PyBorehole: borehole backfill effect on resistivity and induction logs

PyBorehole simulates what a borehole resistivity tool (ERT, PyGIMLi) and an electromagnetic induction tool (FDEM, SimPEG) measure in a backfilled borehole through a layered earth. Both tools are simulated in 3D, with and without the borehole, so you can see how much the backfill distorts each log.

You provide:

• the layered earth model (depths and resistivities),
• the borehole diameter and length, and the backfill resistivity (or several concentric zones such as casing + gravel pack),
• the electrode spacings (ERT) and the coil spacings and frequencies (EM).

You get apparent-resistivity logs for every tool, the no-borehole reference logs, and the borehole effect: the ratio of the two.

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Start here: run_combined.py

run_combined.py is the main file. It runs the complete study: ERT and EM, with and without the borehole, for a sweep of backfill resistivities, and saves all the figures. Usually you only edit its SETTINGS block and run it.

1. Install

PyGIMLi needs the tetgen mesh generator, which is easiest to get from conda:

conda create -n pyborehole -c gimli -c conda-forge python=3.10 pygimli tetgen
conda activate pyborehole
pip install -r requirements.txt

Check that tetgen is found (tetgen -h). If it isn't, ERT meshing fails with No such file or directory: '...-1.node'.

2. Edit the settings

Open run_combined.py and edit the block at the top:

LAYERS = np.array([          # [top_depth, bottom_depth, resistivity]; last layer extends down
    [0.0,   5.0,  100.0],
    [5.0,  10.0,   10.0],
    [10.0, 15.0,  200.0],
])

BOREHOLE_LENGTH = 15.0
BOREHOLE_DIAM   = 0.20
BACKFILL_RES    = 10.0                       # always included in the sweep
BACKFILL_SWEEP  = [1.0, 10.0, 100.0, 1000.0]  # one combined_log_backfill_<rho>.png each

ERT_SHORT_SPACING = 0.5
ERT_LONG_SPACING  = 1.0
ERT_MEAS_SPACING  = 0.1

EM_COIL_SPACINGS  = [0.5, 1.0]
EM_FREQUENCIES    = [400.0, 8000.0]
EM_MEAS_SPACING   = 0.1
EM_MESH_TYPE      = "cylindrical"

OUTPUT_DIR        = "results"               # all figures are written here

Setting	Meaning
LAYERS	One row per layer: top depth (m), bottom depth (m), resistivity (Ohm.m). Depths are positive downward from the surface. The first layer must start at 0; the last layer continues to the bottom of the model.
BOREHOLE_LENGTH, BOREHOLE_DIAM	Borehole depth and diameter (m). The borehole is vertical and starts at the surface.
BACKFILL_SWEEP	Backfill resistivities (Ohm.m) to simulate. Each value costs one ERT solve (about 1–2.5 min).
BACKFILL_RES	A backfill value that is always simulated, even if it isn't in the sweep.
ERT_SHORT_SPACING, ERT_LONG_SPACING	Wenner electrode spacings a (m). The array is A–M–N–B with spacing a between neighbours, so it is 3a long.
ERT_MEAS_SPACING	Electrode step along the hole (m). Spacings must be multiples of it.
EM_COIL_SPACINGS	Transmitter–receiver distances (m) of the coaxial induction coils.
EM_FREQUENCIES	EM frequencies (Hz).
EM_MEAS_SPACING	Step of the EM tool along the hole (m).
EM_MESH_TYPE	"cylindrical" (axisymmetric, a full log in seconds; use this for a centred tool) or "tree" (full 3D octree, ~10 s per tool position and frequency; for an off-centre tool, see the user guide).
OUTPUT_DIR	Folder for the figures.

3. Run

conda activate pyborehole
python run_combined.py

With the default settings this takes about 8–12 minutes and needs about 12 GB of RAM, almost all of it for the 3D ERT solves. The EM simulations take seconds.

What the script does:

1. ERT: builds one 3D mesh (borehole, layers and electrodes are built into it), then solves for the no-borehole reference and for every backfill value in the sweep, reusing the mesh.
2. EM: solves the same model on an axisymmetric cylindrical mesh, with and without the borehole, for every backfill value.
3. Plots: one log figure per backfill value, plus one summary figure of the borehole effect.
4. Table: prints the borehole effect at the middle of every layer.

4. Read the results

All figures go to results/:

File	What it shows
combined_log_backfill_<rho>.png	One per backfill value. Left: the model. Right: apparent resistivity versus depth for every tool. Solid = with the backfilled hole, dotted = the same tool without a hole, dashed black = true formation resistivity.
backfill_effect.png	One panel per tool, one curve per backfill value: ρa(with hole) / ρa(no hole). 1 = no effect; below 1 the hole makes the tool read too low, above 1 too high.

