Interference of imperfect wells

In the case of interference of wells imperfect on penetration degree for a flow under Darcy’s law, at first we determine flow rate of the perfect wells with radiuses r_w using the formulas of the theory of interference for flat flow to sinks and sources, and then we increase filtration resistance of each well by the value of the factors of imperfection C.(i =1,..., 4). If the factors of filtration resistance A_h and B_h are determined either by the above mentioned analytical method or directly by well testing, then it is possible to use the method of equivalent filtration resistance to study interference of imperfect wells, including the case of quadratic filtration law. For this purpose quadratic law should be presented as follows:

Ap = AQ + p"Q, (12.15)

where p” = BQ - p"(Q) is considered as a nonlinear resistance. This resistance is added to internal resistance p which is determined by the distance between wells in the ring battery.

For example, in the schematics of filtration resistance for linear law of filtration, it is necessary to replace internal resistance p with the sum P + p"(Q), where pQ) = BQ for each well. The subsequent calculation is conducted as before by means Ohm’s and Kirchgoff’s laws, but the system of the equations becomes nonlinear, containing quadratic equations that results in complication of calculations.

Interaction of wells in non-uniformly permeable and anisotropic reservoirs

When developing oil or gas fields frequently there occur conditions wherein permeability of out-of-contour area is less than permeability inside a contour.

Let permeability inside a circle of radius R_o is equal to k,, and permeability in aring R Q is k₂. Let R_c » a where a there is the radius of the battery of wells (figure 12.4).

The flow to n to production wells goes from a circle of radius R_o. In the second area the flow is radial-plane from contour R_c up to the aggregate well with radius R_o and well’s flow rate is G₂ = Gin.

Taking into consideration constancy of к within each zone, we shall write the potential as follows: cp = кФ+С, where

Substituting the given expression for (p into expression for flow rates

and excluding Ф₍₎ we shall obtain:

G' = G_i=G₂

1 i К

In к, na^nlr

(12.16)

Figure 12.4. Ring battery of wells in a reservoir with step-wise varying permeability

Calculations carried out by (12.16) show, that in non-uniform reservoir at к_} I k₂ = b <1 the factor of total interaction

(relation of total flow rate of jointly operating groups of wells to the flow rate of a single well) is always higher than total interaction factor of the battery working under the same conditions in a homogeneous reservoir

(/?=1). If/?> 1 then (/will be less than its value in a homogeneous reservoir. At the same values of ft the interaction of wells will grow with the growth of the area occupied by less permeable area of a reservoir.

In anisotropic reservoirs, i.e. where non-uniformity has some orientation, wells interact in much the same way as in anisotropic reservoir. Interaction becomes weaker if permeability is lesser in the direction of line of wells’ placement in comparison with permeability in a perpendicular direction. Strengthening effect of interaction occurs in the opposite case. Thus, to reduce the effect of interaction at projecting new wells it is necessary to choose the direction, in which reservoir is least penetrable.

Influence of radius of a well on its productivity

Calculations under the given formulas result in the following summary. If Darcy’s law is obeyed then at radial plane flow the influence of radius of a well on flow rate is insignificant. We need to increase radius of a well 10 times to obtain 20% growth of flow rate. If filtration is non-linear then the influence of rw on G grows. Underreaming promotes increase in productivity. Bottom-hole torpedoing, hydraulic fracture of a reservoir and other ways of bottom-hole zone treatment give rise to fractures that promotes both increase in permeability in the near bottom-hole zone and violation of law Darcy’s, hence, it leads to the strengthening of influence of radius of a well on liquid inflow to it.

With increase in number of production wells of the ring battery the influence of their radius on flow rate decreases, if there is no injection of a liquid into a reservoir. If there is an injection well in the center of the battery, then the influence of the radius of a well on the flow rate will be greater compared to the case when there is no the central injection of a liquid into a reservoir. Thus, at joint operation of production and injection wells, their interaction raises the influence of radius of wells on the flow rate.

Lecture conclusion

In the lecture a notion of imperfect well is introduced, the types of wells’ imperfections are discussed. The methods to account for well imperfection when calculating flow rate are shown. They consist in application of reduced well radius and additional filtration resistance. Flow of a liquid under Darcy’s law to an imperfect well and flow of real gas under quadratic law to an imperfect well are also considered.

Test questions

• 1. Describe the types of well imperfection.
• 2. What is a reduced radius?
• 3. How is the factor of imperfection determined?
• 4. Describe Shchurov’s diagrams.