Determination of filtration properties of reservoir by non-stationary methods of well testing
Hydrodynamic methods of reservoirs and wells study are linked with measuring reservoir and bottom-hole pressures in active wells and hence they are referred to as piezometric. After start-up or after shut-in of a well a redistribution of pressure occurs. It is possible to record this redistribution to obtain pressure drawdown or pressure buildup curves. The form of the given curves is affected by reservoir properties that enables to use these curves to determine reservoir permeability and piezo-conductivity. To determine these parameters they use mathematical transformations reducing the complex form of piezometric curves to the rectilinear one.
The method of determining reservoir properties based on the data about bottom-hole pressure buildup in shut-in wells in semi-logarithmic coordinates (Ap, Inf) is the most widely applied. We shall derive a formula to construct dependence between bottom-hole pressure and time at startup of a standing idle well. For this purpose we shall take dependence (13.24) and we shall write down it for the well bottom-hole:
A pQq f - n ____
= Л + flnf,
Apw=pc-pw=—-V In —-0,5772
- 4tt hk rw
- 2,246k .
Л = /1п-2—г—; / = 0,1832?—
The equation (14.9) can be considered as the equation of bottom-hole pressure variation after start-up of a well with constant flow rate. The given dependence is represented by a straight line in coordinates Inf - pw (figure 14.1). Bottom-hole pressure buildup at shut-in of a well operating in unsteady regime is described by the dependence (14.8).
Figure 14.1. Diagram of bottom hole pressure build-up
If a well till shut-in has been operating during such a long time that distribution of pressure in a reservoir can be considered as a steady-state one then it is possible to apply superposition method. So, the draw-down pressure at the steady-state, flow may be determined by Dupuit’s formula:
Pressure buildup after shut-in of the well operating with constant flow rate may be calculated by (13.24):
Total pressure drawdown in a well will be determined according to superposition method by the following expression (14.12): r2 In-5—In/ = Л-/1п/, Kt 1
Др, = A?; -M =pc-p, =0,1832^1
A = i In—, / = 0,1832^ e" .
The process of pressure buildup after start-up of an injection well proceeds much as the process of its reduction do. Formula (14.9) can be used to calculate pressure buildup.
The equations (14.9), (14.12) represent straight lines in coordinates (Apc, In /), and the factor i is determined as a tangent of an angle of its slope (p to Int axis and the factor A is an intercept of the pressure axis cut by continuation of a straight line (figure 14.1).
If these factors are known then it is possible to determine filtration properties of a reservoir:
the factor i is used to determine hydraulic conductivity of a reservoir:
If liquid viscosity m in reservoir conditions and the thickness of reservoir h are known then it is possible to find a coefficient of permeability of a reservoir:
If angular factor i~tg(p is known and well radius rw is given then it is possible to determine the factor of piezo-conductivity of a reservoir using the coefficient A:
The specified methods of interpreting results of oil wells testing are limited by the conditions at which the formula (14.9) holds, namely: the well is considered as a source of constant intensity in an infinite homogeneous reservoir and an instant cease of fluid inflow to a well is possible. In the case of a bounded reservoir the change of the pressure caused by well shut-in reaches reservoir border, pressure build-up curve starts to be deformed, and after a sufficiently long time interval it goes on horizontal asymptotic line which describes stationary distribution of pressure. Therefore the length of a rectilinear segment on pressure buildup curve is limited, besides the well cannot be stopped instantly. After closing well-head valve an inflow of a fluid from reservoir continues for some time because of elasticity of liquids and gases filling a well. Time of converging to asymptotic line should, obviously, exceed the time of additional inflow duration. Therefore, a rectilinear segment at pressure buildup curve can appear after a significantly long time interval, or it may not exist at all.
Nowadays, there are developed methods to determine reservoir parameters at the unsteady regimes free of the specified lacks and taking into account both the time of operation up to the instant the well is shut-in and also inflow of fluid to a well after shutting it in.
In the lecture we considered the major formula of compaction drive theory that was derived in Lecture 13. We analyzed its properties and draw some important conclusions. After that interaction of wells was studied. To obtain necessary results superposition method was applied. Methods to determine reservoir parameters by means of well testing are also discussed.
- 1. What areas of formation are admissible for application of the basic formula of formation compaction drive?
- 2. What time intervals are admissible for application of the basic formula of formation compaction drive?
- 3. When and where is the stationary filtration velocity achieved?
- 4. Describe the method of superposition for the case of non-stationary filtration.
- 5. Describe the methods to determine reservoir parameters by means of well testing.