The interlayer interference is very serious in the process of water flooding development, especially when the reservoir adopts commingling production. The implementation of various interlayer interference mitigation measures requires that the production performance parameters and remaining oil distribution of each layer of the reservoir should be clearly defined, and the accurate production splitting of oil wells is the key. In this paper, the five-spot pattern is simplified to a single well production model of commingled production centered on oil well. The accurate production splitting results are obtained through automatic history matching of single well production performance. The comparison between the calculation results of this method and that of reservoir numerical simulation shows that the method is simple, accurate, and practical. In the field application, for the multilayer commingled production reservoir without accurate numerical simulation, this method can quickly and accurately realize the production splitting of the reservoir according to the development performance data.

Due to the influence of reservoir physical property difference among layers, the interlayer interference is serious in the process of water flooding development when the reservoir adopts commingling production, and the difference of producing degree among layers is large. The implementation of measures to reduce the interlayer interference, such as separate layer water injection and formation reorganization, needs to make clear the production performance of each layer and the distribution of remaining oil. Accurate oil well production splitting is the basis of dynamic analysis, development effect evaluation, and remaining oil research of each layer. At present, the commonly used methods of production splitting include liquid production profile coefficient method [13], effective thickness method [4], static land splitting coefficient (KH) method [57], and dynamic splitting equation method [810]. These methods only use effective thickness, effective permeability, and other formation parameters to split production, and seldom consider the influence of remaining oil saturation difference between layers at different time on production splitting, which cannot truly reflect the actual production situation of each underground layer, resulting in low accuracy of production splitting results. Although the reservoir numerical simulation method can obtain more accurate liquid and oil production of each layer at different times, it needs to adjust the parameters manually with long cycle and heavy workload. Tian et al. and Gu et al. [11, 12] introduced the catastrophe method into the production splitting of multilayer combined production wells. They established a new production splitting method of multilayer combined production gas wells considering reserve characteristics, development characteristics, and geological characteristics and applied it in various oilfields [1315]. However, this method does not fully consider the impact of development dynamics on production splitting and has some limitations.

The automatic history matching method can realize the parameter inversion and adjustment of the numerical simulation model and ensure the accuracy of the numerical simulation to the development performance prediction after effective test and correction. In this paper, the production splitting method of multilayer commingled production reservoir based on automatic history matching is established. Based on automatic history matching, the production of multilayer oil wells is further divided, and more accurate production splitting results are obtained.

2.1. Establishment of Mathematical Model

Five-spot pattern is a common well pattern in oilfield development, and it can be regarded as a plane radial flow model with the oil well as the center and four water injection wells as a supply boundary. Suppose an oil well develops n layers at the same time, and there are differences in reservoir physical parameters such as permeability and effective thickness of each layer. Regardless of capillary force and gravity and the change of viscosity of oil and water, both oil and water flow are governed by Darcy’s law. The multilayer commingled production model can be regarded as composed of n one-dimensional radial flow layers, and the mathematical model of one layer is as follows.
  • (1)

    Filtration equation

(1)1rrrρoKKroμoPor+qo=tρoϕSo,(2)1rrrρwKKrwμwPwr+qw=tρwϕSw,
where K is the absolute permeability (μm2), Kro and Krw are the relative permeability of oil phase and water phase, μo and μw are the viscosity of oil phase and water phase (mPa·s), Po and Pw are the pressure of oil phase and water phase (101MPa), So and Sw are the saturation of oil phase and water phase, ρo and ρw are the density of oil phase and water phase (kg/m3), ϕ is the formation porosity, qo is the volume flow rate of oil injected or produced in the unit in unit time under standard ground conditions (m3/day), and qw is the volume flow rate of water injected or produced in the unit in unit time under standard ground conditions (m3/day).
There are four unknowns in the above two partial differential equations, which are Pw, PO, Sw, and SO. Therefore, two auxiliary equations are needed to solve the equations.
  • (2)

    Auxiliary equation

(3)Sw+So=1,(4)Pcow=PoPw.
  • (3)

    Initial condition

(5)Px,tt=0=P0x,(6)Swx,tt=0=Swc,
where p0x and Swc are the original formation pressure and irreducible water saturation under the initial conditions, respectively.
  • (4)

    Boundary condition

The inner boundary is constant liquid production, and the outer boundary is constant pressure.
(7)qvrw,t=qv,Prε,t=Pε,
where qv is the total volume flow of liquid injected or produced per unit time.

