A numerical solution to the well resistivity-sounding problem in the axisymmetric case

Figures & data

Figure 1. The noise-free and noisy potential functions with different m, δ = 0.03 and z_max = 5.

Figure 2. Influence of grid point numbers on the solution to the inverse problem.

Figure 3. Data interpretation for the monotone conductivity function with a contrast ratio equal to 10.

Table 1. Relative errors δ in C-norm and corresponding iterations number.

Case	Noise- free data	Noisy data γ = 1.25%	Noisy data γ = 2.5%	Noisy data γ = 3.7%	Noisy data γ = 5%
1. σ(r) is increasing (Figure 3, right);	0.11;	0.36;	0.43;
A1 = 10, A2 = 0, c1 = 2, ϑ = 1	9602	10621	13,517
2. σ(r) is decreasing (Figure 3, left);	0.04;	0.10;	0.19;	0.22;	0.63;
A1 = 10, A2 = 0, c1 = 0, ϑ = 1	2581	4041	1006	222	217
3. σ(r) has maximum (Figure 4, left);	0.04;	0.15;	0.18;	0.9;	–
A1 = 1, A2 = 0, c1 = 0.4, ϑ = 0.5	10001	12560	4000	8005
4. σ(r) has minimum (Figure 4, right);	0.28;	0.64;	0.14;	0.36;	0.18;
A1 = −0.8, A2 = 0, c1 = 0.4, ϑ = 0.5	13501	26000	37,000	3601	22503
σ(r) has minimum; 1 ≤ r ≤ 1.5	0.07	0.08	0.07	0.07	0.11
A1 = −0.8, A2 = 0, c1 = 0.4, ϑ = 0.5
Note: The last line corresponds to the same case as line 4, but relative error is calculated for 1 ≤ r ≤ 1.5 only.

Figure 4. Data interpretation for a conductivity function maximum and minimum near the well zone, at the distance equal to 0.4.

Figure 5. Data interpreation for a conductivity function having distant maximum and for a wavy conductivity distrbution with different noise level.

Figure 6. The comparison between solutions on uniform and logarithmic grids.