Critical Parameters of a Horizontal Well in a Bottom Water Reservoir Department: Southwest Petroleum University Address: No.8, Xindu Road, Chengdu, P. R. China Dr. Yue Ping

Layout  Background  Experimental Program and Results  Numerical Simulation Results  Theoretical critical parameters models  Startup conditions of bottom water cresting/ Coning  Critical parameters of homogeneous reservoir  Critical parameters of heterogeneous reservoir Impermeable barrier Semi-permeable barrier Semi-permeable barrier considering thickness  Conclusions

Fig. Water cresting in horizontal well The oil-water interface will deform and rise during the oil well producing in the bottom water reservoir (pressure drop and material balance relation) Water coning & water cresting Bottom water zone Oil zone Water coning in vertical wells Water cresting in horizontal wells Background

Formation with a barrier Normally, the calculated horizontal well critical rate is much smaller than the actual oil production rate ----This difference results due to two main reasons Firstly, new critical (startup) conditions (Yue et al. 2009, 2012) should be applied Secondly, geological conditions are far more complicated than assumptions of homogeneous models. reservoir barriers have great impact on the calculated results of critical rate. It has been reported that reservoir barriers can significantly increase critical rate and delay water breakthrough time No quantitative investigations on barriers affecting the bottom water cresting in horizontal wells Background

 Water coning experiment No water coning Developing water coning 突破井底 Experimental Program and result Developed water coning Water coning process According to the description of the dynamic process of the water cresting in bottom water oil reservoir produced by horizontal wells, it is revealed that there is a startup state before the water can rise. Produce rate is small, no water coning Produce rate is bigger enough, developing water coning

Numerical simulation of Water coning & cresting Water cresting develops fast Water will flood the well quickly higher than initial critical rate smaller than initial critical rate Water rises step by step in z direction Small amount remained oil on both side VS

Theoretical critical parameters models Startup conditions of bottom water cresting/ Coning Critical parameters of homogeneous reservoir Critical parameters of heterogeneous reservoir

Fig. Stress balance of water cresting  Former scholars’ Critical condition Derivation form static mechanics equilibrium Former Critical condition Startup conditions of bottom water cresting/ Coning Former Critical rate z CC Bottom water ∆z∆z p1p1 p2p2 w Eq.(1) Eq.(2)

z CC Bottom water ∆z∆z p1p1 p2p2 w Derivation from seepage mechanics integral The former critical condition are only fit for the calculation about the maintaining of certain water cresting height after the formation of water cresting Also can get the Former Critical condition Fig. Stress balance of water cresting Startup conditions of bottom water cresting/ Coning Eq.(1)

No matter when W is at the highest point of N, or at least M, The condition of static equilibrium of the mass of a particle is  Theory of Startup Condition ——New Critical condition Water Cresting h zwzw CC M N bottom water 油区 rwrw y z W New critical condition focus on the condition of bottom water is beginning to rise New critical condition Which? Former Startup conditions of bottom water cresting/ Coning Eq.(4) Eq.(5) Eq.(3)

Theoretical critical parameters models Startup conditions of bottom water cresting/ Coning Critical parameters of homogeneous reservoir Critical parameters of heterogeneous reservoir

The potential function of the of horizontal well production in bottom water reservoir is Set y=0 and Derivate above equation, the potential gradient function below the well borehole axis can be obtained z zw 0 Potential distribution on YZ profile Potential gradient changes in different points on line MN  Potential and potential gradient function of horizontal well Figs. Show the minimum value of potential gradient at M point M N Critical parameters of homogeneous reservoir M N Eq.(6) Eq.(7)

Combine equation (5) with (7) and let z = 0, the startup rate will be obtained: New critical rate ： Critical potential difference: Critical parameters of homogeneous reservoir  Critical parameters of horizontal well Eq.(5) Eq.(7) Eq.(8) We know productivity ： Critical pressure difference: here

Theoretical critical parameters models Startup conditions of bottom water cresting/ Coning Critical parameters of homogeneous reservoir Critical parameters of heterogeneous reservoir

h zw CC M N Water Oil pay y z c b a N’N’ M’M’  impermeable barrier The entire flow process of this model is also divided into two stages. First stage, the fluid on the left side flows from the bottom to the MN plane. Equivalent well radius ： The productivity of the image wells M or M΄ is ： A parameter γ is introduced to describe the relationship between the flow rate of the equivalent well Q1 and the total flow rate of the actual well Q2 with The sketch of a horizontal well with a barrier in a bottom water reservoir Critical parameters of heterogeneous reservoir Set two virtual wells M and M΄, Eq.(9) Eq.(10)

h zw CC M N Water 油区油区 y z c b a N’ M’ ， With Second stage, the fluid flows from the MN plane into the wellbore. The rectangle region MNN΄M΄ is selected as the study area, which is equivalent to an Bottom-water-drive reservoir Rate of second stage Eq.(12) Eq.(11) New critical rate ： Critical parameters of heterogeneous reservoir How can we get this empirical formula?

b=80m, c=20m, R k =0 h zw CC M N Water Oil y z c b a N’ M’ Critical parameters of heterogeneous reservoir Did amount of numerical simulations

c=20m, b =0 ~100m b=60m, c = 5 ~ 25m c=5-25m, b=60m Use this empirical formula to match the simulation results Critical parameters of heterogeneous reservoir Numerical result Theory result Numerical result Theory result Numerical result Theory result

