Last week I bumped into an interesting paper, SPE 178840, published last year by Aas et al. The authors cemented 3 joints of 7" tubing (that's 36 m, or 120 ft) inside 9⅝" casing, let it set for a week and then measured how much water they could squeeze through. Their main goal may have been to prove that you can set abandonment plugs through the tubing, leaving control lines in place; but the data they collected is great to validate models of flow through microannuli, especially when they affect cement plugs.
Well Integrity Consultant, Germany
Part of the paper's conclusions were that:
The calculated microannuli are relatively small and also probably non-uniform, and should not open for large leakage rates in real P&A operations because of the axial length of the cemented annulus.
I'm not sure I fully agree with this statement, especially the part in bold. But let's proceed with order. First a semantic disclaimer: I don't like jargon much; even less obscure abbreviations that are not even leavened by witty acronyms. If you find any word you can't make sense of, please don't hesitate to shoot a comment: it's not you being dumb, it's me being unable to explain properly. So, as an apology, a "microannulus" is a tiny gap, of the order of tens of micrometers (μm, or one thousandth of a millimeter), created when the interface between cement and pipe - or rock - debonds.
Second, if you're not really into geeky thermo-mechanical stuff, or at least not today, just scroll down to the conclusions.
The third point, which may sound like me whining, is that the paper is frugal with data: this is understandable, since SPE conference papers, are not mauled by peer review watchdogs. I'm certainly the last to throw stones, having been guilty of depriving readers of important pieces of information in the past. Nonetheless some essential parameters have to be guessed at by reading between the lines and peering at figures. For instance, I'm pretty sure the 7" casing weighed 32 ppf (pound per foot), and the 9⅝" casing 53.5 ppf.
But now let's get to how you model a leak through 36 m of an almost horizontal abandoned well.
Cement mechanical properties
The first thing we need to figure out is the elastic properties of cement at the time of the pumping test. Helpfully, the authors measured how temperature evolved during the 6.6 days cement was left to set.
Cement hydration is reasonably easy to model, using the approach of Lin & Meyer. If we consider concentric casings, a layer of 25 mm of Rockwool insulation (a thickness recommended by the producer), natural convection from a horizontal pipe, and we play around a bit with a couple of hydration coefficients, we get a pretty good match, as shown in the plot below.
This first modeling exercise tells us two things:
- The degree of hydration at the end of the temperature recording was 71.4%, not far from its ultimate value. This suggests Young's modulus E=16.4 GPa and Poisson's coefficient ν=0.285 for cement.
- The activation energy is around 50 kJ mol-1, slightly higher than the range computed by Lin & Meyer and the value of 33 kJ mol-1 that one would get from the phase composition of Dyckerhoff class G. The affinity coefficient k is also around 0.6 hour-1, compared to an expected value of 0.41 h-1. These values are even more surprising if we look at the inversion of distributed temperature sensing (DTS) fiber optic measurement, which yields rather Ea=8.6 kJ mol-1 and k=0.12 hour-1. If I had to go out on a limb, I would bet that the difference is due to the fast-reacting oxide phases (C3A and C4AF, i.e. aluminate and ferrite): higher temperatures and longer mixing may mean that some hydration is actually happening before the slurry reaches its final destination in the well (but not in a short yard setup). Bogue's formulas may also be less precise for class G cement. But all this is a different story, and a boring one at that, so I'll drop the subject here.
Among the trivia, it is interesting to note that cement started setting 4.1 hours after placement; that is a reasonable value when the outside temperature is less than 10°C (50°F). The high temperature peak also tells us that cement must have been placed both inside the 7" casing and in the 7"-9⅝" annulus (although I must admit this is kind of obvious when one looks at the pictures of the cut casing).
Microannuli are funny beasts: as pointed out by Lecampion et al. and Dusseault et al., pressurized fluids open their own leak paths and move upwards (or along in our case) in a way resembling a hydraulic fracture.
The authors of SPE 178840 suggest that chemical shrinkage caused cement to debond from the 9⅝" casing, but...
- As we'll see shortly, there is little evidence of a pre-existing microannulus: elastic opening is sufficient to explain flow rates at higher pressures.
- The interaction between chemical shrinkage and thermal expansion during the early stages of hydration is anything but simple. There is no intuitive reason why shrinkage should cause cement to debond from the outer interface. In fact, there is little evidence from field and large-scale lab setups that shrinkage (as opposed to drying and desaturation) plays a major role in bonding.
So why should the third interface (counting outwards from the inner wall of the 7" casing) debond preferentially? Actually, the farther away from the axis of the well you move, the more compliant interfaces become. So an outer microannulus is always favored over an inner one. Aha. So why do we see a lot of casing-cement microannuli in the field, and much less casing-rock microannuli? Well, that's because of creeping formations and, in some cases, cement bonding to porous rocks - part of the cement "glue" sets within the formation pores.
