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Since the dual laterolog response is affected by various environmental factors, the instrument response is difficult to reflect the true resistivity of the formation. Plate correction is a common method of resistivity logging environmental correction. Three-dimensional layered media model was established, and the dual laterolog response in directional wells was simulated by three-dimensional finite element method. The influence of environmental factors such as wellbore, well deviation, surrounding rock and mud intrusion on the response of dual laterolog is analyzed emphatically. Therefore, the environmental factors correction plate with different well deviations is constructed, and the plates can be used to guide the dual laterolog environment correction work for better reservoir identification and oil and gas evaluation.

Dual laterolog is one of the important logging methods to obtain formation resistivity information, widely used in oil and gas field exploration and development [

In this paper, the response of dual laterolog is simulated by finite element method. The finite element method (FEM) is a widely used numerical simulation method, and its basic steps are shown in

To determine the response of the dual laterolog is to ask for a continuous and smooth potential function [

∇ ⋅ ( 1 R ∇ μ ) = 0 (1)

In the Formula (1), R is the regional dielectric resistivity, Ω・m; and µ is the potential function, V.

The three-dimensional finite element method can be used to calculate the response of the dual laterolog, and the problem can be reduced to the extreme value problem of the general function ϕ [

ϕ = 1 2 ∫ ∫ ∫ Ω ( ( ∂ μ ∂ x ) 2 + ( ∂ μ ∂ y ) 2 + ( ∂ μ ∂ z ) 2 ) d x d y d z − ∑ E I E μ E (2)

In the Formula (2), I_{E} is the current sent by the electrode, μ_{E} is the potential on the electrode, the integral interval is the space surrounded by the instrument surface and the infinite boundary, E is the number of electrodes, and the sum is for all the electrodes.

The boundary conditions based on:

1) The first type of boundary conditions: μ is known constant in the constant voltage electrode; μ is unknown constant in the constant current electrode.

2) The second type of boundary conditions: the constant current electrode surface to meet:

∫ σ m ∂ u ∂ x ⋅ n d τ = I (3)

In the Formula (3), τ represents the surface of the entire electrode, the vertical electrode is normal, and I represents the current emitted by the electrode, σ_{m} indicates the slurry conductivity.

3) On the insulated boundary:

∂ μ ∂ n = 0 (4)

In the actual simulation process, the electric field of the two detection modes of the dual laterolog can be formed by three sub-field superposition [

The three subfields are only A_{0} emission potentials, only A_{1}, A ′ 1 transmit unit current, only A_{2}, A ′ 2 emit unit current, the formation of the electric field.

μ z = μ 0 + C 1 μ 1 + C 2 μ 2 (5)

Among them, μ z is the total electric field in each detection mode, V; μ 0 , μ 1 , μ 2 is the corresponding electric field, V; C_{1}, C_{2} is the weighting coefficient of each sub-field. Taking the shallow lateral logging response as an example, the potential and current conditions according to the shallow lateral logging mode are:

μ M 1 = μ M 2 (6)

1 + I 1 + I 2 = 0 (7)

In the Formula (6), μ M 1 , μ M 2 _{ }are the potentials on the supervisory electrodes M_{1}, M_{2}, V; I_{1} and I_{2} are the currents flowing out of the electrodes A 1 ( A ′ ) and A 2 ( A ′ ) respectively.

μ 0 ( M 1 ) + C 1 μ 1 ( M 1 ) + C 2 μ 2 ( M 1 ) = μ 0 ( M 2 ) + C 1 μ 1 ( M 2 ) + C 2 μ 2 ( M 2 ) (8)

1 + C 1 + C 2 = 0 (9)

According to (8)-(9), C_{1} and C_{2} can be obtained and then the shallow lateral logging response value can be acquired corresponding to the total electric field distribution.

R S = K S ⋅ μ S ( M 1 ) I 0 (10)

The apparent resistivity values in the shallow lateral logging mode can be obtained using the Equation (10).

The forward model uses the model shown in _{t} and the mud resistivity (R_{m}) is 1 Ω・m. The abscissa of the wellbore correction plate is the ratio of the resistivity of the target layer to the mud resistivity. The vertical axis of the well correction plate is the borehole correction coefficient Rllc/Ra, and the caliper size is the modulus. The caliper size is incremented from 6 in to 22 in, with an increase of 2 in each time.