So far, Equations (1)–(7) constitute a complete one-dimensional radial oil-water two-phase mathematical model. By using the numerical method, the formation pressure and oil saturation at any point in the seepage model can be obtained, as well as their changes with time.

2.2. The Solution of Mathematical Model

2.2.1. Well Treatment Method

The treatment of the well in the mathematical model is the treatment of the boundary conditions in the model, and the production mode of constant liquid production is adopted in the mathematical model of multilayer commingled production in this paper. For the mathematical model, when a well passes through multiple layers vertically, the given liquid production at the wellhead is the sum of the production of all the layers that the well passes through. Therefore, we need to allocate the given liquid production to each layer, that is, liquid rate splitting.

Suppose that the model has n layers, each layer is divided into m grids, and the grid step size is dx. Considering the oil-water two-phase flow, the flow resistance of the k layer can be expressed as follows:
(8)Rk=i=1mdxA1λwi+λoi=i=1mdxA1KkKrwSwi/μwi+KkKroSwi/μoi,
where Kk is the absolute permeability of the k layer (μm2), Krw is the relative permeability of water phase, Kro is the relative permeability of oil phase, A is the flow cross-section area of the k layer (m2), λw is the mobility of water, λo is the mobility of oil, Sw is water saturation, and i is the i-th grid.
According to the law of equivalent percolation resistance, the total percolation resistance of multilayer commingled production reservoir model can be regarded as the result of parallel connection of all layers’ percolation resistance. Then, the total percolation resistance of all layers is as follows:
(9)Req=1k=1n1/Rk.
Assuming that the total liquid production is Qt, the liquid production of k layer is as follows:
(10)Qvk=ReqRkQt.
The proportion of water in total liquid volume is water cut.
(11)fw=QwkQwk+Qok=11+μw/μoKo/Kw.
According to the relationship between water cut and liquid production, the oil and water production of each layer can be calculated.
(12)Qwk=fwQv=Krw/μwKro/μo+Krw/μwQvk,Qok=1fwQv=Krw/μwKro/μo+Krw/μwQvk.

2.2.2. Solution of Multilayer Commingled Production Reservoir Model

There are four unknowns (Sw, So, Pw, and Po) in the mathematical model of multilayer reservoirs, and the saturation term in the equation should be eliminated first when solving the pressure. In this paper, the implicit pressure and explicit saturation [1618] (IMPES) method is used to establish the difference equations and the numerical method is used to solve them.

Based on the single well production mathematical model in multilayer commingled production reservoir, the optimization theory is introduced, the fitting parameters and fitting indexes are reasonably selected, the objective function of single well automatic history fitting is established, and the appropriate algorithm is selected to solve the problem, so as to complete the single well automatic history fitting.

3.1. The Establishment of Objective Function

The purpose of single well automatic history fitting is to minimize the difference between the actual observed data of model production performance prediction value at each time, so that the production data calculated by the model in the whole prediction period can be consistent with the single well production observation value in the actual well history data as much as possible. According to the optimization theory, the fitting index is selected as water cut, and the objective function of single well automatic history fitting is established:
(13)J=1Nj=1NλjFwjobsFwjcal21/2,
where N is the total number of time steps, Fwjobs is the actual observation value of single well water cut in the j time step, Fwjcal is the single well water cut calculated by the model in the j time step, and λj is the weight factor which refers to the proportion of the j time step in the total time (if the time steps are of equal length, it is the fixed value).

3.2. Selection of Fitting Parameters

Injection-production well spacing is the most important fitting parameter. According to the actual dynamic production data of oil wells, the control reserves of single well are calculated by using water drive curve method, and then, the control area of single well is obtained, which is taken as the constraint condition of fitting injection-production well spacing.

