A new way to calculate productivity of fivespot pattern at high water cut stages
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Abstract
This paper presents a new way to calculate productivity of fivespot pattern at high water cut stages. In low permeability reservoirs, it is difficult for the reservoir fluid to flow, and the recovery factor is very low. Being based on the nonDarcy flow in low permeability reservoir, taking threshold pressure gradient into consideration, a new method to calculate the well pattern was built. This method is based on the twoparameter continuous model, and the method of flow line integrals and the numerical integration were used to build the productivity model. Productivity for the new method is validated by comparing with the actual data from field, and both the seepage theory and the reservoir engineering illustrate the correctness of the new method in this paper. The fracture length and pressure difference between the injection well and the production well have a significant effect on the productivity of the fractured fivepoint well pattern. The greater the length of the fracture is, the higher the productivity of the fractured fivepoint well pattern is. The greater the pressure difference is, the higher the productivity of the fractured fivepoint well pattern is.
Keywords
Lowpermeability reservoirs Element analysis method Flow tube integration method Heterogeneity of remaining oil ProductivityIntroduction
Water flooding is an efficient method to maintain reservoir pressure, and it has been widely used to enhance oil recovery. Currently, most waterflooding reservoirs have been in a high water cut period in China. Due to the reservoir heterogeneity, different locations of the reservoir have different water flooding spread area, which results in that the distribution of remaining oil at high water cut stages are not the same.
At present, several major oil companies in China face with the problem of low permeability reservoir development. However, because of low porosity and low permeability, low permeability reservoirs have some differences with middle and high permeability reservoirs. A large number of studies have shown that (Deng and Liu 2001; Zhu et al. 2010; Yang et al. 2011; Sun 2010): the reservoir fluid in low permeability reservoirs does not follow Darcy’s law any more, and it is shown as a lowspeed non Darcy percolation (Li et al. 2008; Zhang and Wang 2011; Fu and Ge 2002; Zhu 2007; Yang et al. 2007). Previous classical reservoir engineering methods have some limitations in the application and guidance of low permeability reservoir development. It is mainly showed in the following aspects:
 1.
The seepage model is too idealized, and only singlephase flow is considered, and the distribution of remaining oil in the reservoir and the effect of water flooding are not fully considered;
 2.
The calculation unit is often concentrated in one injection and production unit, and there is still a certain gap with the actual low permeability reservoir development well network;
 3.
The derivation process is complicated and inconvenient to calculate.
Therefore, it is necessary to establish an engineering calculation model for low permeability reservoir based on nonDarcy seepage.
At present, low permeability reservoirs are mainly exploited by area well pattern. In view of the productivity calculation of area well pattern, many researches have been done by predecessors. Using the principle of pressure superposition and the mirror inversion method, Luo et al. (2010) studied the pseudosteady flow productivity and injection production pressure difference model was built for low permeability reservoirs with closed boundary. By combining the seepage characteristics of low permeability reservoirs and using the equivalent percolation resistance method, Du Dianfa et al. (2012) gives the expression of the area well network productivity. Based on the productivity formula of the area well network in the ultralow permeability reservoir, Xu et al. (2014) used the superposition principle and the equivalent well diameter model to build the productivity expression for the joint area of horizontal well and straight well. He et al. (2009) adopt the flow pipe method to establish the production model of the low permeability rectangular reservoir fracturing straight well. Using the similarity principle of hydropower Zhao et al. (2008) derived the productivity model of different types of well pattern.
Previous studies are summarized in three aspects: starting pressure gradient model, productivity splitting model and quasi starting pressure gradient model. Based on the previous research results, this paper uses the streamline integration method (Munseok 2000; Higgins and Leighton 1961; Ji et al. 2008; He et al. 2009a, b), based on the nonDarcy flow and the twoparameter model, an area well network productivity model considering the starting pressure gradient is built, which makes the production calculation and the well network optimization closer to the actual production. By applying element analysis, an injectionproducing unit of a fractured fivespot pattern is divided into four subunits, each of which is divided into three calculation units in accordance with streamline distribution characteristics. On this basis, the injection production pressure difference and the well spacing are analyzed. The startup coefficient is introduced to evaluate the utilization degree of low permeability reservoir.
Physical model for fractured fivespot patterns at high water cut stages
The schematic illustration in Fig. 1a shows an injectionproducing unit of a fractured fivespot pattern. The fracture halflength of each injection well is the same which equals to \({L_{{\text{fw}}}}\). The fracture halflength of production well equals to \({L_{{\text{fo}}}}\). Well spacing and spacing of well array equal to \({L_1}\) and \({L_2}\), respectively. By applying element analysis, an injectionproducing unit of a fractured fivespot pattern is divided into four SUs, each of which is divided into three CUs in accordance with streamline distribution characteristics. Therefore, an injectionproducing unit of a fractured fivespot pattern is divided into 12 CUs totally, as shown in Fig. 1b.
The assumptions are as follows:

The saturation distributed in the injectionproducing unit is not uniform at the high water cut stage. Various SUs have different saturation. However, Saturation for a SU is the same.

Before seeing water in the oil well, the permeability is constant;

In an injection unit, the starting pressure gradient is constant;

The twophase unstable percolation occurs in the fluid flow in the stratum, and the fluid viscosity is constant.

The effects of gravity and capillary forces are ignored;

In the injection unit, no gas is dissolved or escaped.

Fractures of injection well and production well have infinite conductivity
Productivity of an injection–production unit for a fracture fivespot pattern equals to the sum of productivities for all CUs. Method of productivity for every CU is introduced in the next part.
There are many methods for describing mathematical models of nonlinear seepage at present (Ruina and Xiaodong 2011; Baoquan et al. 2011; Xu et al. 2018), as shown in Table 1. The seepage model in Table 1 is mainly divided into the proposed pressure gradient model, segmented model and continuous model. At present, the nonlinear seepage is mainly based on phenomenological method. The reason is that the classical Darcy’s law cannot describe the percolation characteristics of the lowpermeability reservoirs. The lowpermeability reservoirs have the starting pressure, boundary layer and microscale flow effects, and the starting pressure is not a constant which increases the difficulty and accuracy of describing the equation of motion of the true seepage law of lowpermeability reservoir fluids.
Nonlinear mathematical models
Types of models  Equation  Model description 

