Bayesian Decline Curve Analysis with eXMC
Setup
# CPU only — no GPU required
System.put_env("EXLA_CPU_ONLY", "true")
System.put_env("CUDA_VISIBLE_DEVICES", "")
Mix.install([
{:exmc, path: Path.expand("../", __DIR__)},
{:exla, "~> 0.10"},
{:kino_vega_lite, "~> 0.1"}
])
Application.put_env(:exla, :clients, host: [platform: :host])
Application.put_env(:exla, :default_client, :host)
Nx.default_backend(Nx.BinaryBackend)
Nx.Defn.default_options(compiler: EXLA, client: :host)
[]Why This Matters
An oil well’s production declines over time. How fast? The answer determines whether the field is worth $50 million or $200 million. The traditional approach: fit an Arps decline curve to 12 months of production data, extrapolate 30 years, report a single EUR (Estimated Ultimate Recovery). The board makes a $100 million development decision based on that one number.
The problem: 12 months of noisy data admits many plausible decline curves. The exponential fit says 2 million barrels. The hyperbolic says 3.5 million. Both fit the data equally well. Which one do you report?
Neither. You report the posterior distribution over decline parameters, integrated into a predictive distribution over future production. Not one EUR — a distribution of EURs, with probability masses on the outcomes that matter: P(EUR > 2M), P(EUR > 3M), P(breakeven).
The Arps Decline Model
Every producing oil or gas well declines over time. The Arps hyperbolic decline equation models this behavior with three parameters:
$$q(t) = \frac{q_i}{(1 + b \cdot D_i \cdot t)^{1/b}}$$
- q_i — initial production rate (bbl/day)
- D_i — initial decline rate (per month)
- b — decline exponent (0 = exponential, 1 = harmonic, 0-2 typical)
alias VegaLite, as: Vl
# Show how the decline exponent b changes the curve shape
t_months = Enum.to_list(0..60)
qi = 800.0
curves =
for b <- [0.0, 0.5, 1.0, 1.5], t <- t_months do
di = 0.12
q =
if b == 0.0 do
qi * :math.exp(-di * t)
else
qi / :math.pow(1.0 + b * di * t, 1.0 / b)
end
%{"month" => t, "rate_bpd" => q, "b" => "b = #{b}"}
end
Vl.new(width: 700, height: 350, title: "Arps Decline Curves (q_i=800, D_i=0.12)")
|> Vl.data_from_values(curves)
|> Vl.mark(:line, stroke_width: 2)
|> Vl.encode_field(:x, "month", type: :quantitative, title: "Months")
|> Vl.encode_field(:y, "rate_bpd", type: :quantitative, title: "Production (bbl/day)")
|> Vl.encode_field(:color, "b", type: :nominal, title: "Exponent"){"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"b":"b = 0.0","month":0,"rate_bpd":800.0},{"b":"b = 0.0","month":1,"rate_bpd":709.536349373726},{"b":"b = 0.0","month":2,"rate_bpd":629.3022888532428},{"b":"b = 0.0","month":3,"rate_bpd":558.1410608568249},{"b":"b = 0.0","month":4,"rate_bpd":495.0267134449127},{"b":"b = 0.0","month":5,"rate_bpd":439.04930887522113},{"b":"b = 0.0","month":6,"rate_bpd":389.40180476797735},{"b":"b = 0.0","month":7,"rate_bpd":345.36841874326376},{"b":"b = 0.0","month":8,"rate_bpd":306.3143087800897},{"b":"b = 0.0","month":9,"rate_bpd":271.6764205159513},{"b":"b = 0.0","month":10,"rate_bpd":240.9553695297617},{"b":"b = 0.0","month":11,"rate_bpd":213.70824157268032},{"b":"b = 0.0","month":12,"rate_bpd":189.54220694569742},{"b":"b = 0.0","month":13,"rate_bpd":168.10885696061177},{"b":"b = 0.0","month":14,"rate_bpd":149.09918083152797},{"b":"b = 0.0","month":15,"rate_bpd":132.23911057726926},{"b":"b = 0.0","month":16,"rate_bpd":117.28556970428012},{"b":"b = 0.0","month":17,"rate_bpd":104.02296870274073},{"b":"b = 0.0","month":18,"rate_bpd":92.26009683045001},{"b":"b = 0.0","month":19,"rate_bpd":81.82736537242998},{"b":"b = 0.0","month":20,"rate_bpd":72.57436263153001},{"b":"b = 0.0","month":21,"rate_bpd":64.36768539962596},{"b":"b = 0.0","month":22,"rate_bpd":57.089015645108866},{"b":"b = 0.0","month":23,"rate_bpd":50.63341468771259},{"b":"b = 0.0","month":24,"rate_bpd":44.90781026730698},{"b":"b = 0.0","month":25,"rate_bpd":39.82965469429116},{"b":"b = 0.0","month":26,"rate_bpd":35.32573473575429},{"b":"b = 0.0","month":27,"rate_bpd":31.331116079189663},{"b":"b = 0.0","month":28,"rate_bpd":27.78820715579085},{"b":"b = 0.0","month":29,"rate_bpd":24.645928826200862},{"b":"b = 0.0","month":30,"rate_bpd":21.858977957834057},{"b":"b = 0.0","month":31,"rate_bpd":19.387174276552898},{"b":"b = 0.0","month":32,"rate_bpd":17.194881076071937},{"b":"b = 0.0","month":33,"rate_bpd":15.250491433289309},{"b":"b = 0.0","month":34,"rate_bpd":13.525972522164222},{"b":"b = 0.0","month":35,"rate_bpd":11.996461456382162},{"b":"b = 0.0","month":36,"rate_bpd":10.639906833955013},{"b":"b = 0.0","month":37,"rate_bpd":9.436750815801258},{"b":"b = 0.0","month":38,"rate_bpd":8.369647154741443},{"b":"b = 0.0","month":39,"rate_bpd":7.423211109651795},{"b":"b = 0.0","month":40,"rate_bpd":6.583797639216024},{"b":"b = 0.0","month":41,"rate_bpd":5.839304677430866},{"b":"b = 0.0","month":42,"rate_bpd":5.178998654631524},{"b":"b = 0.0","month":43,"rate_bpd":4.593359748523362},{"b":"b = 0.0","month":44,"rate_bpd":4.073944634159356},{"b":"b = 0.0","month":45,"rate_bpd":3.613264754090136},{"b":"b = 0.0","month":46,"rate_bpd":3.204678353672336},{"b":"b = 0.0","month":47,"rate_bpd":2.8422947249770893},{"b":"b = 0.0","month":48,"rate_bpd":2.5208892787555532},{"b":"b = 0.0","month":49,"rate_bpd":2.235828220029475},{"b":"b = 0.0","month":50,"rate_bpd":1.9830017413330867},{"b":"b = 0.0","month":51,"rate_bpd":1.758764770434025},{"b":"b = 0.0","month":52,"rate_bpd":1.5598844182760965},{"b":"b = 0.0","month":53,"rate_bpd":1.3834933694857257},{"b":"b = 0.0","month":54,"rate_bpd":1.2270485434595715},{"b":"b = 0.0","month":55,"rate_bpd":1.088294430038315},{"b":"b = 0.0","month":56,"rate_bpd":0.9652305711664323},{"b":"b = 0.0","month":57,"rate_bpd":0.8560827197116835},{"b":"b = 0.0","month":58,"rate_bpd":0.7592772596326982},{"b":"b = 0.0","month":59,"rate_bpd":0.6734185187028392},{"b":"b = 0.0","month":60,"rate_bpd":0.5972686467013438},{"b":"b = 0.5","month":0,"rate_bpd":800.0},{"b":"b = 0.5","month":1,"rate_bpd":711.9971520113919},{"b":"b = 0.5","month":2,"rate_bpd":637.7551020408163},{"b":"b = 0.5","month":3,"rate_bpd":574.5475438092503},{"b":"b = 0.5","month":4,"rate_bpd":520.2913631633714},{"b":"b = 0.5","month":5,"rate_bpd":473.3727810650887},{"b":"b = 0.5","month":6,"rate_bpd":432.5259515570935},{"b":"b = 0.5","month":7,"rate_bpd":396.7466772465781},{"b":"b = 0.5","month":8,"rate_bpd":365.2300949598247},{"b":"b = 0.5","month":9,"rate_bpd":337.325012649688},{"b":"b = 0.5","month":10,"rate_bpd":312.49999999999994},{"b":"b = 0.5","month":11,"rate_bpd":290.3178980984178},{"b":"b = 0.5","month":12,"rate_bpd":270.41644131963227},{"b":"b = 0.5","month":13,"rate_bpd":252.4933720489837},{"b":"b = 0.5","month":14,"rate_bpd":236.2948960302458},{"b":"b = 0.5","month":15,"rate_bpd":221.606648199446},{"b":"b = 0.5","month":16,"rate_bpd":208.24656393169514},{"b":"b = 0.5","month":17,"rate_bpd":196.05920988138416},{"b":"b = 0.5","month":18,"rate_bpd":184.91124260355028},{"b":"b = 0.5","month":19,"rate_bpd":174.68774565464238},{"b":"b = 0.5","month":20,"rate_bpd":165.2892561983471},{"b":"b = 0.5","month":21,"rate_bpd":156.62933667475923},{"b":"b = 0.5","month":22,"rate_bpd":148.63258026159335},{"b":"b = 0.5","month":23,"rate_bpd":141.2329637737448},{"b":"b = 0.5","month":24,"rate_bpd":134.37248051599033},{"b":"b = 0.5","month":25,"rate_bpd":128.0},{"b":"b = 0.5","month":26,"rate_bpd":122.0703125},{"b":"b = 0.5","month":27,"rate_bpd":116.5433249810617},{"b":"b = 0.5","month":28,"rate_bpd":111.38338159946538},{"b":"b = 0.5","month":29,"rate_bpd":106.55868719697372},{"b":"b = 0.5","month":30,"rate_bpd":102.04081632653063},{"b":"b = 0.5","month":31,"rate_bpd":97.80429360848943},{"b":"b = 0.5","month":32,"rate_bpd":93.82623381497469},{"b":"b = 0.5","month":33,"rate_bpd":90.08603216071349},{"b":"b = 0.5","month":34,"rate_bpd":86.56509695290859},{"b":"b = 0.5","month":35,"rate_bpd":83.24661810613942},{"b":"b = 0.5","month":36,"rate_bpd":80.11536612722318},{"b":"b = 0.5","month":37,"rate_bpd":77.15751707110067},{"b":"b = 0.5","month":38,"rate_bpd":74.36049970255802},{"b":"b = 0.5","month":39,"rate_bpd":71.7128617017462},{"b":"b = 0.5","month":40,"rate_bpd":69.20415224913495},{"b":"b = 0.5","month":41,"rate_bpd":66.82481873767917},{"b":"b = 0.5","month":42,"rate_bpd":64.56611570247934},{"b":"b = 0.5","month":43,"rate_bpd":62.420024343809494},{"b":"b = 0.5","month":44,"rate_bpd":60.37918125830215},{"b":"b = 0.5","month":45,"rate_bpd":58.43681519357196},{"b":"b = 0.5","month":46,"rate_bpd":56.586690810321414},{"b":"b = 0.5","month":47,"rate_bpd":54.82305857843809},{"b":"b = 0.5","month":48,"rate_bpd":53.14061005420343},{"b":"b = 0.5","month":49,"rate_bpd":51.53443788811874},{"b":"b = 0.5","month":50,"rate_bpd":50.0},{"b":"b = 0.5","month":51,"rate_bpd":48.533087432357},{"b":"b = 0.5","month":52,"rate_bpd":47.12979545668772},{"b":"b = 0.5","month":53,"rate_bpd":45.78649756186901},{"b":"b = 0.5","month":54,"rate_bpd":44.499822000711994},{"b":"b = 0.5","month":55,"rate_bpd":43.26663061114116},{"b":"b = 0.5","month":56,"rate_bpd":42.083999663328015},{"b":"b = 0.5","month":57,"rate_bpd":40.94920251428103},{"b":"b = 0.5","month":58,"rate_bpd":39.85969387755102},{"b":"b = 0.5","month":59,"rate_bpd":38.813095538434666},{"b":"b = 0.5","month":60,"rate_bpd":37.807183364839325},{"b":"b = 1.0","month":0,"rate_bpd":800.0},{"b":"b = 1.0","month":1,"rate_bpd":714.2857142857142},{"b":"b = 1.0","month":2,"rate_bpd":645.1612903225806},{"b":"b = 1.0","month":3,"rate_bpd":588.2352941176471},{"b":"b = 1.0","month":4,"rate_bpd":540.5405405405405},{"b":"b = 1.0","month":5,"rate_bpd":500.0},{"b":"b = 1.0","month":6,"rate_bpd":465.1162790697675},{"b":"b = 1.0","month":7,"rate_bpd":434.7826086956522},{"b":"b = 1.0","month":8,"rate_bpd":408.16326530612247},{"b":"b = 1.0","month":9,"rate_bpd":384.6153846153846},{"b":"b = 1.0","month":10,"rate_bpd":363.6363636363636},{"b":"b = 1.0","month":11,"rate_bpd":344.82758620689657},{"b":"b = 1.0","month":12,"rate_bpd":327.8688524590164},{"b":"b = 1.0","month":13,"rate_bpd":312.5},{"b":"b = 1.0","month":14,"rate_bpd":298.5074626865672},{"b":"b = 1.0","month":15,"rate_bpd":285.7142857142857},{"b":"b = 1.0","month":16,"rate_bpd":273.972602739726},{"b":"b = 1.0","month":17,"rate_bpd":263.1578947368421},{"b":"b = 1.0","month":18,"rate_bpd":253.1645569620253},{"b":"b = 1.0","month":19,"rate_bpd":243.90243902439025},{"b":"b = 1.0","month":20,"rate_bpd":235.29411764705884},{"b":"b = 1.0","month":21,"rate_bpd":227.27272727272728},{"b":"b = 1.0","month":22,"rate_bpd":219.7802197802198},{"b":"b = 1.0","month":23,"rate_bpd":212.76595744680853},{"b":"b = 1.0","month":24,"rate_bpd":206.18556701030928},{"b":"b = 1.0","month":25,"rate_bpd":200.0},{"b":"b = 1.0","month":26,"rate_bpd":194.17475728155338},{"b":"b = 1.0","month":27,"rate_bpd":188.67924528301887},{"b":"b = 1.0","month":28,"rate_bpd":183.48623853211012},{"b":"b = 1.0","month":29,"rate_bpd":178.57142857142856},{"b":"b = 1.0","month":30,"rate_bpd":173.91304347826087},{"b":"b = 1.0","month":31,"rate_bpd":169.49152542372883},{"b":"b = 1.0","month":32,"rate_bpd":165.28925619834712},{"b":"b = 