The first swells of a mammoth new refugee flood are starting to overwhelm international relief efforts in northern Iraq, as more than a million residents in and around the Iraqi city of Mosul are beginning to flee before a vicious battle is joined between ISIS and forces trying to displace the terrorist forces.

They are the latest, enormously disruptive chapter of what President Obama termed a global refugee crisis of “epic proportions” at a special, 32-nation summit he convened on Tuesday at the United Nations.

The effects of the growing tide of desperate “internally displaced persons,” or IDPs, as the U.N. calls them, rising around the northern Iraqi city are still unpredictable. But they already seem to be well beyond the over-stretched and underfunded, U.N.-coordinated humanitarian efforts that are still being put in place to contain and help them.

The coming challenge exodus “is beyond my ability to articulate,” warns Su’ad Jarbawi, Iraq country director for Mercy Corps, one of the world’s largest non-government relief organizations, which has worked in Iraq since 2003.

“The order of magnitude we are going to see is tremendous,” she told Fox News from her base of operations in the nearby Iraqi city of Erbil. “Do we know how it is going to unfold? No.”

Relief planners like Jarbawi are still trying frantically, juggling funds and reworking plans to preposition supplies and camps for as many as 1 million to 1.5 million civilians who could be fleeing Mosul within weeks, or even days, when U.S.- and NATO-backed Iraqi government forces assault the city to retake it from the ISIS jihadists who have held it since 2014.

Where exactly they will flee is also not known.

“I’m looking at version 12 of a Mosul planning map,” one coordinator told Fox News. “It’s not going to be the last.”

“We plan with a very high degree of uncertainty.”

Even as the relief contingents scramble, about 135,000 Iraqi residents of small towns and villages south of Mosul have fled preliminary Iraqi government advances in the so-called “Mosul corridor” that were preliminaries to the main drive on the city. The larger assault could happen in a few weeks, or even days, depending on the Iraqi government’s coherence and determination.

The early departures, which one humanitarian expert described as only a “trickle,” are forcing additional grim forms of relief triage on aid organizations in northern Iraq that are struggling to deal with massive earlier displacements caused by ISIS and recent government counter-offenses in neighboring Anbar province.

“We’ve had to stop actions in other areas,” one relief coordinator told Fox News. “This is only a taste of what we are going to get.”

As newcomers arrive, they also force humanitarian organizations to reduce, or even eliminate, relief supplies and cash handed over to earlier refugees, who are supposed to transit, in many cases, the relief from the Iraqi government.

But the government, in turn, is increasingly swallowed in a financial crisis brought on, atop the costs of war and human upheaval, by dramatically shrinking international oil prices.

Moreover, as one aid worker put it, “Iraqi [government] ministries are absolutely overwhelmed. They don’t have systems to carry the load, along with the support, discipline and training.”

The outcome: no money. “We don’t have enough to cover our needs today,” one humanitarian official told Fox News. That is especially true of life-preserving supplies like sanitary water tanks.

A U.N.-coordinated “humanitarian response plan” for Iraq this year that asked international donors to provide $861 million for relief efforts is only about 60 percent funded. The previous year’s $704 million humanitarian response plan was 74 percent funded, but support trailed off significantly in the second half of the year.

For Mosul, the news is worse: an additional $284 million “flash appeal” for the impending relief effort, announced in January, is only 48 percent covered. Virtually every cent of the money so far comes from the U.S., which announced a week ago that it would contribute $181 million overall in new aid for Mosul.

Along with relief supply shortages there are other fast-growing needs, including doctors and health workers, teachers to keep displaced children in school, and other social services.

“Nobody is taking up the slack,” one relief worker told Fox News.

Some 3.3 million civilians overall are already displaced in Iraq as a result of years of war and the U.N. considers that 10 million overall are in need of assistance. U.N. humanitarian appeals predict that the number in need could rise to 12 million to 13 million by the end of the year.

The full weight of the Mosul exodus, when it arrives, will offer real competition to Syria as a refugee catastrophe -- although Syria, which nearly 5 million residents fleeing abroad and 6.5 million internally displaced, is still in a league of its own.

