This method is based on an analytical model which was developed in a sequence of papers starting with Refs [33-36].

In the assumed model, the reservoir is considered a ‘layer cake’ of homogeneous, non-communicating layers. The injected tracer pulse is distributed among the layers in accordance with the flow conductivity (permeability and thickness) of each layer. The tracer material in a layer moves in the reservoir toward the producer wells and is broadened by longitudinal dispersion in the direction of movement. The combined tracer response from all the layers makes up the response curve of tracer concentration as a function of the cumulative volume of water injected or produced. The peak height, the breakthrough time and the shape of the produced tracer response curve can be computed from the quantity of tracer injected, the formation properties and the well pattern geometry.

The initial model was expanded by Abbaszadeh-Dehghani and Brigham, who presented analytical solutions of tracer breakthrough curves for a number of balanced patterns with a rigorous treatment of the effects of tracer dispersion [35].

The tracer response curves from all homogeneous and balanced patterns analysed by the authors can be correlated into a single curve using a dimensionless pore volume parameter (VpD) given by:

V Vp — VPBT

PD _ 1 — VpBT

Firstly, Abbaszadeh-Dehghani and Brigham found an analytical expression for the displacing fluid cut ‘fD’ for all the balanced patterns [35]:

When a ‘slug’ of tracer with a pore volume equal to VpT and an initial concentration of C0 is injected into a pattern, the effluent tracer concentration profile from the reservoir is the difference between two pattern breakthrough curves (in the absence of mixing or any other transport process). That is:

C

TT _ /d(f„)- /d([7]p — Vpt) (51)

C 0

The mixing of a tracer with reservoir fluid during its transport through a porous medium is due partially to molecular diffusion and partially to mechanical or hydrodynamic dispersion. On the basis of experimental results, the effect of molecular dispersion on mixing in field tracer tests can be neglected with a good approximation. Also, it is possible to neglect the transverse mixing. Furthermore, to simplify the derivation of tracer mixing expressions without losing too much accuracy, mixing is considered to be related in a linear fashion to the interstitial pore velocity, u, so that the dispersion coefficient is:

K = au (52)

Considering convection and hydrodynamic dispersion as the dominant transport processes, it is possible to write the tracer mass balance equation as:

d 2C _ ЭС ЭС

2 dx dt

For a small slug tracer injection, whose length is infinitesimal compared with the distance between wells and defining a coordinate, 5, along each streamline (5 replace x), the resulting solution is:

where As is the width of a stream tube occupied by an undiluted tracer slug at the distance, — and a is the variance of the tracer distribution profile that includes changes for mixing and the geometry of the stream tubes, where:

(55)

For a given pattern geometry the tracer response curve from a homogeneous layer is a function of the Peclet number, ala, where a is the distance between producers and a is the dispersivity of the formation. As an example, the concentration for a balanced five spot pattern is expressed by:

(56)

The Y(W) term is a hyperelliptical integral that results from the mixing integral and VpBT(F) is the pore volume injected at breakthrough of the streamline, W.

The Fr term is the tracer size expressed as a fraction of displaceable pattern pore volume: V

model [35, 36], it is possible to derive how far the true situation deviates from the ideal one. Nevertheless, some additional considerations may be easily included in the model. The effects of adsorption and radioactive decay were analysed by Abbaszadeh-Dehghani [37]. Tracer partition between the water and hydrocarbon phases was considered by Tang [38].