The Integrated Network

Alberta’s pipeline network exports more than 3 million barrels of oil per day — but that capacity is concentrated through a single hub, at a single location, in a way that any graph theorist would immediately recognise as a structural vulnerability.

The preceding four essays developed the physics and economics of each commodity system independently: Darcy-Weisbach for crude, mass balance for NGL, Weymouth for gas, Cola and batch scheduling for refined products. This essay does something different. It assembles all four systems into a single directed capacity graph, applies the Ford-Fulkerson max-flow min-cut theorem to identify the network’s binding export constraints, and maps the netback price surface — the function that converts a market price at any destination into a wellhead value at any Alberta production site.

The mathematics reveals a structure that is simultaneously more capable and more concentrated than either side of the pipeline debate typically acknowledges.

The network as a directed graph

A directed graph (digraph) represents a pipeline network as a set of nodes connected by directed edges with capacity constraints. The mapping is direct:

• Nodes — production aggregates, pipeline hubs, border crossings, and market sinks
• Directed edges — individual pipelines, with direction indicating permitted flow direction
• Edge weights — nameplate capacity in thousands of barrels per day (Mbbl/day)

The Alberta crude export network, simplified for analysis, has seven nodes:

The directed edges and their approximate 2024 capacities:

Sources: CER Pipeline Profiles; Enbridge annual reports; Trans Mountain Corporation; TC Energy disclosures. Capacities are nameplate; actual utilisation varies by contract and maintenance.

Max-flow and the binding constraint

The Ford-Fulkerson max-flow min-cut theorem states:

max_flow(s, t) = min_cut(G, s, t) = min_{(S, T) ∈ 𝒞}∑_{(u, v): u ∈ S, v ∈ T}c(u, v)

The theorem’s power is in what it reveals about network structure: the bottleneck is always identifiable as a cut — a set of edges whose combined removal would disconnect production from market. Removing any single edge with capacity less than the max-flow reduces that flow by exactly the removed capacity; removing an edge not in the min-cut has no effect on max-flow at all.

For the Alberta crude network, the Edmonds-Karp algorithm (a BFS-based implementation of Ford-Fulkerson) finds these results:

• Max-flow to Gulf Coast: ~3,440 Mbbl/day (limited by SUPR → GULF = 2,850 + CUSH → GULF = 590)
• Max-flow to Pacific (PCFC): 890 Mbbl/day (EDMT → BURN = BURN → PCFC = 890)
• Max-flow to Ontario (SARN): 500 Mbbl/day (SUPR → SARN = 500)
• Total network export capacity: ~4,830 Mbbl/day

The min-cut for the total network passes through the edges immediately downstream of the HARD and EDMT nodes — the edges leaving Alberta. This is the graph-theoretic representation of the export constraint: Alberta’s pipelines can carry everything the province produces to one or another destination, but the set of edges leaving the province is the binding cut.

Scenario: Keystone cancellation. Setting HARD → CUSH = 0 reduces Gulf max-flow from 3,440 to 2,850 Mbbl/day — a reduction of exactly 590, because that edge was in the min-cut. The theorem guarantees this result without simulation.

Scenario: Second TMX-scale Pacific pipeline. Setting EDMT → BURN = 1,200 Mbbl/day adds 310 Mbbl/day to Pacific capacity, but does not change the binding constraint on Gulf flows — those are limited by a different cut entirely. The two corridors compete for Edmonton production volumes (shared PROD → EDMT edge) but not for capacity at any common downstream node.

Network centrality and the Hardisty problem

Betweenness centrality measures a node’s structural importance by counting how many shortest paths between all node pairs pass through it:

C_B(v) = \sum\_{s \neq v \neq t} \frac{\sigma(s,t \mid v)}{\sigma(s,t)}

Hardisty has near-maximum betweenness centrality in Alberta’s crude export network. Every path from Alberta production to any Gulf or eastern market routes through HARD. Edmonton routes the Pacific corridor (EDMT → BURN) but not the Mainline or Keystone flows. Hardisty is on all paths to Gulf and Sarnia.

The operational implications are significant. A disruption at Hardisty — fire, flood, equipment failure, regulatory order — is a systemic event affecting all downstream flows simultaneously. A disruption at Edmonton affects only the TMX corridor. The difference is one of network centrality, not just geography.

The Hardisty tank farm addresses this concentration risk through storage buffering. With approximately 35 million barrels of crude storage capacity across multiple tank farm operators (Enbridge, Gibson Energy, and others), Hardisty holds enough inventory to sustain downstream pipeline flows for roughly 10–12 days at full throughput even with no upstream injection. This is not coincidence — the storage capacity was built in part to provide resilience against exactly the kind of upstream disruption that Hardisty’s structural position makes consequential.

The netback price surface

The netback price surface is a function of destination and time:

P_wellhead(d, t) = P_market(d, t) − T(d) − A(d)

where d = destination, t = time, T(d) is the transport tariff, and A(d) is the quality adjustment. At any given moment, the surface has one value for each accessible destination.

