chosen such that: Δu = (u +1/ 2 u 1/ 2 ) = constant or ln r +1/ 2 = constant, r 1/ 2 whch for a system of N grd blocks and well and external rad of r w and r e, respectvely, mples that N ln r +1/ 2 = ln r e r 1/ 2 r w or r +1/ 2 = r 1/ N e = constant. r 1/ 2 r w r -1 r -1 1/2 r -1/2 Queston 5 (4+4+6 ponts) r r +1/2 r +1 r +1 1/2 In the followng 2-dmensonal cross-secton of a reservor (one flud only), a well s producng at a constant rate Q (st. vol. ol/unt tme) and perforates the grd blocks 4, 8, 11, 14, 17 and 21 n the x-z grd system shown: 1 1 5 9 13 17 21 j 2 3 2 6 10 14 18 22 3 7 11 15 19 23 4 4 8 12 16 20 24
Fnal Exam page 10 of 15 The (unknown) bottom hole pressure P bh s specfed at a reference depth d ref. Assume that hydrostatc pressure equlbrum exsts nsde the well tubng. a) Wrte the expresson for ol rate from each perforated block (n terms of productvty ndces, moblty terms, pressure dfferences and hydrostatc pressure dfferences) b) Wrte the expresson for the total ol flow rate for the well (group the constants nto parameters A, B, C, D, F, G, H, representng a constant term and the contrbuton to flow from the 6 grd block pressures nvolved) c) The standard pressure equaton for ths grd system, wthout the well terms, s: e, j P, j 1 + a, j P 1, j + b, j P, j + c, j P +1, j + f, j P, j +1 = d, j =1,...,N 1, j =1,...,N 2 Sketch the coeffcent matrx for ths system, ncludng the well. Indcate how the coeffcent matrx s altered by the well (approxmately, wth x s and lnes labeled wth the approprate coeffcent name). Soluton a) Wrte the expresson for ol rate from each perforated block (n terms of productvty ndces, moblty terms, pressure dfferences and hydrostatc pressure dfferences) b) Wrte the expresson for the total ol flow rate for the well (group constants nto parameters A, B, C, D, E, F, G, H) Q tot = or q j = PI j λ j P j Pbh (d j d ref )ρg e. q 4 = PI 4 λ 4 (P 4 P bh Δd 4 ρg) q 8 = PI 8 λ 8 (P 8 P bh Δd 8 ρg)...... q j perf blocks = A + BP 4 + CP 8 + DP 11 + EP 14 + FP 17 + GP 21 + HP bh BP 4 + CP 8 + DP 11 + EP 14 + FP 17 + GP 21 + HP bh = d 25 c) The standard pressure equaton for ths grd system, wthout the well terms, s: e, j P, j 1 + a, j P 1, j + b, j P, j + c, j P +1, j + f, j P, j +1 = d, j =1,...,N 1, j =1,...,N 2 Sketch the coeffcent matrx for ths system, ncludng the well. Indcate how the coeffcent matrx s altered by the well (approxmately, wth x s and lnes labelled wth the approprate coeffcent name).
Fnal Exam page 11 of 15 b f c e b f c e b f c e b c x a b f c a e b f c a e b f c a e b c x a b f c a e b f c a e b c x a b f c a e b f c a e b f c x a e b f c a e b c a b f c x a e b f c a e b f c a e b c a b f x a e b f a e b f a e b B C D E F G H Queston 6 (3+2+3+5 ponts) The dscretzed form of the ol equaton may be wrtten as Txo +1/ 2 (P o +1 P o ) + Txo 1/ 2 (P o 1 P o ) q o = C po (P o P t t o ) + C so (S w S w ) a) What s the physcal sgnfcance of each of the 5 terms n the equaton? Usng the followng transmssblty as example, Txo 1 / 2 = 2k 1 / 2λ o 1 / 2 Δx ( Δx + Δx 1 ) b) What type of averagng method s normally appled to absolute permeablty between grd blocks? Why? Wrte the expresson for average permeablty between grd blocks (-1) and (). c) Wrte an expresson for the selecton of the conventonal upstream moblty term for use n the transmssblty term of the ol equaton above for flow between the grd blocks (-1) and (). d) Make a sketch of a typcal Buckley-Leverett saturaton profle resultng from the dsplacement of ol by water (e. analytcal soluton). Then, show how the correspondng profle, f calculated n a numercal smulaton model, typcally s nfluenced by the choce of mobltes between the grd blocks (sketch curves for saturatons computed wth upstream or average moblty terms, respectvely). Soluton a) Txo +1/ 2 (P o +1 P o ) = flow between grd blocks and +1 Txo 1/ 2 (P o 1 P o ) = flow between grd blocks and -1 q o = producton term C po (P o P t o ) = fluds compresson/expanson term t C so (S w S w ) = volume change due to saturatons
