One of the practical questions in any waterflood is how much of a producer's rate is actually supported by injection, and how much would have been produced anyway.
Capacitance Resistance Models (CRM) tackle that problem using only historical injection and production rates. No reservoir simulator, no geomodel, and no pressure matching required.
This example applies CRM to producer F-14 from the public Volve field dataset, using injectors F-4 and F-5 to estimate connectivity and quantify injection support.
Equinor Volve, daily rates, Sep 2007 – Dec 2016. One producer (F-14), two injectors (F-4 and F-5). Both injectors share a manifold, it matters later in the analysis.
Figure 1: Volve Field — Production & Injection Overview. F-14 produces 100–170k Sm³/month of liquid for most of the field life; F-4 and F-5 inject through the same manifold.
Figure 2: Well Locations, Volve Field (schematic)
CRM treats a producer as a system responding to nearby injectors. Some injectors have a strong influence, others barely matter, and the response is never instantaneous.
The model boils that behaviour down to three quantities:
The producer's rate is then modelled as a combination of its previous rate, the contribution from surrounding injectors, and any pressure-driven effects:
The unknown parameters are adjusted until the model reproduces the historical production rate. Once calibrated, CRM provides an estimate of injector-producer connectivity and the role each injector plays in supporting production.
CRMP+BHP was fitted to F-14 using average daily rates over the full production history.
| Parameter | Value |
|---|---|
| τ | 1.19 months |
| f₄ (F-4 → F-14) | 0.224 |
| f₅ (F-5 → F-14) | 0.521 |
| J | ≈ 0 |
| R² | 0.77 |
The model captures the main production trends with an R² of 0.77 and reproduces most of the large rate swings visible in the field data.
Three observations stand out.
J collapsed to essentially zero. With J ≈ 0 and yet f₄+f₅ ≈ 0.75, the injectors are mainly providing sweep, whereas the aquifer provides majority of pressure support.
τ ≈ 1.2 months — a fast response time. The reservoir between the injectors and F-14 transmits pressure changes in weeks rather than months, consistent with high-permeability Hugin Sandstone.
F-5 exerts more than twice the influence of F-4 — 52% of the injection response versus 22%. The physical interpretation is the subject of the next section.
Figure 3: F-14 — CRMP+BHP History Match
Figure 4: F-14 — Parity Plot (R²=0.77)
F-5 dominates. With f₅ = 0.52 versus f₄ = 0.22, F-5 provides more than twice the support to F-14 compared with F-4. Weijermars (2024) reached a similar conclusion from water breakthrough analysis, estimating that roughly 70% of F-14's produced water originates from F-5. Different methods, same story.
Not all production is injection-supported. The two injectors account for about 75% of the response (f₄ + f₅ = 0.745), leaving roughly a quarter unexplained by injection. In Volve, that missing support is most likely coming from the aquifer. The field sits on a tilted fault block with a connected basal water leg that helps maintain reservoir pressure. The J ≈ 0 result points in the same direction: changes in producer BHP have little measurable impact because the aquifer is already doing most of the pressure-support work.
The individual split is uncertain. F-4 and F-5 share a manifold and their injection rates move together, making it difficult for the regression to separate their individual contributions. Refitting different time windows produced f₄/f₅ splits of 0.22/0.52, 0.54/0.23, and 0.69/0.10. The individual coefficients move around considerably, but the total connectivity remains stable at roughly 0.75–0.79. That's the number worth trusting.
Two tests, both using only injection rates as forward inputs. That's standard CRM practice: injection is a controlled variable, production is the response.
Blind 2-year forecast. Train on 2008–2013. Forecast 2014–2015.
| MAPE | 13.0% |
| Bias | +10.2% |
| Mean observed | 3,535 Sm³/day |
Figure 5: F-14 Forecast — Blind 2-Year Test (2014–2015)
The forecast comes out about 10% too high. Most of that is because the training years were the well's best years, and the model expects a bit more from F-14 than it actually delivers later on.
Walk-forward refit. In practice, you don't fit a CRM model once and then forecast for years. You refit it every month as new production data comes in, and you only forecast a few months ahead. That's walk-forward: refit, forecast 1–6 months, advance a month, repeat.
Running that on the 2014–2015 test period:
| Horizon | MAPE |
|---|---|
| 1-month | 7.8% |
| 3-month | 12.5% |
| 6-month | 14.2% |
| Blind 2-year | 13.0% |
Figure 6: F-14 Forecast Error vs Horizon (Walk-Forward Refitting)
One-month-ahead is sharp (7.8%). Six months out is noticeably worse, and importantly, no better than the 2-year blind forecast. Refitting stops helping past about six months.
That cut-off isn't arbitrary. It's set by τ ≈ 1.2 months. After roughly 5τ (six months), the influence of the last observed rate has faded to under 1% — the model has effectively "forgotten" where it started, and the forecast is driven by injection alone. From that point, knowing last month's rate adds nothing. τ tells you in advance how far ahead CRM can usefully look.
Volve data licensed by Equinor under the Volve Data Village licence.