The complex fault-block reservoirs in the southern Bohai Sea are characterized by rapid changes in planar physical properties, multiple vertical layers, strong vertical and planar heterogeneity. Irregular well pattern with directional well is used to develop this type of reservoirs. Due to the influence of reservoir properties, fluid properties and well pattern, the vertical and horizontal contradictions always lead to the disequilibrium displacement in the water flooding development process of oilfields. And the disequilibrium displacement leads to the rapid increase of water cut and decline in production. Equilibrium displacement to development of reservoir can achieve maximum oil recovery and economic benefits. Separated layer injection adjustment and plane production system optimization are important ways to improve the effect of water flooding in reservoir development.
Many studies about separated layer injection adjustment and plane production system optimization have been carried on. For the separated layer injection adjustment, methods based on static parameters and field experiences are commonly used. WU Jiawen (2006) proposed an optimization method for layered water injection project based on the distribution of remaining oil . Different strategies of water injection are established and fuzzy mathematics method is used to optimize the various injection schemes. SHI Xiaoqu (2008) studied a dynamic splitting equation considering static reservoir parameters and dynamic development data to establish the injection of separated layer . CUI Chuanzhi (2017) proposed an rational sectional water injection allocation method based on equilibrium displacement . He classified the layers according to the differential absorption index as the dividing standard. Based on Buckley-Leverett displacement theory, the method of calculating the water allocation of each section was established to achieve equilibrium displacement. But some of these methods are only suitable for middle and low water cut stage, some are difficult to obtain calculation parameters, and some do not really achieve the goal of internal equilibrium displacement. On the other hand, serval methods have been proposed to conduct production system optimization to realize plane equilibrium displacement. WANG Delong (2011) studied the effect of permeability range on well spacing and production, and established the optimum well spacing and production for heterogeneous reservoirs using numerical simulation method . Aiming at the problem of injection-production imbalance in plane heterogeneous reservoirs, YAN Ke (2015) proposed a balanced water drive adjustment method based on multi-parameter quantitative calculation by using numerical simulation and reservoir engineering methods . CUI Chuanzhi (2015) established a production optimization method in fault reservoir using non-pistion water flooding theory . The method considered two factors of the reservoir physical properties heterogeneity and remaining oil saturation. WANG Xiang (2018) proposed the design formula of a well pattern for a heterogeneous reservoir to realize equilibrium displacement . The heterogeneity of the reservoir and the non-piston displacement characteristics of the injected water were considered. The relationship between well spacing and reservoir dynamic and static parameters were derived. Their studies either only considered the displacement between injection and production wells, or were complicated to calculate.
J. I. Bingyu (2008) studied the method of oil production calculation for areal well pattern of low-permeability reservoir with non-Darcy seepage flow . The multi-flow tube method (MTM) was firstly used in the calculation. And on this basis, GUO Fenzhuan (2010), ZHU Shengju (2013) and HE Congge (2015) all studied the calculation method of plane sweep coefficient for different well pattern in low-permeability reservoir   . But the application of production system optimization to realize plane equilibrium displacement with maxim sweep coefficient was not realized in their study.
On the basis of previous studies, using non-piston water flooding theory, this paper studies the calculation method of injection allocation for realizing balanced recovery of each section. Irregular well pattern is divided into different triangular injection-production seepage units in plane, and the optimization method of injection and production pressure difference is obtained by MTM to realize production system optimization. According to the calculation results of water injection allocation volume of each layer, separate injection of the injection wells are adjusted, and according to the calculation results of the plane injection production pressure difference, the oil nozzles or pump frequency of the production wells are adjusted, then injection and production system adjustment method based on balanced displacement is obtained by combining the two methods. The method has been applied in oil field, and the effect of oil stabilization and water control is achieved.
2. Injection-Production System Adjustment Theory
2.1. Method of Sectional Water Injection Calculation
The purpose of sectional water injection is to alleviate the contradiction between layers and reduce the interlayer interference caused by the difference of reservoir and fluid physical properties. By adjusting the injection volume of each section, the oil recovery of each section is consistent at a certain time, then the vertical balanced displacement is realized eventually. According to the degree of recovery at different time, the cumulative injection can be calculated, and the sectional injection is obtained under constant speed injection.
