Data processing method of noise logging based on cubic spline interpolation

Noise data is a two-dimensional data with 128 channels of spectral channels, measured by Sondex's NTO009 noise logging tool. The instrument can efficiently detect the fluid inside and outside the cased hole, detect the sound of gas or liquid flowing in the channel or perforation and transmit the detection data to the instrument in real time. The instrument makes use of a very sensitive acoustic sensor, which can measure a wide frequency range in order to effectively detect leaks of various gases, water or oil.

For a set of discrete data nodes (x_i,y_i), i = 1,2,···N, y_i = y(x_i), if the function S (x) meets the following three conditions: (1)

For every x_i, there is S (x_i) = y_i;

For the interval endpoints, there are three types of boundary conditions as follows: (1)

S (x₁) = y₁, S (x_N) = y_N;

The depth value is defined as y, which forms a node (x_i,y_i) with the noise measurement value x. For the boundary conditions, since the depth keeps increasing, only the first or second type of boundary condition can be selected. This article selects the first type of boundary condition and combines the three conditions in Section 2.1. Assuming M_i = s^″(x_i),i = 0,1,2,···, N, the following takes place: s″(x)=x-xi-hiMi-1+x-xi-1hiMis(x)=Mi-16hi(xi-x)3+Mi6hi(x-xi-1)3+c1x+c2 \matrix{{s{''}(x) = {{x - {x_i}} \over {- {h_i}}}{M_{i - 1}} + {{x - {x_{i - 1}}} \over {{h_i}}}{M_i}} \cr {s(x) = {{{M_{i - 1}}} \over {6{h_i}}}{{({x_i} - x)}^3} + {{{M_i}} \over {6{h_i}}}{{(x - {x_{i - 1}})}^3} + {c_1}x + {c_2}} \cr} where c₁,c₂ can be calculated by the condition s(x_i) = y_i,s(x_i−1) = y_i−1, and hence: s(x)=Mi-16hi(xi-x)3+Mi6hi(x-xj-1)3+(yi-1hi-Mi-16hi)(xi-x)+(yihi-Mi6)(x-xi-1)xi-1≤x≤xi,i=1,2,⋯,N \matrix{{s(x) = {{{M_{i - 1}}} \over {6{h_i}}}{{({x_i} - x)}^3} + {{{M_i}} \over {6{h_i}}}{{(x - {x_{j - 1}})}^3} + ({{{y_{i - 1}}} \over {{h_i}}} - {{{M_{i - 1}}} \over 6}{h_i})({x_i} - x) + ({{{y_i}} \over {{h_i}}} - {{{M_i}} \over 6})(x - {x_{i - 1}})} \cr {{x_{i - 1}} \le x \le {x_i},i = 1,2, \cdots ,N} \cr}

