Orthogonal optimization of horizontal well fracturing method

Transcription

1 Vol.2, No.9, (2010) hp://dx.doi.org/ /ns Naural Science Orhogonal opimizaion of horizonal ell fracuring mehod Chi Dong*, Kao-Ping Song, Xiao-Qi Chen Key Laboraory of Minisry of China on Enhanced Oil and Gas Recovery, Da Qing, China; *Corresponding Auhor: Received 18 May 2010; revised 29 June 2010; acceped 5 July ABSTRACT In many applicaions, sreamline simulaion shos paricular advanages over finie-difference simulaion. By he advanages of sreamline simulaion such as is abiliy o display pahs of fluid flo and acceleraion facor in simulaion, he descripion of flooding process ges more visual effec. The communicaion beeen ells and flooding area has been represened appropriaely. In lo permeabiliy reservoirs horizonal ells have been applied o simulae. And some horizonal ells have been fracured. Bu differen fracuring mehods are folloed by differen effecs. Through sreamline simulaion his paper presens ha he fracures improve boh he produciviy of horizonal ell and he efficiency of he flooding uni. And he synagmaic relaion of disance beeen arificial fracures, fracure lengh and fracure conduciviy has been given by orhogonal experimen. This mehod has been applied o he D160 reservoir in Daqing Oilfield. The producion hisory of his reservoir is abou 2 years. The reservoir is mainained above bubble poin so he simulaion mees he sligh compressibiliy assumpion. Ne horizonal ells are fracured folloed from his rule. Keyords: Sreamline Simulaion; Horizonal Well; Opimizaion 1. INTRODUCTION Using horizonal ells o simulae and enhance oil recovery, has been applied idely all over he orld. I is necessary o use horizonal ells o develop lo permeabiliy reservoirs. In he long process of pracice, i as discovered ha he advanages of horizonal ell ould no be made full use of hen only he horizonal ell drilling echniques no he combinaion ih he acual developmen condiions has be paid aenion o. In lo permeabiliy reservoirs he horizonal ells need o be fracured o simulae. As he increase of he lengh of horizonal secion, he number of arificial fracures should be increased oo. And he synagmaic relaion of disance beeen fracures, fracure lengh and fracure conduciviy is necessary o be opimized. As he applicaion of high-resoluion geologic models in simulaions, more aenion had been paid o he use of simulaion in oil field developmen. Finie difference mehod hich applied idely canno mee he needs such as high compuaional efficiency and being visually appealing. So he 3D sreamline simulaion approach has been applied o simulaion as complemenary o finie difference simulaion echniques [1-4]. Sreamline simulaion is in he advanced and maured sage in slighly compressible sysem. I calculaes he sauraion via 1D sreamline insead of sauraion field [5]. Boh compuaional efficiency and maching high resoluion geologic models becomes possible. No like he uniy of he sauraion in finie difference grid cell, sreamline simulaion provides several values of he sauraion in one grid cell, hen he fineness of he soluion ges elevaed. Taking advanage of sreamline simulaion s paricular capabiliy, he synagmaic relaion of disance beeen arificial fracures, fracure lengh and fracure conduciviy has been opimized by orhogonal experimen mehod. 2. STREAMLINE SIMULATION THEORY 2.1. Velociy Field Soluion and Sreamline Tracing The sreamline model is a model ih he assumpions as follos: 1) Considering he effecs of graviy and capillary; 2) There are oil phase and aer phase in he reservoir; 3) The fluid flo in he reservoir is incompressi-

