Page 311 - Geology and Geochemistry of Oil and Gas

272 MATHEMATICAL MODELING IN PETROLEUM GEOLOGY

pore volume minus the residual oil saturation S w;2 ¼ ð12S o;r Þ. A dynamic model
S w ¼ f ðtÞ with a S w range between S w;r and (1–S o,r ) was constructed using Eq. 11.76
at f ðxÞ ¼ xð12xÞ, where x ¼ S w . With the initial and final conditions taken into
account, the solution of Eq. 11.76 is as follows:

S w ¼ ½ð1 S o;r ÞS w;r expðð1 S o;r Þ tފ=½ð1 S o;r Þ S w;r þ S w;r expðð1 S o;r Þ tފ
(11.107)
where t is the duration of oil displacement by water. The change of water saturation
versus time is shown in Fig. 11.38.
The above-described dynamic model allows one to achieve two goals:
(1) To follow closely the changes in reservoir saturation occurring during oilfield
development.
(2) To apply the model for observation of the water invasion into the reservoir and
of movement of the oil–water contact (based on well logs and oil and water
production information).
The specific feature of the model is that it takes into consideration the so-called
‘‘technological’’ duration of the process of reservoir development. The model
(Eq. 11.107), which describes the dynamics of water encroachment in oil-saturated
reservoir in the process of oil production, can be changed into a model of lateral
migration of oil (if the water is replaced by the oil) in ‘‘geologic’’ time scale.

Fig. 11.38. Results of simulation of water saturations S w vs time. (1) G ¼ petrophysical parameter,
G o ¼ initial value of G, 2 ¼ acoustic transit time nt, 3 ¼ anisotropic true senstivity R ta , 4 ¼ isotropic
resistivity R t . (After Buryakovsky et al., 2001, p. 399.)