Fnal Exam page 12 of 15 b) Harmonc average s used, based on a dervaton of average permeablty of seres flow, assumng steady flow and Darcy s equaton k 1/ 2 = Δx + Δx 1 Δx 1 + Δx k 1 k c) λo 1/ 2 = λo f 1 Po 1 Po λo f Po 1 < Po d) Queston 7 (5x2 ponts) Normally, we use ether a Black Ol flud descrpton or a compostonal flud descrpton n reservor smulaton. a) What are the components and the phases used n Black Ol modelng? b) What are the components and the phases used n compostonal modelng? c) Wrte the standard flow equatons for the components requred for Black Ol modelng (one dmensonal, horzontal, constant flow area). d) Wrte the standard flow equatons the components requred for compostonal modelng (one dmensonal, horzontal, constant flow area). Let C kg = mass fracton of component k present n the gas phase C ko = mass fracton of component k present n the ol phase. e) A Black Ol flud descrpton may be regarded as a subset of a compostonal flud descrpton. Defne the pseudo-components requred n order to reduce the compostonal equatons to Black Ol equatons (one dmensonal, horzontal, constant flow area) Soluton a) Components: ol and gas, phases: ol and gas b) Components: hydrocarbons (C 1 H 4, C 2 H 6, C 3 H 8,...) and non- c) hydrocarbons ( CO 2, H 2 S, C 2,...), phases: ol and gas x kk rg B g µ g P g x + kk ror so µ o P o x = φ S g + S or so B g B g
Fnal Exam page 13 of 15 kk ro P o = x µ o x φs o d) x C kk rg P g kgρ g µ g x + C kk ro P o koρ o µ o x = [ φ ( C kgρ g S g + C ko ρ o S o )], k =1,N c e) The Black Ol model may be consdered to be a pseudo-compostonal model wth two components. Defne the components and the fractons needed to convert the compostonal equatons to Black-Ol equatons. component 1: ol k=o x C kk rg P g ogρ g µ g x + C kk ro P o ooρ o µ o x = component 2: gas k=g x C kk rg P g ggρ g µ g x + C kk ro P o goρ o µ o x = [ φ ( C ogρ g S g + C oo ρ o S o )] [ φ ( C ggρ g S g + C go ρ o S o )] Queston: what are the fractons needed to get the Black Ol equatons: kk rg P g x B g µ g x + kk ror so P o µ o x = φ S g + S or so B g B g kk ro P o = φs o x µ o x Answer: Queston 8 (6x2 ponts) fracton of "gas n gas": C gg =1 fracton of "ol n gas": C og = 0 fracton of "gas n ol": C go = ρ gs R so ρ o fracton of "ol n ol": C oo = ρ os ρ o Normally, we use ether a conventonal model or a fractured model n smulaton of a reservor. a) Descrbe the man dfferences between a conventonal reservor and a fractured reservor, n terms of the physcs of the systems.
Fnal Exam page 14 of 15 b) How can we dentfy a fractured reservor from standard reservor data? c) Explan brefly the prmary concept used n dervng the flow equatons for a dualporosty model. d) Wrte the basc equatons (one-phase, one-dmenson) for a two-porosty, two-permeablty system a two-porosty, one-permeablty system e) In terms of the physcs of reservor flow, what s the key dfference between the two formulatons n queston d)? f) How s the flud exchange term n the flow equatons n queston d) represented? What are the shortcomngs of ths representaton? Soluton a) Conventonal: One porosty, one permeablty system, wth one flow equaton for each component flowng. Fractured: Two porostes, two permeabltes system, wthmost of the fluds n the matrx system, and most of the transport capacty n the fracture system. Requres two flow equatons for each component flowng. b) K core K welltest c) The matrx system supples fluds to the fracture system, by whatever mechansms present (depleton, gravty dranage, mbbton, dffuson,...), and the fracture system transports the fluds to the wells. Som transport may also occure n the matrx system, from block to block, provded that there s suffcent contact. d) Dual porosty, dual permeablty model: k P x µb x f + q mf = φ B f k P x µb x m q mf = φ B m One porosty, one permeablty model (fracture eqn.): k P x µb x f + q mf = φ B f Conventonal fracture models represents the exchange term by q mf = σλ ( P m P f ) where σ s a geometrc factor, λ s the moblty term, and P m and P f represent matrx and fracture pressures, respectvely. e) In the dual porosty, dual permeablty model, flud flow may occure from one matrx block to another. In the one porosty, one permeablty model, all flow occurs n the fracture system f) The exchange term s conventonally defned as
Fnal Exam page 15 of 15 q mf = σλ ( P m P f ) where σ s a geometrc factor, λ s the moblty term, and P m and P f represent matrx and fracture pressures, respectvely. Obvously, ths term cannot adequately represent the flow mechansms present, such as depleton, gravty dranage, mbbton, dffuson,... In addton, an average pressure for the matrx block s used n the exprexxon, so that pressure gradents nsde the block s not accounted for.