The basic formula of A-type water drive curve is as follows [19]:
(14)logWp=a1+b1Np,
where Wp is the cumulative water production (104t), Np is the cumulative oil production (104t), a1 is the intercept of the straight line segment (which is a constant related to rock properties or fluid properties), and b1 is the slope of straight line section (which is a constant related to geological conditions, well pattern arrangement, and oilfield management measures).

In the linear regression of A-type water drive curve, the occurrence time of the straight line should be considered. In the early stage of production, the first half of A-type water drive curve drawn by production data is not linear. Only when the oilfield enters the stage of comprehensive water drive development and the oil well reaches a certain water cut, the water drive curve at this stage is approximate to the inclined straight line, which has a certain representativeness.

The relationship between water cut fw and cumulative oil production Np of type a water drive curve is as follows:
(15)Np=lgfw/1fwc1b1,c1=a1+lg2.303×b1.
When the reservoir reaches the water cut limit (fw=98%), the formula of recoverable reserves derived by dynamic method can be obtained from the above formula:
(16)NR=1.6902a1+lg2.303×b1b1.
Another method to calculate recoverable reserves is to use static method to determine recovery factor and then calculate recoverable reserves according to recovery factor and geological reserve formula:
(17)NR=NER=100AHϕSoiρoscBoiER,
where N is original oil in place (104t), A is oil bearing area (m2), H is average effective thickness (m), ϕ is average effective porosity, Soi is average original oil saturation of reservoir, ρosc is average surface crude oil density (t/m3), Boi is volume coefficient of crude oil under original formation pressure (m3/m3), and ER is oil recovery.
The oil recovery of multilayer reservoirs can be expressed by Equation (18) [20].
(18)ER=NNorN=Ahϕ1Swc/BoiAhϕSor/BoAhϕ1Swc/Boi=1Sor1SwcBoiBo,
where Swc is irreducible water saturation, Sor is residual oil saturation, and Bo is the volume coefficient of crude oil under formation pressure at the end of water flooding (m3/m3).
If water injection can keep formation pressure unchanged, then BoiBo, so Equation (18) can be changed into the following:
(19)ER=1Sor1Swc=1SwcSor1Swc.
By introducing Equations (16) and (19) into Equation (17), the calculation formula of single well control area can be deduced:
(20)A=1.6902a1+lg2.303×b1b1Boi100HϕSoiρosc1Swc1SwcSor.
When the mathematical model of multilayer commingled production reservoir is equivalent to the radial flow model of n layers, the control area of a single oil well in the center can be equivalent to a circle; then, the injection-production well spacing (fitting parameters) can be expressed as follows:
(21)L=Aπ.

In the automatic history fitting of single well, the interlayer heterogeneity becomes the main factor affecting the water cut of single well when the production well is producing with constant liquid rate. The interlayer heterogeneity is mainly reflected in the permeability difference of each small layer. Therefore, the permeability of each small layer is selected as one of the fitting parameters of single well history fitting.

Finally, the permeability and injection-production well spacing of each layer are determined as fitting parameters, and the adjustment of all fitting parameters of single well automatic history fitting can be expressed by vector M.
(22)M=K1,K2,,Kn,L,
where Ki is the permeability of the i layer (μm2) and n is the total number of all layers.

3.3. Solution of Optimization Model

After establishing the objective function of single well automatic history fitting, it needs to choose the appropriate algorithm to solve. In this paper, the genetic algorithm [2124] is chosen which has strong compatibility, fast solving speed, and is suitable for large-scale parallel computing.

The operation process of genetic algorithm is a typical iterative process. Combined with the single well production mathematical model and single well automatic history fitting optimization model studied in this paper, the algorithm flow is shown in Figure 1.

The termination condition is whether the maximum iteration number of the genetic algorithm is achieved or whether the stagnation iteration number of the genetic algorithm is achieved. When the adjustable parameter M (Equation (22)) is in a certain range, a group of parameters Mc is found to minimize the objective function J (Equation (13)), and the solution Mc is the optimal solution of the objective function of single well automatic history fitting.

