[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5

Size: px

Start display at page:

Download "[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5"

Isabel Poppy Hill
5 years ago
Views:

1 TPG460 Reservor Smulaon 08 page of 5 DISCRETIZATIO OF THE FOW EQUATIOS As we already have seen, fne dfference appromaons of he paral dervaves appearng n he flow equaons may be obaned from Taylor seres epansons We shall now proceed o derve appromaons for all erms needed n reservor smulaon Spaal dscrezaon Consan grd block szes We showed ha he appromaon of he second dervave of pressure may be obaned by forward and backward epansons of pressure: o yeld 3 ( ( P( +, = P(, + P (, + P (, + P (, +!! ( ( ( P(, = P(, + P (, + P (, + 3 D D D D P (, +!! P P P + P ( = ( + + O( D, whch apples o he followng grd sysem: + Varable grd block szes Δ A more realsc grd sysem s one of varable block lenghs, whch wll be he case n mos smulaons Such a grd would enable fner descrpon of geomery, and beer accuracy n areas of rapd changes n pressures and sauraons, such as n he neghborhood of producon and njecon wells For he smple onedmensonal sysem, a varable grd sysem would be: + Δ Δ Δ + he Taylor epansons become (droppng he me nde: P + [( + / ] [( + / ] ( + / D D D D = P + P + P +!! P P = P + (Δ + Δ /! P [ ] + (Δ + Δ /! [ ]3 P + (Δ + Δ / P o yeld P = 4 Δ + Δ Δ + Δ + + Δ P + P Δ + + Δ + Δ + Δ + + Δ P + O(Δ (Δ + Δ + ( Δ + Δ orwegan Unversy of Scence and Technology Deparmen of Peroleum Engneerng and Appled Geophyscs 08

2 TPG460 Reservor Smulaon 08 page of 5 An mporan dfference s now ha he error erm s of only frs order, due o he dfferen block szes However, normally he flow erms n our smulaon equaons wll be of he ype ( ú, where f ( ncludes permeably, mobly and flow area Therefore, we wll nsead derve a cenral appromaon for he frs dervave, and apply wce o hs flow erm and whch yelds é P ù P / f f f P ( / ( ( ( ê û ú = é ê ù û ú é ù! ú! +/ é P ù P / f f f P ( / ( ( ( ê û ú = é ê ù û ú + D é ù! ú + D! / é P ù P f ( f ( ( ú = ê û ú é ê ù û ú Smlarly, we may oban he followng epressons: and æ Pö P + P è ø + / ( + + / æ Pö P P è ø / ( + / + / / + O( D + O( D + O( D é P ù f ( ú + é P ù f ( ú + As we can see, due o he dfferen block szes, he error erms for he las wo appromaons are agan of frs order only By nserng hese epressons no he prevous equaon, we ge he followng appromaon for he flow erm: ( P + P ( P P f ( + / f ( / ( ( ( O( + D Boundary condons We have seen earler ha we have wo ypes of boundary condons, Drchle, or pressure condon, and eumann, or rae condon If we frs consder a pressure condon a he lef sde of our slab, as follows: P Δ hen we wll have o modfy our appromaon of he frs dervave a he lef face, = /, o become a forward dfference nsead of a cenral dfference: æ Pö P P è ø / ( / Δ + O( D, and he flow erm appromaon hus becomes: orwegan Unversy of Scence and Technology Deparmen of Peroleum Engneerng and Appled Geophyscs 08

3 TPG460 Reservor Smulaon 08 page 3 of 5 ( P P ( P P f ( / f ( / ( ( ( O( D Wh a pressure P R specfed a he rgh hand face, we ge a smlar appromaon for block : P f ( = f( + / (P R P (Δ For a flow rae specfed a he lef sde (njecon/producon, f ( / (P P (Δ + Δ Δ + O(Δ Q Δ Δ we make use of Darcy's equaon: or Q = æ Pö ç µ B è ø æ Pö Q è ø / / µ B Then, by subsung no he appromaon, we ge: f ( ( ú = / ( P P + Q ( + µ B + O( D Wh a rae Q R specfed a he rgh hand face, we ge a smlar appromaon for block : µ Q B R ( ú = ( P P f ( / ( + + O( D For he case of a noflow boundary beween blocks Q R and Q R, he flow erms for he wo blocks become: ( P + P ( P P f ( + / f ( / ( ( ( O( + D ( P + P ( P P f ( + / f ( / ( ( ( O( + D Tme dscrezaon We showed earler ha by epanson backward n me: P(, = P(, + Δ + Δ! P (, + Δ + ( Δ! P (, + Δ + ( Δ 3 he followng backward dfference appromaon wh frs order error erm s obaned: P (, + Δ + orwegan Unversy of Scence and Technology Deparmen of Peroleum Engneerng and Appled Geophyscs 08

4 TPG460 Reservor Smulaon 08 page 4 of 5 P P P ( = + O( D D An epanson forward n me: P(, = P(, P (, + (D P (, + (D3 P (, +!! yelds a forward appromaon, agan wh frs order error erm: P P P ( = + O( D D Fnally, epandng n boh drecons: P(, = P(, P (, + (D P (, + (D3 P (, +!! P(, = P(, P (, + (D P (, + (D3 P (, +!! we ge a cenral appromaon, wh a second order error erm: ( P + D = P +D P + O(D D The me appromaon used as grea nfluence on he soluons of he equaons Usng he smple case of he flow equaon and consan grd sze as eample, we may wre he dfference form of he equaon for he hree cases above Eplc formulaon Here, we use he forward appromaon of he me dervave a me level Hence, he lef hand sde s also a me level Q R, and we can solve for pressures eplcly: P + P + P» ( fµc k P +D P, for =,, D As dscussed prevously, hs formulaon has lmed sably, and s herefore seldom used Implc formulaon Here, we use he backward appromaon of he me dervave a me level Q R, and hus lef hand sde s also a me level Q R : + Δ P + P + Δ Δ + Δ + P = ( φµc k P + Δ P Δ ow we have a se of equaons wh unknowns, whch mus be solved smulaneously, for nsance usng he Gaussan elmnaon mehod The formulaon s uncondonally sable orwegan Unversy of Scence and Technology Deparmen of Peroleum Engneerng and Appled Geophyscs 08

5 TPG460 Reservor Smulaon 08 page 5 of 5 Crankcholson formulaon Fnally, by usng he cenral appromaon of he me dervave a me level Q R, and hus lef hand sde s also a me level Q R : P + P + P Δ + P +Δ + + P Δ + P Δ + Δ = ( φµc k P + Δ P Δ The resulng se of lnear equaons may be solved smulaneously jus as n he mplc case The formulaon s uncondonally sable, bu may ehb oscllaory behavor, and s seldom used orwegan Unversy of Scence and Technology Deparmen of Peroleum Engneerng and Appled Geophyscs 08

P R = P 0. The system is shown on the next figure:

P R = P 0. The system is shown on the next figure: TPG460 Reservor Smulaon 08 page of INTRODUCTION TO RESERVOIR SIMULATION Analycal and numercal soluons of smple one-dmensonal, one-phase flow equaons As an nroducon o reservor smulaon, we wll revew he smples