For the deep lateral logging, it can be shown from _{t}/R_{m}) is less than 20, the wellbore correction coefficient of the deep lateral log is greater than 1, gradually increasing with the R_{t}/R_{m} increasing. When the ratio of the target layer resistivity to the mud resistivity (R_{t}/R_{m}) is greater than 20, the wellbore correction coefficient of the deep lateral logging is greater than 1, but it decreases with the increase of R_{t}/R_{m}. It can be shown from _{t}/R_{m}; when the well diameter is less than 12 in, the correction coefficient decreases with the increase of R_{t}/R_{m} and the correction coefficient is less than 1. When the well diameter is more than 12 in, the correction coefficient increases with the increase of R_{t}/R_{m}, indicating the wellbore shunt effect obvious and the shallow lateral logging greatly influenced by the borehole.

The stratum is modeled with a three-dimensional stratigraphic model with a focal point of −0.5 and 0.5, the thickness of target layer (H) is 1 m, the resistivity of target layer (R_{a}) is 20 ohms, the thickness of an upper and lower surrounding rock is infinite and resistivity is 5 ohms, the instrument is centered, and the mud resistivity (R_{m}) is 1 ohm. The results of the calculation are shown in

It can be shown from

For the correction of the thickness and the surrounding rock, it is necessary to make a map of the surrounding rock/layer thickness with different inclination. The model of inclined well is established, and the influence of mud intrusion is not taken into account. A three-dimensional finite element numerical simulation method was used to calculate the multi sets of different types of surrounding rock and layer thickness forward model. Figures 8-11 are surrounding rock/layer thickness correction plates when the inclination angle is 0˚, 30˚, 60˚ and 90˚ respectively. The deep laterolog correction coefficient (R_{lldc}/R_{t}) and the shallow laterolog correction coefficient (R_{llsc}/R_{t}) are the ordinate, the formation thickness is the abscissa, and the ratio of the true resistivity and the resistivity of the surrounding rock is modulo.

It can be shown from Figures 8-11 that the difference of the surrounding rock/layer thickness correction plate at different inclination angles is large. The deep laterolog resistivity correction coefficient varies with the increase of the thickness of the target layer (H), and its variation depends on the ratio of the resistivity of the surrounding rock to the resistivity of the target layer. When the resistivity of the surrounding rock is greater than the target layer resistivity, the correction coefficient is greater than 1, indicating that the corrected resistivity is greater than the original apparent resistivity; When the resistivity of the surrounding rock is less than the resistivity of the target layer, the correction coefficient is less than 1, indicating that corrected resistivity is lower than the original apparent resistivity. With the thickness of the target layer increasing, the deep laterolog correction coefficient curve gradually reduced, when the thickness of the layer up to 20 m, the correction coefficient is close to 1. Layer thickness correction is not necessary at this time. When the well angle is small, such as less than 30˚, the correction coefficient does not change much, and the correction coefficient of different layer thickness is not large; with the increase of the inclination of the well, the correction coefficient gradually increases. When the angle is 60º, the correction factor changes greatly. This shows that the greater the deviation, the bigger influence on resistivity, the larger corresponding correction coefficient. In short, the greater the inclination and layer thickness, the greater the correction coefficient.

The data obtained from the forward modeling numerical simulation is carried out with wellbore, well deviation, surrounding rock/thickness correction, which can be considered without the influence of borehole, well deviation, surrounding rock and eccentricity.

In this paper, the response of dual laterolog is studied by finite element numerical simulation technique. The influence of borehole, well deviation, surrounding rock and mud intrusion on logging results is studied by finite element method. The following conclusions can be drawn from the calculation and analysis.

1) The borehole size has a large effect on the dual laterolog, in which the shallow lateral logging response is particularly affected by the borehole size.

2) When the inclination angle is less than 15˚, the deviation of the well deviation has little effect on the dual laterolog. When the angle of the well is greater than 30˚, the deviation of the well deviation has a great influence on the dual laterolog. With the increase of the inclination angle, the stratification ability of the dual laterolog will be weakened, and the stratified interface cannot be effectively identified.

3) The greater the difference between the resistivity of surrounding rock and the resistivity of the target layer, the greater the effect on the apparent resistivity. When the depth of mud invasion is less than 0.75 m, the dual laterolog is sensitive to the change of mud invasion depth.

Ni, X.-W., Xu, S.-H., Xu, G.-Y., Ao, X.-F., Feng, J.M. and Liu, D.-R. (2017) Numerical Simulation of Dual Laterolog Response in Directional Wells and Development of Correction Plate. Open Journal of Yangtze Gas and Oil, 2, 237-248. https://doi.org/10.4236/ojogas.2017.24019