4.1. Model Parameter Setting

The mathematical model of multilayer commingled production reservoir is established, and P1 oil well is selected as the research object. There are five small layers in the well, and the basic parameters of the physical properties of the model reservoir are set as follows: viscosity of crude oil is 15 mPa·s, viscosity of formation water is 0.5 mPa·s, crude oil density is 850 kg/m3, and formation water density is 1000 kg/m3. The production time and liquid production of P1 well are shown in Table 1. At the same time, the reservoir numerical simulation model is established for simulation calculation, and the accuracy of this method is verified by the reservoir numerical simulation results.

According to the production data of P1 well, the cumulative oil production and cumulative water production data of the well are used to draw the A-type water drive curve, as shown in Figure 2.

According to the linear regression of A-type water drive curve, the calculated parameters and results are shown in Table 2. The control area of single well in the table can be used as one of the limiting conditions for fitting injection production well spacing. The well spacing calculated from the single well control area in the table is taken as the initial value of fitting.

4.2. Calculation Results

The objective function is solved by genetic algorithm, and the relevant genetic parameters are set as follows. The number of genetic independent variables (nvars) is 6, and the upper and lower limits of genetic variables are given. Fitness function (fitnessfun) of genetic algorithm is the objective function established in this paper. The generation of genetic algebra is set to 50. StallGenLimit is set to 50.

The automatic history fitting process and fitting results of P1 well are shown in Figure 3. It can be seen from the figure that in the process of applying genetic algorithm to single well automatic history fitting, the observed data of daily oil production, daily water production, and water cut of oil wells will approach the actual production data with the process of genetic variation. It can be seen from the figure that the water cut curve of P1 oil well obtained by the model simulation calculation after fitting has little difference with the actual water cut dynamic change of oil well, which can verify the feasibility and practicability of the single well automatic history fitting method.

Figure 4 shows the daily oil production comparison between the actual model and the model after single well automatic history fitting. It can be seen from the figure that the production performance of oil wells in multilayer reservoir after single well automatic history fitting is highly consistent with the production performance of actual model, which proves that the permeability of each layer after fitting is quite close to the real permeability value of each layer.

The daily oil production of each small layer during the development period was calculated by using the fitted multilayer combined production single well flow model and compared with the numerical simulation results to verify the split production results of P1 well. Figures 57 show the daily oil production comparison of layer 1, layer 3, and layer 5, respectively. In the diagram, the red curve represents the time-varying daily oil production of layer calculated by numerical simulation, and the blue curve represents the daily oil production of layer after fitting. It can be seen that in the whole production and development stage, the daily oil production situation of each layer is almost consistent with the actual daily oil production situation of each layer, which proves that the splitting result of the fitted model is highly accurate and can be used as a reliable basis for clarifying the distribution of remaining oil in each layer. At the same time, it provides help for the design of production and injection allocation plan or the improvement of remaining oil tapping measures in the later stage of production and development.

  • (1)

    The mathematical model of single well production in multilayer commingled production reservoir is established, and the optimization theory is introduced. The objective function of single well automatic history fitting for multilayer coproduction reservoir is established, and the genetic algorithm is selected to solve the objective function

  • (2)

    A production splitting method based on automatic history fitting of single well is proposed. Based on single well production mathematical model of multilayer commingled production reservoir, automatic history fitting of single well is carried out, and the reservoir parameters of each layer near the single well are accurately reflected, which makes the production splitting result more accurate

  • (3)

    Through the example verification, it can get more accurate daily oil production of layer by using single well automatic history fitting for a production well in multilayer commingled production reservoir and then split the production. In the field application, this method can quickly and accurately realize the production splitting of the reservoir according to the development performance data

All the data are in the paper.

The authors declare no conflicts of interest.

This work was supported by the National Natural Science Foundation of China with No. 51974343 and Project funded by China Postdoctoral Science Foundation with No. 2021M703588. The first author would like to acknowledge the College of Petroleum Engineering at China University of Petroleum (Qingdao) for providing experimental equipment.

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