Connecting Wellbore and Reservoir Simulation Models Seamlessly Using a Highly Refined Grid

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1 PROCEEDINGS, 43rd Worksop on Geoermal Reservoir Engineering Sanord Universi, Sanord, Caliornia, Februar 1-14, 018 SGP-TR-13 Connecing Wellbore and Reservoir Simulaion Models Seamlessl Using a Higl Reined Grid Misuo Masumoo Eploraion & Producion Deparmen, Idemisu Kosan Co., Ld., Marunouci, Cioda-ku, Toko , Japan misuo.masumoo@idemisu.com Kewords: reservoir simulaion, wellbore low ABSTRACT Te auor demonsraes an approac or connecing wellbore and reservoir simulaion models. Tis approac connecs ese models seamlessl using a igl reined local grid around e wellbore. Appling is approac, we can direcl compare observed and simulaed boomole pressure values a a lowing well wiou assuming a sead-sae low in e vicini o e wellbore. Tis advanage is useul in speciicall suding and modeling canges in boomole pressure aeced b e local disribuion o ransmissivi near e wellbore. We consider e ransien drop o reservoir pressure caused b a producing wellbore inerceping a racured zone orming a planar reservoir. Te inner boundar o e ring-saped local grid is along e wellbore surace. Te condiion o is boundar describes e mass balance a e wellbore surace. Te ouer boundar reers o e pressure disribuion o e global grid dnamicall, and e global grid reers o e producion rae deermined b solving e coupled problem o mass lows in e wellbore and local grid. Using is approac, we successull simulae an analic soluion a assumes a consan producion rae. Finall, an applicaion o a coupled problem o wellbore and reservoir simulaion models is demonsraed. 1. INTRODUCTION Coupled models o mass lows in a wellbore and reservoir are necessar o simulae e producion o geoermal luids roug wells under an appropriae condiion suc as consan wellead pressure. Te ke poin o ese coupled problems is o deermine boomole pressure accurael. Because numerical grids generall applied o reservoir simulaions are oo roug o simulae a drop in reservoir pressure in e vicini o e wellbore, some ecniques are required or deermining boomole pressure. One pracical and successul approac is o assume a sead-sae mass low near e wellbore as adoped in e reservoir simulaor TOUGH (Pruess e al., 1999). In is approac, e producivi inde relaes e mass low rae o pressure dierence beween e boomole and reservoir proporionall. Te auor as developed anoer approac a simulaes emporal and spaial canges in pressure in e vicini o e wellbore sricl. For is simulaion, we consider a igl reined local grid deined in e vicini o e wellbore. Te canges in pressure on is local grid are simulaed eicienl b eending e numerical ecniques proposed b Masumoo (015). Because low veloci in e vicini o e wellbore is ig, e quasi-uniorm disribuion o speciic enalp on e local grid could be deermined b inerpolaing e values a grid poins o e global grid. Tereore, we mus solve onl e diusion equaion in erms o pressure derived rom e conservaion equaions o mass and enalp. Tis sud describes e maemaical model and numerical ecniques used in is approac aer a brie review o Masumoo (015). A comparison beween numerical and analic soluions a a consan producion rae is also provided or validaion purposes. Finall, an eension o a coupled problem o wellbore and reservoir simulaion models is described.. BRIEF REVIEW OF THE FUNDAMENTAL MODEL AND TECHNIQUES Te maemaical model and numerical ecniques described in is sud are eensions o ose proposed b Masumoo (015). Troug ese eensions, we can avoid numerical diiculies in simulaing pressure canges on e igl reined local grid suc as ver large Couran numbers and an eplosive increase in compuaional load..1 Maemaical model Adoping a Caresian coordinae ssem, we describe mass and enalp conservaion in a single-pase planar (wo-dimensional) reservoir model as ollows: U V QM M U M V M M, (1) U V T T QH H U H V H H. () 1

