Rock strength, pore pressure and temperature at the bit

This note derives how DrillPyze sets the rock strength seen by the bit (DrillPyze.formation) and how it models the wellbore temperature that feeds it (DrillPyze.thermal). The bit models of bit-models.md all scale with the drilling strength \(S = \zeta\varepsilon\). When BitParameters.strength_source = 'formation', the intrinsic specific energy is taken from the rock’s confined compressive strength:

\[\varepsilon = \frac{\text{CCS}_\text{eff}}{\eta},\qquad S = \frac{\zeta\,\text{CCS}_\text{eff}}{\eta}.\]

\(\eta\) is a cutting efficiency, equal to 1 for a sharp cutter. Two lines of evidence support linking \(\varepsilon\) to compressive strength:

scratch tests show that \(\varepsilon\) is well correlated with UCS for a sharp cutter in ductile mode [Mitaim et al. 2004];
field MSE methods equate bit specific energy with CCS divided by a mechanical efficiency [Teale 1965; Pessier & Fear 1992; Caicedo et al. 2005].

\(\text{CCS}_\text{eff}\) combines three effects, described in §1–3. All depths are in m, pressures in Pa and temperatures in °C.

1. Confinement by the overbalance (P1)

Mohr–Coulomb strength under a confining stress equal to the differential pressure \(\Delta p\) across the rock at the bit gives [Caicedo, Calhoun & Ewy 2005, eq. 2]

\[\text{CCS} = \text{UCS} + \Delta p\,\frac{1+\sin\varphi}{1-\sin\varphi} = \text{UCS} + \Delta p + \frac{2\,\Delta p\,\sin\varphi}{1-\sin\varphi}.\]

Permeable rock (drained)

Pore pressure has time to equalise at the cutting face:

\[\Delta p_\text{HP} = p_\text{bh} - p_p .\]

Low-permeability, low-porosity rock (undrained)

When the bit removes the overburden from the rock beneath it, the rock is unloaded from \(\sigma_v\) to \(p_\text{bh}\) faster than pore fluid can flow. The pore pressure then drops by Skempton’s response to the change in mean stress [Skempton 1954]. Following Caicedo et al. (eqs. 11–13), the effective pore pressure is

\[p_p^{*} = p_p - B\,\frac{\sigma_v - p_\text{bh}}{3}, \qquad \Delta p_\text{LP} = p_\text{bh} - p_p^{*},\]

with \(B = 1\) in their formulation. That is the default here, and it can be adjusted when lab \(B\) values are available. For FORGE, OU measured Skempton and Biot coefficients on 58-32 and 78-32 cores [GDR submission 1464].

Blending by porosity

Effective porosity \(\phi_e\) is the proxy for drainage, and the strength is blended linearly [Caicedo et al. 2005]:

\(\phi_e\)	CCS used
\(\le 0.05\)	\(\text{CCS}_\text{LP}\)
\(\ge 0.20\)	\(\text{CCS}_\text{HP}\)
in between	linear blend of the two

Granite has very low permeability (FORGE matrix permeability below 30 µD) and very low porosity (matrix porosity below 0.5%) [Utah FORGE review 2026]. It is therefore treated as fully undrained. For FORGE-like conditions this raises CCS well above the drained value. For example, at 2,500 m TVD with hydrostatic pore pressure and a 25 kPa/m overburden gradient, \(\Delta p_\text{LP}\) is about 12 MPa larger than \(\Delta p_\text{HP}\).

Context. Detournay & Atkinson (2000) showed that in shear-dilatant, low-permeability rock, cutting at drilling speeds is undrained. Detournay & Tan (2002) found that shale specific energy then depends on \(p_\text{bh}\) and not on \(p_\text{bh} - p_p\): \(\varepsilon = \varepsilon_0 + m\,p_\text{bh}\). The Skempton construction above is the engineering counterpart. Both say the far field pore pressure matters little in tight rock. Whether FORGE granite follows the drained or the undrained limit can be tested directly against the field ROP data.

2. Thermal weakening (T1)

\[\text{UCS}(T) = \text{UCS}_0\,\bigl[1 - \beta\,\max(T - T^{*}, 0)\bigr],\]

where \(T\) is the undisturbed formation temperature at the bit, from the geothermal gradient. The rock ahead of the bit has not yet been cooled [Wang et al. 2023, assumption 6].