The mammoth size of the Syria cataclysm has, in turn, also helped to overshadow the full dimensions of the calamity in Iraq.

Relief experts, however, say that Mosul is likely to be the “single largest, most complex” humanitarian operation in the world this year. Part of the complexity will be the strains it may cause between Iraq’s still unstable central government and the Kurdish regional government in the north, where distrust of Baghdad is acute.

Kurdistan is already hosting more than 1 million Iraqi internal refugees, as well as some 239,000 Syrians.

Another part of that complexity is the effort required even when ISIS has been driven out of areas, as it already has been elsewhere in Iraq. What is left behind is often ruins, frequently mined and booby-trapped, that will require months of painstaking and dangerous effort to reclaim.

The refugees may well be unable to return home, as one official put it, “for an indeterminate period.”

The looming Mosul crisis could quickly help to drown the added multinational efforts at refugee financing, retraining and resettlement that Obama extolled as the outcome of his summit earlier this week, and which were intended as an antidote to donor fatigue and climbing unhappiness about immigrant absorption in the region and well beyond.

Participants at the Obama summit claimed that they will collectively add $4.5 billion beyond 2015 levels to U.N. and other humanitarian appeals -- with about $1 billion more to come from the U.S. The new funds also are intended to lift some of the strain from neighboring countries, such as Jordan and Lebanon, that have been reeling under the impact of refugee flows from Syria.

More controversially, in terms of U.S. presidential elections, the summiteers said they would “roughly double” the number of refugees they would resettle or otherwise admit to their own countries -- and specifically bump U.S. resettlement numbers for refugees from 70,000 in 2015 to 85,000 this year to 110,000 in 2017.

Mosul got no specific mention in a joint statement by the U.S. and its summit co-hosts, which included Canada, Germany, Jordan and Mexico. Nor did it figure in Obama’s own remarks at the meeting, though the $181 million in recently announced Mosul aid doubtless was part of the announced $1 billion in additional U.S. assistance.

CLICK HERE FOR THE LEADERS’ STATEMENT

CLICK HERE FOR OBAMA’S REMARKS

Nor, moreover, did Mosul figure in a sprawling and wafty “New York Declaration for Refugees and Migrants,” passed by U.N. members at yet another summit convened a day earlier by U.N. Secretary General Ban Ki-moon.

The 22-page document, with annexes, offered a variety of empty promises (“we will ensure a people-centered, sensitive, humane, dignified, gender-sensitive and prompt reception for all persons arriving in our countries…”) alongside a variety of windy profundities (“statelessness can be a root cause of forced displacement; in turn, forced displacement can lead to statelessness”).

It also promised that signatories would “commit” to implementing a 6-page Comprehensive Refugee Response Framework developed by the United Nations High Commissioner for Refugees (UNHCR), whose actions in Syria and overall organizational abilities have been criticized harshly, even inside the U.N. itself.

Accompanying the Comprehensive Refugee Response Framework is supposed to come a Global Compact for Safe, Orderly and Regular Migration, which would “deal with all aspects of international migration, including the humanitarian, developmental, human rights-related and other aspects of migration.”

It would, a declaration assures the world, make “an important contribution to global governance and international cooperation on migrants and human mobility.”

All of that will be chewed over “through a process of intergovernmental negotiations, for which preparations will begin immediately.”

The negotiations -- presumably including governments whose barbarities are causing mass migrations in the first place -- will “culminate” at yet another intergovernmental conference in 2018.

And that, in turn, will be followed by a High-Level Dialogue on International Migration and Development, to be held in New York “no later than 2019.”

CLICK HERE FOR THE NEW YORK DECLARATION

A non-government humanitarian official who observed the U.N. meeting -- only governments could participate -- called it a “wasted opportunity” to reform the U.N.’s, and the world’s, ponderous and under-funded humanitarian relief system.

“They could have taken the money” involved in holding the conference, he said, “and spent it on humanitarian supplies.”

George Russell is Editor-at-Large of Fox News. He is reachable on Twitter at @GeorgeRussell and on Facebook at Facebook.com/George.Russell