Depth on all plots is the centre of the tool: the midpoint of the Wenner array, or the midpoint between the transmitter and receiver coils.

The printed table looks like this (default model, D = 0.2 m):

 layer_rho  depth  backfill  ERT a=0.5  ERT a=1  EM s=0.5  EM s=1
     100.0    2.5       1.0      0.512    1.199     0.210   0.549
     100.0    2.5     100.0      1.000    1.000     1.000   1.000
     200.0   12.5       1.0      0.315    0.887     0.135   0.445
     ...

What these default results show:

• A conductive backfill (1 Ohm.m) has a large effect on the short induction coils: s = 0.5 m reads 5–7 times too low. Longer coils are much less affected.
• The ERT Wenner array reads too low at short spacing and can overshoot the true value at longer spacing (ratio > 1). This is real Wenner behaviour in a conductive hole and matches the analytic solution.
• A resistive backfill (1000 Ohm.m) changes ERT by 4–8% and EM by only 1–5%.
• When the backfill equals the formation, every ratio is exactly 1, as it should be.

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Realistic tools: run_realistic.py

run_combined.py uses idealised tools: point electrodes on the borehole axis in a Wenner array, and a plain two-coil EM tool. run_realistic.py repeats the backfill study with tools that behave like real logging sondes:

	run_combined.py	run_realistic.py
ERT electrodes	points on the axis	rings on an insulating tool body (default 50 mm)
ERT arrays	Wenner (A, M, N, B all downhole)	normal (A–M on the sonde, B and N remote; 16" and 64") and lateral (A, M, N on the sonde, B remote)
ERT geometric factor	analytic half-space formula	calibrated like a real tool: the sonde reads the true resistivity in a homogeneous earth
EM tool	two-coil pair	two-coil pair and a focused tool with a bucking receiver

Settings (top of the script):

Setting	Meaning
TOOL_DIAM	Diameter of the insulating ERT tool body (m); None = no tool body.
NORMAL_SPACINGS	AM distances of the normal arrays (m). Default 16" (0.41 m) and 64" (1.63 m).
LATERAL_SPACINGS, LATERAL_MN	AO distance(s) (A to the midpoint of MN) and MN distance of the lateral (m). The defaults are an example; the classic oilfield lateral is AO = 5.69 m (18'8"), MN = 0.81 m.
ERT_MEAS_SPACING	Tool step along the hole (m).
EM_COIL_SPACINGS, EM_BUCKING	Main transmitter–receiver distance per EM tool, and the bucking receiver distance for each (None = plain two-coil).
LAYERS, BOREHOLE_*, BACKFILL_SWEEP, EM_FREQUENCIES, OUTPUT_DIR	As in run_combined.py.
REUSE_SAVED_RESULTS	True redraws all figures from results/realistic_logs.csv without simulating again (seconds instead of ~30 minutes).

With the defaults it takes about 30 minutes and up to ~17 GB of RAM: ring electrodes on a tool body need a finer ERT mesh (~1.2M cells). Setting ERT_MEAS_SPACING = 0.2 roughly halves both.

Outputs in results/:

• realistic_log_backfill_<rho>.png: the model, a to-scale close-up of the tools in the hole at mid-depth (ERT sonde with its tool body and ring electrodes A, M, N; EM tool with transmitter, bucking and main receiver; the window is shaded on the model), and all tool logs for that backfill (dotted = no borehole).
• realistic_backfill_effect.png: with-hole / no-hole ratio per tool.
• realistic_logs.csv: every simulated value (tool, depth, apparent resistivity, backfill; empty backfill = no-borehole reference), plus a printed summary table.

Replace the example spacings and tool diameter with those of your sondes. The physics of these tools is described in docs/user_guide.md.