If injection water doesn’t breakthrough in the jth section, according to Bukley-Leverett’s non-piston water flooding theory , the accumulated water in section can be expressed as
where Wj is the accumulated water injection in the jth section, T is the cumulative water injection time, Qj is the water injection volume in section, t is the injection time, Xfj is the oil-water front position in the jth section, is reservoir porosity, A is the seepage section area, Swfj is the water saturation of oil-water front in section, is the derivative of the water cut corresponding to the water saturation of oil-water front.
Under the condition that the amount of injection remains constant, the average injection volume can be obtained according to the following formula.
where is the allocation of water injection in the jth section, Δt is allocation of water injection time, L is the distance of injection to prodution, Sw2j is the water saturation at the outlet of the jth section after water breakthrough, is the water content derivative corresponding to water saturation at the outlet after water breakthrough. The allocation water injection rate cannot be solved directly, but can be obtained iteratively by setting the predicted outlet saturation.
Before water breakthrough, the recovery of reservoir in the jth section can be obtained by cumulative injection water:
After water breakthrough, the recovery of reservoir in the jth section can be obtained by average water saturation:
where Rj is the oil recovery factor, Swi is the irreducible water saturation, fw(Sw2j) is the outlet water content in the jth section.
If injection water has breakthrough in the jth section, the cumulative injection volume required for the water saturation from Swj to Sw2j at the outlet of the jth section can be expressed as
where Swj is the outlet water saturation of the jth section before allocation, is the water content derivative corresponding to water saturation at the outlet after water breakthrough.
Assuming that the injection allocation of each section remains constant in Δt time, the injection allocation of the jth section can be obtained by
When this method is used to segmental injection allocation, the recovery factor of each section can be obtained according to the accumulated water volume of each layer. By setting allocation time and target recovery factor, the accumulated water volume of each segment is calculated, and then the injection allocation amount of each section can be obtained.
2.2. Method of Planar Production System Optimization
The injection-production well pattern is divided into different triangular seepage units (1/2 of the maximum area can be affected by water injection wells). The two vertices of the triangle seepage unit are water injection well and corresponding oil production well. The triangle seepage unit is assumed to be homogeneous and equal thickness reservoir. The oil and water well are composed of flow tubes. The seepage rule in the flow tubes conforms to the oil-water two-phase piston displacement. The compressibility of formation rocks and fluids, the capillary force and gravity are neglected, and the injection-production pressure difference is constant. As shown in Figure 1, each flow tube between injection and production well has only one turning point O. Geometric relation formula can be expressed as
Figure 1. Flow tube diagram of injection-production seepage unit.
where α is the angle between streamline AO and injection-production connection line, β is the angle between streamline BO and injection-production connection line, α0 is the angle between sweep maximum area triangle side AC and injection-production connection line, β0 is the angle between sweep maximum area triangle side BC and injection-production connection line.
In the case of oil and water two-phase flow, the length of the water phase displacement of any angle at any time t is 
where C is a constant(C = 0.0864), pinj is the bottom hole pressure of injection well, pwf is the bottom hole flow pressure of production well, μo is the viscosity of reservoir crude oil, μw is formation water viscosity, Ko is the oil phase permeability, Kw is the water phase permeability.
According to the distribution of oil and water at different times, the whole displacement process can be divided into four periods: when t is less than (α = 0, the time when the oil-water front reaches the turning point O of AB), the oil-water fronts in flow tubes don’t reach the turning point of the flow tube. When t is greater than and less than (α = 0, the time when the oil-water front reaches point B), part of the oil-water fronts in flow tubes have reached the turning point, but not reached the production well. When t is greater than and less than (α = α0, the time of the oil-water front reaching C point), most of the oil-water fronts in flow tubes have crossed the turning point, and some of them reach the production well. When t is greater than and less than (α = α0, the time of the oil-water front reaching B point), the oil-water fronts in flow tubes have crossed the turning point, and reached the production well.