2 C. Dong e al. / Naural Science 2 (2010) ble; 4)The fluid flo obeys Darcy s La; 5)The flo is under an isohermal condiion. On he basis of hese assumpions, and uilizing he coninuiy equaion and flo equaion, he pressure equaion of sreamline model can be esablished as follos: P D q P v 0 co 0 (1) here λ o, λ are, respecively, mobiliy of oil and aer, μm 2 /Pa s; λ is oal mobiliy, μm 2 /Pa s; γ o, γ are, respecively, graviies of oil and aer, dimensionless; q vo, q v are volume flo rae of producer and injecor, m 3 /s; q v is he oal flo rae of injecion or producion ells, m 3 /s; D is he deph from he reference level, is direcion is he same o graviy acceleraion, m; p co is he capillary force beeen oil phase and aer phase, Pa. Eq.1 can be solved by finie difference mehod. And he pressure disribuion, i.e. pressure field, can be derived. According o he pressure field, sreamline racing is implemened by using Pollock mehod hich is defined ih he folloing assumpions: he velociy field in a grid cell as a linear inerpolaion, and he velociy in each direcion is only a funcion of he coordinae in ha direcion. In a 2D grid sysem, ih he pressure a he face, he poenial a he face of he grid cell can be calculaed. Then he velociy in each direcion is derived, he TOF can be calculaed also. The face a hich he sreamline exis is he one corresponding o he minimum posiive exi ime. The exi poin ill be he enry poin of he adjacen grid cell, repeaing his calculaion process unil he resul converges o he grid cell hich a producion ell is in. Connecing he exi poins in chronological, an inegraed sreamline can be derived. In a 3D sysem, he exi poins can be obained by uilizing Pollock mehod o calculae he values in he z-direcion. By connecing hese poins, producion and injecion ells, he sreamline can be derived. The advanages of his mehod: being analyical; and i consrucs a velociy field ha saisfies he flux conservaion condiion Sauraion Field Afer he pressure filed is solved and he sreamlines are raced by uilizing finie difference mehod, he nex sep is o solve he sauraion along he sreamlines by inroducing he ime-of-fligh concep. The ime-of-fligh is defined as he ime required for a paricle o ravel a disance along he sreamline, x 1 s d (2) 0 u In sreamline racing, he ime for sreamlines across each grid cell, Δ e,i, can be calculaed using Pollock mehod. The ime of fligh can be expressed as follo: nblocks ei, (3) i1 here nblocks designaes he number of grid cells hen a paricle ravel a disance as s; Δ e,i is he ime for a paricle o ravel hrough he i h cell, s. A. Disribuion of volumeric flux of sreamline. Tracing sreamline from he cell of injecion ell o he cell of producion ell. For a grid cell ih source or sink, as he sreamline is no linear in segmens, he sreamline generalizes from he face of he grid cell insead of generalizing from he cenre of he grid cell. For simpliciy, disribuion of flux is based on changing he number of sreamlines ih differen flo rae of injecion ell, and he volumeric flux along each sreamline is consan. The higher injecion flo rae, he more sreamlines. Similarly, much more sreamlines can be raced in a high flo rae region. In grid cell (i,j,k) ih a source, he volumeric flux a he inerface (i ± 1/2,j,k) of grid cell (i,j,k) and grid cell (i ± 1,j,k) is Q i ± 1/2, j,k. The volumeric flux along each sreamline is q sli ± 1/2, j,k a he inerface(i ± 1/2,j,k). The number of sreamlines generalized from his face is n i ± 1/2, j,k can be expressed as follo: n Q q (4) here Q i ± 1/2, j,k is given as: i12, j, k i12, j, k sli12, j, k i12, jk, i12, jk, i, jk, i1, jk, Q TX P P (5) here TX i ± 1/2,j,k represens he ransmissibiliy in he x-direcion of oil phase and aer phase beeen he grid cell (i ± 1,j,k) and he grid cell (i,j,k); P i,j,k is he pressure in he grid cell (i,j,k). Similarly, he number of sreamlines on oher faces can be obained. B. Esablishing he sauraion equaion in sreamline model. Considering capillary effec, he fluid flo equaion can be expressed as follo: vi P D 0 Pco (6) Combining ih sauraion equaion and mass conservaion equaion, he sauraion equaion of sreamline model for oil-aer o phases can be derived: S S o D Pco qv τ (7)

3 1032 C. Dong e al. / Naural Science 2 (2010) Simplified as: here S f 1 q G τ G is defined as: o G D P co C. Numerical soluion for sauraion equaion of sreamline model. Because he derived equaion is a complicaed convecive diffusion equaion, for simpliciy, solve he convecion erm, graviy and capillary iem in a sauraion equaion by uilizing he echnique of operaor separaion. The convecive diffusion equaion can be divided o o pars, one is a nonlinear hyperbolic equaion describing he convecion erm, he oher one is a parabolic equaion describing he effecs of graviy and capillary. As he sreamlines generae from he grid cell face of injecion ell and dissolved a he grid cell face of he producion ell, he field along sreamlines is sourcefree. Decoupling (8) by uilizing operaor separaion echnique, o equaions are obained as follos: S f 0 S,0 S0 (10) S 1 G τ 0 S,0 S 0 (8) (9) (11) Eq.10 is a 1D sauraion equaion, and can be solved by 1D numerical soluion along sreamlines (ransformaion of he 3D equaion o 1D equaion). The soluion for (10) is assumed as Sf(). Eq.11 is a sauraion equaion considering he effecs of graviy and capillary, and can be solved only by 3D numerical soluion. The soluion for (11) is assumed as H(). Finally, he soluion a ime n ( n = nδ) is derived as: f n, S n H S S 2.3. Sreamline Updae 0 (12) For he acual developmen process in he oil field, ell paern and producion sysem is no fixed. Especially for he oil filed ih very long producion periods, e ofen need o esablish some mehods o simulae, such as paern adjusmen, shu-in, isolaion of individual zones and aer shu off, fracuring and acidizing, infilling, hich ill change he sreamline disribuion. So e have o updae he sreamlines immediaely afer hose condiions o accuraely represen he displacing dynamic informaion Compare ih Finie Difference Simulaion Compare he aer cu of sreamline simulaion and finie difference simulaion, i is easy o kno ha he sreamline simulaion s resul is more close o he hisorical daa a iniial sage. This is because he improvemen on sauraion calculaion. So sreamline simulaion is more suiable for sudy on flooding. 3. OPTIMIZATION OF FRACTURING 3.1. Simulaion of Fracured Horizonal Wells The flo field of flooding uni in combined ell paern of horizonal and verical ells is dre basing on he resul of sreamline simulaions as Figure 1. Respecively, he siuaion ih fracured horizonal ell and he one ih unfracured horizonal ell are simulaed. As shon in Figure 1, he fracures inflec he flo field around he middle of horizonal secion. As he former sudy, he inflo rae is loes in he middle of horizonal secion. So he fracures improve boh he produciviy of horizonal ell and he efficiency of he flooding uni Orhogonal Experimenal Design In flooding uni conains fracured horizonal ell, he main facors affecing he seep efficiency are disance beeen fracures, fracure half lengh and fracure conduciviy. In orhogonal design experimens, influence facors are called facors, and he daa poins of facors MIN MAX Figure 1. Sreamlines represening oil sauraion.

