RFour Energy · Field Notes

Geomechanics from 1D to 4D

Q: What is a 1D mechanical earth model?

A 1D mechanical earth model is a depth profile of the geomechanical state along a single wellbore: overburden stress from integrated density, pore pressure, minimum and maximum horizontal stress, and rock elastic and strength properties, all as curves against depth. It is built mostly from wireline logs calibrated to direct measurements such as leakoff tests and image-log breakouts. Its main deliverable is the safe mud weight window, along with sanding onset and optimal well trajectory at that location.

Q: What is the difference between 3D and 4D geomechanics?

A 3D geomechanical model describes the stress state throughout a rock volume at one moment in time, usually the initial or pre-production state, including faults and the overburden and sideburden around the reservoir. A 4D model adds time by coupling that volume to reservoir flow simulation, so pore pressure changes from production or injection feed back into stress, strain, compaction and permeability. 3D answers where; 4D answers what happens next.

Q: Why does depletion change stress rather than just pressure?

Rock is poroelastic: the grain framework and the fluid share the load. When pore pressure falls, more of the total load transfers onto the grain framework, so effective stress rises and the reservoir compacts. Because the reservoir is laterally constrained, the horizontal total stress also falls by a fraction of the pressure drop, typically around half to two-thirds. That path of changing stress with depletion is what drives compaction, fault reactivation, permeability loss and casing damage.

Q: When is a 1D model enough, and when do you need 3D or 4D?

A 1D model is enough for drilling a well in a structurally simple setting where the question is the mud weight window at that location. Move to 3D when structure, faults or strong lateral property changes mean the stress at one well cannot be assumed at another, or when the decision is field-wide such as fault reactivation or well placement. Move to 4D only when the decision depends on how stress evolves with production: compaction and subsidence, permeability loss, caprock integrity in storage or injection, seismic timeshifts, or induced seismicity.

Every geomechanical study answers the same question — how close is this rock to failing, and what happens if we push it? — but the dimension you build in decides which version of that question you are allowed to ask.

RFour Energy · Geoscience & Rock Mechanics Series

Geomechanics has a ladder. A 1D mechanical earth model describes the stress state along one wellbore as curves against depth. A 2D model stretches that description across a section, so structure and lateral change enter the picture. A 3D model fills a volume, with faults as real surfaces and the overburden carrying load around the reservoir. A 4D model couples that volume to flow simulation and lets pore pressure change, so stress becomes a function of time. Each rung buys a class of question the rung below simply cannot answer — and each rung costs data, time and a great deal more that can go quietly wrong. The engineering skill is not building the biggest model. It is knowing which rung the decision actually requires.

What follows walks the ladder in order: what each dimension contains, what it demands as input, which decisions it supports, and — the part usually left out — where it stops being honest.

FIG. 01The ladder. Each rung keeps everything below it and adds one axis: 1D adds depth, 2D adds lateral position along a section, 3D adds the full volume with faults and surrounding rock, 4D adds time through coupling to flow simulation.

What Every Dimension Is Trying to Compute

Before the ladder, the physics. At any point in the subsurface the state of stress is a tensor — three principal magnitudes and their orientations. In most sedimentary basins one principal stress is close to vertical, which lets the problem be written with the vertical stress S_v, the minimum horizontal stress S_hmin, and the maximum horizontal stress S_Hmax. Their relative ranking defines the faulting regime — normal, strike-slip or reverse — a classification that goes back to Anderson and still organises the whole subject.^[1]

Rock does not respond to total stress. It responds to effective stress: the load carried by the grain framework once the pore fluid has taken its share, σ′ = S − αP_p, with α the Biot coefficient. This is the hinge of the entire discipline, and it comes from poroelasticity.^[5] It explains why drawing pressure down makes rock feel more stress without anyone adding load at the surface, and why the same rock can be perfectly stable one year and failing the next.

Pore pressure and stress are not two separate subjects. They are one subject, and effective stress is where they meet.

So every model, at every dimension, is chasing five things: the vertical stress, the pore pressure, the two horizontal stresses, and the rock's elastic and strength properties. What changes up the ladder is not the list — it is where those five are known, and whether they are allowed to change.

1D — The Mechanical Earth Model at a Well

The 1D MEM is the foundation, and the majority of practical geomechanics never leaves it. It is a set of curves against measured depth at one location, each built from logs and pinned to whatever direct measurements exist.

