Geothermal doublet design: well spacing, flow rate and thermal breakthrough

Geothermal doublet design comes down to three choices: how far apart the injector and the producer are, how much water you circulate, and how hard you may push water into the injector. Spacing and rate decide when cooled reinjected water reaches the producer, the thermal breakthrough. The stress in the rock caps the rate. And because nobody knows the rock between the wells in detail, each answer should come as a range, a P10 to P90, not a single number.

What a doublet is, and what it has to deliver

A doublet is two wells into the same hot layer or fractured rock mass. The producer brings hot water up; a heat exchanger takes heat out; the injector returns the cooled water to the reservoir to keep the pressure up and dispose of the brine. Around the injector a cold zone grows. The rock heats the water as it flows towards the producer, but the rock near the injector slowly gives up its heat, so the cold zone spreads along the flow paths.

For years the producer delivers its starting temperature. Then the leading edge of the cold zone arrives and the produced temperature starts to fall. A design is good when the temperature stays high enough for the plant, for long enough to repay the wells, at a rate the reservoir and the stress allow.

Geothermal well spacing and thermal breakthrough time

The classic starting point is the analytical solution of Gringarten and Sauty for heat extraction from an aquifer by a well pair [1]. For a confined layer with no regional flow and no heat drawn from the layers above and below, the cold front reaches the producer after roughly

tB = (π / 3) · (ρca / ρcw) · h · D² / Q

where D is the spacing at reservoir depth, h the thickness that takes the flow, Q the volumetric rate, ρca the heat capacity of the saturated rock per unit volume and ρcw that of water. The ratio of heat capacities is why the thermal front moves more slowly than the water itself: the rock has to be cooled as well.

Three rules of thumb follow directly from that expression:

• Spacing counts twice. Breakthrough time grows with the square of the spacing. Doubling D gives four times longer before the cold arrives.
• Rate counts once. Doubling the rate halves the breakthrough time. More rate means more heat now, and less later.
• Thickness is a lever you do not control. What matters is the thickness that actually carries flow, not the gross interval. A sand body that is half shale, or a fractured granite where flow takes a few fractures, behaves as a much thinner layer.

What the simple formula leaves out

The formula is a check, not a design. It ignores heat conducted into the cooled zone from the rock above and below, which delays and softens the decline. It ignores regional groundwater flow, which Gringarten and Sauty treat separately in the same paper [1] and which can push the cold zone towards or away from the producer. It assumes uniform rock. And it says nothing about what happens after breakthrough, which is often what decides the project: a slow decline of a degree or two a decade is very different from a steep one.

Sedimentary aquifers and enhanced geothermal systems

In hot sedimentary aquifers, spacing has often been set generously. Willems and co-authors describe a standard of 1,000 to 1,500 m in the West Netherlands Basin, note that no thermal breakthrough had been reported in 40 years of production in the Paris Basin, and estimate that reducing the spacing from 1,400 m to 1,000 m could raise the net present value by up to 15 % [2]. Their facies-based models also show that homogeneous models underestimate the uncertainty in lifetime and overestimate the energy produced, because clay bodies and baffles change the flow paths [2].

In an enhanced geothermal system (EGS), the picture changes. The rock is hot but tight, and flow follows fractures that stimulation opened or created. The U.S. Department of Energy's FORGE field laboratory in Utah exists to learn how to create and sustain those fracture networks in basement rock [3]. Here the spacing is set by how far stimulation connects the wells, the flowing thickness may be a few fractures, and the cold front can travel fast along them. In KAI's simulated study of the FORGE well pair, about 100 m apart, cold water reached the producer after 1.26 ± 0.13 years at 10 kg/s; the details are in what 30 years of the FORGE doublet look like.

Injection pressure limit and minimum horizontal stress

Rate is not only a heat question. Pushing more water through the same rock needs more pressure at the injector, and the pressure the rock can take has a ceiling.

