The moment analysis method was originally developed for closed reactor vessels [25, 26], but has been applied to the more general conditions of open boundaries [27] for characterization of fractured media under continuous tracer reinjection [28, 29] and for estimation of flow geometry [30, 31]. Certain restrictions are inherent in the calculation, for example, steady state conditions and conservative tracer behaviour are assumed a priori. Nevertheless, the method has a rigorous mathematical basis and has been extensively validated analytically and experimentally.

The governing equations used in moment analysis are based on knowledge of the residence time distribution of the tracer, defined as:

where C is the collected sample tracer concentration (ppm, ppb, Bq/L, etc.), and Qinj and Minj are the injection water flow rate (m3/d) and the injected tracer quantity (kg, g, Bq, etc.), respectively.

If the tracer experimental curve is not fully recorded (which is very frequently experienced because of limited the sample collection and analysis), it is important to employ some criteria for extrapolating.

The parameters of the experimental residence time distribution curve are calculated by the moment method. The nth moment of a residence time distribution curve is defined as:

Jtn ■ E(t) • dt

tn = (37)

J E (t) ■ dt

0

The zero moment (equivalent to the fractional cumulative recovery of the tracer) is:

■ dt (38)

The first moment (equivalent to the residence mean time) is:

J t ■ E(t) ■ dt

t* = ——————- (39)

J E (t) ■ dt

0

IV. 2.1.2. Pore volume determination

Pore volume determination is based on the knowledge of the zero and the first moments. It is calculated from:

The calculated pore volume represents only the watered pore volume. The complementary volume occupied by the oil cannot be reached by a passive tracer.

IV. 2.1.3. Calculating flow geometry

It has been proposed that the flow and the storage (pore volume) geometry of the formation can be estimated directly from a tracer test [30, 31]. The cumulative flow capacity at any streamline ‘i’ of a formation (F) is the sum of the contribution of each streamline that has a velocity greater than the ‘i’ and is normalized by the ensemble properties. Darcy’s law gives:

X j

}=i J N kA

X Xj

J =1 J

The cumulative storage capacity of these streamlines (Фг) is simply the sum of their individual pore volumes:

XVp

F, = j——————————————————————————————— (42)

j=1

These can be estimated from a tracer test, where Фг — is the incremental first moment calculated at the time t and normalized by the true first moment:

0

The cumulative flow capacity is simply the cumulative tracer recovery at time t normalized by the complete recovery:

(44)

FIG. 102. (Е, Ф)-рШ showing a hypothetical experiment and uniform flow cases.

Flow and storage capacities are most often plotted in a (F Ф)-р1оі The shape of a (FФ)^Ы is useful as a diagnostic tool indicating what fraction of the pore volume contributes to what fraction of the fluid flow. The (F Ф)-p1ots are widely used in oil reservoir engineering. Figure 102 illustrates the (F Ф)^Ы showing experimental values compared with the case of uniform flow (parallel equidistant streamlines).