 # FLAT PROBLEMS OF FILTRATION THEORY

Lecture Outline

• 10.1. System of equations for potentials of a group of wells
• 10.2. A flow from an injection well to a production well
• 10.3. Inflow to a well located near to impenetrable rectilinear border
• 10.4. Inflow to a well in a reservoir with a remote charge contour of an arbitrary form

When developing oil and gas fields two types of problems arise:

• 1. The values of wells’ flow rates are given and it is necessary to determine bottom-hole pressures and formation pressure at any point of a reservoir. In this case the value of flow rate is determined by the limiting level of pressure drop-down for the reservoirs at which they remain not destructed or by strength parameters of wells’ equipment.
• 2. The bottom hole pressures are given and it is necessary to determine the flow rates of wells and formation pressure at any point of the formation. Such problem frequently occurs in practice of oil and gas fields’ development. The value of bottom-hole pressure is determined by conditions of operation. For example, pressure should be greater than the pressure of saturation to prevent oil degassing in a reservoir or it should be greater than the pressure at which the condensate in a reservoir drops out when developing gas-condensate fields. If sloughing of sand from a reservoir to the well bottom-hole is expected then the velocity in the well should be sufficient to carry out sand to the surface.

When operating a group of wells in identical conditions, i.e. with identical bottom-hole pressure, the total production of the whole field grows more slowly than grows the number of the wells (figure 10.1).

fSQ

n

Figure 10.1. Total field production vs number of wells

## SYSTEM OF EQUATIONS FOR POTENTIALS OF A GROUP OF WELLS

To solve the above mentioned problems we shall study the problem of flat interference (interaction) of wells. We shall assume, that a formation is unlimited, horizontal, it has constant thickness and an impenetrable bottom and roof. The formation is developed by hydrodynamic perfect wells and the formation is filled with a homogeneous liquid or gas. The motion of a liquid is steady, it follows Darcy’s law and it is flat.

The potential function generated by all sinks (sources) is calculated by algebraic addition of the values of potential function of each sink (or source). Resulting velocity of a filtration is determined as the vector sum of the velocities of filtration, caused by the operation of each well (figure 10.2 b).

Let n sinks with positive mass flow rate G and sources with negative flow rate operate in an unlimited formation (figure. 10.2a). Their potential can be written as follows:

^=-?-to,+C„ (10.1)

LTt n

where i is the number of a well; r. is the distance between some point of a formation and the center of the ith well.  Figure 10.2. The schematic of filtration velocity at the point M -when sources and sinks operate at an unlimited plane (a) and the resulting vector of filtration velocity at the point M (b)

Using the method of superposition, we shall determine the potential of a complex flow as follows:

Ф^Ф.^-^^+С, (Ю.2)

2.Я П

where

c=?c,.

/=1

The formula (10.2) is the basic one in solving problems of wells’ interference.

Relation (10.2) physically means, that filtration flows caused by the operation of each source or sink are stacked together.

To define the equations of equipotential surfaces (or isobars) it is necessary to remember, that in all the points of these curves the value of potential (or pressure) must remain constant. Thus, equating (10.2) to some constant, we shall obtain:

(Ю.З)

i

where П is a sign on product; Cx is a constant.

If flow rates of all the wells are the same then:

П^')=С1- (Ю.4)

i

Flow lines form a family of curves, orthogonal to isobars.

The method of superposition can be used not only in infinite reservoirs, but also in the reservoirs with a charge contour located close to wells or with an impenetrable border. In this case fictitious sinks or sources are entered at the outside of a reservoir to meet the conditions on the borders. Fictitious wells in aggregate with the real ones provide necessary conditions on borders and the problem is reduced to solving problem for joint operation of real and fictitious wells in an unlimited reservoir. The given method is called the reflection of sources and sinks technique. 