The usual ceiling is the minimum horizontal stress, Shmin, the smallest compressive stress in the rock where faulting is normal or strike-slip, as at FORGE; where faulting is reverse, the vertical stress is the smallest and sets the ceiling instead. If the injector pressure exceeds it, fractures can open and grow beyond the zone you planned, and the risk of induced earthquakes rises. Staying below it is necessary but not sufficient: fractures already close to slipping can move at lower pressures. Cooling the rock around the injector also reduces the stress there, so the margin can shrink over the life of the doublet. The DOE induced-seismicity protocol explains that fractures slip in response to rising pore pressure and temperature change during injection, and that how easily they slip depends on the rock's strength and the stresses acting on it [4].

Two practical points follow. First, check the injector pressure against Shmin for the whole operating life, not only on day one: as the cold zone grows, the colder, more viscous water needs more pressure to move. Second, treat Shmin itself as uncertain. At FORGE, with the stress measured by minifracs in well 16B and carried down to the injector 16A, KAI's simulated highest safe rate is about 4.9 kg/s (P50); with the regional GDR 1222 stress gradient it is 9.8 kg/s, as the Utah FORGE case shows. A stress test in the reservoir section is often the cheapest way to settle a design.

Geothermal reservoir modelling: what to simulate

Once the quick checks are done, a numerical model answers the questions the formula cannot. A useful doublet model:

• solves water flow and heat together, with water viscosity and density that change with temperature;
• represents the rock between the wells as an ensemble of plausible models, not one average;
• uses cells small enough to resolve the cold front, or upscales the flow properties carefully. In the FORGE study, the usual coarse model was about 11 °C too optimistic about the 30-year temperature, against about 2 °C when the flow was upscaled from the fine model (simulated);
• is checked against exact solutions and against an independent simulator before it is trusted. The open-source simulator open-DARTS, from TU Delft, is one such reference for geothermal and CO₂ problems [5].

P10, P50 and P90: carrying the uncertainty

Every input that matters for a doublet, permeability, flowing thickness, connectivity, stress, starting temperature, is uncertain. Run the ensemble and report the spread of the produced temperature, heat output and injector pressure each year.

Say which way the labels run. In KAI's results, P10 is the value that 10 % of the ensemble members fall below, so the P10 temperature is the pessimistic one. Petroleum reserves reporting labels the low case P90 instead, the value exceeded with 90 % probability. Either is fine if it is written down.

Then turn the spread into a decision rule set before the run: for example, "the producer must stay above the plant's minimum temperature for 30 years in at least nine members out of ten". A design that passes only at the P50 is a gamble, and the people funding it should know that.

A geothermal feasibility study, step by step

1. Temperature at depth. Build the temperature model and check it against measured well logs, not only against regional gradients.
2. Flowing thickness and permeability. From logs, cores, well tests and, for EGS, the stimulated fracture network. Keep the heterogeneity.
3. Stress. The Shmin gradient and its uncertainty, ideally measured in the reservoir section.
4. Quick screening. Breakthrough-time estimates over a grid of spacings and rates.
5. Ensemble simulation. 30 years of produced temperature, heat and injector pressure for the short list, with P10 to P90.
6. Calibration. If the wells exist, match circulation, pressure and tracer tests, and report the misfits you could not remove.
7. Decision. Spacing and rate that meet the temperature target at the chosen confidence without exceeding the stress limit.

What KAI offers, and what it does not

KAI-GeoEnergy is a live preview. The Reservoir Lab lets you place an injector and a producer, set the rates and see simulated produced temperature, heat power, injector pressure and breakthrough as P10 to P90 bands, computed in your browser. The browser solver is a first look: it leaves out temperature-dependent water properties, gravity, separate fracture paths and stress, and on uniform rock it breaks through 17 to 21 % earlier than the KAI engine and ends 6 to 11 °C cooler after 30 years. Signed in, you can queue the full run on the KAI engine from your workspace.

The engine is checked against the Gringarten and Sauty doublet solution and against open-DARTS on a 3D doublet, where breakthrough came at 4.906 against 4.924 years (0.4 %). For a site study we build the model from your wells, logs and tests and give you the forecast with its P10 to P90 and the misfits we could not remove. All of these results are simulated.

KAI does not drill, hire out equipment or run field acquisition such as stress tests or circulation tests. Fault-slip, subsidence and caprock checks are still in build, so for seismic hazard you will still need a specialist assessment alongside the flow and heat model.