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Using the classes directly

run_combined.py is built on two classes you can also use on their own:

import numpy as np
from geometry_class import Geometry, EMGeometry

layers = np.array([[0, 5, 100], [5, 10, 10], [10, 15, 200]], dtype=float)

# ERT
geo  = Geometry(borehole_length=15, borehole_diameter=0.2, layer_1d_geometry=layers,
                backfill_resistivity=10.0, long_spacing=1.0)
plc  = geo.make_basic_geometry()
data = geo.make_array(plc, abmn_order=(1, 4, 2, 3), short_spacing=0.5,
                      long_spacing=1.0, measuring_spacing=0.1)
mesh = geo.make_mesh(plc)
with_hole = geo.run(mesh, data)                        # pygimli DataContainer, 'rhoa' column
no_hole   = geo.run(mesh, data, include_borehole=False)
geo.set_borehole(backfill_resistivity=1000.0)          # new backfill, same mesh

# EM
em = EMGeometry(borehole_length=15, borehole_diameter=0.2, layer_1d_geometry=layers,
                backfill_resistivity=10.0, frequencies=[400, 8000], coil_spacings=[0.5, 1.0])
log    = em.run()                                      # pandas DataFrame
log_nh = em.run(include_borehole=False)

Instead of a single backfill you can give concentric zones (outer_radius, resistivity[, top, bottom]), for example a water-filled PVC casing inside a gravel pack:

zones = [(0.045, 20.0),            # r < 4.5 cm: water inside the casing
         (0.050, 1e6),             # 5 mm PVC wall (insulating)
         (0.100, 30.0, 2.0, 15.0)] # gravel pack, 30 Ohm.m, from 2 to 15 m depth
geo = Geometry(..., borehole_zones=zones)
em  = EMGeometry(..., borehole_zones=zones)

(A closed PVC wall blocks the ERT current, as in reality; galvanic tools only see the formation through open hole or a slotted screen.)

See docs/user_guide.md for all parameters, outputs, accuracy and run-time settings, the physics, and troubleshooting.

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Validation

The simulators are checked against exact solutions (pytest tests/; add -m "not slow" to skip the 3D ERT tests, which take about 5 minutes):

Check	Reference	Result
ERT Wenner in a 0.2 m borehole, backfill 1 and 1000 Ohm.m in 100 Ohm.m, a = 0.1–1 m	Fourier–Bessel solution for an infinite borehole	within 0.1%
ERT in a layered model (100/10/200 Ohm.m), with and without the hole	1.7× finer mesh	within 0.04%
ERT without borehole	homogeneous half-space	exact
EM coaxial coils in a full space (1–100 Ohm.m), 400 and 8000 Hz	analytic dipole field	quadrature and apparent resistivity within ~1.5%
EM coaxial coils in a 0.2 m borehole, backfill 1–1000 Ohm.m	Fourier–Bessel solution for an infinite borehole	quadrature within 0.1–1.7% (cylindrical), 0.1–1.5% (3D octree)
EM coplanar coils in a full space, 3D octree	analytic dipole field	within 0.7%
ERT normal (16", 64") and lateral on the axis, 1 Ohm.m backfill	Fourier–Bessel solution	within 0.3%
ERT ring electrodes on a 50 mm insulating tool, normal and lateral, backfill 1 and 1000 Ohm.m	Fourier–Bessel solution with a mandrel	within 0.3%
EM focused tool (bucking receiver), backfill 1–1000 Ohm.m	Fourier–Bessel solution	within 0.2–1.3%

Limitations

• The borehole is vertical and the layers are horizontal.
• The ERT tool is centred in the hole; its body is an insulating cylinder over the whole borehole length (the real sonde is a few metres long, with a thinner cable above). The EM tool can be off-centre (tool_offset, 3D octree mesh).
• Ring electrodes are modelled as the average over 4 points around the tool, which is exact for a centred tool (checked against the analytic solution); remote electrodes are ideal poles at infinity.
• Focused induction tools are represented by one bucking receiver per coil pair; commercial tools use their own (often proprietary) coil sets.
• The default EM solver is axisymmetric (exact for a centred coaxial tool in flat layers). The 3D octree solver handles off-centre tools and transverse coils but takes ~10 s per tool position and frequency, so it is meant for selected depths rather than full logs.
• EM real, amplitude and phase are not meaningful at these low induction numbers (the in-phase secondary field is ~1e-8 of the primary); use the quadrature (imag) or the apparent resistivity.

Repository layout

run_combined.py               main script: ERT + EM backfill study  <- start here
run_realistic.py              the same study with realistic tools (tool body, normal/lateral, focused EM)
geometry_class/
    geometry.py               ERT: 3D mesh, electrodes, PyGIMLi forward model
    em_geometry.py            EM: cylindrical / octree mesh, SimPEG forward model
    borehole_model.py         shared resistivity model (layers, borehole zones, air)
    analytic.py               exact reference solutions used by the tests
    plotting.py               drawing helpers (tools-in-the-hole close-up)
tests/test_validation.py      validation against the analytic solutions
docs/user_guide.md            full documentation
results/                      figures written by run_combined.py
demo.ipynb                    older ERT-only notebook demo