Since S^′ (x) is continuous, M_i can be calculated as follows: γiMi-1+2Mi+αiMi+1=βi,i=1,2,⋯,N-1 {\gamma _i}{M_{i - 1}} + 2{M_i} + {\alpha _i}{M_{i + 1}} = {\beta _i},i = 1,2, \cdots ,N - 1 where αi=hi+1hi+hi+1βi=6hi+hi+1(yi+1-yihi+1-yi-yi-1hi)γi=1-αi {\alpha _i} = {{{h_{i + 1}}} \over {{h_i} + {h_{i + 1}}}}{\beta _i} = {6 \over {{h_i} + {h_{i + 1}}}}({{{y_{i + 1}} - {y_i}} \over {{h_{i + 1}}}} - {{{y_i} - {y_{i - 1}}} \over {{h_i}}}){\gamma _i} = 1 - {\alpha _i} , and with the help of the first type of boundary condition, leads to the following: [2α0γ12α1γ2⋱α3γ3⋱⋱⋱⋱αN-2γN-12αN-1γN2] [M0M1⋮⋮⋮MN-1MN]=[β0β1⋮⋮⋮βN-1βN] \left[ {\matrix{2 & {{\alpha _0}} & {} & {} & {} & {} & {} \cr {{\gamma _1}} & 2 & {{\alpha _1}} & {} & {} & {} & {} \cr {} & {{\gamma _2}} & \ddots & {{\alpha _3}} & {} & {} & {} \cr {} & {} & {{\gamma _3}} & \ddots & \ddots & {} & {} \cr {} & {} & {} & \ddots & \ddots & {{\alpha _{N - 2}}} & {} \cr {} & {} & {} & {} & {{\gamma _{N - 1}}} & 2 & {{\alpha _{N - 1}}} \cr {} & {} & {} & {} & {} & {{\gamma _N}} & 2 \cr}} \right]\;\left[ {\matrix{{{M_0}} \cr {{M_1}} \cr \vdots \cr \vdots \cr \vdots \cr {{M_{N - 1}}} \cr {{M_N}} \cr}} \right] = \left[ {\matrix{{{\beta _0}} \cr {{\beta _1}} \cr \vdots \cr \vdots \cr \vdots \cr {{\beta _{N - 1}}} \cr {{\beta _N}} \cr}} \right]

Since the coefficient matrix is strictly dominant on the main diagonal, this matrix equation has a unique solution. There are many ways to solve this tridiagonal matrix equation. This article will use the catch-up method to solve it. The algorithm steps are as follows: 1

Let A=[2α0γ12α1γ2⋱α3γ3⋱⋱⋱⋱αN-2γN-12αN-1γN2] X=[M0M1⋮⋮⋮MN-1MN], B=[β0β1⋮⋮⋮βN-1βN] A = \left[ {\matrix{2 & {{\alpha _0}} & {} & {} & {} & {} & {} \cr {{\gamma _1}} & 2 & {{\alpha _1}} & {} & {} & {} & {} \cr {} & {{\gamma _2}} & \ddots & {{\alpha _3}} & {} & {} & {} \cr {} & {} & {{\gamma _3}} & \ddots & \ddots & {} & {} \cr {} & {} & {} & \ddots & \ddots & {{\alpha _{N - 2}}} & {} \cr {} & {} & {} & {} & {{\gamma _{N - 1}}} & 2 & {{\alpha _{N - 1}}} \cr {} & {} & {} & {} & {} & {{\gamma _N}} & 2 \cr}} \right]\;\,X = \left[ {\matrix{{{M_0}} \cr {{M_1}} \cr \vdots \cr \vdots \cr \vdots \cr {{M_{N - 1}}} \cr {{M_N}} \cr}} \right],\,B = \left[ {\matrix{{{\beta _0}} \cr {{\beta _1}} \cr \vdots \cr \vdots \cr \vdots \cr {{\beta _{N - 1}}} \cr {{\beta _N}} \cr}} \right] , then.

Triangulate A to get, then the original system of equations is equivalent to the following equation: LY=B,UX=Y LY = B,UX = Y

(1) Mean sampling of noise data at single depth point

The data of well D1 is selected for processing. The measured interval of this well is 5,690.0–5,790.0 m, and the noise data type is point data. There are 29 depth points in total, and each depth point corresponds to a noise data file that changes with time. Therefore, before interpolation, it is necessary to sample the noise data at these 29 depths, and use the sampled noise data to replace the noise characteristics of this depth point.

This article chooses mean sampling, that is, summing the noise data on each channel at each depth point and then taking the average, and finally a data with 1 row and 128 columns can be obtained. Figure 1 shows the curve formed by the noise data before sampling. It can be observed from the image that the curve distribution is relatively clear. Figure 2 is obtained after mean sampling. Comparing the two figures, it can be seen that the curve after sampling can reflect the noise characteristics of the current depth point.