2 Masumoo Te smbols appearing in all equaions are deined in e nomenclaure a e end o is paper. Tis nomenclaure also gives e maemaical deiniions o some smbols. For simplici, e second erm sown in e rig-and side o () neglecs spaial canges in ermal conducivi. Te componens o low veloci U and V obeing Darc s law are deined as ollows: U k P g, (3) V k P g. (4) Equaions (1) (4) are maemaicall idenical wi e model adoped in e reservoir simulaor HYDROTHERM (Kipp e al., 008) bu ave dieren orms or appling general numerical scemes in solving advecion problems (e.g., upsream dierence, oal variaion diminising, and consrained inerpolaion proile (CIP) scemes). Te ermodnamic uncions and viscosi o luid are compued using e equaions o sae and empirical equaion developed b e Inernaional Associaion or e Properies o Waer and Seam.. Numerical ecniques Using e ime-spliing meod, we advance e advecion and non-advecion erms o (1) and () separael. I e CIP sceme is adoped, we also solve e equaions derived wen diereniaing (1) and () regarding and as described b Masumoo (015). Te advecion erms are irs advanced, ollowed b e non-advecion erms. We en consider e ollowing epressions derived rom removing e advecion erms rom (1) and (): U V QM M M, (5) U V T T QH H H. (6) Te discreized orms o ese equaions using e inie dierence meod are advanced semi-implicil o improve numerical sabili. Appling e cain rule, we derive e diusion equaion regarding pressure rom (5) and (6) as ollows: were K k P g K k P g 3 1 P K K, (7) P M H P P H M P K1, (8) K K M H P H M P, (9) M P T T Q M H P Q H M P 3. (10) Te pressure value a eac grid poin is deermined b advancing (7) implicil. Reerring o ese advanced pressure values, (5) and (6) are advanced. Finall, we deermine e remaining unknown variables suc as speciic enalp, emperaure T, luid densi, and viscosi a eac grid poin. For simplici, we assume a e properies o rock marices included in e deiniions o M and H (porosi, densi m, and speciic enalp c m ) are consan. 3. EXTENSIONS Te aoremenioned numerical ecniques are eended o simulae emporal and spaial canges in pressure eecivel in e vicini o a producing wellbore. Hig low veloci in e vicini o e wellbore sould resul in a quasi-uniorm disribuion o speciic enalp unless a wo-pase region appears and generaes e mass lues o waer and seam a cange independenl as a resul o relaive permeabili. Tereore, ocusing on simulaing pressure sricl and esimaing speciic enalp rougl using inerpolaion is reasonable. In is eension, we adop a igl reined local grid a epands eponeniall as e disance rom e well ais increases. 3.1 Pressure cange in e vicini o a wellbore We appl (7) (10) o emporal and spaial canges in pressure in e vicini o a wellbore using a polar coordinae ssem as ollows: K P K r rk P g K r k P r g 1 r r r. (11) Equaion (11) assumes a ea low resuling rom ermal conducion is negligible because a due o advecion is ig. We also assume a no sources eis a generae in- or oulow in e vicini o e wellbore. Tese assumpions resul in K 3 0.