\(T^{*} = 193\) °C and \(\beta = 1.41\times10^{-3}\) /K are a least-squares fit (RMS error 3.8%) to the residual UCS of granite heated to 250–650 °C and air cooled [Xu et al. 2019]:

T (°C)	250	350	450	550	650
measured loss	12%	14%	38%	53%	63%
fitted loss	8%	22%	36%	50%	64%

At FORGE temperatures (about 200–230 °C) the effect is small, 1–5%.

Caveat: these are residual strengths measured after cooling. Strength measured at temperature can differ, and so can the effect of repeated quenching.

3. Cooling-induced thermal stress at the hole bottom (T2)

Circulating mud is cooler than the rock at the bit by \(\Delta T = T_\text{rock} - T_\text{mud,bit}\). A cooled surface layer that is constrained laterally carries an in-plane thermoelastic tensile stress [e.g. Timoshenko & Goodier 1970]:

\[\sigma_\text{th} = \frac{\alpha E\,\Delta T}{1-\nu}.\]

For FORGE granite (\(\alpha = 6\times10^{-6}\) /K, \(E = 55\) GPa, \(\nu = 0.26\)) this is about 0.45 MPa/K, or 45 MPa per 100 K. That is comparable to the tensile strength (about 11 MPa) and to the overbalance term.

The cooled layer is thin. Between blade passes (\(t_p = 2\pi/(n\omega)\), with \(n\) blades) heat diffuses to a depth

\[\delta = \sqrt{\pi\,D_\text{th}\,t_p}, \qquad D_\text{th} = \frac{k}{\rho c}\approx 0.9\ \text{mm}^2/\text{s},\]

while each blade cuts to depth \(d_b = \text{ROP}\cdot t_p\). Only the cooled fraction of the cut is weakened:

\[\text{CCS}_\text{eff} = \max\!\Bigl(\text{CCS} - \kappa\,\sigma_\text{th}\,\min(1, \delta/d_b),\ \ f_\text{floor}\,\text{UCS}(T)\Bigr).\]

\(\kappa \in [0, 1]\) is the efficiency of the thermal stress. It is the one parameter that has to be calibrated; the default is 0.5.
\(f_\text{floor} = 0.1\) keeps the strength physical.
The term makes the thermal benefit depend on ROP, RPM and blade count. It is largest at low ROP, high RPM and with many blades.

Status: this formulation is DrillPyze’s own, built from standard thermoelasticity and conduction. SPE-217718 studies the effect of rapid cooling on cutting hot dry rock experimentally and is the natural source for calibrating \(\kappa\).

4. Wellbore temperature: the growing-wellbore drift-flux model

The mud temperature at the bit and the bottomhole pressure come from the model of Wang, Khaled, Luu, Ashok & van Oort (2023), SPE-214836-MS. That model couples thermohydraulic flow to a wellbore that grows as it is drilled, and it was validated against Utah FORGE 16(78)-32. It builds on Evje & Fjelde (2002), Fallah et al. (2019) and Khaled et al. (2023c).

DrillPyze implements it in the single-phase liquid form (gas volume fraction zero), which is the case for water-based mud.

Conservation equations

For each flow path (inside the drillstring, flowing down; in the annulus, flowing up), the mixture energy equation (paper eq. 4) becomes, per cell of length \(\Delta x\):

\[\rho c A\,\Delta x\,\frac{dT}{dt} = \rho c Q\,(T_\text{upstream} - T) + \dot Q_\text{wall}\,\Delta x + \dot m_\text{cut} c_r\,(T_\text{rock} - T)\,\delta_{k,\text{bottom}} .\]

\(Q\) is the pump rate.
The mass and momentum equations (eqs. 1, 3) reduce to \(v = Q/A\). Assumption 1 of the paper, ROP ≪ mud velocity, justifies this.
Mud density follows paper eq. 2:

\[\rho(T,p) = \bigl(1 - (T - T_\text{ref})\,c_\text{exp}\bigr)\rho_\text{ref} + \frac{p - p_\text{ref}}{v_\text{sound}^2}.\]

It is integrated up the annulus, together with the frictional loss (Fanning, Newtonian), to give \(p_\text{bh}\) and the ECD used in §1.