1.0","month":33,"rate_bpd":161.29032258064515},{"b":"b = 1.0","month":34,"rate_bpd":157.48031496062993},{"b":"b = 1.0","month":35,"rate_bpd":153.84615384615384},{"b":"b = 1.0","month":36,"rate_bpd":150.37593984962405},{"b":"b = 1.0","month":37,"rate_bpd":147.05882352941177},{"b":"b = 1.0","month":38,"rate_bpd":143.88489208633095},{"b":"b = 1.0","month":39,"rate_bpd":140.84507042253523},{"b":"b = 1.0","month":40,"rate_bpd":137.93103448275863},{"b":"b = 1.0","month":41,"rate_bpd":135.13513513513513},{"b":"b = 1.0","month":42,"rate_bpd":132.4503311258278},{"b":"b = 1.0","month":43,"rate_bpd":129.87012987012986},{"b":"b = 1.0","month":44,"rate_bpd":127.38853503184714},{"b":"b = 1.0","month":45,"rate_bpd":125.00000000000001},{"b":"b = 1.0","month":46,"rate_bpd":122.69938650306749},{"b":"b = 1.0","month":47,"rate_bpd":120.48192771084338},{"b":"b = 1.0","month":48,"rate_bpd":118.34319526627219},{"b":"b = 1.0","month":49,"rate_bpd":116.27906976744187},{"b":"b = 1.0","month":50,"rate_bpd":114.28571428571429},{"b":"b = 1.0","month":51,"rate_bpd":112.35955056179775},{"b":"b = 1.0","month":52,"rate_bpd":110.49723756906077},{"b":"b = 1.0","month":53,"rate_bpd":108.69565217391305},{"b":"b = 1.0","month":54,"rate_bpd":106.95187165775401},{"b":"b = 1.0","month":55,"rate_bpd":105.26315789473685},{"b":"b = 1.0","month":56,"rate_bpd":103.62694300518135},{"b":"b = 1.0","month":57,"rate_bpd":102.04081632653062},{"b":"b = 1.0","month":58,"rate_bpd":100.50251256281408},{"b":"b = 1.0","month":59,"rate_bpd":99.00990099009901},{"b":"b = 1.0","month":60,"rate_bpd":97.5609756097561},{"b":"b = 1.5","month":0,"rate_bpd":800.0},{"b":"b = 1.5","month":1,"rate_bpd":716.4215628872685},{"b":"b = 1.5","month":2,"rate_bpd":651.724500794643},{"b":"b = 1.5","month":3,"rate_bpd":599.8962858657379},{"b":"b = 1.5","month":4,"rate_bpd":557.2768758041947},{"b":"b = 1.5","month":5,"rate_bpd":521.4999282653351},{"b":"b = 1.5","month":6,"rate_bpd":490.9618691430698},{"b":"b = 1.5","month":7,"rate_bpd":464.53420110486405},{"b":"b = 1.5","month":8,"rate_bpd":441.3977375255928},{"b":"b = 1.5","month":9,"rate_bpd":420.94219314867206},{"b":"b = 1.5","month":10,"rate_bpd":402.7027846894224},{"b":"b = 1.5","month":11,"rate_bpd":386.3187711532278},{"b":"b = 1.5","month":12,"rate_bpd":371.5055149996345},{"b":"b = 1.5","month":13,"rate_bpd":358.0351584986975},{"b":"b = 1.5","month":14,"rate_bpd":345.72295097830835},{"b":"b = 1.5","month":15,"rate_bpd":334.4173781128726},{"b":"b = 1.5","month":16,"rate_bpd":323.9929075401236},{"b":"b = 1.5","month":17,"rate_bpd":314.3445713574713},{"b":"b = 1.5","month":18,"rate_bpd":305.3838616817531},{"b":"b = 1.5","month":19,"rate_bpd":297.0355801982901},{"b":"b = 1.5","month":20,"rate_bpd":289.2353911027419},{"b":"b = 1.5","month":21,"rate_bpd":281.9278996738687},{"b":"b = 1.5","month":22,"rate_bpd":275.0651284929548},{"b":"b = 1.5","month":23,"rate_bpd":268.60529790093995},{"b":"b = 1.5","month":24,"rate_bpd":262.5118416597306},{"b":"b = 1.5","month":25,"rate_bpd":256.7526062055319},{"b":"b = 1.5","month":26,"rate_bpd":251.29919449172812},{"b":"b = 1.5","month":27,"rate_bpd":246.12642465340448},{"b":"b = 1.5","month":28,"rate_bpd":241.21188056250526},{"b":"b = 1.5","month":29,"rate_bpd":236.5355364560455},{"b":"b = 1.5","month":30,"rate_bpd":232.07944168063898},{"b":"b = 1.5","month":31,"rate_bpd":227.82745453770957},{"b":"b = 1.5","month":32,"rate_bpd":223.7650164729659},{"b":"b = 1.5","month":33,"rate_bpd":219.87895960278684},{"b":"b = 1.5","month":34,"rate_bpd":216.15734193431567},{"b":"b = 1.5","month":35,"rate_bpd":212.5893057074338},{"b":"b = 1.5","month":36,"rate_bpd":209.16495513377973},{"b":"b = 1.5","month":37,"rate_bpd":205.87525048177082},{"b":"b = 1.5","month":38,"rate_bpd":202.71191599576912},{"b":"b = 1.5","month":39,"rate_bpd":199.66735957142953},{"b":"b = 1.5","month":40,"rate_bpd":196.73460246030288},{"b":"b = 1.5","month":41,"rate_bpd":193.90721756220984},{"b":"b = 1.5","month":42,"rate_bpd":191.1792750971349},{"b":"b = 1.5","month":43,"rate_bpd":188.54529463984508},{"b":"b = 1.5","month":44,"rate_bpd":186.0002026582916},{"b":"b = 1.5","month":45,"rate_bpd":183.53929482755595},{"b":"b = 1.5","month":46,"rate_bpd":181.15820249975525},{"b":"b = 1.5","month":47,"rate_bpd":178.8528628010044},{"b":"b = 1.5","month":48,"rate_bpd":176.61949190248757},{"b":"b = 1.5","month":49,"rate_bpd":174.45456107654624},{"b":"b = 1.5","month":50,"rate_bpd":172.3547752025507},{"b":"b = 1.5","month":51,"rate_bpd":170.31705343290562},{"b":"b = 1.5","month":52,"rate_bpd":168.33851176823663},{"b":"b = 1.5","month":53,"rate_bpd":166.41644732375923},{"b":"b = 1.5","month":54,"rate_bpd":164.54832409697514},{"b":"b = 1.5","month":55,"rate_bpd":162.73176007094546},{"b":"b = 1.5","month":56,"rate_bpd":160.96451550809277},{"b":"b = 1.5","month":57,"rate_bpd":159.2444823073081},{"b":"b = 1.5","month":58,"rate_bpd":157.5696743125281},{"b":"b = 1.5","month":59,"rate_bpd":155.93821847426457},{"b":"b = 1.5","month":60,"rate_bpd":154.34834677711902}]},"encoding":{"color":{"field":"b","title":"Exponent","type":"nominal"},"x":{"field":"month","title":"Months","type":"quantitative"},"y":{"field":"rate_bpd","title":"Production (bbl/day)","type":"quantitative"}},"height":350,"mark":{"strokeWidth":2,"type":"line"},"title":"Arps Decline Curves (q_i=800, D_i=0.12)","width":700}Synthetic Well Data
We generate 24 months of production data from a simulated Eagle Ford shale well with known “true” parameters. In practice, these parameters are unknown — that’s what Bayesian inference recovers.
