BASIC RELATIONSHIPS OF UNIDIMENSIONAL FILTRATION

Lecture Outline

• 8.1. Flow of incompressible liquid in a non-deformable formation
• 8.2. Flow of incompressible liquid in a fractured deformable formation
• 8.3. Flow of ideal gas in a non-deformable formation
• 8.4. Filtration of real gas in a non-deformable formation

FLOW OF INCOMPRESSIBLE LIQUID IN A NON-DEFORMABLE FORMATION

To study practically filtration flows that obey Darcy’s law, it is necessary to know physical parameters: the values of pressure, velocity, etc.

Radial-plane flow is the most important type of flows applied in the theory of oil and gas fields’ development. Therefore we shall obtain relationships for the above-mentioned parameters for radial-plane flow.

Let’s list the initial equations:

a) for inflow:

G = 2nh^; (8.1)

lnr_c

In this case к = const, r = const, h = const:

(p — — p + c.

Substituting the last equation to (8.1), we shall obtain Dupuit’s formula for volumetric flow rate:

„ 2тг hk

(8.4)

Q =----

Inr ’

It follows from (8.4) that flow rate is proportional to pressure drawdown &P_C = Pc ~ Р» • The diagram of dependence of Q on Ap_c = p_c- p_w is given at figure 8.1. The ratio of flow rate to pressure drawdown is the well productivity index:

k=-°-

(8.5)

Relation between volumetric flow rate and radiuses of a well and of charge contour is weak because the relation of these radiuses enters into the formula of volumetric flow rate under the natural logarithm sign.

Q

Figure 8.1. Indicator curve of incompressible fluid flow under Darcy’s law for radial-plane flow

Substituting parameters of formation and liquid in (8.2), we obtain an expression for distribution of pressure in a formation:

p = p_c-fl₁-ln—,

where

(8-6)

ln^

Equation (8.6) indicates that the pressure in reservoir is distributed logarithmically. Thus, if the value of radius is close to the source contour radius, then the pressure changes in a small extent, but on approaching the well the pressure changes drastically (figure 8.2).

Figure 8.2. Distribution ofpressure in radial-plane filtration flow

Rotating the curve about the well’s axis we obtain the surface of pressure distribution. This surface is called the depression cone.

Isobars are concentric cylindrical surfaces, orthogonal to trajectories.

Similarly we find expressions for gradient of pressure in a formation: and for filtration velocity:

_Q/ _^{k 1 u} /in hr r/ r

• (8-7)
• (8-8)

Figure 8.3. Gradient ofpressure and filtration velocity vs radius

The gradient of pressure and velocity of filtration are inversely proportional to distance (figure 8.3) and form a hyperbole with sharp increase of their values at the approach to the well bottom hole.

Such behavior of reservoir pressure and filtration velocity is physically understandable. Indeed, the same volume of the incompressible fluid flows through any cylindrical surface concentric relative to the well (Q = const). The side surface area near the charge contour is very high, so velocity there is small. On approaching the well, the area gradually declines, and velocity increases. To ensure that, the pressure gradient must increase.

Motion of a particle is described by the equation:

dr _ Q

dt mhr

If we integrate the given equation on time from 0 up to t and on distance from R_q up to r, where R_o is an initial position of a particle of a fluid, we shall obtain:

тг mh(R^ -r²)

^/= Q

Time of extraction of all liquid from a round formation is determined by the formula:

_г_

Q

kyer&ge pressure in a formation will be determined by the formula:

(8-9) por

To determine average pressure let’s write formulas for porous volume of formation and for its differential:

^vpor=^ (^rc-^rw)'^h'^m’ dV_por = 27t hmrdr.

Substituting into (8.9) and integrating on r from r_w to r we shall obtain approximate expression for average pressure:

p = p_c-0,5a₁. (8.10)