Anyway, now we have geometry and mechanical properties, so we can go ahead, build a model and pump water through it. For reasons related to masochism and braggadocio, I considered water a compressible fluid; that's easy, but not really needed at pressures of <10 MPa (1,450 psi) and constant temperature.
The authors reports three tests where they measure flow rate at the end of the setup and three where they collected water mid-way along it. The setup schematic is very idealized, so we don't know exactly the distance from the inlet to either outlet. Let's however first look at what we get with concentric pipes and, respectively, 36 m and 18 m of microannulus.
At the higher pumping pressure, 9.4 MPa, the model predicts 40 milliliter per minute flow rate for the longer distance and 79 ml min-1 for the shorter one. <15% error is not bad for a first pass blind simulation, and luckily my smug grin doesn't transpire in this post.
At lower pressures, however, the model predictions are smaller than the observed flow rates. As a next step, we can increase the model's accuracy in two ways:
- First, it is unlikely that the microannulus length is exactly 36 m and 18 m. We can try and shorten the distance between inlet and outlet so we get the exact flow rate at higher pressure, and see how much that would increase the rates at around 6 MPa. The distances thus calculated are, respectively, 33.31 m and 14.42 m.
- The 7" casing is not perfectly centered in the 9⅝" casing but rather lies on the collars, leaving a lower gap of 11 mm, corresponding to a standoff SO=56%. Simulating eccentered casings in a Lamé problem requires higher mathematics or a finite element code. Out of laziness I chose the second and looked at the microannulus opening in a more realistic configuration; the microannulus flow model is anyway agnostic with respect to the source of the law relating pressure and defect geometry. Luckily, moving the inner casing off-center doesn't change much the shape of the microannulus, as the figure below reveals (microannulus opening is plotted for a pressure of 20 MPa, and the angle is 0 pointing downwards).
Considering a shorter and eccentered annulus doesn't increase the flow rate at lower pressure much. This is puzzling, considering that there are a lot of reasons why the model should overestimate actual leaks, but far less to suggest why the model is too pessimistic. The fact that the model predicts almost exactly one third of the observed rate leads one to wonder if a calculation error affected the values in the paper (or, alternatively, mine).
Could chemical shrinkage explain the higher rates? Hardly: we would need 12 μm (a relatively large gap) to obtain the rates observed at around 6 MPa, but then 9.4 MPa would result in 81.2 ml min-1, 89% higher than what was observed. Robbing Peter to pay Paul is a good enough reason to invoke the sharp razor of Occam and cutting off the shrinkage assumption.
At any rate, the good news is that eccentering doesn't have a major effect on microannulus leaks, even though other factors (e.g. hole ovalization and stress anisotropy) might. Simple and faster model are good enough - pretty good, I dare say.
An axisymmetric microannulus model is able to reproduce well the amount of water squeezed through an abandoned plug. Indeed, it is easier to model debonding inside the casing than outside it, since in the annulus one has to deal with far-field stresses and often with uncertainties in rocks' mechanical properties and borehole geometry.
Leaks through a system of abandonment plugs need not be zero as implied by the paper's authors. 36 m of cement let through only about 23 t y-1 (tons per year) of water, but simulations of realistic abandonment scenarios with compressible fluids, a mixture of natural gas and crude oil, suggest that rates can reach hundreds of tons per year. Part of the issue is that shallower, wider plugs oppose less resistance to flow: often the deepest plug is the one determining the overall leak rate.
How about the other materials that make up a well? Say, cast iron of a bridge plug or carbon steel of a casing? The standard NORSOK D-010 is not particularly helpful in this instance, when it says that
Permanently abandoned wells shall be plugged with an eternal perspective
Eternity can be a long time to wait: it may not be difficult to design a metal and cement structure to last 1,000 years, and the nuclear waste crowd aims for 1,000,000 years or so. But eternity is longer than that, and you can safely assume that all steel has rusted away in the meanwhile.
When dealing with corrosion, leakage simulations have to be wrapped in a probabilistic approach (visualize Monte Carlo method) that handles the uncertainties in corrosion rate. Sadly, very slow degradation processes, e.g. biologically-induced corrosion (BIC), or corrosion in slightly acidic brines with natural convection, is as little studied as the ageing and settling of drilling mud - another topic of utmost interest for abandoned wells. One is left with sparse literature, hints, experience and uncomfortably wide probability distribution.
Oh, and what about the subject that most appeared to worry the paper's authors, namely whether you can safely cement cables outside the casing? The experience I have is with a DTS cable sheath that was cemented in the annulus in a carbon dioxide storage well in Germany. The figure below displays a USIT map (on the right): the tool mostly sees what is in direct contact with the casing, and the log duly shows a well cemented casing with little azimuthal variation. However an map of reflectors within cement that was extracted from the Isolation Scanner log (on the left) clearly images the thin pipe snaking up the annulus, confirming it is very well cemented along its length.