REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS
Generally speaking, flow equations for flow in porous materials are based on a set of mass, momentum and energy conservation equations, and constitutive equations for fluids and the porous material. For simplicity, we will in the following assume isothermal conditions, so that we not have to involve an energy conservation equation. However, in cases of changing reservoir temperature, such as in the case of cold water injection into a warmer reservoir, this may be of importance.
Below, equations are described for linear, one-dimensional systems, but can easily be extended to two and three dimensions, and to other coordinate systems.
Conservation of mass
Again we will consider the following one dimensional slab of porous material:
Mass conservation may be formulated across a control element of the slab, with one fluid of density is flowing through it at a velocity u:
The mass balance for the control element is then written as:
Dividing by , and taking the limit as goes to zero, we get the conservation of mass, or continuity equation:
For constant cross sectional area, the continuity equation simplifies to:
Conservation of momentum
Conservation of momentum is governed by the Navier-Stokes equations, but is normally simplified for low velocity flow in porous materials to be described by the semi-empirical Darcy's equation, which for one dimensional, horizontal flow is:
Alternative equations are the Forchheimer equation, for high velocity flow:
where n is proposed by Muscat to be 2, and the Brinkman equation, which applies to both porous and non-porous flow:
Brinkman's equation reverts to Darcy's equation for flow in porous media, since the last term then normally is negligible, and to Stoke's equation for channel flow because the Darcy part of the equation then may be neglected.
In the following, we assume that Darcy's equation is valid for flow in porous media.
Constitutive equation for porous materials
To include pressure dependency in the porosity, we use the definition of rock compressibility:
Keeping the temperature constant, the expression may be written:
Normally, we may assume that the bulk volume of the porous material is constant, i.e. the bulk compressibility is zero.
Constitutive equation for fluids
Recall the familiar fluid compressibility definition, which applies to any fluid at constant temperature:
Equally familiar is the gas equation, which for an ideal gas is:
and for a real gas includes the deviation factor, Z:
The gas density may be expressed as:
where the subscript S denotes surface (standard) conditions. These equations are frequently used in reservoir engineering applications. However, for reservoir simulation purposes, we normally use either so-called Black Oil fluid description, or compositional fluid description. For now, we will consider the Black Oil model, and get back to compositional models later on.
The standard Black Oil model includes Formation Volume Factor, B, for each fluid, and Solution Gas-Oil Ratio, Rso, for the gas dissolved in oil, in addition to viscosity and density for each fluid. A modified model may also include oil dispersed in gas, rs, and gas dissolved in water, Rsw. The definitions of formation volume factors and solution gas-oil ratio are:
The density of oil at reservoir conditions is then, in terms of these parameters and the densities of oil and gas, defined as:
Typical pressure dependencies of the standard Black Oil parameters are:
Flow equation
For single phase flow, in a one-dimensional, horizontal system, assuming Darcy's equation to be applicable and that the cross sectional area is constant, the flow equation becomes:
Boundary conditions
As discussed previously, we basically have two types of BC's; pressure conditions (Dirichlet conditions) and rate conditions (Neumann conditions). The most common boundary conditions in reservoirs, including sources/sinks, are discussed in the following.
Dirichlet conditions
When pressure conditions are specified, we normally would specify the pressures at the end faces of the system in question. Applied to the simple linear system described above, we may have the following two pressure BC's at the ends: For reservoir flow, a pressure condition will normally be specified as a bottom-hole pressure of a production or injection well, at some position of the reservoir. Strictly speaking, this is not a boundary condition, but the treatment of this type of condition is similar to the treatment of a boundary pressure condition.
Neumann condition
Alternatively, we would specify the flow rates at the end faces of the system in question. Using Darcy's equation at the ends of the simple system above, the conditions become:
For reservoir flow, a rate condition may be specified as a production or injection rate of a well, at some position of the reservoir, or it is specified as a zero-rate across a sealed boundary or fault, or between non-communicating layers.
Initial condition (IC)
The initial condition specifies the initial state of the primary variables of the system. For the simple case above, a constant initial pressure may be specified as:
The initial pressure may be a function of position. For non-horizontal systems, hydrostatic pressure equilibrium is normally computed based on a reference pressure and fluid densities:
Multiphase flow
A continuity equation may be written for each fluid phase flowing:
and the corresponding Darcy equations for each phase are:
However, the continuity equation for gas has to be modified to include solution gas as well as free gas, so that the oil equation only includes the part of the oil remaining liquid at the surface:
where represents the part of the oil remaining liquid at the surface (in the stock tank), and the part that is gas at the surface.
Thus, the oil and gas continuity equations become:
After substitution for Darcy's equations and Black Oil fluid properties, and including well rate terms, the flow equations become:
The oil equation could be further modified to include dispersed oil in the gas, if any, similarly to the inclusion of solution gas in the oil equation.
Non-horizontal flow
For one-dimensional, inclined flow,
the Darcy equation becomes:
or, in terms of dip angle, , and hydrostatic gradient:
where
is the hydrostatic gradient of the fluid.
Multidimensional flow
The continuity equation for one-phase, three-dimensional flow in cartesian coordinates, is:
and the corresponding Darcy equations are:
Coordinate systems
Normally, we use either a rectangular coordinate system or a cylindrical coordinate system in reservoir simulation.
In operator form, the continuity and the Darcy equations for one-phase flow may be written:
where the operators for rectangular coordinates are defined as:
(divergence)
(gradient)
for cylindrical coordinates
Boundary conditions of multiphase systems
The pressure and rate BC's discussed above apply to multiphase systems. However, for a production well in a reservoir, we normally specify either an oil production rate at the surface, or a total liquid rate at the surface. Thus, the rate(s) not specified must be computed from Darcy's equation. The production is subjected to maximum allowed GOR or WC, or both. We will discuss these conditions later.
Initial conditions of multiphase systems
In addition to specification of initial pressures, we also need to specify initial saturations in a multiphase system. This requires knowledge of water-oil contact (WOC) and gas-oil contact (GOC). Assuming that the reservoir is in equilibrium, we may compute initial phase pressures based on contact levels and densities. Then, equilibrium saturations may be interpolated from the capillary pressure curves. Alternatively, the initial saturations are based on measured logging data.
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