Fig. 1

Fig. 2

(2) Interpolation processing of noise data at full depth

After step (1), the noise data of 29 depth points is obtained. Because it is not continuous in depth, it is impossible to form a continuous noise image. Therefore, the noise data needs to be interpolated. Before interpolation processing, first sample the noise data of each depth point, get the noise data y of each depth point after the sampling is completed, and then use each sampled noise data and depth point data as the node(x_i,y_i); the depth interpolation step is 0.125 m. Enter the data into the cubic spline interpolation algorithm program for calculation, and the output value is the result of each interpolation. Then five-point average filtering is performed on it, and finally 128 pairs of node data are obtained. As shown in Figure 3, it is the flow chart of the cubic spline interpolation processing program.

Fig. 3

The interpolated noise data is imported into MATLAB. Figure 4 shows the noise data of a certain track: the blue circles in the figure are the original noise data, and the red curve is the interpolation result on the depth point (abscissa). It can be seen from the figure that the interpolation curve is smooth everywhere. Because the cubic spline interpolation function has continuous first and second derivatives, there are no discontinuities in the processed noise data, and it has good mathematical properties. In order to detect the accuracy of the interpolation data, the noise data is imaged.

Fig. 4

The imaging effect of the noise data depends on the appropriate colour code system, that is, the noise data and the colour code are on one-to-one correspondence. This article takes the maximum and minimum values of the noise data as the endpoints of the colour code interval, and linearly maps the noise data to the RGB colour code space, and finally get the noise imaging picture. Figure 5 is a comparison imaging diagram of linear interpolation and cubic spline interpolation. It can be seen that cubic spline interpolation improves the resolution of noise data while maintaining the linear value structure, and has a better display effect in the local areas.

Fig. 5

In order to test whether the cubic spline interpolation is feasible in the logging interpretation process, this paper selects Well D1, combined with the production profile data, and performs imaging logging interpretation on the noise data.

Well D1 is located in the DH area. After consulting the production profile data, the main production zone of this well is from below 5,785.0 m. The water holdup count and fluid density curve of this well both show the flow pattern of oil and water in the wellbore: the oil production is 1.71 m³/d and water production is 20.8 m³/d. The secondary production zone is at 5,755.0–5,777.0 m. The logging curve of this section shows the gradient change in the well temperature curve; the water holdup curve and the fluid density curve also change significantly, and the calculation results of the turbine flow combined with noise show that the section produces oil and water, where the oil production is 8.26 m³/d and water production is 12.49 m³/d.

For the noise data, through research, it is believed that 1–200 Hz corresponds to the fluid flow characteristics of the wellbore area, 300–600 Hz to the oil jacket annulus area (packer sealing conditions can be judged), 600–2,000 Hz to the perforation and transition section, 3,000–8,000 Hz to the formation fluid and 10,000–12,000 to the dense pore fluid. Hence, noise logging interpretation can be analysed from these five frequency intervals.

The noise data is two-dimensional data divided into 128 channels, where the column label is 1–128, its meaning is frequency, and the magnification is 100. The noise data body is the counting rate at the corresponding frequency, which shows the distribution of the counting rate (the distribution of the noise signals).

Figure 6 is the noise production profile of Well DH1, in which the frequency channel is the waveform of cubic spline interpolation data, the amplitude channel is the cumulative count rate curve corresponding to the five frequency intervals and the last channel is the liquid production. Two obvious production layers can be observed intuitively from the figure, which correspond to the main production layer below 5,785.0 m and the secondary production layer in 5,755.0–5,777.0 m, respectively. Besides, there are also a few middle- and high-frequency noise signals near the 5,882 m. Therefore, the noise data processed by cubic spline interpolation not only completely represent the original output layer noise signal, but also monitor the fluid flow in other parts. It can be seen from the imaging channel in Figure 6 that the colour of the rear part of the noise image at 5,882 m is lighter, which is the mid- and high-frequency noise signal. This is because the depth here is just in the transition region between the main production layer and the measured production layer.

Fig. 6