Lecture Outline

  • 6.1. Initial and boundary conditions
  • 6.2. Fluid density and viscosity dependence of reservoir pressure
  • 6.3. Rock porosity and permeability dependence of reservoir pressure


In initial conditions we specify the value of potential at the time moment that is considered as initial time instance:

(p = (pQ (x, y, z)&t = 0. (6.1)

If at t = 0 formation is not disturbed, then:

(p = (pQ = const. (6.2)

Boundary conditions are given on borders of a formation (external boundary conditions) and at well bottom hole (internal boundary conditions). Let’s write the basic boundary conditions on external and on internal border.

External border:

1) constant potential:

ф (Г, /) = фк = const, (6.3)

i.e. the border is a charge contour of reservoir;

2) an infinite formation:

Lim ф (Г, Р) = фк = const. (6.4)

X->00 y->00

The case of internal border:

1) constant potential at the bottom hole of a well of radius rw:

w = const; (6.5)

1) constant mass flow rate (Darcy’s law is obeyed):


G = pufw = 2nrwh-^ = const (6.6)

To obtain closed system of equations it is necessary to add the equations of dependences of r, m, k, h on pressure.

Fluid density and viscosity dependence of reservoir pressure

We will study three types of liquids:

a) incompressible liquid.

Then the density remains constant:

p = const. (6.7)

b) elastic liquid.

We encounter with such liquids in non-stationary processes of oil production at the expense of oil expansion caused by decrease of pressure:

P=p/*(p-p.)> (6.8)

where 0 is the factor of volumetric expansion of a liquid,


  • (6.9)
  • * dp JT pdp’

here V*. is the volume of a liquid; /7^= (7 30)-1010/Pa for oil and /3U = (2,7 -5- 5)-1010/Pa for formation waters;

c) compressible liquid-gas mixture.

This situation can be observed when developing gas and gascondensate fields. For pressures up to p^ < 9 MPa and Np < 1 MPa it is possible to use the equation of ideal gas law:

p=pRT, (6.10)

where R is gas constant, T is temperature.

Ideal gas is a model of gas where the molecules have no volume and do not interact with each other.

At isothermal process (T = const) we use the equation:

p=p„plp„- (6.11)

If prock > 9 MPa then it is necessary to use the generalized equation for real gas state:

P = zrRT, (6.12)

where z is the factor of super compressibility, z is a function of pressure at isothermal flow

z = zoe_ai(p_po). (6.13)

If the pressure is less than the pressure of saturation then the viscosity is supposed to be independent of pressure, and at big values of pressure the viscosity is determined by the formula:

77=r7oe’4'(lPo). (6.14)

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