Radial heat transfer (paper eqs. 6–9)

\[\dot q_\text{ds} = \frac{T_\text{ann} - T_\text{ds}}{R_\text{ds}},\qquad \dot q_\text{ann} = \frac{T_\text{for} - T_\text{ann}}{R_\text{for}} + \frac{T_\text{ds} - T_\text{ann}}{R_\text{ds}} .\]

\(R_\text{ds}\) is inside convection, plus conduction through the pipe wall, plus outside convection.
\(R_\text{for}\) is wall convection, plus conduction through the casing where cased, plus conduction to the first formation node.
Each convective resistance is \(1/(h\pi D)\), with \(h = \text{Nu}\,k/D_h\).
The paper does not state its Nusselt correlation. DrillPyze uses Dittus–Boelter, \(\text{Nu} = 0.023\,\text{Re}^{0.8}\text{Pr}^{0.4}\) [Dittus & Boelter 1930], for turbulent flow and \(\text{Nu} = 3.66\) for laminar flow.

Formation

Each axial cell has 12 radial conduction cells, log-spaced out to 5 m beyond the wall, solved implicitly. The outer boundary is held at the undisturbed geothermal temperature (paper boundary conditions; Khaled et al. 2023c).

Growing wellbore (paper eqs. 10, 14–20; Figs. 1–3)

Bottom cell. The bottom cell lengthens at the ROP. When it reaches 1.5 cell lengths it is cut in two, and both halves keep the state of the original cell (“grow-cut-grow”).
Cuttings. Cuttings enter the bottom annulus cell at the undisturbed formation temperature. DrillPyze uses the rock’s density and heat capacity for them; the paper assumes mud properties.
Bottom formation cells. As the hole deepens, they relax toward the undisturbed temperature:

\[T^\text{bottom}_{t+\Delta t} = T^\text{bottom}_t + \frac{\text{ROP}\,\Delta t}{L_\text{bottom}}\bigl(T^\text{bottom,formation}_t - T^\text{bottom}_t\bigr).\]

Moving string. The string geometry is volume-averaged over each cell at every update (Fig. 1). The ROP-scaled area-change terms of eqs. 11–13 are of order ROP/\(v\) and are omitted.

Numerics

Each step uses backward Euler for the fluid and the formation, coupled by three Gauss–Seidel iterations: drillstring downstream, annulus upstream from the bit, then a tridiagonal solve for each formation layer. The time step is at most 2 s, and the model is updated from the drilling simulation every 2 s of simulated time.

Verification against SPE-214836-MS

examples/forge_thermal_spe214836.py and tests/test_thermal_rock.py use the paper’s FORGE case: about 5,900 ft MD, 550 gpm, 40 °C surface temperature and 69.1 °C/km, with Table 2 properties. The inlet temperature is not given in the paper; 35 °C is assumed.

quantity	SPE-214836	DrillPyze
downhole temperature at pump start → after 30 min	≈ 310 → 210 °F (Fig. 6)	325 → 209 °F
downhole rise from drilling after 30 min, 40 / 60 / 72 ft/h	≈ 1.3 / 2.2 / 2.7 °F (Fig. 7)	1.25 / 1.87 / 2.24 °F
rise linear in ROP	yes	yes
ROP → 0 converges to the static wellbore	yes	yes (test)
static well stays at the geothermal profile	–	exact (test)
energy balance while circulating	–	within 3% (test)

Open item. The time history of the drilling-induced rise differs:

In the paper it jumps to most of its final value within a minute and then plateaus.
In DrillPyze it builds steadily. About 60% of it comes from the bit advancing into hotter rock along the 69 °C/km gradient; cuttings and the freshly exposed wall give a prompt rise of about 0.4 °F at 60 ft/h.
Possible causes include the unstated inlet temperature, the paper’s mesh, and details of its eqs. 10 and 19 (for example, whether the cuttings enthalpy is taken relative to the mud temperature or to a reference temperature).
The result is insensitive to the DrillPyze cell size (2–30 m).

5. Parameters and FORGE defaults

parameter	default	source
UCS	110 MPa	FORGE core 91–126 MPa [Ye et al. 2022, via Utah FORGE review 2026]
friction angle φ	36°	FORGE granitoid model value [Finnila & Podgorney 2020]; core 32–48°
porosity	0.005	matrix < 0.5% [Utah FORGE review 2026]
pore pressure gradient	9.79 kPa/m	hydrostatic
overburden gradient	25 kPa/m	assumed
Skempton B	1.0	Caicedo et al. 2005
surface T, gradient	40 °C, 69.1 °C/km	SPE-214836 Table 2
rock ρ, c, k	2800 kg/m³, 930 J/kg K, 2.31 W/m K	SPE-214836 Table 2
α, E, ν	6×10⁻⁶ /K, 55 GPa, 0.26	Podgorney et al. 2021; Utah FORGE review 2026
T*, β	193 °C, 1.41×10⁻³ /K	fit to Xu et al. 2019
κ	0.5	to be calibrated (§3)
mud c, k, expansion, viscosity	3750 J/kg K, 0.75 W/m K, 10⁻⁴ /K, 20 mPa·s	SPE-214836 Table 2

