Homogeneous model: Horizontal pipe and horizontal well. Flow loops can't duplicate field conditions. Daniel D. Joseph. April 2001

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1 Homogeneous model of producton of heavy ol through horzontal ppelnes and wells based on the Naver-Stokes equatons n the ppelne or the well and Darcy's law n the reservor Homogeneous model: Danel D. Joseph Aprl 1 The gas and ol can be descrbed as a mture n whch the gas content s represented at every pont of the mture by a volume fracton φ of gas. I do not epect that the homogeneous model wll work well f slugs are the flow type and certanly t won't work f the gas s connected n streams as n stratfed or annular flow. The homogeneous model should work best n the case that all the gas (or most of t) s dspersed n small bubbles that move wth the ol, whch may be the case for foamy ols, heavy ol lke Zuata, Cerro Negro and other heavy ols n the Ornoco belt, Canada, Albana, etc. Emlo Guevara told me that n hs eperence wth multphase pumps t was always much easer to montor the flow of foamy ol because the gas was dspersed. I tred on my last vst n early March 1 to fnd out f the flow type was slugs or homogeneous n horzontal ppes and wells n reservors of heavy ol. No one that I asked seemed really to know the answer to ths mportant queston. Horzontal ppe and horzontal well. There s no flow through the wall of the ppe; all the ol s ntroduced at the entrance of the ppe. The well has lner wth many slots that allow the ol to enter the well by over n the reservor; probably ths can be modeled by the contnuty of velocty of the ol and gas normal to the lner wall. It s probable that the contnuty of velocty normal to the lner s modeled well by the contnuty of the normal component of velocty gven by Darcy's law n the reservor and Naver-Stokes n the ppe. Flow loops can't duplcate feld condtons In horzontal wells and ppe lve ol at saturaton s drven nto the ppe by a created by the drawdown at the pump. The gas evolves by outgassng of dssolved gas as the drops by gradents n the drecton of the pump. In flow loops the gas s ntroduced by pumpng n gas and ol and outgassng s not an mportant factor. Of course f a lot of dssolved gas s released by outgassng t mght collect nto slugs whch could be modeled n a flow loop.

2 Mathematcal model of homogeneous flow () Pressure P vs dspersed gas fracton φ (all of the gas s dspersed nto bobbles whch more or less move wth the flud). Aran Kamp and I derved the equaton of state (called a solublty sotherm) for dspersed as a functon of βφ P P (1) 1 φ P where β(t) depends on temperature and P s the saturaton or bubble pont. When P P there s no dspersed gas, φ. We computed β 3 for Cerro Negro and values dfferent but near 3.4 for Lloydmnster and Lndbergh heavy ol. Obvously the most severe outgassng occurs at the pump where P P ρ s the smallest. Usng (1) we may estmate φ 1 β 1 / P where P PT. If P 1 ps and P p 9 ps, the φ 1/3.6. If P 7 ps, P p 4 ps, then φ 1 /.6. These are the worst condtons; n the ppe or well φ s much smaller. It certanly appears that n many cases not very much gas wll come out ecept near the pump. Ths supports the dea that the homogeneous model mght hold mostly anywhere away from the pump. It wll be convenent to work wth the drop Then (1) 1 may be wrtten as π P - P. ρ βφ π 1 φ π P (1) () The vscosty (φ ) and densty ρ(φ) of the mture depend on φ. The vscosty s assumed n the separable form (φ) () f (φ) f (φ) () There are may emprcal formulas for f (φ ) but we really don't know the correct f (φ ) n the case of foamy ol. On the other hand, the mture densty can be consdered to be accurately descrbed by ρ (φ ) ρ (1-φ ) ρ g φ ρ (1-φ ) (3) Prnted 4/1/1 DDJ/1/homogeneous/model-NS-Darcy.doc

3 () The substantal dervatve n a flud and n a porous meda u φ Dt t n a pure flud (4) α q φ Dt t n a porous meda wth porosty α (5) (v) Darcy's Law q s the seepage velocty of the mture n the porous meda and t satsfes Darcy's law q π k( f φ) (6) where k s the permeablty (v) contnuty equaton Dρ( φ) d ρ α ρ (7) Dt dφ Dt Dt From Dρ ρ dv u Dt (8) we get Dt ( 1 φ ) dvu n the flud (9) Dt ( 1 φ ) dvq porous meda (1) The stress Ths doesn't est n the porous meda T ( u u ) 1 D[ u] (11) In the Naver-Stokes equaton we have a term correspondng to the vscosty part of the stress S( φ, u) dv ( D[ u] ) dv ( φ) D[ u] dv S( φ, u) T [ ( u u )] (1) Prnted 4/1/1 3 DDJ/1/homogeneous/model-NS-Darcy.doc

4 Naver-Stokes equaton for dspersed flow ρ u t (13) ( φ ) y y π S( φ, ) 1 u Horzontal well and ppelne: Boundary condtons H q - k π D R n a round hole P p s pump Naver-Stokes n the well or ppe P(H) s the bubble pont at a heght H L Fgure 1. H. We are gong to assume that there s one bubble pont P whch doesn t depend on (1.) Horzontal ppe. u at r R. The porous meda determnes the at the end of the ppe at L. P p s the lowest n the system. Flow s from reservor to ppe entrance; then from ppe entrance to pump. (.) Horzontal well. The normal component of velocty s contnuous across the lner from the porous meda nto the well The tangental component on the ppe lner vanshes (3.) Pressure at the nlet end L of a long ppe. k π u er q er at r R (14) r u e at r R (15) It can be shown that f the ppe s long the P L at L s very nearly the reservor P L P (16) Prnted 4/1/1 4 DDJ/1/homogeneous/model-NS-Darcy.doc

5 (4.) Reynolds number n a long ppe. Assume that the ol s homogeneous and n Poseulle flow n a long ppe. The mean velocty s gven by u R 8 L where s the vscosty The Reynolds number s P P p. Re 3 ur ρ R v 8 L Numercal estmate: P P L 5 ft / L 1 dynes/cm R 1n.5cm (.5) u (.5) Re 3 p 1 ps ρ cm 6 dynes/cm The Reynolds number s very small Re << 1 for heavy ol (5.) Concluson. The Reynolds number for heavy ol > 1 pose s very small. Neglect the nerta (left sde) n (13). Prnted 4/1/1 5 DDJ/1/homogeneous/model-NS-Darcy.doc

6 π P P π P -P < p βφ π 1 φ π P ( φ) p Pressure dfference prescrbed whch drves the flow Solublty sotherm Vscosty dfference The porous meda does not enter ths problem P p pump r u at r R u φ ( 1 φ) dvu t π Stokes flow Contnuty equaton. We can neglect / t. P reservor Fgure. Horzontal ppe (length L, Radus R r). The unknowns are φ and u; π may be elmnated. φ and u depend on or r. All the equatons n the horzontal ppe hold n the horzontal well. In addton, we need k q π Darcy law reservor equatons: α q φ (1 φ) dvq Contnuty equaton t P p pump r u e The equatons n the well are the same as n the ppe but the boundary condtons are dfferent. π k π u er q er at r R r P reservor Fgure 3. Horzontal well. The porous meda s very mportant. Flow s drven by the reservor reference π p toward the pump through the lner. The equatons n fgures and 3 should be made dmensonless n order to dentfy the controllng parameters. Qute frankly, I am not at all clear on how to do ths. It s easy to make the equatons dmensonless, but choces of scales must be made and I do not yet know the optmal choce. You could try your hand at gettng a good dmensonless formulaton. Prnted 4/1/1 6 DDJ/1/homogeneous/model-NS-Darcy.doc