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Selectively completed horizontal wells (SCHWs) can significantly reduce cost of completing wells and delay water breakthrough and prevent wellbore collapse in weak formations. Thus, SCHWs have been widely used in petroleum development industry. SCHWs can shorten the effective length of horizontal wells and thus have a vital effect on production. It is significant for SCHWs to study their rate decline and flux distribution in naturally fractured reservoirs. In this paper, by employing motion equation, state equation, and mass conservation equation, three-dimension seepage differential equation is established and corresponding analytical solution is obtained by Laplace transform and finite cosine Fourier transform. According to the relationship of constant production and wellbore pressure in Laplace domain, dimensionless rate solution is gotten under constant wellbore pressure in Laplace domain. Dimensionless pressure and pressure derivate curves and rate decline curves are drawn in log-log plot and seven flow regimes are identified by Stehfest numerical inversion. We compared the simplified results of this paper with the results calculated by Saphir for horizontal wells in naturally fractured reservoirs. The results showed excellent agreement. Some parameters, such as outer boundary radius, storativity ratio, cross-flow coefficient, number and length of open segments, can obviously affect the rate integral and rate integral derivative log-log curves of the SCHWs. The proposed model in this paper can help better understand the flow regime characteristics of the SCHWs and provide more accurate rate decline analysis of the SCHWs data to evaluate formation.

In the early age, since horizontal wells can increase productivity, they became a popular method to develop oil and gas. Compared with vertical wells, horizontal wells can control severe water or gas coning problems, increase the connecting area with the reservoir, and reduce wellbore turbulence [

Although horizontal wells show a number of advantages, increasing wellbore length may lead to production imbalance along the wellbore, which can lead to water coning and decreased production. Certainly, the uneven rate distribution can lead to bottom-water break through [

To analyze the wellbore pressure and rate response of SCHWs, some engineers tend to use an effective horizontal well length to replace the open length of the horizontal well. This treatment assumes that the open length of the horizontal well is continuous instead of interval distribution. An analytical model was developed in real domain to predict the inflow performance of SCHWs and selectively completed vertical wells (SCVWs) [

In order to analyze the rate decline curve of SCHWs, a mathematical model considering difference between horizontal and vertical permeability of SCHWs is established in naturally fractured reservoirs. Based on point source and the superposition principle, pressure analytical solution of the SCHWs under the condition of constant production in impermeable top and bottom boundary and lateral impermeable boundary by Laplace transform and finite cosine Fourier transform. Log-log curves of pressure and pressure-derivate and rate decline are drawn in naturally fractured reservoirs by employing Stehfest numerical inversion. Seven flow regimes, according to the characteristic of pressure-derivate curve, are identified and every flow regime characteristic is described in detail. This paper discusses that relevant parameters (storativity ratio, flow coefficient, number and length completed horizontal sections, etc.) have effect on pressure and rate decline curves. Corresponding solutions can be useful in completion design and rate decline in field practice.

Horizontal wells are located in naturally fractured reservoirs with impermeable top and bottom boundary and lateral impermeable boundary and parallel to the upper and lower impermeable boundary. Horizontal well consists of

The fluid flow in the reservoir obeys Darcy’s law and law of isothermal percolation.

Flow is single phase and the fluid has constant and small compressibility and constant viscosity.

Formation permeability is anisotropic with three major directional permeability

Formation initial pressure is

Horizontal well consists of

The length of the open segments and completed segments may be unequal, and each open segment may have a different skin effect and production rate.

Schematic of the SCHWs in naturally fractured oil reservoirs.

Schematic of fluid flowing path for SCHW in naturally fractured oil reservoirs.

In this paper, we follow the point source theory adopted by Gringarten and Ramey [

where

The initial pressure is assumed to be equal and is represented by original formation pressure in naturally fractured oil reservoirs; thus

It is assumed that production rate of point source is

Corresponding outer boundary conditions can be expressed as for a laterally impermeable boundary, top, and bottom boundaries being

According to dimensionless variables definition in Table

Dimensionless variables definition.

Variables | Dimensionless definition |
---|---|

Dimensionless pressure of fracture system | |

Dimensionless pressure of matrix system | |

Dimensionless wellbore pressure | |

Dimensionless production time | |

Dimensionless distance | |

Dimensionless reservoir thickness | |

Dimensionless radius of impermeable circle boundary | |

Dimensionless coordinate | |

Dimensionless x-y-z coordinate of point source | |

Dimensionless length of open segment | |

Dimensionless wellbore radius | |

Dimensionless mid-point of | |

Dimensionless continuous production | |

Dimensionless infinitesimal vertical distance | |

Dimensionless infinitesimal radial distance | |

For convenience in derivation, by adopting Laplace transform with respect to

where

By finite cosine transform with respect to

In deriving

Equation

where

According to the properties of modified Bessel’s functions and outer boundary condition, the coefficient B can be expressed by

Hence, (

Combining with inner boundary condition, the coefficient A in (

Substituting (

Equation (

Taking the SCHWs shown in Figure

It is noted that (

According to geometric relations shown in Figure

Schematic of geometric relationship mid-point of

With (

where

However, it is also required that the sum of the flow rates for each open segment be equal to the total flow rate; that is,

Combining with (

The dimensionless wellbore flow rate for the constant-pressure production in naturally fractured reservoirs can be determined by dimensionless pressure with the constant-rate production in the Laplace domain [

In order to be consistent with current literature, we use the Fetkovich [

In a similar fashion, the

The rate integral and rate integral derivative functions introduced by McCray [

And the dimensionless rate integral derivative function,

To verify the model and solutions derived in the above section, a relatively particular case is considered and pressure and pressure-derivate curves generated by our solution are compared to well-test stimulator Saphir. Fluid flow into wellbore is treated as infinite conductivity, but rate distribution in wellbore is no-uniform. Therefore, based on different dimensionless variable definition between this paper and well-test stimulator, we can set ^{−8} with

Comparison of the results of this paper with that of well-test simulator.