# True parameters (unknown to the model)
true_qi = 850.0
true_di = 0.13
true_b = 0.85
true_sigma = 0.18
# Generate noisy monthly production
months = Enum.to_list(1..24)
rng = :rand.seed_s(:exsss, 42)
{observed, _rng} =
Enum.map_reduce(months, rng, fn t, rng ->
q_true = true_qi / :math.pow(1.0 + true_b * true_di * t, 1.0 / true_b)
{noise, rng} = :rand.normal_s(rng)
q_obs = q_true * :math.exp(true_sigma * noise)
{{t, q_obs, q_true}, rng}
end)
# Plot observed vs true
obs_data =
Enum.flat_map(observed, fn {t, q_obs, q_true} ->
[
%{"month" => t, "rate" => q_obs, "series" => "Observed"},
%{"month" => t, "rate" => q_true, "series" => "True decline"}
]
end)
Vl.new(width: 700, height: 350, title: "Eagle Ford Well — Observed Production")
|> Vl.data_from_values(obs_data)
|> Vl.encode_field(:x, "month", type: :quantitative, title: "Month")
|> Vl.encode_field(:y, "rate", type: :quantitative, title: "Production (bbl/day)")
|> Vl.encode_field(:color, "series", type: :nominal)
|> Vl.layers([
Vl.new()
|> Vl.mark(:point, size: 60)
|> Vl.transform(filter: "datum.series == 'Observed'"),
Vl.new()
|> Vl.mark(:line, stroke_dash: [6, 3], stroke_width: 2)
|> Vl.transform(filter: "datum.series == 'True decline'")
]){"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"month":1,"rate":834.6646072287461,"series":"Observed"},{"month":1,"rate":751.3939176983965,"series":"True decline"},{"month":2,"rate":477.3045621347843,"series":"Observed"},{"month":2,"rate":672.0483165512962,"series":"True decline"},{"month":3,"rate":608.5518325063268,"series":"Observed"},{"month":3,"rate":606.9252353369016,"series":"True decline"},{"month":4,"rate":785.7325267282438,"series":"Observed"},{"month":4,"rate":552.5873927913246,"series":"True decline"},{"month":5,"rate":365.8638012150139,"series":"Observed"},{"month":5,"rate":506.61244779397475,"series":"True decline"},{"month":6,"rate":577.2772088214894,"series":"Observed"},{"month":6,"rate":467.2461067053963,"series":"True decline"},{"month":7,"rate":578.7030011061236,"series":"Observed"},{"month":7,"rate":433.1879286538675,"series":"True decline"},{"month":8,"rate":474.1613721880113,"series":"Observed"},{"month":8,"rate":403.4542914289607,"series":"True decline"},{"month":9,"rate":356.909933579503,"series":"Observed"},{"month":9,"rate":377.2880028380391,"series":"True decline"},{"month":10,"rate":374.07745210872446,"series":"Observed"},{"month":10,"rate":354.0970733452601,"series":"True decline"},{"month":11,"rate":317.43917296347536,"series":"Observed"},{"month":11,"rate":333.41226174988225,"series":"True decline"},{"month":12,"rate":303.3349824568508,"series":"Observed"},{"month":12,"rate":314.8570224448178,"series":"True decline"},{"month":13,"rate":250.0648237281361,"series":"Observed"},{"month":13,"rate":298.12583522959716,"series":"True decline"},{"month":14,"rate":255.0687853317099,"series":"Observed"},{"month":14,"rate":282.9683184477741,"series":"True decline"},{"month":15,"rate":326.37666857668586,"series":"Observed"},{"month":15,"rate":269.17740642931216,"series":"True decline"},{"month":16,"rate":334.927257731246,"series":"Observed"},{"month":16,"rate":256.58043120818803,"series":"True decline"},{"month":17,"rate":298.1628738557441,"series":"Observed"},{"month":17,"rate":245.03231127918775,"series":"True decline"},{"month":18,"rate":335.301015506736,"series":"Observed"},{"month":18,"rate":234.41029030658044,"series":"True decline"},{"month":19,"rate":216.01543618594238,"series":"Observed"},{"month":19,"rate":224.60983054576397,"series":"True decline"},{"month":20,"rate":213.726781634412,"series":"Observed"},{"month":20,"rate":215.54137662703423,"series":"True decline"},{"month":21,"rate":171.38396147434122,"series":"Observed"},{"month":21,"rate":207.12778248193072,"series":"True decline"},{"month":22,"rate":222.1863622394998,"series":"Observed"},{"month":22,"rate":199.3022485972225,"series":"True decline"},{"month":23,"rate":173.0822178243466,"series":"Observed"},{"month":23,"rate":192.0066556545304,"series":"True decline"},{"month":24,"rate":199.15104510726954,"series":"Observed"},{"month":24,"rate":185.19020872366164,"series":"True decline"}]},"encoding":{"color":{"field":"series","type":"nominal"},"x":{"field":"month","title":"Month","type":"quantitative"},"y":{"field":"rate","title":"Production (bbl/day)","type":"quantitative"}},"height":350,"layer":[{"mark":{"size":60,"type":"point"},"transform":[{"filter":"datum.series == 'Observed'"}]},{"mark":{"strokeDash":[6,3],"strokeWidth":2,"type":"line"},"transform":[{"filter":"datum.series == 'True decline'"}]}],"title":"Eagle Ford Well — Observed Production","width":700}Bayesian Model
We define priors on the four unknown parameters and a custom likelihood that computes the log-probability of the observed data given the Arps model.
Priors:
-
log_qi ~ Normal(log(800), 0.5)— median 800 bbl/day, 95% CI: ~300-2100 -
log_Di ~ Normal(log(0.12), 0.4)— median 0.12/month, allows 0.05-0.30 -
b ~ Beta(5, 3)— mode ~0.67, range 0.2-1.0 -
sigma ~ HalfCauchy(0.3)— observation noise scale
Likelihood: Lognormal noise on production rates.
alias Exmc.{Builder, NUTS.Sampler, Dist}
# Prepare data tensors
t_data = Nx.tensor(Enum.map(observed, fn {t, _, _} -> t * 1.0 end))
q_obs_list = Enum.map(observed, fn {_, q, _} -> q end)
log_q_obs = Nx.tensor(Enum.map(q_obs_list, &:math.log/1))
# Custom logpdf: computes full DCA log-likelihood
dca_logpdf = fn _x, params ->
qi = Nx.exp(params.log_qi)
di = Nx.exp(params.log_di)
b = params.b
sigma = params.sigma
t = params.t_data
log_q_observed = params.log_q_obs
# Arps model: log(q) = log(qi) - (1/b) * log(1 + b*Di*t)
log_q_pred =
Nx.subtract(
Nx.log(qi),
Nx.multiply(Nx.divide(1.0, b), Nx.log(Nx.add(1.0, Nx.multiply(Nx.multiply(b, di), t))))
)
# Lognormal log-likelihood: -0.5 * sum((log_obs - log_pred)^2 / sigma^2) - N*log(sigma)
residuals = Nx.subtract(log_q_observed, log_q_pred)
n = Nx.tensor(Nx.size(log_q_observed) * 1.0)
Nx.subtract(
Nx.sum(
Nx.negate(Nx.divide(Nx.pow(residuals, 2), Nx.multiply(2.0, Nx.pow(sigma, 2))))
),
Nx.multiply(n, Nx.log(sigma))
)
end
# Build the model IR
ir = Builder.new_ir()
# Priors (working in log-space for qi and Di to ensure positivity)
ir = Builder.rv(ir, "log_qi", Dist.Normal, %{mu: Nx.tensor(:math.log(800.0)), sigma: Nx.tensor(0.5)})
ir = Builder.rv(ir, "log_di", Dist.Normal, %{mu: Nx.tensor(:math.log(0.12)), sigma: Nx.tensor(0.4)})
ir = Builder.rv(ir, "b", Dist.Beta, %{alpha: Nx.tensor(5.0), beta: Nx.tensor(3.0)}, transform: :logit)
ir = Builder.rv(ir, "sigma", Dist.HalfCauchy, %{scale: Nx.tensor(0.3)}, transform: :log)
# Custom likelihood
dca_dist = Dist.Custom.new(dca_logpdf, support: :real)
ir =
Dist.Custom.rv(ir, "ll", dca_dist, %{
log_qi: "log_qi",
log_di: "log_di",
b: "b",
sigma: "sigma",
t_data: t_data,
log_q_obs: log_q_obs
})
ir = Builder.obs(ir, "ll_obs", "ll", Nx.tensor(0.0))
IO.puts("Free parameters: #{inspect(Enum.sort(Map.keys(ir.nodes) -- ["ll", "ll_obs"]))}")
irFree parameters: ["b", "log_di", "log_qi", "sigma"]%Exmc.IR{
nodes: %{
"b" => %Exmc.Node{
id: "b",
op: {:rv, Exmc.Dist.Beta,
%{
alpha: #Nx.Tensor<
f32
5.0
>,
beta: #Nx.Tensor<
f32
3.0
>
}, :logit},
deps: [],
shape: nil,
dtype: nil
},
"ll" => %Exmc.Node{
id: "ll",
op: {:rv, Exmc.Dist.Custom,
%{
b: "b",
t_data: #Nx.Tensor<
f32[24]
[1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0]
>,
log_q_obs: #Nx.Tensor<
f32[24]
[6.727029800415039, 6.168154716491699, 6.4110822677612305, 6.666616439819336, 5.902261257171631, 6.358322620391846, 6.3607892990112305, 6.161547660827637, 5.877483367919922, 5.924462795257568, 5.760286331176758, 5.714837551116943, 5.5217204093933105, 5.541533470153809, 5.788052082061768, 5.813913345336914, 5.6976399421691895, 5.815028667449951, 5.375349998474121, 5.36469841003418, 5.143906593322754, 5.4035162925720215, 5.153766632080078, 5.294063568115234]
>,
log_qi: "log_qi",
log_di: "log_di",
sigma: "sigma",
__dist__: %Exmc.Dist.Custom{
logpdf_fn: #Function<41.81571850/2 in :erl_eval.expr/6>,
support: :real,
transform: nil,
sample_fn: nil
}
}},
deps: ["b", "log_qi", "log_di", "sigma"],
shape: nil,
dtype: nil
},
"ll_obs" => %Exmc.Node{
id: "ll_obs",
op: {:obs, "ll",
#Nx.Tensor<
f32
0.0
>, %{}},
deps: ["ll"],
shape: nil,
dtype: nil
},
"log_di" => %Exmc.Node{
id: "log_di",
op: {:rv, Exmc.Dist.Normal,
%{
sigma: #Nx.Tensor<
f32
0.4000000059604645
>,
mu: #Nx.Tensor<
f32
-2.1202635765075684
>
}},
deps: [],
shape: nil,
dtype: nil
},
"log_qi" => %Exmc.Node{
id: "log_qi",
op: {:rv, Exmc.Dist.Normal,
%{
sigma: #Nx.Tensor<
f32
0.5
>,
mu: #Nx.Tensor<
f32
6.684611797332764
>
}},
deps: [],
shape: nil,
dtype: nil
},
"sigma" => %Exmc.Node{
id: "sigma",
op: {:rv, Exmc.Dist.HalfCauchy,
%{
scale: #Nx.Tensor<
f32
0.30000001192092896
>
}, :log},
deps: [],
shape: nil,
dtype: nil
}
},
outputs: [],
ncp_info: %{},
data: nil
}MCMC Sampling
Run the NUTS sampler with 500 warmup + 500 draws.
