Mathematics of Enhanced Oil Recovery
Challenge of improving efficiency of recovery of hydrocarbons, crude oil and gas, from natural petroleum reservoirs, is an important and urgent global problem facing mankind. Many branches of science and technology contribute to its solution. Nowadays, co
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Challenge of improving efficiency of recovery of hydrocarbons, crude oil and gas, from natural petroleum reservoirs, is an important and urgent global problem facing mankind. Many branches of science and technology contribute to its solution. Nowadays, computer modeling becomes a key element of integration of all multifarious knowledge and decision-making tool. Here, an effort is made to present basic mathematical models of enhanced oil recovery and some mathematical problems emerging from their study.
1 Two-Phase Flow and Oil Displacement From point of view of fluid mechanics the oil recovery process is a multi phase multicomponent flow within porespace of natural porous rocks. It is described by the set of conservation laws for mass of components, phase flow rules and constitutive relations necessary for the model closure. For example, one of the most frequently used model corresponds to two-phase flow of two immiscible phases: displacing (water) and displaced (oil), and is described by relations
m
kfi(s)
8(1- s) 8t
Ui = ---Y'(pi J.li
P2- Pl
+ Y'·u2 = O·'
+ Pigz);
= Pc(s).
(1)
(2)
(3)
Here, m is the medium porosity, fractional volume of pores in rock; s is the first phase (water) saturation, ratio of the water filled pore volume to the total pore volume, ui, i = 1, 2 are phase flow (seepage) velocities; Pi are phase pressures which are generally different due to the action of capillary forces at porescale level, Pi and J.li are phase densities and viscosities, g is acceleration due to gravity, z is a height over a datum; k is the medium permeability and fi (s) are relative phase permeabilities. The model material functions fi and Pc ("the A. Buikis et al. (eds.), Progress in Industrial Mathematics at ECMI 2002 © Springer-Verlag Berlin Heidelberg 2004
32
V.M. Entov
LL
s
b
a
Fig. 1. (a) Relative phase permeabilities and capillary pressure vs saturation;
(b). Fractional flow curve
capillary pressure curve") are assumed to be specified functions of saturation s, which, in principle, are derived from the experiment. Their characteristic shape is shown in Fig. 1. The main features of this model become more clear if we consider "frontal displacement", one-dimensional flow along horizontal x-direction which can be reduced to a single equation
as+ U aF(s) at m ax F_ !I(s) . - !I(s) + J.Lh(s)'
J.L
=
~ (A(s) as); ax
A(s)
= J.L1;
J.L2
(4 )
ax
= _
kP~(s)fi(s)h(s) J.L2!I(s) + J.Lih(s) ·
(5)
Here, U is the total flow rate of two phases, and "fractional flow curve" F(s) expresses water fraction in the overall flow rate as a function of the local water saturation s (Fig. 1, b). Equation (4) is a nonlinear counterpart of the convection-diffusion equation, in which "diffusion" term expresses an effect of capillarity-driven flow. In analyzing large-scale phenomena, such as oil displacement from a petroleum reservoir of characteristic length L ,...., 10 2 - 103 m, the capillary flux can be apparently neglected, and Eq. (4) becomes just usual transport equation
as+ U aF(s) at m ax
=
O
(6)
which
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