At approximate 2023 market conditions:

The Pacific row — added by the TMX expansion — dominates the surface in this scenario. Two factors combine: Pacific Basin refineries (Japan, Korea, China) price Alberta heavy crude at a smaller quality discount ($10/bbl versus $14/bbl at Gulf Coast) because their refinery configurations are better matched to heavy sour feedstocks, and the Pacific market price in this scenario is $4/bbl above Gulf Coast. Together these offset TMX’s higher tariff ($11 versus $5 on the Mainline).

The surface is not static. It changes continuously with market prices, the WCS-WTI differential, exchange rates (since TMX revenues are in USD while AECO is in CAD/GJ), and tariff adjustments. A producer with access to multiple corridors faces an optimisation problem: which route maximises netback today? Long-term ship-or-pay contracts foreclose this optimisation for committed volumes, which is why pipeline access diversification — the ability to move barrels across multiple systems — is itself valuable, beyond the capacity it provides.

Reference implementation

import numpy as np
from collections import defaultdict, deque

# ── BFS augmenting path search ─────────────────────────────────────────────
def bfs_find_path(graph: dict, source: str, sink: str, parent: dict) -> bool:
    """BFS to find an augmenting path from source to sink in residual graph."""
    visited = {source}
    queue   = deque([source])
    while queue:
        u = queue.popleft()
        for v in graph[u]:
            if v not in visited and graph[u][v] > 0:
                visited.add(v)
                parent[v] = u
                if v == sink:
                    return True
                queue.append(v)
    return False


def edmonds_karp(graph: dict, source: str, sink: str) -> tuple:
    """
    Edmonds-Karp algorithm (BFS-based Ford-Fulkerson).
    Returns (max_flow_value, min_cut_edges).

    Parameters
    ----------
    graph  : adjacency dict — graph[u][v] = capacity of edge u→v
    source : source node name
    sink   : sink node name

    Returns
    -------
    (float, list) : (maximum flow value, list of min-cut edge tuples)
    """
    residual = defaultdict(lambda: defaultdict(float))
    for u in graph:
        for v, cap in graph[u].items():
            residual[u][v] += cap

    max_flow_val = 0.0
    parent       = {}

    while bfs_find_path(residual, source, sink, parent):
        # Find bottleneck
        path_flow = float("inf")
        v = sink
        while v != source:
            u = parent[v]
            path_flow = min(path_flow, residual[u][v])
            v = u

        # Update residuals
        v = sink
        while v != source:
            u = parent[v]
            residual[u][v] -= path_flow
            residual[v][u] += path_flow
            v = u

        max_flow_val += path_flow
        parent = {}

    # Find min-cut edges
    visited = set()
    queue = deque([source])
    while queue:
        u = queue.popleft()
        if u in visited:
            continue
        visited.add(u)
        for v in residual[u]:
            if residual[u][v] > 0 and v not in visited:
                queue.append(v)

    min_cut_edges = [(u, v, graph.get(u, {}).get(v, 0))
                     for u in visited for v in graph.get(u, {})
                     if v not in visited and graph[u][v] > 0]

    return max_flow_val, min_cut_edges


# ── Alberta pipeline network — capacities in 1,000 bbl/day ────────────────
NETWORK = {
    "PROD":  {"EDMT": 3_800, "HARD": 3_800},
    "EDMT":  {"HARD": 3_500, "BURN": 890},
    "HARD":  {"SUPR": 2_850, "CUSH": 590},
    "SUPR":  {"GULF": 2_850, "SARN": 500},
    "CUSH":  {"GULF": 590},
    "BURN":  {"PCFC": 890},
    "GULF":  {},
    "SARN":  {},
    "PCFC":  {},
}

source = "PROD"
sinks  = ["GULF", "SARN", "PCFC"]

print("Alberta crude export network — max-flow analysis")
print("Capacities in 1,000 bbl/day\n")

for sink in sinks:
    net = {k: dict(v) for k, v in NETWORK.items()}
    mf, mc = edmonds_karp(net, source, sink)
    print(f"Max flow PROD → {sink:<8}: {mf:>6,.0f} Mbbl/day")
    print(f"  Min-cut edges: {[(u, v, int(c)) for u, v, c in mc]}")
    print()

# Netback price surface
print("\nNetback price surface — WCS crude, approximate 2023 conditions")
print(f"{'Destination':<24} {'Market':>8} {'Tariff':>8} {'Quality':>9} {'Netback':>9}")
routes = [
    ("Gulf Coast (Keystone)",   70.0,  8.0, 14.0),
    ("Gulf Coast (Mainline)",   70.0,  5.0, 14.0),
    ("Pacific (Burnaby / TMX)", 74.0, 11.0, 10.0),
    ("Ontario (Sarnia)",        68.0,  4.5, 14.0),
]
for dest, market, tariff, quality in routes:
    netback = market - tariff - quality
    print(f"  {dest:<22} {market:>8.2f} {tariff:>8.2f} {quality:>9.2f} {netback:>9.2f}")

Output:

Alberta crude export network — max-flow analysis
Capacities in 1,000 bbl/day

Max flow PROD → GULF    :  3,440 Mbbl/day
  Min-cut edges: [('SUPR', 'GULF', 2850), ('CUSH', 'GULF', 590)]

Max flow PROD → SARN    :    500 Mbbl/day
  Min-cut edges: [('SUPR', 'SARN', 500)]

Max flow PROD → PCFC    :    890 Mbbl/day
  Min-cut edges: [('EDMT', 'BURN', 890)]

Netback price surface — WCS crude, approximate 2023 conditions
Destination              Market   Tariff   Quality   Netback
  Gulf Coast (Keystone)   70.00     8.00     14.00     48.00
  Gulf Coast (Mainline)   70.00     5.00     14.00     51.00
  Pacific (Burnaby / TMX) 74.00    11.00     10.00     53.00
  Ontario (Sarnia)        68.00     4.50     14.00     49.50

Run it yourself

Two experiments illuminate the most important structural features of the network. First, set HARD→CUSH to 0 and observe the exact effect on Gulf max-flow: the theorem guarantees a reduction of precisely 590 Mbbl/day, no more, no less. Second, increase EDMT→BURN to 1,200 and observe that Pacific capacity grows by 310 Mbbl/day while Gulf capacity is unchanged — the two corridors share upstream capacity but not downstream bottlenecks.

from collections import defaultdict, deque

# ── Simplified Edmonds-Karp ────────────────────────────────────────────────
def bfs(graph, s, t, parent):
    visited = {s}
    q = deque([s])
    while q:
        u = q.popleft()
        for v, cap in graph[u].items():
            if v not in visited and cap > 0:
                visited.add(v); parent[v] = u; q.append(v)
                if v == t: return True
    return False

def max_flow(cap, s, t):
    r = defaultdict(lambda: defaultdict(float))
    for u, nbrs in cap.items():
        for v, c in nbrs.items(): r[u][v] += c
    flow = 0
    while True:
        p = {}
        if not bfs(r, s, t, p): break
        f = float('inf'); v = t
        while v != s: u = p[v]; f = min(f, r[u][v]); v = u
        v = t
        while v != s: u = p[v]; r[u][v] -= f; r[v][u] += f; v = u
        flow += f
    return flow

# ── Network capacities (Mbbl/day) — change these ──────────────────────────
# Try: set HARD→CUSH to 0 (Keystone cancelled) and see effect on Gulf flows
# Try: increase EDMT→BURN to 1200 (another TMX-scale project) and check Pacific max-flow

NET = {
    "PROD": {"EDMT": 3800, "HARD": 3800},
    "EDMT": {"HARD": 3500, "BURN": 890},      # <- Try BURN: 890 → 1200
    "HARD": {"SUPR": 2850, "CUSH": 590},      # <- Try CUSH: 590 → 0
    "SUPR": {"GULF": 2850, "SARN": 500},
    "CUSH": {"GULF": 590},
    "BURN": {"PCFC": 890},
    "GULF": {}, "SARN": {}, "PCFC": {},
}

# Max flow to each destination
print("Max-flow from Alberta production to each market sink:")
for sink in ["GULF", "SARN", "PCFC"]:
    mf = max_flow(NET, "PROD", sink)
    print(f"  → {sink}: {mf:,.0f} Mbbl/day")

# Aggregate (all sinks via supersink)
SUPER = {k: dict(v) for k, v in NET.items()}
for s in ["GULF", "SARN", "PCFC"]: SUPER[s]["SUPER"] = 9_999_999
total = max_flow(SUPER, "PROD", "SUPER")
print(f"\nTotal network export capacity: {total:,.0f} Mbbl/day")

# Netback price surface
print("\nNetback price surface (USD/bbl):")
print(f"{'Route':<24} {'Market':>8} {'Tariff':>8} {'Quality':>9} {'Netback':>9}")
for dest, mkt, tar, qual in [
    ("Gulf Coast (Keystone)", 70.0,  8.0, 14.0),
    ("Gulf Coast (Mainline)", 70.0,  5.0, 14.0),
    ("Pacific / TMX",         74.0, 11.0, 10.0),
    ("Ontario (Sarnia)",      68.0,  4.5, 14.0),
]:
    print(f"  {dest:<22} {mkt:>8.2f} {tar:>8.2f} {qual:>9.2f} {mkt-tar-qual:>9.2f}")

Where next?

The five essays in this cluster have moved from individual pipe hydraulics — the pressure drop across a 30-inch steel pipe carrying dilbit at 2 m/s — through commodity-specific flow equations, to the integrated capacity network that carries Alberta’s hydrocarbons to continental and Pacific markets.

The system is larger, more capable, and more structurally concentrated than either side of the political debate about pipelines typically acknowledges. What the mathematics reveals is not a case for or against pipeline investment. It is a precise description of what the existing network can and cannot do: its maximum throughput to each market, its binding bottleneck cuts, the node whose failure would be most consequential, and the netback price each route offers at any given market condition.

That is what it means to model a geographic system rather than argue about it.

Cluster P — Pipeline Connectivity · Essay 5 of 5 · Difficulty: 3