3 Masumoo Te pressure canges in is polar coordinae ssem are simulaed on e igl reined local grid. To generae is grid, we deine psical and compuaional spaces and adop coordinae conversion beween em. Te disance rom e well ais r in e psical space is convered o a coordinae in e compuaional space. Te angle is deined commonl in bo spaces. Te relaionsip beween r and is deined as ollows: r r rw r sin. (1) Using is relaionsip, a uniorm grid deined in e compuaional space is equivalen o a non-uniorm grid a epands eponeniall as r increases in e psical space. Te same approac is adoped or emporal discreizaion. Te relaionsip beween e psical ime and compuaional ime is deined as ollows: Subsiuing sin. (13) r and or r and, respecivel, (11) is redeined in e compuaional space and ime as ollows: K P K r rk P g K r k P r g. (14) 1 r r r Equaion (14) is advanced using e inie dierence meod. Te grid sizes are uniorm in bo e and direcions. Te grid size in e direcion is uni and a in e direcion is 4. Te consans r and r in (1) are derived rom e grid size r a e wellbore surace ( 0 ) and e pair o reerence values, r and. Tese wo relaionsips ield wo non-linear equaions wi respec o r and r rom (1). We can solve em using e Newon-Rapson meod. Te compuaional ime-sep leng is also uniorm a uni. Te consans and in (13) are deermined using e same procedures as ose o deermine r and r. Noe a e emporal discreizaion using is approac is also applied o (7) a is deined in e global grid. Tus, e simulaed canges in pressure in bo e local and global grids are sncronized. 3. Connecing e local grid o e wellbore and global grid Te ring-saped local grid is oulined b inner and ouer boundaries, along wic r are consan. Te inner boundar corresponds o e wellbore surace. In erms o e mass balance a is surace, we obain e inner boundar condiion b e ollowing: q M kwr P (15) w r r r w or q M kwrw r P. (16) 0 Te pressure disribuion along e ouer boundar is deermined b reerring o e pressure disribuion on e global grid dnamicall. Figure 1 illusraes e ouer boundar o e local grid superimposed on e global grid. Te size o e global grid is pariall uniorm a Δ in bo e and direcions in e superimposed region. Te wellbore in e global grid is represened b a poin source corresponding o a single grid poin. Te ouer boundar o e local grid orms a circle wose radius is Δ. Aer e cener o is circle a e grid poin i, j corresponding o e poin source in e global grid is locaed, e pressure value a eac grid poin along e ouer boundar o e local grid is deermined based on e values a e grid poins in e global grid as ollows: P 0 P i, j, (17) 6 4 P i1, j1 3 4 Pi, j1 Pi 1, j 3 Pi, j P, (18) 1 P, (19) P i, j 6 4 P i1, j1 3 4 Pi, j1 Pi 1, j 3 Pi, j P, (0) 3 P 4 P i, j, (1) 3

4 Masumoo 6 4 P i1, j1 3 4 Pi, j1 Pi 1, j 3 Pi, j P, () 5 P, (3) 6 P i, j 6 4 P i1, j1 3 4 Pi, j1 Pi 1, j 3 Pi, j P, (4) 7 were bilinear inerpolaion is adoped o calculae P 1, P 3, P 5, and P 7. Te values o speciic enalp a all grid poins in e local grid are deermined based on ose a e grid poins in e global grid and b adoping bilinear inerpolaion. I e emporal cange in e mass low rae q M is alread known beore beginning a run, e global grid reers o noing in e local grid. In oer words, simulaions on e global grid can be perormed independenl. However, i e mass low rae is deermined dnamicall based on e boomole pressure and speciic enalp using a wellbore simulaion model, e global grid reers o e mass low rae a eac ime sep. Te boomole values o e pressure and speciic enalp are equal o e means o e values a e grid poins along e inner boundar o e local grid. For simplici in is sud, we do no consider skin eecs. I we were o consider ese eecs, we could coose one o wo approaces: add e pressure si based on e skin acor o e aoremenioned boomole pressure, or deine e local disribuion o permeabili eplicil in e vicini o e wellbore. Figure 1: Ouer boundar o e local grid superimposed on e global grid. Te solid and open poins denoe grid poins in e local and global grid, respecivel. Te pressure values corresponding o e numbered grid poins in e local grid are compued using (17) (4). Figure : Reservoir model o e numerical soluions. 4