Proposed starting pressure model  \(v=\left\{ \begin{gathered} \frac{k}{\mu }\left( {\nabla p  G} \right)\quad \nabla p \geqslant G \hfill \\ 0\quad \nabla p<G \hfill \\ \end{gathered} \right.\)  When the displacement pressure is less than G, the seepage velocity cannot be calculated 
Power exponential form  \({v^n}=\frac{k}{\mu }\nabla p,\quad n \geqslant 1\)  There is high simulation error of simulating the linear segment 
Full description  \(v=\frac{k}{\mu }\nabla {p^{1+{n_d}}},{n_d}=1+\frac{2}{{1+\frac{{{\raise0.7ex\hbox{${\Delta p}$} \!\mathord{\left/ {\vphantom {{\Delta p} L}}\right.\kern0pt}\!\lower0.7ex\hbox{$L$}}}}{G}}}\)  It is very difficult to solve the connection position between the linear area and nonlinear 
Threeparameter continuous model  \(v\left( {{a_1}+\frac{{{a_2}}}{{1+bv}}} \right)=  \nabla p\)  It does not reflect the actual seepage 
Twoparameter continuous model  \(v=  \frac{k}{\mu }\nabla p\left( {1  \frac{1}{{a+b\left {\nabla p} \right}}} \right)\)  It can reflect the seepage flow of underground fluid very well 
Where \(v\) is the seepage velocity for a flow tube, sm/day; \(\nabla p\)is the displacement pressure between the injection well and the production well, MPa; \(k\) is the effective permeability for fluid flow, 10^{− 3} µm^{2}; \({a_1},{a_2},a,b\) are the model parameters and their values are related to the actual reservoir; \(\mu\) is the fluid viscosity, mPa s; \(G\) is the starting pressure gradient, MPa/m; \(L\) is the distance from injection well along the centerline for a flow tube, m.
Productivity for every CU
The well pattern productivity equals to the sum of the production for all flow tubes which are distributed in the CU. Therefore, it is necessary for the production well to calculate the production of a flow tube.
From Table 1, the twoparameter continuous model reflect the seepage flow of underground fluid well, and it is also very easy to get the twoparameter in this model, so this paper chooses the twoparameter continuous model as the basic seepage model to build the production model for the flow tube.
Production for a flow tube
The injection unit interface diagram is shown in Fig. 2. By rewriting the twoparameter continuous model and taking the oil–water twophase flow into consideration, the production of a crosssectional of a flow tube is given by the following equation,
where \(M=\frac{{\Delta q}}{{\sum\nolimits_{{i=1}}^{2} {A({\xi _i})} }}\frac{\mu }{k}\).
Production for a triangular CU
When the two phases of oil and water reach pseudosteady state flow, the production of each triangular CU is the same. To simplify the model, this paper just takes the CU 1 as an example to calculate the production of a triangular CU, which is shown in Fig. 3.
By the way, \(A(\xi )\) is also can be obtained:
where \({Q_{{\text{o}}1}}\) is the production of CU 1, sm^{3}/day.
Production for a quadrilateral CU
When the two phases of oil and water reach pseudosteady state flow, the production method for each quadrilateral CU is the same. To simplify the model, this paper just takes the CU 1 as an example to calculate the production of a triangular CU, which is shown in Fig. 4.
Transient productivity method for fractured fivespot patterns
By combining the Eqs. (18)–(21), and by following the operation of ‘six step method’, the transient productivity of fractured fivespot patterns can be obtained.
The method is present as the following (Yao et al. 2014; Yoa and Wang 2004):
 1.
① The initial reservoir parameters and saturation for each SU are known;
 2.
 3.
 4.
④ Calculate the production for fractured fivespot patterns at the beginning moment using Eq. (16);
 5.
⑤ Calculate the saturation for each SU at the next moment using Eq. (17);
 6.
⑥ Update the saturation for each CU in the steps ①, Repeat steps ②–⑥, one can get the production for fractured fivespot patterns at the next moment; Finally, we get the transient productivity.
Reservoir basic parameters and productivity method validation
Reservoir basic parameters
The Lasa Xing oil field is located in the Changshu anticlinal structural belt in the Central Depression of the Songliao Basin, and it is also located in the most favorable area for oil production and oil storage in the Songliao basin. The sedimentary system is a large riverdelta body, and the lithology is mainly fine sand and siltstone. The air permeability of the main oil layer is 0.5 µm^{− 2} or less, and the permeability of the thin sand layer is between 0.02 and 0.08 µm^{− 2}, and the permeability of the offsheet reservoir is between 0.001 and 0.05 µm^{− 2}, which is an anticline structured reservoirs and also a low permeability reservoir. The Lasa Xing oil field is now in the high water cut stage. This paper takes the Lasa Xing oil field as an example to show the method of calculating productivity for fivespot well pattern.
The input value which are necessary for the model to calculating the productivity is shown in Table 2.
The value of the input parameters
Parameters  Value  Parameters  Value 