In the first part, partial pore space of the whole formation provides permeability of K1. In the second part, the remaining part of the pore space provides zero permeability in the barrier area; while the permeability is K-K1 in the other area of the formation except the barrier area. Critical parameters of heterogeneous reservoir The pore space can be taken as the superposition of two parts pore spaces Fig. Sketch of the total formation divided into two parts  Semi-permeable barrier ( ignoring thickness of barrier )

The critical rate  Semi-permeable barrier ( ignoring thickness of barrier ) Critical parameters of heterogeneous reservoir First part can be treated as the bottom water reservoir with permeability K1 Second part can be treated as the bottom water reservoir with an impermeable barrier and the permeability of the formation is K-K1. Superposition and coupling method can be used to determine the total critical potential difference for bottom water reservoir with semi-permeable barrier. Eq.(13) Eq.(14) Eq.(15) Eq.(16) Where RK = K1/K,

 When K1=K, Eq. (16) becomes the critical rate of horizontal well in bottom water reservoir with no barriers.  When K1=0, Eq. (16) degrades to critical rate of horizontal well in bottom water reservoir with an impermeable barrier.  Eq. (7) is better to satisfy the formula for critical production. The critical rate Critical parameters of heterogeneous reservoir Eq.(16)  Semi-permeable barrier ( ignoring thickness of barrier )

Fig. 2. The sketch of the total formation divided into two parts (Considering thickness of barrier ) If considering the thickness of the barrier, we found that Eq. (17) was more suitable when parameter d was restricted in 0.2 to h/4. This means that the larger the thickness of the barrier, the higher is the parameters γ. It also means the larger the R K (the same to K 1 ), the greater the thickness influence extent to parameters γ. Eq. (17)  Semi-permeable barrier ( Considering thickness of barrier ) Critical parameters of heterogeneous reservoir The critical rate

Fig.. Formation with a thick semi-permeable barrier (b = 75m, c = 20m, d = 6m, R k = 0.5) Critical parameters of heterogeneous reservoir Water rising process while the well produces below the initial critical rate Water rising process while the well produces above the initial critical rate Also did a lot of numerical simulation  Semi-permeable barrier ( Considering thickness of barrier )

Comparison between simulation and theory results Fig. 12. Simulation and theory Q m comparison when c = 20m, d = 1m, R k = 0 and 0.5, b range 0 to 100m Fig. 13. Simulation and theory Q m comparison when b = 60m, d = 1m, R k = 0, c range 5 to 25m Fig. 14. Simulation and theory ΔФ m comparison when b = 60m, d=1m, R k = 0, c range 5 to 25m Use empirical formula Eq. (17) to match the simulation results Contrast theory and simulation results Eq. (17)

Forecast, Analysis and Discussion Fig. 16. Critical rates with different R K and b Fig. 16 shows that the critical rate monotonically linear decreases with the increase of RK when d is 1 m, but does not satisfy the linear decrease when d is 4 m. Fig. 16 also presents that when R K = 1, all values are converged to the m 3 /d, which is the critical rates of the formation without barrier ItemValue Horizontal well length: L, m350 Formation thickness: h, m40 Distance between wellbore and bottom boundary: z w, m 30 Reservoir oil density: ρ o, kg/m Reservoir water density: ρ w, kg/m Reservoir oil viscosity: μ o, mPa · s 15 Reservoir water viscosity: μ w, mPa · s 1 Reservoir oil volume factor: B o, dimensionless 1.05 Isotropic permeability: K, μm Wellbore radius: r w, m0.1 Radius on Y section: r e, m80 Distance from barrier to oil-water contact: c, m 15 and 20 Ratio of permeability: R K, dimensionless 0 to 1 Semi-permeable barrier width: b, m0 to 100 Semi-permeable barrier thickness: d, m0.2 to 10 Table. Basic information of reservoir, well and barrier

Fig. 17. Comparison of critical rates with different R K, d and b All the calculations show that the increase in b leads to better prevention of water cresting. When b is close to zero, all values (Fig. 17) are converged to the m3/d, which is the critical rate of the formation without barrier. Forecast, Analysis and Discussion

Fig. 18 Comparison of critical rates with different R K, c and b Both the Fig. 18 shows all curves for c = 20 are higher than the correspondent curves for c = 15 at the same value of R K. This means that the closer the location of the barrier to the wellbore, the bigger the critical rate is. The same to the critical potential difference Forecast, Analysis and Discussion

Conclusions  New critical condition focus on the condition of bottom water is beginning to rise, and the result more reasonable.  Barriers have significant impact on the production performance of horizontal wells developed in a bottom water drive reservoir.  The bigger size and thickness, smaller permeability, higher position of semi-permeable barrier, the better the performance to prevent or resist the water cresting in a horizontal well producing from a reservoir with strong bottom water drive.

Acknowledgement This work was supported by The National Natural Science Foundation of China (No ) The State Key Laboratory of Petroleum Resources and Prospecting of CUP (No. PRP/open-1501) Sichuan provincial science and Technology Department ( No. 2015JY-0076) Sichuan Provincial Education Department (No.14ZB0045).

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