The area sweep coefficient of triangular seepage unit is the ratio of sweep area to unit area, and the water content is the ratio of flow rate in flow tubes that the front of water drives to the well to all flow rate in flow tubes.
where fw is the water cut of production well, Ea is the area sweep coefficient, m is the total number of flow pipes, n is the number of flow pipes that the front of the water reaches the production well, Qi, Qj are the flow rates of the ith and jth flow pipe, Si is the area of the ith flow pipe, S is the area of ΔABC.
The area sweep coefficient and water cut of production well are calculated by judging the distribution in each flow pipe at different time. Setting the target area sweep coefficient of the adjustment time, the injection-production pressure difference of the corresponding seepage unit is inversely calculated. Through the adjustment of production system optimization, the eventual balanced displacement can be achieved.
2.3. Adjustment Method of Injection-Production System
Based on the above theory, a method of injection-production system adjustment for offshore fault-block reservoirs is established. According to cumulative water injection and water content of each section vertically, the recovery degree and water saturation at the outlet can be obtained by Equation (1) or Equation (5). The target water saturation at the outlet end can be calculated by Equation (4) after setting the target recovery degree of each section. And the average injection allocation of each section in the target time is calculated according to Equation (2) and Equation (6). On the plane, injection-production triangle seepage units are divided. And fluid properties and well pattern parameters of each unit are obtained. According to the current water content of production well in the unit, the area sweep coefficient and the oil-water front distribution are derived. The area sweep coefficient of each seepage unit is compared, and the seepage unit with smaller plane area sweep coefficient needs to increasing injection-production pressure difference to improve water flooding effect. By setting the sweep coefficient of the target area of each seepage unit, the injection-production pressure difference is calculated according to Equation (8). The production system of production wells is adjusted according to injection-production pressure difference. Equilibrium displacement can be achieved through vertical and horizontal injection-production adjustment, thus the development effect of water flooding can be improved. The calculation procedure is shown in Figure 2.
Figure 2. Flow chart of injection and production system adjustment.
3. Field Applications
The injection-production adjustment of S1 block in BZ oilfield has been conducted by using this method, as shown in Figure 3. The production layers of the block are N1ml I-1 layer and I-2 layer. The development mode of commingled production by directional production wells and sectional water injection by directional water injection wells. The injection and production system adjustment of well I1-P1 is shown in this paper. The well spacing from I1 well to P1 well and P2 well is respectively 590 m and 420 m, and the well spacing between P1 well and P2 well is 745 m. The reservoir and fluid parameters are shown in Table 1.
Figure 3. Development well pattern of S1 block in BZ oilfield. (a) I-1 layer of S1 block in BZ oilfield; (b) I-2 layer of S1 block in BZ oilfield.
Table 1. Reservoir and fluid parameter of S1 block in BZ oilfield.
3.1. Vertically Sectional Injection Adjustment
The test results of production profile of well P1 before allocating shows the water cut in I-1 layer is 45.0% and the water content in I-2 layer is 60.8%. Calculated by Equation (5), the recovery factor of layer I-1 is 13.8%, the recovery factor in layer I-2 is 17.7%. The calculation result shows that due to good reservoir physical properties of I-2 layer, the injected water front advances rapidly along this layer. The recovery of layer I-1 is lower than that of layer I-2. Water injection allocation is needed to tap the remaining oil potential. The allocation time is set to 300 days, and the target recovery of the two layers is 20.0%. The average injection allocation of layer I-1 is 96 m3/d and that of layer I-2 is 68 m3/d, which is calculated by Equation (6).
