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VA – GP - OA: Numerical Multiphase PTA
p 12/29
The classical interpretation can hence provide 3 results from the derivative curve:
The initial and water mobilities.
The injection radius
Let us further analyze Figure 14, which shows the analytical run for a composite radius of 27.3
ft, equivalent to the cumulative injection after 11 h. Fall-off 2 shows the same behavior as the
standard case:
A first transient period can be seen, where the mobility corresponds to the water zone. In
the analytical case, the mobility was calculated assuming a maximum invasion of water, at
Sw = 1 – Sorw. In fact, the water saturation upstream of the front progressively varies
from the Buckley-Leverett frontal saturation (Swf) to this maximum value.
The last transient period corresponds to the mobility of the original oil zone, with initial
water saturation Swi.
This comparison validates the interpretation in the two-phase model of the fall-off curve.
Although not clear here, in some cases (usually with weak mobility contrasts) one may notice
that the transition period is smoother in the numerical model than in the analytical prediction.
This indicates that the saturation profile sharpness of the Buckley-Leverett model is not
respected, as it is in the composite analytical model. This is due to some numerical dispersion,
and can be corrected using a finer simulation grid.
One can also notice that the 3 fall-off derivative curves begin to show small oscillations at late
times (figures 13 and 14). These oscillations are different in nature from the previously
observed ones (i.e. when the front was moving), and are due to numerical precision. This can
be explained by recalling that the numerical mechanism is an iterative calculation which stops
when a given convergence criterion is reached (here, a local material balance error). This
implies pressure results are erratic inside a very small interval, but when the pressure
variation of a time step reaches the same order of magnitude - as it is the case when the
pressure completely stabilizes in the reservoir - the derivative starts to reflect this erratic
behavior. This is not a significant problem, since the pressure variation itself is negligible in
this period, and the length of the fall-off periods does not correspond to real cases. Note that
in any case, these oscillations can be completely damped by increasing the numerical precision
(i.e. by reducing the numerical mass balance error criterion in the numerical settings).
2.5. Conclusions for injection
The injection of water into an oil or gas reservoir (without mass transfer between the phases)
can be simulated in transient mode using the numerical model in Saphir NL, with some specific
observable behavior:
The fall-off and injection periods exhibit a first part representative of the in place water
mobility, which can be hidden by the storage effect. This corresponds to water saturation
between the Buckley-Leverett front saturation and the maximum water saturation: 1-
Sorw
.
The fall-off and injection periods exhibit a second part representative of the initial in place
fluid mobility, at the connate water saturation.
The injection period exhibits a third part representative of a mobility coming back to the
level of the first part.
The injection period is subject to oscillations due to the discretized nature of the model.
The oscillation level increases with low mobility ratio (gas or light oil), or with large cells in
the radial flow direction, and with the non-linearity of the total mobility curve. The
oscillation level can be reduced by reducing the radial cell size with the gridding
progression ratio, and by using a correction based on pseudo-relative permeability curves.
This correction is automatically activated by default in Ecrin, in the case of water injection.