The pull-down spatial pressure solution for point A of the double-well production, Eq. 1, can be obtained according to the Duhamel superposition principle (Wang, 2015): P ∼ A = P ∼ 1 A + P ∼ 2 A , (1)

where P ∼ A is the pressure solution at point A when both wells are producing, P ∼ 1 A is the pressure solution at point A when well P-1 is producing alone, and P ∼ 2 A is the pressure solution at point A when well P-2 is producing alone.

When L tends to 0, the pressure response of the unsteady seepage stage at point A is equivalent to the pressure response of well P-1 when it produces individually at a constant production rate q i + q 2 , and the unsteady flow stage exhibits radial flow characteristics. When L is large, q i = q 2 , and P ∼ 1 A > P ∼ 2 A , the flow characteristics of the unstable seepage stage at point A are dominated by the flow characteristics of the unstable seepage stage at point A when well P-1 is producing alone.

Figure 2 shows a conceptual diagram of the r_eD2 solution. Overall, in Figure 2, the flow stage at point A, when the two wells are producing, can be simplified as the infinite radial flow stage, the proposed steady-state stage. In turn, the pressure solution at point A when well P-2 is producing alone can be simplified as the pressure solution for one well in the centre of a circular closed reservoir with constant production, while the simplified well control radius expression for well B is given in this paper.

The average of the distances from point B to the circular closure boundary at the simplified drain radius is as follows: r e D 2 = ∫ 0 2 π r e D ⁡ sin ⁡ θ 2 + r e D ⁡ cos ⁡ θ + r 1 2 d θ 2 π , (2) r 1 = L r w , (3) r e D = r e r w , (4)

where r e D is the radius of the circular closed reservoir and r w is the radius of the wellbore.

From Eqs 2–4, we can derive r e D 2 ≥ r e D (take the equal sign when L = 0), so the reservoir reaches the proposed steady-state flow stage later when P-2 is producing alone than when well P-1 is producing alone, which is in line with the actual flow condition.

Table 1 lists the equivalent well control radius error rating table. After obtaining the simplified well control radius using this method, the corresponding equivalent well control area is further calculated and compared with the actual well control area. It is found that the simplified well control area is more accurate at r 1 < 0.5 r e D .

The fixed solution problem for p-1 wells can be described as follows: ∂ 2 p D ∂ r D 2 + 1 r D ∂ p D ∂ r D = ∂ p D ∂ t D , (5) p D t D = 0 = 0 , (6) r D ∂ p D ∂ r D r D = 1 = − 1 , (7) ∂ p D ∂ r D r D = r e D = 0 , (8) p 1 A = p D 1 , t D . (9)

The fixed solution problem for p-2 wells can be described as follows: ∂ 2 p D ∂ r D 2 + 1 r D ∂ p D ∂ r D = ∂ p D ∂ t D , (10) p D t D = 0 = 0 , (11) q B = q 2 q i , (12) r D ∂ p D ∂ r D r D = 1 = − q B , (13) ∂ p D ∂ r D r D = r e D = 0 , (14) p 2 A = p D r 1 , t D . (15)

The dimensionless variables are defined as follows: p D r D , t D = k h 1.842 × 10 − 3 B q i μ p i − p r , t , t D = 3.6 k t ∅ μ C t r w 2 ,   r w = r r w ,   r e D = r e r w , r e D 2 = ∫ 0 2 π r e D ⁡ sin ⁡ θ 2 + r e D ⁡ cos ⁡ θ + r 1 2 d θ 2 π ,

where r is the distance from any point to the well in mm,   r w is the wellbore radius in mm, r e is the drain radius in mm, k is the reservoir permeability in µm², θ is the reservoir porosity with no causality, C t is the total reservoir compression factor in MPa⁻¹, h is the thickness of the reservoir in mm, μ is the viscosity of the crude oil in MPa·s, B is a crude oil volume factor with no factorisation, q is the surface production of the well in m³/d, p i is the original stratigraphic pressure in MPa, p is the formation pressure at a point in the reservoir at a given time in MPa, and t is production time in h.