3.2. Planar Production System Optimization
The well group is divided into two seepage units as shown in Figure 3. The angle θ1 is 35 degrees, and the angle θ2 is 53 degrees. The area sweep coefficient is calculated for each seepage unit respectively by applying this method. The area sweep coefficient of seepage unit (1) is 0.52. The area sweep coefficient of seepage unit (2) is 0.78. In order to achieve equilibrium displacement, the sweep area of seepage unit (1) needs to be increased. The target time is 300 days and the target area sweep coefficient is 0.80. The injection-production pressure difference in well I1-P1 direction is optimized. By increasing the production frequency of pump in well P1, the injection-production pressure difference is increased from 10.0 MPa to 11.7 MPa.
3.3. Application Result
After adjustment, the effect of water control and oil increase is obvious, as shown in Figure 4. The daily oil production increase 24 m3, and the water cut remains stable. This method has been applied to adjust injection-production system of other well groups in this block, and the effect is remarkable. The daily oil increase is 110 m3, and the annual production decline rate is only 1.5%, which improves the development effect of water flooding.
Figure 4. Well group production curve.
It should be figured out that absolute equilibrium displacement cannot be achieved due to the heterogeneity of actual reservoirs. In addition, the adjustment of injection-production system is a dynamic process. A relatively balanced displacement needs to be adjusted in time according to the development dynamics and combined with the corresponding technological measures.
1) Aiming at the prominent contradiction between layers in multi-layer development of offshore fault-block reservoirs, based on Buckely-Leverett non-piston water flooding theory, a sectional injection allocation method is established. By setting the unified target recovery of each section, the vertical balanced displacement can be achieved.
2) In order to maximize the plane sweep area of water flooding, injection-production well group is divided into triangle seepage units. By using multi-flow pipe method, the adjustment method of production system optimization for offshore fault-block reservoirs with irregular pattern development is established to realize horizontal balanced displacement.
3) Combining the vertical sectional injection allocation method and planar production system optimization method, the injection-production system adjustment method of fault block reservoir is established. This method is applied to guide the injection-production system adjustment in block S1 of BZ oilfield. The practical application results show that this method has obvious effect on water control and oil production increase. And the annual production decline rate of the S1 block is only 1.5%. This method has achieved good application effect, which can conduct the injection-production system adjustment of offshore fault block reservoirs.
4) It should be pointed out that the method of sectional water injection calculation proposed in this paper does not consider the influence of plane well pattern. Moreover, the hypothesis of triangular seepage unit division with planar production system optimization is limited to the reservoir with strong heterogeneity. Further study is needed to improve the above limitations.
Wj Accumulated water injection in the jth section, m3
T Cumulative water injection time, d
Qj Water injection volume in section, m3
t Injection time, d
Xfj Oil-water front position in the jth section, m
Reservoir porosity, dimensionless
A Seepage section area, m2
Swfj Water saturation of oil-water front in section, m
Derivative of the water cut corresponding to the water saturation of oil-water front, dimensionless
Allocation of water injection in the jth section, m3
Δt Allocation of water injection time, d
L Distance of injection to prodution, m
Sw2j Water saturation at the outlet of the jth section after water breakthrough, dimensionless
Water content derivative corresponding to water saturation at the outlet after water breakthrough, dimensionless
R Oil recovery factor, dimensionless
Swi Irreducible water saturation, dimensionless
fw(Sw2j) Outlet water content in the jth section, dimensionless
Swj Outlet water saturation of the jth section before allocation, dimensionless
Water content derivative corresponding to water saturation at the outlet after water breakthrough, dimensionless
pinj Bottom hole pressure of injection well, MPa
pwf Bottom hole flow pressure of production well, MPa
μo Viscosity of reservoir crude oil, mPa∙s
μw Viscosity of formation water, mPa∙s
Ko Oil phase permeability, dimensionless
Kw Water phase permeability, dimensionless
fw Water cut of production well, dimensionless
m Total number of flow pipes, dimensionless
n Number of flow pipes that the front of the water reaches the production well, dimensionless
Qi Flow rates of the ith flow pipe, dimensionless
Ea Area sweep coefficient, dimensionless
S Area of triangle seepage unit, m2
Si Area of the ith flow pipe, m2
j Serial number of layer
i Serial number of flow tube
w water phase
o oil phase
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