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The classification method of relative permeability curves is rarely reported, when relative permeability curves are applied ; if the multiple relative permeability curves are normalized directly, but not classified, the calculated result maybe cause a large error. For example, the relationship curve between oil displacement efficiency and water cut, which derived from the relative permeability curve in LD oilfield is uncertain in the shape of low water cut stage. If being directly normalized, the result of the interpretation of the water flooded zone is very high. In this study, two problems were solved: 1) The mathematical equation of the relationship between oil displacement efficiency and water cut was deduced, and repaired the lost data of oil displacement efficiency and water cut curve, which solve the problem of uncertain curve shape. After analysis, the reason why the curve is not available is that relative permeability curves are not classified and optimized ; 2) Two kinds of classification and evaluation methods of relative permeability curve were put forward, the direct evaluation method and the analogy method ; it can get the typical relative permeability curve by identifying abnormal curve .

There are few reports on the classification of relative permeability curves, and the static parameter method represented by J function is common, but the relative permeability curve is the core of static and dynamic transformation. Static parameters alone are not sufficient to describe the full meaning of the curve, such as the law of water cut rising, etc. Based on the relative permeability curve of water phase, Wang Guoxian and Wang Lijun [

In the water flooding experiment, when the water is produced, the water content is higher, so there is no experimental data in the low water cut stage (

Generally, the lower the viscosity of crude oil is, the slower the increase rate of water cut is [

need to be optimized and corrected [

Based on the actual production of oilfields, two classification methods are proposed to optimize the relative permeability curves, and the two methods are combined; recognition gives the abnormal curve, thus obtaining the representative curve of oil field, and then guides the development and production of oil fields.

The oil-water two-phase relative permeability curve expression [

K r o K r w = a e b S w (1)

where K_{ro} is relative permeability of oil phase, K_{rw} is relative permeability of water phase, S_{w} is water saturation.

According to the fractional flow equation, under the condition of ignoring the effects of capillary force, gravity and dissolved gas, the water cut can be expressed as:

f w = Q o Q w + Q o = 1 1 + μ w ⋅ K r o μ o ⋅ K r w (2)

where f_{w} is water content. Q_{o} is daily oil production, the unit is m^{3}/d. Q_{w} is daily water production, the unit is m^{3}/d. μ_{o} is oil viscosity, the unit is mPa・s. μ_{w} is water viscosity, the unit is mPa・s.

The oil displacement efficiency expression is:

η = S w i 1 − S w i (3)

where η is oil displacement efficiency. S_{wi} is original oil saturation.

The relationship between oil displacement efficiency and water content can be obtained by simultaneous solving the above three equations:

f w = 1 1 + a e b ( 1 − S w i ) η + b S w i (4)

In the above formula, a, b and S_{wi} are constants, let A = a , B = b ( 1 − s w i ) and C = b ⋅ s w i , so simplify it to:

f w = 1 1 + A e B η + C (5)

Take the logarithm:

ln ( 1 f w − 1 ) = B η + C + ln ( A ) (6)

In the above formula, B and C + ln(A) are constants, let α = B , β = C + ln ( A ) , so they can be simplified to:

ln ( 1 f w − 1 ) = α η + β (7)

The above formula is the mathematical expression of oil displacement efficiency and water content, and it can be seen from the expression that oil displacement efficiency and the logarithm of water content are linear. By fitting curve, the exact expression of the displacement efficiency and water cut can be determined, and then the curve form is completed.

As the largest oilfield in Bohai sea, SZ oilfield has accumulated a lot of experience in the interpretation of water-flooded layers. In order to verify the correctness of the derived mathematical equation of oil displacement efficiency and water content, it is applied to SZ oil field first, and then to guide the application of the equation in LD oil field.

The relative permeability curve of SZ oilfield is shown in

According to

ln ( 1 f w − 1 ) = − 19.353 η + 3.353 (8)

The relation curve between oil displacement efficiency and water content can be calculated by using the above formula, and its calculated value and actual value are compared as shown in

However, LD oilfield is the case of missing data points in front (

and water content can be obtained by fitting the relationship between oil displacement efficiency and water content (

ln ( 1 f w − 1 ) = − 18.668 η + 2.987

The curve of the orange curve in

The exponent part of Formula (5) is:

y = e b η + C (10)

The above formula is drawn as

By observation, in

points can be estimated based on the triangle similarity principle. A simple diagram is shown in

and the maximum error is only about 5%, which can be used for the interpretation of the water flooded layer. The correctness of the mathematical equation of oil displacement efficiency and water content is proved.

There are 4 relative permeability curves in LD oilfield. After correction, as shown in

Therefore, the reason for the high interpretation result of the water flooded layer is that the relative permeability curve has not been classified and optimized, and the water content has increased faster than the heavy oil field.