Overburden S_v — integrate bulk density from surface to depth. Cheap, but only as good as the shallow density, which is exactly where logs are usually missing and must be extrapolated.
Pore pressure P_p — from the departure of sonic or resistivity from a normal compaction trend (Eaton-style), or from an effective-stress/velocity relation that handles unloading (Bowers-style). Calibrated against direct measurements: RFT/MDT, mud weight history, kicks.
Minimum horizontal stress S_hmin — the one horizontal stress that can be measured, from leakoff and extended leakoff tests, minifracs and step-rate tests. Everything else is inference.
Maximum horizontal stress S_Hmax — never measured directly. Bounded by the frictional strength of the crust and constrained by wellbore failure: the width of breakouts and the presence of drilling-induced tensile fractures on image logs. Orientation comes from the same features.^[1]
Elastic and strength properties — dynamic moduli from compressional and shear slowness with density, converted to static moduli through a calibrated correction, then to unconfined compressive strength and friction angle through local correlations anchored on core.^[2]

FIG. 02The 1D deliverable. Pore pressure sets the lower bound (below it the well flows), the collapse limit sets the practical lower bound for hole stability, and the fracture gradient — anchored on S_hmin — sets the upper bound. The shaded gap is the safe mud weight window; where it pinches, you set casing. Circles mark the direct measurements the curves are pinned to.

What a 1D model earns you is concrete: the mud weight window and casing points, the safest trajectory through a given interval, an estimate of the drawdown at which sand production begins, and a first view of whether the interval will fracture where you want it to. It is also, by a wide margin, the most cost-effective geomechanics ever done.

Where it stops. A 1D model assumes the world is layer-cake around that well. It has no opinion about the fault two hundred metres away, no way to represent a dipping structure, and — critically — no time axis. It describes the rock as drilled. Ask it what the stress will be after five years of depletion and it has nothing to say.

2D — The Section

The second dimension is the one most often skipped, and it is worth defending. A 2D geomechanical model extends the profile along a plane — typically a structural cross-section through a seismic line, or a plane through a planned well path.

Two distinct things arrive with it. First, structure: dipping beds, a growing anticline, a fault plane with real geometry rather than a name in a report. Stress rotates and concentrates around structure, and a section shows it. Second, and independently, the wellbore becomes two-dimensional: the stress concentration around a hole of finite radius, described analytically since Kirsch, is what produces breakouts on one azimuth and tensile fractures ninety degrees away.^[1][2][10] That near-wellbore cross-section is the reason a deviated well in the same rock needs a different mud weight than a vertical one.

FIG. 03The stress polygon. Since S_Hmax is never measured directly, it is bracketed: the polygon's outer bounds come from the frictional strength of the crust (drawn for μ = 0.6, giving f = 3.12), its lower bound is the 1:1 diagonal where S_Hmax = S_hmin, the two dashed lines at S_v separate the three Andersonian regimes, and observations — a measured S_hmin, a breakout width, the presence or absence of tensile fractures — collapse the allowable region toward a single state.

In practice many teams treat 2D as a stepping stone rather than a destination, and that is fair. Its honest role is as a reasoning tool: cheap enough to run many scenarios, rich enough to show whether structure matters at all. If a 2D section shows stress barely rotating across the field, the case for 3D weakens considerably — which is a useful thing to learn before committing to a volume build.

3D — The Field-Wide Volume

A 3D geomechanical model is a finite-element volume covering the reservoir and the rock around it: overburden above, underburden below, sideburden around the flanks. That surrounding rock is not padding. It carries load, and leaving it out is the most common way a 3D model gives confident wrong answers.

Building one means populating elastic and strength properties through the volume — typically from seismic inversion tied to the 1D models at every available well — then meshing the structural framework with faults as discrete surfaces with their own friction, and finally equilibrating the model so it reproduces the measured in-situ stress before anything is changed. That initialisation step is where most of the effort goes, and where a model earns or loses its credibility: if it cannot reproduce the stresses you measured at the wells you have, its predictions elsewhere are decoration.

Dimension	Core question it answers	Typical inputs	Decision it supports
1D	How close to failure is the rock at this well, as drilled?	Logs, LOT/minifrac, image logs, core, pressure tests	Mud weight window, casing points, trajectory, sanding onset
2D	Does structure change that answer along a section?	1D models + a seismic section, structural interpretation	Whether 3D is justified; near-wellbore failure vs. trajectory
3D	What is the stress state everywhere, including on the faults?	Seismic volume & inversion, structural framework, all 1D MEMs, FE mesh	Well placement, drilling in undrilled areas, fault reactivation risk, caprock integrity
4D	How does that state evolve as pressure changes?	3D model + reservoir simulation model + production/injection history	Compaction & subsidence, permeability loss, casing integrity, induced seismicity, storage containment

The characteristic 3D deliverables are the ones that are meaningless at a point: a map of slip tendency across every fault surface, the stress rotation around a salt body or a steep flank, safe drilling windows in blocks where no well has yet been drilled, and the integrity of a seal considered as a surface rather than a depth.