4 C. Dong e al. / Naural Science 2 (2010) called levels. There are hree facors hich effec developmen, so using L9 (33)-ype orhogonal o arrange experimens. Basing on he orhogonal experimenal design able, orhogonal experimen is designed as Table Simulaion of Orhogonal Experimen Simulae projecs and record he seep efficiency hen aer cu is 98%. The resuls of sreamline simulaion sho ha NO.6 projec has he highes seep efficiency ih he number 88.57%. Bu he resuls of simulaion canno disinguish he sequence of prioriy of facors. To sudy he impac on seep efficiency by facors he range of facors should be analyzed Analysis of Facors According o he size of range, he impac of facor on he size of he experimenal resuls can be deermined. In orhogonal experimen mehod, range is he difference beeen he maximum and minimum values a differen levels of he same facor. The bigger range a facor has, he more i influences he size of he experimenal resul. Before he analysis of he sequence of prioriy of facors, he experimenal resul should be processed as follos: 1) Sum he values of resuls in same level of one Table 1. Facors and levels. Level A B C Fracure Half Lengh (m) Fracure Conduciviy (D cm) Disance beeen Fracures (m) Table 2. Orhogonal experimen. Projec No. A B C Fracure Half Lengh (m) Fracure Conduciviy (D cm) Disance beeen Fracures(m) facor. As here are hree levels in one facor, K1, K2 and K3 are used o indicae he sums respecively. 2) Average he values of resuls in same level of one facor. Use k1, k2 and k3 o indicae he averages respecively. 3) Solve he size of range of each facor, and use R o indicae i. According o he above rules,he ranges is calculaed as Table 4. I shos, he range of facor A is bigges ih Table 3. Resuls of simulaion. Level Seep Efficiency Table 4. The range of experimenal resul. Projec No. A B C Measuremen Fracure Half Lengh (m) Fracure Conduciviy (D cm) Disance beeen Fracures (m) Seep efficiency K K K k k

5 1034 C. Dong e al. / Naural Science 2 (2010) k R he value In consequence, he main facor effecing he seep efficiency is he fracure half lengh, folloed by disance beeen fracures and he eakes hich is fracure conduciviy. The sequence of prioriy of facors is as follo: A > B > C. 4. CONCLUSIONS In his paper, he sreamline model considering fracured horizonal ell is derived. The sreamlines provide us ih a clear picure of flo field of flooding uni in combined ell paern of horizonal and verical ells. The sreamline simulaion resuls indicae ha he fracure plays an imporan role in enhancing seep efficiency and simulaion. The sequence of prioriy of disance beeen fracures, fracure lengh and fracure conduciviy has been deermined. REFERENCES [1] Grinesaff, G.H. and Caffrey, D.J. (2000) Waerflood managemen: A case sudy of he norhes faul block area of Prudhoe Bay, Alaska, using sreamline simulaion and radiional aerflood analysis. Paper SPE Presened a he 2000 SPE Annual Technical Conference and Exhibiion, Dallas, 1-4 Ocober. [2] Chakravary, A., Liu, D.B. and Meddaugh, S. (2000) Applicaion of 3D sreamline mehodology in he Saladin reservoir and oher sudies. Paper SPE Presened a he 2000 SPE Annual Technical Conference and Exhibiion, Dallas, 1-4 Ocober. [3] Lolomari, T., Braved, K., Crane, M., Milliken, W.J. and Tyrie, J.J. (2000) The use of sreamline simulaion in reservoir managemen: Mehodology and case sudies. Paper SPE Presened a he 2000 SPE Annual Technical Conference and Exhibiion, Dallas, 1-4 Ocober. [4] Grinesaff, G.H. (2000) Waerflood paern allocaions: Quanifying he injecor o producer relaionship ih sreamline simulaion. Paper SPE Presened a he 2000 SPE Wesern Regional Meeing, Anchorage, May. [5] Thiele, M.R. (2001) Sreamline Simulaion. Paper Presened a he 6h Inernaional Forum on Reservoir Simulaion, Schloss Fuschl, 3-7 Sepember.