Rate distribution along the wellbore in different time.

In order to study the flow regimes of SCHWs in naturally fractured oil reservoirs more graphically, type curves of pressure response and production rate performance are illustrated in Figures

Important basic data for SCHWs.

Parameters (unit) | Value |
---|---|

Wellbore radius (m) | 0.1 |

Outer boundary radius (m) | 10000 |

Horizontal well length (m) | 400 |

Length of each open segment (m) | 25 |

Length of each completed segment (m) | 100 |

Number of open segments (dimensionless) | 4 |

Number of completed segments (dimensionless) | 3 |

Reference length (m) | 40 |

Storativity ratio (dimensionless) | 0.2 |

Flow coefficient (dimensionless) | 0.01 |

Pressure and pressure derivative responses of SCHWs with 4 open segments.

Rate, rate integral, and rate integral derivative responses of SCHWs with 4 open segments.

Period I is the first radial (FR) flow period. During this period, the flow regime is radial flow around open segment in vertical direction (see Figure

Schematic of flow stage for SCHWs with 4 open segments.

First radial flow

First linear flow

Second pseudo-radial flow

Second linear flow

Late pseudo-radial flow

Period II is first linear (FL) flow period, in which fluid flow in the reservoir is parallel to the upper and lower boundary of the reservoir and each open segment is independent during first linear flow stage (see Figure

Period III is second pseudo-radial (SPR) flow, in which the pressure derivative curve is horizontal line of “0.5/

Period IV is second linear (SL) flow. Pressure wave propagates to drainage area controlled by each open segment, and interference between open segments occurs. After the superposition of the pressure waves, pressure waves propagate continually as time goes. The second linear flow can be formed in natural fracture reservoir (see Figure

Period V is the cross-flow stage, in which fluid flows into natural fracture from matrix firstly when the SCHWs are put into production. The pressure of natural fracture system will gradually decrease, causing pressure difference between natural fracture system and matrix system. Because existence of pressure drop between natural fracture system and matrix system led to cross-flow from natural fracture system to matrix system, the characteristic of pressure derivative during stage is “dip.” Corresponding rate integral derivative curve also exhibits a “dip” in this stage.

Period VI is late pseudo-radial (LPR) flow stage. After cross-flow flow stage, the pressures in natural fracture system and matrix system gradually incline to equilibrium. Pseudo-radial flow around SCHWs is formed in naturally fractured reservoirs (see Figure

Period VII is characteristic of closed boundary. Pressure waves propagate to circular impermeable outer boundary during this stage. Curves of pressure derivative exhibit unite-slope line and corresponding rate integral and derivative curve coincide and exhibit negative unite-slope line.

Figure

The effect of outer boundary radius on

Figure

The effect of storativity ratio on

Figure

The effect of cross-flow coefficient on

Figures

The effect of number of open (or completed) segments on

The flux distribution of each open segment.

Figures

The effect of length of open (or completed) segment on

The flux distribution of each open segment.

In this work, we have developed a solution to compute the rate decline of SCHWs with constant wellbore pressure. According to characteristic of pressure-derivative curves under constant production and rate integral derivative curve under constant wellbore pressure, flow regimes of SCHWs are identified. Specific conclusions can be drawn as follows.

Total compressibility of natural fracture system and oil, atm^{−1}

Total compressibility of matrix system and oil, atm^{−1}

Reservoir thickness, cm

Equivalent permeability, ^{2},

Horizontal permeability of natural fracture system, ^{2}

Vertical permeability of natural fracture system, ^{2}

Permeability of matrix system, ^{2}

Vertical permeability, ^{2}

Length of

Length of

Reference length, cm

Open segment number, dimensionless

Cumulative production, cm^{3}

Initial reservoirs pressure, atm

Pressure of natural fracture system, atm

Pressure of natural matrix system, atm

Wellbore pressure of natural matrix system, atm

Production under constant wellbore pressure, cm^{3}

Decline rate function as defined by Fetkovich, cm^{3}

Decline rate integral as defined by McCray, cm^{3}

Decline rate integral derivative function as defined by McCray

Production rate under the standard conditions, cm^{3}/s

Surface production rate of a point source, cm^{3}/s

Radial distance, cm

Radius of impermeable circle boundary, cm

Wellbore radius, cm

Laplace variables

Decline time, s

x-coordinates, cm

Mid-point of

x-coordinates of a point source, cm

y-coordinates, cm

y-coordinates of a point source, cm

z-coordinates, cm

z-coordinates of a point source, cm

Shape factor of dual-porosity system, cm^{−2}

Infinitesimal radial distance, cm

Infinitesimal vertical distance, cm

Cross-flow coefficient of dual-porosity reservoirs, dimensionless

Viscosity at current reservoir pressure, cp

Reservoir porosity, dimensionless

Storativity ratio of dual-porosity reservoirs, dimensionless

The first kind modified Bessel function, zero order

The second kind modified Bessel function, zero order

The first kind modified Bessel function, first order

The second kind modified Bessel function, first order.

Dimensionless

Natural fracture system

Matrix system.

Laplace domain

Finite cosine transform.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

You-jie Xu and Qi-guo Liu contributed equally to this work (co-first authors).

This article was supported by the National Major Research Programme for Science and Technology of China (Grant No. 2017ZX05009-004 and No. 2016ZX05015-003).