References

Wang, N., Khaled, M.S., Luu, A., Ashok, P. and van Oort, E., 2023. Downhole Temperature Estimation of a Growing High-Temperature Wellbore Using a Modified Drift Flux Modeling Approach. SPE Annual Technical Conference and Exhibition, San Antonio, Texas. SPE-214836-MS. https://doi.org/10.2118/214836-MS
Khaled, M.S., Wang, N., Ashok, P. and van Oort, E., 2023c. Downhole heat management for drilling shallow and ultra-deep high enthalpy geothermal wells. Geothermics 107, 102604. https://doi.org/10.1016/j.geothermics.2022.102604
Evje, S. and Fjelde, K.K., 2002. Relaxation schemes for the calculation of two-phase flow in pipes. Math. Comput. Model. 36(4–5), 535–567.
Fallah, A. et al., 2019. An integrated thermal and multi-phase flow model for estimating transient temperature dynamics during drilling operations. SPE/IADC-194083-MS.
Caicedo, H.U., Calhoun, W.M. and Ewy, R.T., 2005. Unique ROP predictor using bit-specific coefficient of sliding friction and mechanical efficiency as a function of confined compressive strength impacts drilling performance. SPE/IADC-92576-MS. CCS equations as given in US Patent 7,412,331 (https://patents.google.com/patent/US7412331) and the Skempton low-permeability form in US Patent 7,555,414.
Skempton, A.W., 1954. The pore-pressure coefficients A and B. Géotechnique 4(4), 143–147.
Detournay, E. and Atkinson, C., 2000. Influence of pore pressure on the drilling response in low-permeability shear-dilatant rocks. Int. J. Rock Mech. Min. Sci. 37(7), 1091–1101. https://doi.org/10.1016/S1365-1609(00)00050-2
Detournay, E. and Tan, C.P., 2002. Dependence of drilling specific energy on bottom-hole pressure in shales. SPE/ISRM Rock Mechanics Conference, Irving, Texas, 526–535.
Mitaim, S., Dagrain, F., Richard, T., Detournay, E. and Drescher, A., 2004. A novel apparatus to determine the rock strength parameters. 9th National Convention on Civil Engineering, Thailand. https://orbi.umons.ac.be/handle/20.500.12907/1650
Teale, R., 1965. The concept of specific energy in rock drilling. Int. J. Rock Mech. Min. Sci. 2, 57–73.
Pessier, R.C. and Fear, M.J., 1992. Quantifying common drilling problems with mechanical specific energy and a bit-specific coefficient of sliding friction. SPE-24584-MS.
Xu, C., Sun, Q., Pan, X., Zhang, W. and Wang, Y., 2019. Variation on thermal damage rate of granite specimen with thermal cycle treatment. High Temp. Mater. Process. 38, 849–855. https://doi.org/10.1515/htmp-2019-0046
SPE-217718, 2025. Study examines thermal effects on drilling performance in hot dry rock (JPT synopsis). https://jpt.spe.org/study-examines-thermal-effects-on-drilling-performance-in-hot-dry-rock-restricted
Timoshenko, S.P. and Goodier, J.N., 1970. Theory of Elasticity, 3rd ed. McGraw-Hill (thermal stresses).
Dittus, F.W. and Boelter, L.M.K., 1930. Heat transfer in automobile radiators of the tubular type. Univ. Calif. Publ. Eng. 2, 443–461.
Utah FORGE: A Decade of Innovation — Comprehensive Review of Field-Scale Advances (Part 1), 2026. Processes 14(3), 512. https://doi.org/10.3390/pr14030512 (compiles Ye et al. 2022, ARMA-2022-0272; Finnila & Podgorney 2020, SGP-TR-216; Podgorney et al. 2021, Energies 14, 4758).
Utah FORGE 5-2615: Well 58-32 and 78-32 Poroelastic Properties (dataset), GDR submission 1464. https://gdr.openei.org/submissions/1464