Description of the oil recovery process implies creation of an adequate simulation model. One of the such models, the Dynamic Invasion Percolation with Trapping (DIPT) model, has been recently developed [1, 2]. This model is based on two independent steps.

The first step is to describe the oil reservoir saturated and/or unsaturated porous media in terms of the integer lattice models. In these models the lattice sites and bonds are identified with pores and capillary channels, respectively. It was suggested in [1,2] to describe n possible different states (empty, filled with displacing fluid or displaced one, etc.) of each lattice site j (j ⊆ [1, N]) in terms of the time-dependent n-component spinor functions χjm(j)(t), where N is the total number of lattice sites and m(j) ⊆ [l, n]. In such an approach the time-dependent state function ψ(t) of the entire lattice modelling oil reservoir porous medium can be introduced as the following direct product:
Ψ(t)=j=1Nχjm(j)(t).
(1)
The second step is to describe evolution of the oil reservoir subjected to the recovery process. It was done within the DIPT model. In this model it was suggested to describe evolution of the oil reservoir state function ψ(t) with the help of the following Schrodinger-like equation:
Ψt=τHΨ(t),
(2)

Description of the oil recovery process implies creation of an adequate simulation model. One of the such models, the Dynamic Invasion Percolation with Trapping (DIPT) model, has been recently developed [1, 2]. This model is based on two independent steps.

The first step is to describe the oil reservoir saturated and/or unsaturated porous media in terms of the integer lattice models. In these models the lattice sites and bonds are identified with pores and capillary channels, respectively. It was suggested in [1,2] to describe n possible different states (empty, filled with displacing fluid or displaced one, etc.) of each lattice site j (j ⊆ [1, N]) in terms of the time-dependent n-component spinor functions χjm(j)(t), where N is the total number of lattice sites and m(j) ⊆ [l, n]. In such an approach the time-dependent state function ψ(t) of the entire lattice modelling oil reservoir porous medium can be introduced as the following direct product:

Ψ(t)=j=1Nχjm(j)(t).
(1)

The second step is to describe evolution of the oil reservoir subjected to the recovery process. It was done within the DIPT model. In this model it was suggested to describe evolution of the oil reservoir state function ψ(t) with the help of the following Schrodinger-like equation:

Ψt=τHΨ(t),
(2)

where τ is the average time between consecutive displacing acts taking place in neighboring sites and H is the operator which governs evolution of the oil reservoir state function ψ(t). This operator has the form:

H=j=11NHj+ji1NHji,
(3)

where the local term Hj describes microphysics of displacement from a single pore whereas the non-local term Hji; is responsible for correlation between displacements in neighboring pores. The summation in this expression is taken over the subset j1N of the particular sites which are displacement available under given physico-chemical conditions. These sites are supposed to be randomly distributed over the entire lattice. Their relative number P gives the probability that the arbitrary taken site j belongs to the subset When P ≥ Pc there exists percolation phenomenon between any two arbitrary taken sites (Pc is the percolation threshold). Introduction of spinor formalism for the oil reservoir description allows to write the operators Hj and Hji in terms of the (2n+l)-spin operators. The evident form of these operators for the particular case of homogeneous water-oil displacement process can be found in [1, 2]. The operator Hji is constructed as the non-Hermitian one and, thus, equation (2) describes irreversible evolution corresponding to the irreversible nature of the oil recovery process. One of the main advantages of this operator approach is the possibility to describe analytically microphysics of the displacement process. Changing form of the correlation operator Hji it has become possible to describe conventional and chemical two-phase flow in porous media. Under different appropriate conditions this description can reveal both the fractal and hydrodynamic asymptotics.

The main restriction of the outlined above model is the assumption that pressure distribution in displacing fluid is homogeneous. Such a restricted situation can be valid only for the uniform models which are applicable for description of the small scale oil reservoir domains. The introduced generalization of the DIPT model consists in the following: the probability Pj(t) of displacement in every site j is assumed to be dependent on the local pressure gradient which evolves in time. The corresponding master equation describing the probability Pj(t) evolution is:

dPjdt=iWij[Pj(t)Pi(t)],
(4)

where Wji is the probability of the quantity Pj(t) to change due to ji-bond. This kinetic equation takes into account that the total rate of the probability Pj(t) change due to ji-bond has to be dependent on the pressure gradient which is proportional to the difference Pj(t)- Pj(t). In correspondence with the master equation (4) the value Pj(t) increases only when pressure at any neighboring site i is higher than pressure at site j.

In order to solve kinetic equation (4) it is necessary to introduce boundary conditions for the lattice sites identified with the injection j(in) and production j(pr) wells. When pressure in these wells is kept constant in the course of recovery boundary conditions for pressure are given in the form:

Pj(in)(t)=Pj(in)=const,Pj(pr)(t)=Pj(pr)=const.

Otherwise equation (4) acquires the form:

dPj(t)dt=iWij[Pj(t)Pi(t)]++Ij(in)(t)δj,j(in)Ij(pr)(t)δj,j(pr),

where Ij(in)(t) and Ij(pr)(t) are the time-dependent rates of the well pressure change.

In terms of the previously developed DIPT model such a Generalized DIPT (GDIPT) model means that the subset of the displacement available sites becomes time-dependent: ≺j≻tN(t). It gives the unique possibility to develop the self-consistent oil recovery simulation model which automatically accounts the reverse impact of recovery on the pressure distribution in oil reservoir. With the help of such a model it becomes possible to approach the problem of macroscopic oil recovery simulation. Macroscopic scales are characterized by the necessity to take into account oil reservoir pressure inhomogeneities. The GDIPT model allows to do that.

Results of the developed analytical approach were implemented in C language and applied for: (1) improvement of knowledge of the oil recovery physico-chemical mechanism in homogeneous and heterogeneous porous media and (2) simulation of the macroscopic enhanced oil recovery.

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A. Telex, 163245 SPEUT.

Bakurov
V.G.
,
Gusev
V.I.
,
Izmailov
A.F.
and
Kessel
A.R.
: "
Dynamic Percolation Model of Oil Displacement by Water in the Oil Reservoirs
",
J. Phys. A: Math.Gen
., vol.
23
(
1990
)
2507
2521
.
Izmailov
A.F.
and
Kessel
A.R.
: "
Hamiltonian Formalism for Phase Transition in Percolation Lattice
",
paper ThOB6 presented at the 17th IUPAP International Conference on Thermodynamics and Statistical Physics
,
Rio de Janeiro
,
1989
.