init = %{"log_qi" => :math.log(800.0), "log_di" => :math.log(0.12), "b" => 0.7, "sigma" => 0.2}
t0 = System.monotonic_time(:millisecond)
{trace, stats} =
Sampler.sample(ir, init,
num_warmup: 500,
num_samples: 500,
seed: 42,
ncp: false
)
elapsed = System.monotonic_time(:millisecond) - t0
# Compute constrained parameters from log-space
qi_samples = trace["log_qi"] |> Nx.exp() |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
di_samples = trace["log_di"] |> Nx.exp() |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
b_samples = trace["b"] |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
sigma_samples = trace["sigma"] |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
n_div = Enum.count(stats.sample_stats, & &1.divergent)
IO.puts("Sampling complete in #{elapsed}ms")
IO.puts("Divergences: #{n_div}")
IO.puts("Step size: #{Float.round(stats.step_size, 4)}")
IO.puts("Samples: #{length(qi_samples)}")Sampling complete in 4114ms
Divergences: 0
Step size: 0.1577
Samples: 500:okDiagnostics
summary = Exmc.Diagnostics.summary(trace)
# Build a formatted table
rows =
for {name, s} <- Enum.sort(summary) do
true_val =
case name do
"log_qi" -> :math.log(true_qi)
"log_di" -> :math.log(true_di)
"b" -> true_b
"sigma" -> true_sigma
_ -> nil
end
label =
case name do
"log_qi" -> "log(q_i)"
"log_di" -> "log(D_i)"
n -> n
end
true_str = if true_val, do: Float.round(true_val, 4), else: "-"
IO.puts(
"#{String.pad_trailing(label, 10)} " <>
"mean=#{String.pad_trailing(Float.round(s.mean, 4) |> to_string(), 8)} " <>
"std=#{String.pad_trailing(Float.round(s.std, 4) |> to_string(), 8)} " <>
"true=#{true_str}"
)
end
:okb mean=0.6505 std=0.1643 true=0.85
log(D_i) mean=-2.2958 std=0.2371 true=-2.0402
log(q_i) mean=6.7035 std=0.1265 true=6.7452
sigma mean=0.2079 std=0.038 true=0.18:okTrace Plots
trace_data = fn samples, name ->
Enum.with_index(samples, fn val, i -> %{"iteration" => i, "value" => val, "param" => name} end)
end
all_traces =
trace_data.(qi_samples, "q_i (bbl/day)") ++
trace_data.(di_samples, "D_i (per month)") ++
trace_data.(b_samples, "b (exponent)") ++
trace_data.(sigma_samples, "sigma (noise)")
Vl.new(width: 700, height: 120, title: "MCMC Trace Plots")
|> Vl.data_from_values(all_traces)
|> Vl.mark(:line, opacity: 0.6, stroke_width: 0.5)
|> Vl.encode_field(:x, "iteration", type: :quantitative)
|> Vl.encode_field(:y, "value", type: :quantitative)
|> Vl.encode_field(:row, "param", type: :nominal, title: nil,
header: [label_font_size: 11])
|> Vl.resolve(:scale, y: :independent){"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"iteration":0,"param":"q_i (bbl/day)","value":686.2866439642702},{"iteration":1,"param":"q_i (bbl/day)","value":740.7775561018065},{"iteration":2,"param":"q_i (bbl/day)","value":857.1164045468029},{"iteration":3,"param":"q_i (bbl/day)","value":858.1817506064392},{"iteration":4,"param":"q_i (bbl/day)","value":739.1993891126873},{"iteration":5,"param":"q_i (bbl/day)","value":720.8868506949101},{"iteration":6,"param":"q_i (bbl/day)","value":1022.3493828130426},{"iteration":7,"param":"q_i (bbl/day)","value":690.1573918375683},{"iteration":8,"param":"q_i (bbl/day)","value":713.6059007898623},{"iteration":9,"param":"q_i (bbl/day)","value":825.6328325289971},{"iteration":10,"param":"q_i (bbl/day)","value":784.10289984588},{"iteration":11,"param":"q_i (bbl/day)","value":798.1929837610535},{"iteration":12,"param":"q_i (bbl/day)","value":650.466124318969},{"iteration":13,"param":"q_i (bbl/day)","value":846.8151423697443},{"iteration":14,"param":"q_i (bbl/day)","value":833.164406703329},{"iteration":15,"param":"q_i (bbl/day)","value":828.1360113275658},{"iteration":16,"param":"q_i (bbl/day)","value":836.037752553102},{"iteration":17,"param":"q_i (bbl/day)","value":818.3902751557017},{"iteration":18,"param":"q_i (bbl/day)","value":792.2201184529489},{"iteration":19,"param":"q_i (bbl/day)","value":846.4609064032398},{"iteration":20,"param":"q_i (bbl/day)","value":865.3426225256181},{"iteration":21,"param":"q_i (bbl/day)","value":995.676971840165},{"iteration":22,"param":"q_i (bbl/day)","value":846.2138817146184},{"iteration":23,"param":"q_i (bbl/day)","value":865.7779482652811},{"iteration":24,"param":"q_i (bbl/day)","value":673.3848368203167},{"iteration":25,"param":"q_i (bbl/day)","value":648.5589743212969},{"iteration":26,"param":"q_i (bbl/day)","value":621.4101810821873},{"iteration":27,"param":"q_i (bbl/day)","value":853.1891669825524},{"iteration":28,"param":"q_i (bbl/day)","value":751.6014703632977},{"iteration":29,"param":"q_i (bbl/day)","value":851.2978721274714},{"iteration":30,"param":"q_i (bbl/day)","value":717.4901594885479},{"iteration":31,"param":"q_i (bbl/day)","value":837.6947905685863},{"iteration":32,"param":"q_i (bbl/day)","value":648.8042761867332},{"iteration":33,"param":"q_i (bbl/day)","value":705.646237789798},{"iteration":34,"param":"q_i (bbl/day)","value":792.5925156819159},{"iteration":35,"param":"q_i (bbl/day)","value":787.5935335154287},{"iteration":36,"param":"q_i (bbl/day)","value":831.4283755678638},{"iteration":37,"param":"q_i (bbl/day)","value":776.2642687685789},{"iteration":38,"param":"q_i (bbl/day)","value":969.2583649509318},{"iteration":39,"param":"q_i (bbl/day)","value":654.0629345856604},{"iteration":40,"param":"q_i (bbl/day)","value":674.1307742523582},{"iteration":41,"param":"q_i (bbl/day)","value":716.2887220737205},{"iteration":42,"param":"q_i (bbl/day)","value":844.4573186093772},{"iteration":43,"param":"q_i (bbl/day)","value":850.5272664354426},{"iteration":44,"param":"q_i (bbl/day)","value":865.8269234121591},{"iteration":45,"param":"q_i (bbl/day)","value":899.9656496751808},{"iteration":46,"param":"q_i (bbl/day)","value":778.4999898903079},{"iteration":47,"param":"q_i (bbl/day)","value":788.5153583320216},{"iteration":48,"param":"q_i (bbl/day)","value":837.8799510601084},{"iteration":49,"param":"q_i (bbl/day)","value":833.4779538303676},{"iteration":50,"param":"q_i (bbl/day)","value":905.2366003912172},{"iteration":51,"param":"q_i (bbl/day)","value":700.6926887291867},{"iteration":52,"param":"q_i (bbl/day)","value":655.0102628525109},{"iteration":53,"param":"q_i (bbl/day)","value":675.2422894296164},{"iteration":54,"param":"q_i (bbl/day)","value":844.0543172714865},{"iteration":55,"param":"q_i (bbl/day)","value":881.8299856594217},{"iteration":56,"param":"q_i (bbl/day)","value":919.8667988492501},{"iteration":57,"param":"q_i (bbl/day)","value":870.3917872394327},{"iteration":58,"param":"q_i (bbl/day)","value":615.6379152420352},{"iteration":59,"param":"q_i (bbl/day)","value":747.8857226062765},{"iteration":60,"param":"q_i (bbl/day)","value":817.0709366814759},{"iteration":61,"param":"q_i (bbl/day)","value":775.8969211093945},{"iteration":62,"param":"q_i (bbl/day)","value":870.2277194509948},{"iteration":63,"param":"q_i (bbl/day)","value":933.410208456675},{"iteration":64,"param":"q_i (bbl/day)","value":867.2159876291144},{"iteration":65,"param":"q_i (bbl/day)","value":834.2666470779427},{"iteration":66,"param":"q_i (bbl/day)","value":817.855124829216},{"iteration":67,"param":"q_i (bbl/day)","value":1023.2173424622879},{"iteration":68,"param":"q_i (bbl/day)","value":700.7614488030341},{"iteration":69,"param":"q_i (bbl/day)","value":715.520497732395},{"iteration":70,"param":"q_i (bbl/day)","value":690.3589738135778},{"iteration":71,"param":"q_i (bbl/day)","value":787.421190187487},{"iteration":72,"param":"q_i (bbl/day)","value":745.6457771706293},{"iteration":73,"param":"q_i (bbl/day)","value":776.6676830645986},{"iteration":74,"param":"q_i (bbl/day)","value":793.6272849098959},{"iteration":75,"param":"q_i (bbl/day)","value":818.6851080168173},{"iteration":76,"param":"q_i (bbl/day)","value":806.1451667683131},{"iteration":77,"param":"q_i (bbl/day)","value":833.5928054257066},{"iteration":78,"param":"q_i (bbl/day)","value":663.9668967179609},{"iteration":79,"param":"q_i (bbl/day)","value":836.6211378660196},{"iteration":80,"param":"q_i (bbl/day)","value":801.7438295104041},{"iteration":81,"param":"q_i (bbl/day)","value":816.2595246565161},{"iteration":82,"param":"q_i (bbl/day)","value":822.58120657227},{"iteration":83,"param":"q_i (bbl/day)","value":883.5302810450667},{"iteration":84,"param":"q_i 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(noise)","value":0.22200656169512142},{"iteration":499,"param":"sigma (noise)","value":0.22478785476344904}]},"encoding":{"row":{"field":"param","header":{"labelFontSize":11},"title":null,"type":"nominal"},"x":{"field":"iteration","type":"quantitative"},"y":{"field":"value","type":"quantitative"}},"height":120,"mark":{"opacity":0.6,"strokeWidth":0.5,"type":"line"},"resolve":{"scale":{"y":"independent"}},"title":"MCMC Trace Plots","width":700}Parameter Recovery
Posterior distributions vs. true values:
# q_i posterior histogram with true value
qi_hist =
Enum.map(qi_samples, fn v -> %{"q_i" => v} end)
Vl.new(width: 350, height: 200, title: "Posterior: q_i (bbl/day)")
|> Vl.data_from_values(qi_hist)
|> Vl.layers([
Vl.new()
|> Vl.mark(:bar, opacity: 0.7, color: "steelblue")
|> Vl.encode_field(:x, "q_i", type: :quantitative, bin: [maxbins: 30], title: "q_i")
|> Vl.encode(:y, aggregate: :count),
Vl.new()
|> Vl.data_from_values([%{"true" => true_qi}])
|> Vl.mark(:rule, color: "red", stroke_width: 2, stroke_dash: [6, 3])
|> Vl.encode_field(:x, "true", type: :quantitative)
]){"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"q_i":686.2866439642702},{"q_i":740.7775561018065},{"q_i":857.1164045468029},{"q_i":858.1817506064392},{"q_i":739.1993891126873},{"q_i":720.8868506949101},{"q_i":1022.3493828130426},{"q_i":690.1573918375683},{"q_i":713.6059007898623},{"q_i":825.6328325289971},{"q_i":784.10289984588},{"q_i":798.1929837610535},{"q_i":650.466124318969},{"q_i":846.8151423697443},{"q_i":833.164406703329},{"q_i":828.1360113275658},{"q_i":836.037752553102},{"q_i":818.3902751557017},{"q_i":792.2201184529489},{"q_i":846.4609064032398},{"q_i":865.3426225256181},{"q_i":995.676971840165},{"q_i":846.2138817146184},{"q_i":865.7779482652811},{"q_i":673.3848368203167},{"q_i":648.5589743212969},{"q_i":621.4101810821873},{"q_i":853.1891669825524},{"q_i":751.6014703632977},{"q_i":851.2978721274714},{"q_i":717.4901594885479},{"q_i":837.6947905685863},{"q_i":648.8042761867332},{"q_i":705.646237789798},{"q_i":792.5925156819159},{"q_i":787.5935335154287},{"q_i":831.4283755678638},{"q_i":776.2642687685789},{"q_i":969.2583649509318},{"q_i":654.0629345856604},{"q_i":674.1307742523582},{"q_i":716.2887220737205},{"q_i":844.4573186093772},{"q_i":850.5272664354426},{"q_i":865.8269234121591},{"q_i":899.9656496751808},{"q_i":778.4999898903079},{"q_i":788.5153583320216},{"q_i":837.8799510601084},{"q_i":833.4779538303676},{"q_i":905.2366003912172},{"q_i":700.6926887291867},{"q_i":655.0102628525109},{"q_i":675.2422894296164},{"q_i":844.0543172714865},{"q_i":881.8299856594217},{"q_i":919.8667988492501},{"q_i":870.3917872394327},{"q_i":615.6379152420352},{"q_i":747.8857226062765},{"q_i":817.0709366814759},{"q_i":775.8969211093945},{"q_i":870.2277194509948},{"q_i":933.410208456675},{"q_i":867.2159876291144},{"q_i":834.2666470779427},{"q_i":817.855124829216},{"q_i":1023.2173424622879},{"q_i":700.7614488030341},{"q_i":715.520497732395},{"q_i":690.3589738135778},{"q_i":787.421190187487},{"q_i":745.6457771706293},{"q_i":776.6676830645986},{"q_i":793.6272849098959},{"q_i":818.6851080168173},{"q_i":806.1451667683131},{"q_i":833.5928054257066},{"q_i":663.9668967179609},{"q_i":836.6211378660196},{"q_i":801.7438295104041},{"q_i":816.2595246565161},{"q_i":822.58120657227},{"q_i":883.5302810450667},{"q_i":805.0026386005371},{"q_i":903.2316764400821},{"q_i":783.1651913496844},{"q_i":903.6510476210439},{"q_i":953.7590448368393},{"q_i":989.5554708425595},{"q_i":980.7459340521328},{"q_i":712.5521811251855},{"q_i":724.3288308980069},{"q_i":933.4769615849045},{"q_i":930.4573988925356},{"q_i":778.7313433763925},{"q_i":777.1870098258154},{"q_i":899.7449894586094},{"q_i":916.2371286415743},{"q_i":749.2790375611918},{"q_i":839.3881180645342},{"q_i":843.9605139800883},{"q_i":821.1726269153669},{"q_i":913.8131296631641},{"q_i":921.6714653693793},{"q_i":818.7268036941044},{"q_i":917.9528742790853},{"q_i":784.2091917