5 4. NUMERICAL SOLUTIONS 4.1 Comparison wi an analic soluion Masumoo Le us consider producing a a consan rae rom a square and orizonal reservoir aving a side o 000 m as sown in Figure. Te reservoir ickness is 100 m. A poin source represening a wellbore is locaed 1000 m rom eac side. Te iniial values o pressure and speciic enalp in e reservoir are uniorm a 8 MPa and 1000 kj/kg, respecivel. Te pressure and speciic enalp a eac side deining e ouer boundar o e reservoir are mainained a ese iniial values. Te producion rae is consan a 300 /. Te psical properies o e rock are uniorm as summarized in Table 1. Because o ese rock properies and e ermodnamic condiion o e reservoir, e ransmissivi and soraivi are m 3 /Pa/s and m/pa, respecivel. To validae e numerical ecnique, we reer o e analic linear soluion o e diusion equaion wi respec o e pressure derived using ese wo parameers wile assuming a e reservoir epands ininiel (e.g., Dake, 1978). Te size o e global grid covering e enire reservoir is 100 m uniorml. Te oal number o grid poins along eac side is 1, including ose locaed a e boundaries. Te local grid is deined or simulaing emporal and spaial canges in pressure in e vicini o e wellbore represened b e poin source in e global grid. Te inner boundar o e local grid is m rom e 1 poin source ( r [m]). Tis disance represens a wellbore diameer o 8.5 inces. Te ouer boundar is wo grids (00 m) w rom e poin source as illusraed in Figure 1. Te grid size in e r direcion a e wellbore surace is m ( r 110 [m] a 0 ). Te oal number o grid poins in e r direcion is 60 ( r 00 [m] a 60 ). From ese wo condiions and (1), e nonuniorm grid in e r direcion is generaed as sown in Figure 3. Te iniial value o e ime-sep leng is s 13 ( [s] a 0 ). Te psical ime reaces 1 d aer 4000 seps o ime inegraion ( 1 [d] a 4000 ). From ese wo condiions and (13), e ime and ime-sep leng a eac sep are deermined. As a resul, e ime-sep leng epands o d during e irs da o e producion. Te consan ime-sep leng a is value is adoped during e second and laer das. Te producion is erminaed a e end o e en da. Te oal number o ime seps or 10 das o e producion is Table 1: Psical properies o e rock. Permeabili Porosi Termal conducivi Hea capaci o e rock mari m W/m/K 10 6 J/m 3 /K Figure 3: Local grid in e Caresian coordinae ssem drawn on dieren scales. Te 8.5-inces wellbore is locaed a = 0 [m] and = 0 [m]. Te numerical and analic linear soluions o e boomole pressure are sown in Figure 4. Aer an inerval o non-linear cange, e numerical soluion decreases proporionall o logarimic ime. Te superimposed analic linear soluion is consisen wi e numerical soluion. A a ime corresponding o e radius o invesigaion o 1000 m, e numerical soluion begins o diverge rom e analic linear soluion and converges a a consan value. Tis cange is caused b e boundar condiion in wic e pressure a e iniial value is mainained. From is resul, we can prove e validi o no onl e emporal and spaial canges in pressure simulaed in bo e local and global grids bu also e connecion beween ese grids. 5

6 Masumoo Figure 4: Numerical and analic linear soluions o e boomole pressure or e problem o a consan producion rae. 4. Eension o a coupled problem o a wellbore and reservoir We ne consider a variable producion rae based on e boomole pressure as sown in Figure 5a. We assume a is discreized relaionsip is derived rom several observed and/or simulaed lows in a producion well a a consan wellead pressure. Te aoremenioned problem is eended o ake ino accoun is variable producion rae. In is sud, e dependenc o e producion rae on pressure is considered onl or simplici. Ta on speciic enalp can be immediael considered b deining e dependenc on bo pressure and speciic enalp. Currenl, e numerical ecnique o is sud deermines e producion rae eplicil, wereas e pressure value a eac grid poin is advanced implicil. Furer sopisicaed approaces could deermine bo e