4D — Stress Through Time

The fourth dimension is not a bigger grid. It is a coupling: the 3D geomechanical model exchanges information with a reservoir flow simulation, so that pore pressure changes computed by the flow model alter the stress and strain computed by the mechanical model, and — in the fuller schemes — the resulting deformation feeds back to change porosity and permeability.^[6]

The coupling is done at several levels of rigour, and the choice is an engineering decision rather than a matter of taste:

One-way — flow simulation runs, its pressure history is handed to the mechanical model once. Cheap, adequate when deformation does not materially change flow.
Iterative (two-way, staggered) — the two solvers exchange at each timestep until they agree. The workhorse for compacting reservoirs, where permeability really does fall as the rock compacts.
Fully coupled — flow and mechanics solved in one system of equations. Most rigorous, most expensive, and reserved for problems where the two are inseparable.

The stress path is the whole story

Deplete a laterally extensive reservoir and the vertical total stress barely moves — the weight of the overburden is still there. But pore pressure falls, so vertical effective stress climbs and the rock compacts. Because the reservoir cannot expand sideways, the horizontal total stress falls too, by a fraction of the pressure drop commonly in the region of one-half to two-thirds depending on rock stiffness and geometry. That ratio is the stress path, and it governs almost everything that follows.^[1][2]

FIG. 04Depletion moves the effective stress state along a path roughly 2–3 times steeper than 1:1, because Δσ′_v = αΔP_p while Δσ′_h = (α−γ)ΔP_p, with a stress-path coefficient γ of roughly 0.5–0.7. Being steeper than the failure envelope, the path carries the rock toward it. Compaction, permeability loss, fault reactivation, casing damage, a falling fracture gradient and time-lapse seismic timeshifts are not six separate problems. They are six views of this one trajectory.

Because these consequences share a mechanism, 4D models are also uniquely checkable. Compaction predicted at reservoir level implies subsidence at surface, which can be measured by levelling, GPS or satellite InSAR; the classical solution linking a compacting disc to the surface bowl dates to Geertsma.^[4] The same strain field perturbs seismic velocity and geometry, producing time-lapse timeshifts — most diagnostically in the overburden, which stretches as the reservoir compacts.^[7] Building and calibrating geomechanical models against 3D and 4D seismic is now a discipline of its own.^[3]

A 4D model that predicts compaction but cannot match the measured subsidence bowl has told you something — just not about the reservoir.

The application list has broadened well past compaction. Depletion-driven fault reactivation and injection-driven induced seismicity are the same poroelastic problem approached from opposite signs, a link established for extraction by Segall.^[8] Carbon storage and any long-term injection make caprock integrity a time-dependent question by definition. And in unconventional development, the stress change from producing one well is what makes the next well's fracture treatment behave differently — the parent-child problem, which is 4D geomechanics whether or not anyone calls it that.^[9]

Choosing the Rung

The temptation is always to build one dimension higher than the decision requires. Resist it for a specific reason: uncertainty does not fall as dimension rises — it usually grows. A 1D model at a well with a leakoff test and an image log is anchored on measurements. A 3D volume is anchored on those same few wells and then interpolated through hundreds of metres of rock nobody has sampled. A 4D model adds the uncertainty of the flow model on top. More dimensions mean more places for an unconstrained assumption to hide behind a well-rendered result.

Three practical tests:

Does the decision live at a point? Mud weight for one well, sanding drawdown for one completion — 1D, and be done.
Does the answer change across the field? Faults, steep structure, salt, strong lateral property contrast, or wells planned where none exist — then 3D, because a single well's stress is not transferable.
Does the answer change with time? Compaction, subsidence, permeability decline, fault reactivation under depletion or injection, caprock integrity for storage, seismic monitoring — only then 4D, and only with a flow model you already trust.

The last point deserves emphasis. A 4D geomechanical model inherits every weakness of the reservoir simulation it is coupled to. If the flow model's pressure history is not history-matched, coupling it to a mechanical model does not produce geomechanical insight — it produces geomechanically-styled restatements of a bad pressure forecast.

Validation

The framework used here is the standard one. The description of the stress tensor, the Andersonian classification, the constraint of S_Hmax through the stress polygon and wellbore failure, and the treatment of depletion-induced stress change follow Zoback's Reservoir Geomechanics, which organises the subject from the tectonic stress field through to production-induced faulting and subsidence.^[1] The rock-mechanics foundation — elasticity, failure criteria, acoustic properties and the link to log-derived and 4D-seismic observables — follows Fjær and co-authors.^[2] Effective stress and the poroelastic coupling that makes the fourth dimension possible rest on Biot's consolidation theory.^[5] The coupling schemes for reservoir-geomechanical simulation, and their differing costs, follow Settari and co-workers.^[6] Subsidence from a compacting reservoir follows Geertsma's classical solution,^[4] the extraction-induced seismicity link follows Segall,^[8] and the observation and calibration of geomechanical models against time-lapse seismic follows Hatchell and Bourne^[7] and Herwanger and Koutsabeloulis.^[3]