Oil viscosity, µ_{o}  8.04 mPa s  Porosity, □  0.204 
Injection pressure P_{in}  21.696 MPa  Production pressure, P_{pro}  9.6 MPa 
SU 1 permeability, k_{1}  39.24 mD  SU 3 permeability, k_{3}  30.6 mD 
SU 2 permeability, k_{2}  36.12 mD  SU 4 permeability, k_{4}  35.28 mD 
SU 1 water saturation, s_{w1}  0.78  SU 3 water saturation, s_{w3}  0.66 
SU 2 water saturation, s_{w2}  0.72  SU 4 water saturation, s_{w4}  0.684 
Reservoir thickness, h  16.8 m  Wellbore radius, r_{w}  0.1 m 
Injection well fracture length of, L_{fw}  280 m  Well spacing, L_{2}  240 m 
Production well fracture length of, L_{fo}  240 m  Fracture width, w_{f}  0.024 m 
\({k_{{\text{ro}}}}\), \({k_{{\text{rw}}}}\) is the relative permeability of the oil phase and the water phase, respectively; \({s_{\text{o}}}\) is the oil saturation; \(m\) is the fitting parameter.
Productivity method validation
Step 1 Input parameters.
Through the fracture construction design report and fracture monitoring data, the fracture parameters of the injection–production well, such as the length of the fracture, are obtained. It is noted that the fracture halflength of the injection well is 125 m, and the fracture length of the production well is 114 m, which is not equal, so the Eq. (19) is used to obtain the production for a quadrilateral CU.
Step 2 Saturation distribution.
Step 3 Starting pressure gradient.
The phaseinfiltration curve is obtained using the initial oil saturation and the combined formulas (24) and (25).
The relative permeability curves of the reservoir are shown in Fig. 5.
Step 4 The transient productivity can be obtained, which is shown in Fig. 6.
The saturation obtained in the above step and the subunit average permeability inversely combined with the phaseinfiltration curve are sequentially substituted into the corresponding subunit productivity model (28), and then using the iterative method, the transient productivity can be obtained, which is shown in Fig. 6.
Figure 6 shows the calculating results between the new method and the actual data. Figure 6 shows that the data of new method and the actual data have a quite good agreement, which means that the new method in this paper is reliable. Therefore, it is suitable to use this method to predict the transient productivity of fivespot patterns.
Figure 7 shows that it takes only 21.282 s to run the program of transient productivity prediction for a fractured fivespot pattern once, when the computer’s processor model is Intel(R) Core(TM) i76700 CPU @3.4 GHz.
When \(\lambda\) approaches zero, it becomes Darcy’s seepage. The pressure difference between the production well and the injection well is 15 MPa, and the permeability is 0.033 mD, and the thickness of the oil layer is 5 m, and the viscosity is 5 mPa s, and the radius of wellbore is 0.1 m. Calculate the single well output of the deduced fivepoint well network (this paper solution) and compare it with the Muskat (1937) calculation solution (reference solution), and the calculation results comparing between this paper solution and the reference solution is shown in Table 3. Table 3 shows that the relative error is within 0.5%, which shows the method in this paper is feasible and accurate.
calculation results between this paper solution and the reference solution
Well space  Reference solution (m^{3}/day)  This paper solution (m^{3}/day)  Relative error (%) 

200  1.7492  1.7411  0.4680 
225  1.7202  1.7123  0.4631 
250  1.6951  1.6871  0.4727 
275  1.6729  1.6652  0.4652 
300  1.6532  1.6455  0.4674 
325  1.6355  1.6283  0.4437 
350  1.6194  1.6124  0.4375 
375  1.6048  1.5978  0.4334 
400  1.5913  1.5849  0.3997 
Both the seepage theory and the reservoir engineering illustrate the correctness of the new method in this paper.
Sensitivity analysis
In this section, using the single factor variable principle, the impacts of some relevant parameters on the transient productivity of fivespot patterns are analyzed. The relevant parameters are shown in Table 2.
Figures 8 and 9 illustrate the impacts of fracture length on production and cumulative production for a fractured fivespot pattern. Seen from Figs. 8 and 9, the length of fracture has a significant effect on productivity or cumulative production for fracture fivespot patterns. The bigger the fracture length is, the more the production rate will be. From the analysis of flow regime, the flow regime in the area of CU 2, CU 4, CU 6 and CU 8 is linear flow. The flow regime in other CUs is radial flow. The bigger the fracture length, the bigger the area of CU 2, CU 4, CU 6 and CU 8 will be. Therefore, the flow resistance for a fractured fivespot pattern will be small and the productivity rate will be larger, when the fracture is longer.
Figure 10 shows the impacts of pressure difference between injection well and production well on production. Seen from Fig. 10, the bigger the pressure difference is, the more the productivity rate will be.
Conclusions
Using the methods of elemental analysis and flow tube integration, a new productivity of predicting fivespot well pattern is built, and this new method takes the remaining oil heterogeneity into account. This new method can accurately predict the productivity of the fivepoint model of fractured well in high water cut.
The fracture length and pressure difference between the injection well and the production well have a significant effect on the productivity of the fractured fivepoint well pattern. The greater the length of the fracture is, the higher the productivity of the fractured fivepoint well pattern is. The greater the pressure difference is, the higher the productivity of the fractured fivepoint well pattern is.
Notes
Acknowledgements
This work was supported the State Energy Center for Shale Oil Research and Development.
Compliance with ethical standards
Conflict of interest
The authors declare no conflicts of interest regarding the publication of this paper.
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