485493},{"q_i":825.99178123849},{"q_i":677.8662608484473},{"q_i":667.2061380298043},{"q_i":696.5635374459486},{"q_i":775.839521713792},{"q_i":781.0948932463974},{"q_i":852.2445617276794},{"q_i":814.6376298721185},{"q_i":805.5959213753995},{"q_i":706.6039921890465},{"q_i":793.902682578892},{"q_i":694.6367848085159},{"q_i":794.4729214262305},{"q_i":874.4069146380205},{"q_i":746.6766657005276},{"q_i":830.7585399383399},{"q_i":927.281815211733},{"q_i":964.7120538739568},{"q_i":937.6834749479829},{"q_i":764.4041352255477},{"q_i":717.4639536814866},{"q_i":798.5258205466868},{"q_i":831.5867465043533},{"q_i":770.0218717139846},{"q_i":741.1704947639279},{"q_i":735.7615496512431},{"q_i":718.3215030627983},{"q_i":810.094811701588},{"q_i":779.8111479291309},{"q_i":965.8884772786754},{"q_i":786.7412178725464},{"q_i":803.73500611674},{"q_i":771.9747603340554},{"q_i":621.3137321637586},{"q_i":675.7975979716479},{"q_i":807.2288842802426},{"q_i":777.4759190902454},{"q_i":708.2445768578343},{"q_i":820.3395900190991},{"q_i":842.8073208963407},{"q_i":750.1640092279752},{"q_i":869.4226769570171},{"q_i":716.0361064299003},{"q_i":739.2723555823147},{"q_i":817.5703816682878},{"q_i":758.0775014856707},{"q_i":783.7305334729512},{"q_i":868.2807791171033},{"q_i":763.2399757173089},{"q_i":835.5084379906372},{"q_i":808.6215470076908},{"q_i":799.0964633899274},{"q_i":871.8030464612843},{"q_i":1191.9810330252121},{"q_i":1199.4682885327068},{"q_i":827.0347926233921},{"q_i":805.1875858844925},{"q_i":669.1811848337228},{"q_i":674.6131953084941},{"q_i":668.9210290923074},{"q_i":662.454339269066},{"q_i":665.7739530233247},{"q_i":635.5062501319385},{"q_i":759.9126653252433},{"q_i":711.8989259035329},{"q_i":758.1575295566906},{"q_i":838.600825651331},{"q_i":968.7202832514047},{"q_i":882.2025039672116},{"q_i":679.4670122758126},{"q_i":784.4033844649991},{"q_i":930.2230445691473},{"q_i":944.9270357742579},{"q_i":931.1134623645125},{"q_i":912.7634450629891},{"q_i":876.430323557652},{"q_i":901.297092991757},{"q_i":878.5931371226777},{"q_i":853.0289543821646},{"q_i":864.6772935706748},{"q_i":831.742888255897},{"q_i":931.5133214518911},{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q_i (bbl/day)","width":350}# Joint posterior: q_i vs D_i (shows parameter correlation)
joint_data =
Enum.zip(qi_samples, di_samples)
|> Enum.map(fn {q, d} -> %{"q_i" => q, "D_i" => d} end)
Vl.new(width: 400, height: 400, title: "Joint Posterior: q_i vs D_i")
|> Vl.data_from_values(joint_data)
|> Vl.layers([
Vl.new()
|> Vl.mark(:point, opacity: 0.3, size: 8, color: "steelblue")
|> Vl.encode_field(:x, "q_i", type: :quantitative, title: "q_i (bbl/day)")
|> Vl.encode_field(:y, "D_i", type: :quantitative, title: "D_i (per month)"),
Vl.new()
|> Vl.data_from_values([%{"q_i" => true_qi, "D_i" => true_di}])
|> Vl.mark(:point, size: 200, shape: "cross", color: "red", stroke_width: 3)
|> Vl.encode_field(:x, "q_i", type: :quantitative)
|> Vl.encode_field(:y, "D_i", type: :quantitative)
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(bbl/day)","type":"quantitative"},"y":{"field":"D_i","title":"D_i (per month)","type":"quantitative"}},"mark":{"color":"steelblue","opacity":0.3,"size":8,"type":"point"}},{"data":{"values":[{"D_i":0.13,"q_i":850.0}]},"encoding":{"x":{"field":"q_i","type":"quantitative"},"y":{"field":"D_i","type":"quantitative"}},"mark":{"color":"red","shape":"cross","size":200,"strokeWidth":3,"type":"point"}}],"title":"Joint Posterior: q_i vs D_i","width":400}Production Forecast: P10 / P50 / P90
The key deliverable for petroleum engineers. Each posterior sample generates a different forecast curve. We take percentiles at each future time point to get probabilistic production bands.
# Forecast 60 months into the future
forecast_months = Enum.to_list(1..60)
n_draws = min(200, length(qi_samples))
# Sample posterior draws and compute forecasts
forecast_curves =
Enum.zip([
Enum.take(qi_samples, n_draws),
Enum.take(di_samples, n_draws),
Enum.take(b_samples, n_draws)
])
|> Enum.map(fn {qi, di, b} ->
Enum.map(forecast_months, fn t ->
qi / :math.pow(1.0 + b * di * t, 1.0 / b)
end)
end)
# Compute P10, P50, P90 at each time point
percentiles =
Enum.map(Enum.with_index(forecast_months), fn {t, idx} ->
rates = Enum.map(forecast_curves, fn curve -> Enum.at(curve, idx) end) |> Enum.sort()
n = length(rates)
p10 = Enum.at(rates, round(0.1 * n))
p50 = Enum.at(rates, round(0.5 * n))
p90 = Enum.at(rates, round(0.9 * n))
%{month: t, p10: p10, p50: p50, p90: p90}
end)
# Build chart data
band_data =
Enum.flat_map(percentiles, fn p ->
[
%{"month" => p.month, "rate" => p.p10, "band" => "P10 (conservative)"},
%{"month" => p.month, "rate" => p.p50, "band" => "P50 (median)"},
%{"month" => p.month, "rate" => p.p90, "band" => "P90 (optimistic)"}
]
end)
obs_points =
Enum.map(observed, fn {t, q, _} -> %{"month" => t, "rate" => q, "band" => "Observed"} end)
Vl.new(width: 700, height: 400, title: "Production Forecast: P10 / P50 / P90")
|> Vl.data_from_values(band_data ++ obs_points)
|> Vl.layers([
# P10-P90 area band
Vl.new()
|> Vl.data_from_values(
Enum.map(percentiles, fn p ->
%{"month" => p.month, "p10" => p.p10, "p90" => p.p90}
end)
)
|> Vl.mark(:area, opacity: 0.2, color: "steelblue")
|> Vl.encode_field(:x, "month", type: :quantitative, title: "Month")
|> Vl.encode_field(:y, "p10", type: :quantitative, title: "Production (bbl/day)")
|> Vl.encode_field(:y2, "p90"),
# P50 line
Vl.new()
|> Vl.data_from_values(
Enum.map(percentiles, fn p -> %{"month" => p.month, "rate" => p.p50} end)
)
|> Vl.mark(:line, stroke_width: 2, color: "steelblue")
|> Vl.encode_field(:x, "month", type: :quantitative)
|> Vl.encode_field(:y, "rate", type: :quantitative),
# Observed data points
Vl.new()
|> Vl.data_from_values(obs_points)
|> Vl.mark(:point, size: 40, color: "black", filled: true)
|> Vl.encode_field(:x, "month", type: :quantitative)
|> Vl.encode_field(:y, "rate", type: :quantitative)
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Forecast: P10 / P50 / P90","width":700}EUR Distribution
Estimated Ultimate Recovery — total barrels the well will produce over its lifetime.
