Diffusivity equation for real liquids with gas in solution

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SPE
SPE 23740

Diffusivity Equation for Real Liquids with Gas in Solution

INTRODUCI’ION ABsrRAcr A rather comprehensive theory has been developed to describe fluid flow phenomena in porous media. This theory, under the constrain of appropriate simplifying assumptions, ultimately leads to expressions which ettitble the petroleum engineer to characterize reservoir behaviour. inthis context, one frquently meats some form of the diffusivity equation in conjunction with data obtained from well teats to arrive at speeific rcaemoir parameters. , For example, so-cakd pressure transient analysis can be used to arrive at estimates of the formation thicknesspermeability product, the storage capacity, the average reservoir pressure, etc. For liquid flow, the teehnique involvesreducing the diffusivity equation to a form where pressure becomes the independent variable. ‘Ibis requires the assumptions that the liquid is both, ideal and slightly compressible. In this work it has been developed an alternative approach which does not depend itpon the assumptions ciuxl above. An additional in!en~ is to investigate the effects of th~e suppositions on conTputed pressuredistributions in a onedimensional system. IEe mathematical’deaeription of fluid flow through porous media has been known for a long time. However, in the past most solutions to teaewoir problems were limited to simple eases whert! the applications did not involve tttueh eontputation. ‘fhe develo meat of eontputers and new numerical techniques 8 3), have made it possible to treat the more difficult problemsof petroleum engineering. In W34, Hutst@) and Muskat(4~introduced mathematical equations describing flow of a singlephase slightly~mpressible fluid through a homogeneous” porous medium. ~ey presented several solutions for some relatively simple cases, The fins effort m apply these equations to bttildup pressure anaiysii for inmmprcsible fluid flow was made by Muskat@). The advance in methods andeontputers allowed the study of more complicated simations, but in al! of them, it is implicitly assumed that the liquid phases in a reservoir are ideal and that they furthermore arc slightly compressible. When a large amount of gas is disolvcd in a crude under resetvoir conditions, the departure from ideality and slight compressibility may be substantial. The prcaent work was undertaken toevaluate the effects of these assumptions and develop an alternative approach which does not rely upon them. To this end, we confine our attention to z amparison of computed pressure distributions in a onedimensional system. 279

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Diffusivity Equation For Real Liquids With Gas in Solution
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SPE 23740

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iODYIMRtThe oil density equation in this work is derived from the Pm curves of oil formation volume factor and solution gas-oil ratio as functions of pressure. We first assume the existence of some pressure below the bubble point, say Pl, such that the Pm curves for pressures P > PI cart be essenciallity represented by a linear relationship. Titus the pertinent curves WN appear Iike those in Fig. L Wecan therefore write B. = Bob + m(p - pb) % 7 f%+n’(P-PJ (1) (2)

awumulation term on the righ[-hand side of Eq. (4) will carry a minus sign. [n thi zegion we will employ the equation of state given by Eq. (3). Our interest is not centered on pressures in excess of the bubble point since for P > Pb the assumption of slight$y mmpr=sible fluids is probably valid. Eq. (3) can be employed in Eq. (4) toeliminate the pressure (details of this development are in Appendix B). Thus,

for P 2 PI where m and n’ are the appropriate slopes. Above the bubble poin~ the slope m is negative while below it is positive. Similarly n’ is positive for P < Pb and zaro for P > P& & shown in Appendix ~ these relationships can be used to arrive at the following expression for oil density

If permeability...