producion rae and pressure values implicil. However, eplici deerminaion o e producion rae is adopable or e wide range o pracical problems unless ver roug ime-sep lengs are used. Te compuaional load per ime sep o e semi-implici sceme adoped in is sud is muc lower an a o ull implici scemes widel adoped in muli-purpose reservoir simulaors a include solving a large ssem o non-linear equaions. Te simulaed canges in e boomole pressure and producion rae are sown in Figure 5b. Te coninuous cange in e producion rae based on e boomole pressure is deermined using cubic inerpolaion a reers o our pairs o values o e producion rae and boomole pressure sown in Figure 5a. Wiin 1 s rom e beginning o e producion, e producion rae drops rom 373 / corresponding o e iniial boomole pressure (8 MPa) o 50 /. Aer is drop in e earl ime, e producion rae graduall becomes sable. Te boomole pressure becomes approimael proporional o logarimic ime because o is sable producion rae. Finall, bo e producion rae and boomole pressure become consan as a resul o e boundar condiion a mainains e pressure a e iniial value. Figure 5: Numerical soluion o e coupled problem o a wellbore and reservoir: (a) Relaionsip beween e producion rae and boomole pressure assumed in e numerical soluion. (b) Canges in e boomole pressure and producion rae. 5. CONCLUSION In is sud, an approac or connecing wellbore and reservoir simulaion models using a igl reined local grid was demonsraed. Comparing e numerical soluion o is approac wi e analic soluion a a consan producion rae, we successull validaed e emporal and spaial canges in pressure simulaed on e local and global grids as well as e connecion beween ese grids. Te variable producion rae based on e boomole pressure was successull applied o is approac. B deermining e relaionsip 6

7 Masumoo beween e producion rae and boomole pressure based on observed and/or simulaed lows in a producion well, we can connec e wellbore and reservoir simulaion models successull. NOMENCLATURE Lain smbols c m Speciic ea capaci o rock marices, J/kg/K. g, g and componen o gravi acceleraion in m/s. g r, g r and componen o gravi acceleraion in m/s. H Enalp o luid and rock marices per uni volume o rocks in J/m 3, H c T H Enalp o luid per uni volume o rocks in J/m 3, H. i j Speciic enalp o luid, J/kg. Grid inde in direcion. Grid inde in direcion. K 1 Reer o (8). K Reer o (9). K 3 Reer o (10). k Permeabili in m. M Mass o luid per uni volume o rocks in kg/m 3, M. P Pressure in Pa. Q Source low rae o enalp in W/m 3. H Q M Source low rae o mass in kg/s/m 3. q M Mass low rae o a well in kg/s (negaive value or producion). r Coordinae o e polar coordinae ssem in m (disance rom e well ais). r Value o r coordinae a e wellbore surace in m. w r Grid size in r direcion in m, r dr d. T Temperaure in C. Time in s. Time-sep leng in s, d d. U, V and componen o low veloci in m/s. w Tickness o e planar reservoir in m., and coordinae in m. Greek smbols r Consan o coordinae conversion in m. Consan o emporal conversion in s. r Consan o coordinae conversion. Consan o emporal conversion. Δ Grid size in m. Porosi o rocks. Termal conducivi o rocks in W/m/K. Viscosi o luid in Pa s. Coordinae o e polar coordinae ssem (angle). Densi o luid in kg/m 3. m Densi o rock maries in kg/m 3. Compuaional ime. Derivaive o regarding in 1/s, ( d d ). Coordinae in e compuaional space. Derivaive o regarding r in 1/m, ( d dr ). r r 1 m m. REFERENCES Dake, L. P.: Fundamenals o reservoir engineering, Developmens in peroleum science, 8, Elsevier, (1978). 7

8 Masumoo Kipp Jr., K.L., Hsie, P.A., and Carlon, S.R.: Guide o e Revised Ground-Waer Flow and Hea Transpor Simulaor: HYDROTHERM-Version 3, U.S. Geological Surve Tecniques and Meods, 6-A5, (008). Masumoo, M.: Applicaion o e Consrained Inerpolaion Proile (CIP) Sceme o Two-Dimensional Single-Pase Hdroermal Reservoir Simulaions, Geoermics, 54, (015), 10. Pruess, K., Oldenburg, C. and Moridis, G.: TOUGH user s guide, version, Lawrence Berkele Naional Laboraor Repor, LBNL , (1999). 8