For hyperbolic decline with b < 1: EUR = q_i / (D_i * (1 - b)).
# Compute EUR for each posterior draw (to economic limit of 5 bbl/day over 30 years)
eur_samples =
Enum.zip([qi_samples, di_samples, b_samples])
|> Enum.map(fn {qi, di, b} ->
# Cumulative production over 360 months (30 years)
cum =
Enum.reduce(1..360, 0.0, fn t, acc ->
q = qi / :math.pow(max(1.0 + b * di * t, 1.0e-10), 1.0 / max(b, 0.01))
# Monthly volume = daily rate * 30.4 days/month
acc + max(q, 0.0) * 30.4
end)
# Convert to thousands of barrels
cum / 1000.0
end)
|> Enum.filter(&is_number/1)
eur_sorted = Enum.sort(eur_samples)
n = length(eur_sorted)
eur_p10 = Enum.at(eur_sorted, round(0.1 * n))
eur_p50 = Enum.at(eur_sorted, round(0.5 * n))
eur_p90 = Enum.at(eur_sorted, round(0.9 * n))
# Deterministic EUR (single best-fit)
det_eur =
Enum.reduce(1..360, 0.0, fn t, acc ->
q = true_qi / :math.pow(1.0 + true_b * true_di * t, 1.0 / true_b)
acc + q * 30.4
end) / 1000.0
IO.puts("EUR Distribution (kbbl):")
IO.puts(" P10 (conservative): #{Float.round(eur_p10, 0)}")
IO.puts(" P50 (median): #{Float.round(eur_p50, 0)}")
IO.puts(" P90 (optimistic): #{Float.round(eur_p90, 0)}")
IO.puts(" Deterministic: #{Float.round(det_eur, 0)}")
eur_hist = Enum.map(eur_samples, fn v -> %{"EUR_kbbl" => v} end)
Vl.new(width: 600, height: 300, title: "EUR Distribution (thousands of barrels)")
|> Vl.data_from_values(eur_hist)
|> Vl.layers([
Vl.new()
|> Vl.mark(:bar, opacity: 0.7, color: "steelblue")
|> Vl.encode_field(:x, "EUR_kbbl", type: :quantitative, bin: [maxbins: 30], title: "EUR (kbbl)")
|> Vl.encode(:y, aggregate: :count),
# P50 line
Vl.new()
|> Vl.data_from_values([%{"v" => eur_p50}])
|> Vl.mark(:rule, color: "black", stroke_width: 2)
|> Vl.encode_field(:x, "v", type: :quantitative),
# Deterministic point
Vl.new()
|> Vl.data_from_values([%{"v" => det_eur}])
|> Vl.mark(:rule, color: "red", stroke_width: 2, stroke_dash: [6, 3])
|> Vl.encode_field(:x, "v", type: :quantitative)
])EUR Distribution (kbbl):
P10 (conservative): 467.0
P50 (median): 573.0
P90 (optimistic): 682.0
Deterministic: 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(kbbl)","type":"quantitative"},"y":{"aggregate":"count"}},"mark":{"color":"steelblue","opacity":0.7,"type":"bar"}},{"data":{"values":[{"v":572.59140436725}]},"encoding":{"x":{"field":"v","type":"quantitative"}},"mark":{"color":"black","strokeWidth":2,"type":"rule"}},{"data":{"values":[{"v":623.9027790172873}]},"encoding":{"x":{"field":"v","type":"quantitative"}},"mark":{"color":"red","strokeDash":[6,3],"strokeWidth":2,"type":"rule"}}],"title":"EUR Distribution (thousands of barrels)","width":600}Live Updating: Posterior Narrows as Data Arrives
The key eXMC value proposition: as new monthly production data arrives, the posterior updates and uncertainty shrinks. Here we simulate this by running MCMC with 6, 12, and 24 months of data.
run_dca = fn months_of_data ->
subset = Enum.take(observed, months_of_data)
t_sub = Nx.tensor(Enum.map(subset, fn {t, _, _} -> t * 1.0 end))
log_q_sub = Nx.tensor(Enum.map(subset, fn {_, q, _} -> :math.log(q) end))
ir_sub = Builder.new_ir()
ir_sub = Builder.rv(ir_sub, "log_qi", Dist.Normal, %{mu: Nx.tensor(:math.log(800.0)), sigma: Nx.tensor(0.5)})
ir_sub = Builder.rv(ir_sub, "log_di", Dist.Normal, %{mu: Nx.tensor(:math.log(0.12)), sigma: Nx.tensor(0.4)})
ir_sub = Builder.rv(ir_sub, "b", Dist.Beta, %{alpha: Nx.tensor(5.0), beta: Nx.tensor(3.0)}, transform: :logit)
ir_sub = Builder.rv(ir_sub, "sigma", Dist.HalfCauchy, %{scale: Nx.tensor(0.3)}, transform: :log)
dist_sub = Dist.Custom.new(dca_logpdf, support: :real)
ir_sub =
Dist.Custom.rv(ir_sub, "ll", dist_sub, %{
log_qi: "log_qi", log_di: "log_di", b: "b", sigma: "sigma",
t_data: t_sub, log_q_obs: log_q_sub
})
ir_sub = Builder.obs(ir_sub, "ll_obs", "ll", Nx.tensor(0.0))
{trace_sub, _} =
Sampler.sample(ir_sub, init, num_warmup: 500, num_samples: 300, seed: 42, ncp: false)
qi_s = trace_sub["log_qi"] |> Nx.exp() |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
di_s = trace_sub["log_di"] |> Nx.exp() |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
b_s = trace_sub["b"] |> Nx.to_flat_list() |> Enum.filter(&is_number/1)
# Compute EUR distribution
eurs =
Enum.zip([qi_s, di_s, b_s])
|> Enum.map(fn {qi, di, b} ->
Enum.reduce(1..360, 0.0, fn t, acc ->
q = qi / :math.pow(max(1.0 + b * di * t, 1.0e-10), 1.0 / max(b, 0.01))
acc + max(q, 0.0) * 30.4
end) / 1000.0
end)
|> Enum.filter(&is_number/1)
eurs_sorted = Enum.sort(eurs)
n = length(eurs_sorted)
{Enum.at(eurs_sorted, round(0.1 * n)), Enum.at(eurs_sorted, round(0.5 * n)),
Enum.at(eurs_sorted, round(0.9 * n))}
end
# Run with increasing data
IO.puts("Sequential updating — EUR narrows as data arrives:\n")
IO.puts("Months | P10 (kbbl) | P50 (kbbl) | P90 (kbbl) | Range")
IO.puts("-------|------------|------------|------------|------")
for n_months <- [6, 12, 24] do
{p10, p50, p90} = run_dca.(n_months)
range = Float.round(p90 - p10, 0)
IO.puts(
"#{String.pad_leading(to_string(n_months), 6)} " <>
"| #{String.pad_leading(Float.round(p10, 0) |> to_string(), 10)} " <>
"| #{String.pad_leading(Float.round(p50, 0) |> to_string(), 10)} " <>
"| #{String.pad_leading(Float.round(p90, 0) |> to_string(), 10)} " <>
"| #{range}"
)
endSequential updating — EUR narrows as data arrives:
Months | P10 (kbbl) | P50 (kbbl) | P90 (kbbl) | Range
-------|------------|------------|------------|------
6 | 311.0 | 502.0 | 766.0 | 455.0
12 | 396.0 | 548.0 | 732.0 | 337.0
24 | 448.0 | 572.0 | 684.0 | 237.0[:ok, :ok, :ok]Summary
This notebook demonstrated:
- Bayesian DCA recovers the true decline parameters from noisy production data
- Full posterior provides P10/P50/P90 forecasts — not just a single best guess
- EUR distribution gives risk-adjusted reserves estimates for booking and financing
- Live updating — each new month of data narrows the uncertainty automatically
With eXMC’s sample_stream/4, this entire analysis runs as a live dashboard
where posteriors update in real time as new production data arrives from the field.
