In this paper we discuss the problem of the reservoir characterisation from the tracer test data. Oil-water rates and tracer concentration on the productional wells are available. The problem is to determine the permeability profile.
The new pseudorised equations for the displacement of oil by water with tracer in heterogeneous stratified reservoir are derived. Analyses of the tracer wave propagation into the layer cake reservoir during the waterflooding was given on the basis of analytical solution of the direct displacement problem Exact analytical solution of the inverse reservoir characterisation problem provides explicit formulae for the profile of permeability.
The method developed was applied for the interpretation of pilot test on gas injection in the Vuktyl field (Russia, Timan Pechora region). The results obtained from the tracer analyses data fits well with the data of logging and core analyses.
The heterogeneity of oil reservoirs is the main factor that determines the recovery during the waterflooding. The profile of heterogeneity could be obtained from the waterflood data . Use of tracers for the control on waterflood increases the quality of reservoir characterisation. But the inverse problem of the permeability profile determination from the waterflood data as well as from the tracer analyses data does not have the unique solution. Tuning of the reservoir model from a history matching can lead to different pictures of heterogeneity.
The inverse problem of permeability profile determination can be solved in frames of models with the low dimensions. Pseudofunctional flow approach is widely used to reduce the dimension of the reservoir model. It reduces the number of degrees of freedom for the inverse problem as well.
Many oil reservoirs operate under the conditions of vertical equilibrium (VE). The VE assumption (gravity and capillary forces are equal) permits to describe 2D displacement process in layer cake reservoir by one quasilinear hyperbolic equation [2–6].
Viscous dominated case assumes that the lateral pressure gradient is independent of the vertical coordinate. The main assumption is that the displacement of oil by water in different layers is going on in order of decreasing of their permeabilities. This model consists on the one hyperbolic equation as well [7 – 9].
Application of the Welge's method  to each of the mentioned above models allows us to determine the pseudofractional flow curve and profile of permeabilities from the waterflooding data.
The range of validity of different pseudorised models was obtained in [11 – 13] by approaching to zero different dimensionless parameters in frames - of 2D model. 5 pseudorised models were obtained in  as an asymptotical limits of 2D model: two mentioned above models; averaged model of commingled layers, gravity dominated model [14,15] and model for homogeneous reservoir.
All these models permit solution of the inverse problem by the Welge's method. It is necessary to highlight, that actually we do not need the solution of the direct displacement problem to obtain solution of the inverse reservoir characterisation problem. The self-similarity of the displacement problem is enough for existence of the explicit solution of the inverse problem [17,18].
In this work we derive a pseudorised equations for the two-phase flow with tracer In stratified heterogeneous reservoir. The problem of oil displacement by water with tracer in layer cake reservoir is solved. We analyze the propagation of the tracer front in regions with the different permeability. Inverse problem is solved as well. Extension to Welge's method developed allows us to determine more precisely profile of permeabilities from the tracer analyses data.
Discuss the 1D tracer flow with the two-phase oil-water flux.
It is obvious from the Fig. 1 that the tracer front lags behind the front of water, Dc< Df. There are three reasons for the delay between the tracer front and water front:
sorption of tracer on the matrix surface and on the interphase water-oil surface;
solubility of tracer in the oil phase;
presence of connate water in the reservoir before the injection.