References

Zoback, M.D. (2007). Reservoir Geomechanics. Cambridge University Press, Cambridge, 449 pp. ISBN 978-0-521-77069-9. Tectonic stress field, pore pressure at depth, rock failure, wellbore stability and breakouts, the stress polygon, reservoir depletion and production-induced faulting and subsidence.
Fjær, E., Holt, R.M., Horsrud, P., Raaen, A.M. & Risnes, R. (2008). Petroleum Related Rock Mechanics (2nd ed.). Developments in Petroleum Science 53, Elsevier, Amsterdam, 491 pp. ISBN 978-0-444-50260-5. Third edition (2021), Developments in Petroleum Science 72, ISBN 978-0-12-822195-2. Elasticity, failure mechanics, acoustic wave propagation and its link to 4D seismic, reservoir compaction, sand production.
Herwanger, J.V. & Koutsabeloulis, N. (2011). Seismic Geomechanics: How to Build and Calibrate Geomechanical Models using 3D and 4D Seismic Data. EAGE Publications, Utrecht.
Geertsma, J. (1973). Land Subsidence Above Compacting Oil and Gas Reservoirs. Journal of Petroleum Technology 25(6), 734–744; SPE-3730-PA. Companion paper: A basic theory of subsidence due to reservoir compaction: the homogeneous case, Verhandelingen Kon. Ned. Geol. Mijnbouwk. Genootschap.
Biot, M.A. (1941). General Theory of Three-Dimensional Consolidation. Journal of Applied Physics 12(2), 155–164. The poroelastic foundation of effective stress and flow-mechanics coupling.
Settari, A. & Walters, D.A. (2001). Advances in Coupled Geomechanical and Reservoir Modeling with Applications to Reservoir Compaction. SPE Journal 6(3), 334–342. See also Settari, A. & Mourits, F.M. (1998), A Coupled Reservoir and Geomechanical Simulation System, SPE Journal 3(3), 219–226.
Hatchell, P. & Bourne, S. (2005). Rocks Under Strain: Strain-Induced Time-Lapse Time Shifts are Observed for Depleting Reservoirs. The Leading Edge 24(12), 1222–1225.
Segall, P. (1989). Earthquakes Triggered by Fluid Extraction. Geology 17(10), 942–946. See also Segall, P., Induced Stresses Due to Fluid Extraction from Axisymmetric Reservoirs, Pure and Applied Geophysics.
Zoback, M.D. & Kohli, A.H. (2019). Unconventional Reservoir Geomechanics. Cambridge University Press. Stress in unconventional plays, hydraulic fracturing, depletion-induced stress change between wells, and induced seismicity.
Jaeger, J.C., Cook, N.G.W. & Zimmerman, R.W. (2007). Fundamentals of Rock Mechanics (4th ed.). Blackwell Publishing, Oxford, 488 pp. ISBN 978-0-632-05759-7. Analysis of stress and strain, elasticity, poroelasticity, Mohr-Coulomb and related failure criteria, and the stresses around a circular opening.

Frequently Asked Questions

What is a 1D mechanical earth model?

A depth profile of the geomechanical state along one wellbore: overburden from integrated density, pore pressure, minimum and maximum horizontal stress, and rock elastic and strength properties — all curves against depth, built from logs and pinned to direct measurements such as leakoff tests, pressure tests and image-log breakouts. Its main deliverable is the safe mud weight window, plus sanding onset and optimal trajectory at that location.

What is the difference between 3D and 4D geomechanics?

A 3D model describes stress throughout a volume at one moment — usually the pre-production state — with faults as surfaces and the surrounding rock carrying load. A 4D model couples that volume to reservoir flow simulation so pore pressure changes feed back into stress, strain, compaction and permeability. 3D answers where; 4D answers what happens next.

Why does depletion change stress rather than just pressure?

Rock is poroelastic: grains and fluid share the load. When pore pressure falls, more load transfers to the grain framework, so effective stress rises and the rock compacts. Since the reservoir is laterally confined, horizontal total stress also falls, typically by roughly one-half to two-thirds of the pressure drop. That stress path drives compaction, fault reactivation, permeability loss and casing damage alike.

When is a 1D model enough, and when do you need 3D or 4D?

1D is enough when the decision lives at a point in a structurally simple setting — the mud weight window for one well. Move to 3D when structure, faults or lateral property contrast mean one well's stress cannot be assumed elsewhere, or when the decision is field-wide. Move to 4D only when the answer depends on time: compaction and subsidence, permeability decline, caprock integrity under injection, seismic timeshifts, induced seismicity — and only with a flow model you already trust.