A Laplace transformation of Eqs 5–15 yields ∂ 2 p ∼ D ∂ r D 2 + 1 r D ∂ p ∼ D ∂ r D = s p ∼ D . (16)

The boundary conditions within the P-1 well are expressed as ∂ p ∼ D ∂ r D r D = 1 = − 1 s . (17)

The boundary conditions within the P-2 well are expressed as ∂ p ∼ D ∂ r D r D = 1 = − q B s , (18) ∂ p ∼ D ∂ r D r D = r e D = 0 . (19)

Substituting Eq. 16 and Eq. 17 into Eq. 19, we have Eq. 20 p ∼ 1 A = 1 s 3 2 K 0 s I 1 r e D s + I 0 s K 1 r e D s K 1 s I 1 r e D s − I 1 s K 1 r e D s . (20)

Substituting Eq.16 and Eq.18 into Eq. 19, we have Eq. 21 p ∼ 2 A = q B s 3 2 K 0 r 1 s I 1 r e 1 D s + I 0 r 1 s K 1 r e 1 D s K 1 s I 1 r e 1 D s − I 1 s K 1 r e 1 D s . (21)

Eq. 22 can be obtained from the Duhamel superposition principle: p ∼ A = p ∼ 1 A + p ∼ 2 A . (22)

Blasingame-type curve definitions of the dimensionless parameters are formed by introducing normalisation coefficients, α , to the original definition of the dimensionless parameter.

Here, Blasingame-type curve definitions of the dimensionless parameters are as follows:

Define the dimensionless material balance time as Eq. 23 t D d = 1 1 2 r e D 2 − 1 ln ⁡ r e D − 1 2 t D . (23)

Define the dimensionless decline flow rate Eq. 24 q D d = 1 + q B ln ⁡ r e D − 1 2 L − 1 p ∼ A . (24)

Define the dimensionless cumulative production as Eq. 25 N p D d = ∫ 0 t D d q D d τ d τ . (25)

Define the dimensionless decline flow rate integral function as Eq. 26 q D d i = ∫ 0 t D d q D d τ d τ t D d = N p D d t D d . (26)

Define the dimensionless decline flow rate integral derivative as Eq. 27 q D d i d = − d q D d i d ⁡ ln ⁡ t D d = q D d i − q D d . (27)

Figure 4 shows an unstable analysis curve of production from double straight wells in homogeneous reservoirs. From Figure 4, taking r_eD = 5,000 and q_B = 4 as an example, the crude oil flow can be roughly divided into an early radial flow stage, a stage influenced by the interference of neighbouring wells, and a later stage when the influence of the interfering wells is weakened (unsteady flow stage). The following section addresses the effect of well spacing, neighbouring well production, and neighbouring well start time factors on the Blasingame curve.

To regularise the resultant data from numerical model runs,

Calculate the material balance time by Eq. 28 t c = N p q . (28)

Calculate the normalised rate by Eq. 29 q ∆ p = q p i − p w f . (29)

Calculate the normalised rate integral by Eq. 30 q ∆ p i = 1 t c ∫ 0 t c q p i − p w f d x . (30)

Calculate the normalised rate integral derivative by Eq. 31 q ∆ p i d = − t c d q ∆ p i d t c . (31)

Table 2 is the numerical simulation data–production data regularisation table.

Figure 10 shows measured curve fitting results from plotting the three relational curves, q_d-t_c, q_di-t_c, and q_did-t_c, fitting the Blasingame yield instability analysis plate, and recording the fitted values, where the three relational curves can be used simultaneously or individually.

Using the theoretical fit points (8 × 10⁻³, 0.4), the actual fit points can be obtained (1, 0.7).

The dynamic reserve at the centre point can be calculated according to the following Eq 32: N = 1 C t t c t D d M q / ∆ p q D d M 1 − S w (32)

as 3.65 × 10 6 m³, and the numerical model has a reserve of 7.68 × 10 6 m³.

Additionally, the flowing material balance method (FMB) is listed for comparison to verify the accuracy of the method used in this article.

Figure 11 shows the equilibrium curves for the flowing material balance.

FMB is a method similar to the traditional material balance equation. However, it differs from the traditional material balance equation in that it uses dynamic pressure data instead of static pressure data for storage evaluation. Additionally, it differs from the flow material balance method, which uses flow pressure for dynamic storage evaluation. Once the reservoir is producing and has achieved a steady-state flow, the following relationship can be established (Eq. 33): q ∆ p = − N p b p s s N ∆ p C t + 1 b p s s . (33)

The dynamic reserves N can be determined from the intercept results if the q ∆ p — N p ∆ p C t curve is plotted.

From Figure 11, the dynamic reserves calculated for the P-1 well’s reservoir are 3.486 × 10 6 m³. Comparing the results of the two methods, it is found that the method in this paper is more accurate in its calculations.