The common classification method of relative permeability curve is a static

parameter method represented by J function [

Comparing the classification data (

Take the logarithm of both sides of Formula (1):

ln ( K r o K r w ) = b s w + ln a (11)

The above formula can be transformed into:

K r o K r w = e b s w + ln a (12)

The slope and intercept of the Formula (11), and the exponential part of the Formula (12) are related to the shape of the oil-water relative permeability curve, so the slope, intercept and index can quantitatively characterize the morphology of the curve, and they can be used as a basis for the classification of relative permeability curves. It can be seen from mathematical expressions that the ratio of oil-water relative permeability has a linear relationship with water saturation (Formula 11). According to relative permeability curve data, draw a straight line diagram as follows:

The mathematical relationship is as follows:

Rock Sample | Depth (m) | Air Permeability (mD) | Porosity % | K / Φ |
---|---|---|---|---|

Rock Sample 1 | 1558 | 8310 | 31.0 | 16.4 |

Rock Sample 2 | 1571 | 6020 | 32.0 | 13.7 |

Rock Sample 3 | 1578 | 2840 | 29.4 | 9.8 |

Rock Sample 4 | 1584 | 4830 | 29.5 | 12.8 |

Average | 1573 | 5500 | 30.5 | 13.4 |

ln ( K r o K r w ) = − 21.35 s w + 11.94 (13)

ln ( K r o K r w ) = − 22.82 s w + 12.84 (14)

ln ( K r o K r w ) = − 32.89 s w + 19.55 (15)

ln ( K r o K r w ) = − 21.42 s w + 12.74 (16)

The slope, intercept and index statistics of each line are as follows.

It can be seen from

By analogy with the curve of oil field with slightly larger viscosity, the curve can be classified qualitatively according to the rising law of water content. The comparison of each oil displacement efficiency and water cut curve of LD oilfield is shown in

According to the influence of viscosity on the rising rate of water, it can be qualitatively judged from

Water Saturation | Characteristic Value | Rock Sample 1 | Rock Sample 2 | Rock Sample 3 | Rock Sample 4 |
---|---|---|---|---|---|

Slope | −21.35 | −22.82 | −32.89 | −21.42 | |

Intercept | 11.91 | 12.84 | 19.55 | 12.74 | |

0.4 | Index | 3.37 | 3.71 | 6.39 | 4.17 |

0.5 | Index | 1.24 | 1.43 | 3.11 | 2.03 |

0.6 | Index | −0.90 | −0.85 | −0.18 | −0.11 |

0.7 | Index | −3.04 | −3.13 | −3.47 | −2.25 |

As can be seen from

water flooded layer. With the development of oil field into the middle and late period, it is more and more difficult to comprehensive adjustment. This requires more and more fine research by reservoir engineers. By studying the classification of relative permeability curve, the accuracy of application of the curve can be guaranteed.

The relative permeability curves before and after optimization were put into the CMG numerical simulation software to calculate the water cut of a well. The comparison results are shown in

Using the fractional flow Formula (17), combined with the water injection development process, the average water saturation of oil layer can be expressed by the Formula (18). According to the two standardization results of LD oilfield, the theoretical curve of water content and water cut rising rate can be calculated by using Formula (17) and Formula (19), the comparison with the actual production data is shown in

f w = 1 1 + μ w ⋅ K r o μ o ⋅ K r w (17)

S w = N P N ( 1 − S w i ) + S w i (18)

where N_{P} is cumulative oil production, the unit is m^{3}. N_{ }is petroleum geological reserves, the unit is m^{3}.

f ′ w = f w 2 − f w 1 R f 2 − R f 1 ⋅ 1 − S w i 1 − S w i − S o r (19)

where f ′ w is Water content derivative, R_{f} is recovery percent, S_{or} is residual oil saturation.

As can be seen from

1) The mathematical model of oil displacement efficiency and water cut has been deduced by combining the equation of the fractional flow equation, oil displacement efficiency and relative permeability curve. The correctness of the equation is verified by the data of relative permeability curve in SZ oilfield.

2) Two methods of classification and evaluation of relative permeability curves are proposed: direct evaluation and analogy method. The direct evaluation method can classify the curve quantitatively according to the characteristic value, and the analogy method can classify the curves qualitatively according to the law of the effect of viscosity on water cut rise. In the practical application process, it is necessary to combine the two methods to identify the abnormal curve, so as to obtain the representative relative permeability curve of the oilfield. The classification and optimization can ensure the accuracy of the application of the relative permeability curve, especially for the interpretation of the water flooded layer.

3) The classification method of relative permeability curve was applied to LD oilfield, and a representative relative permeability curve was obtained. By using the direct evaluation method, the multiple relative permeability curves were divided into two categories, and then the curve which is more in line with the law of water rising was chosen by analogy. It is proved by practical application that the optimized relative permeability curve is more in line with the actual oil field and more representative.

The authors declare no conflicts of interest regarding the publication of this paper.

Gong, P.Z., Liu, B., Zhang, J.T., Lv, Z. and Zhang, G.H. (2018) Classification Study on Relative Permeability Curves. World Journal of Engineering and Technology, 6, 723-737. https://doi.org/10.4236/wjet.2018.64047