After a decade of rapid growth fueled by a nearly tenfold increase in world oil prices, the energy-dependent Houston economy is faltering under the prospect of a continued slump in international petroleum markets.

Regional economists now predict that Houston, once the unchallenged leader of the rapidly growing Southwest, will be outperformed by other Sun Belt cities, in part because it is so closely tied to the oil industry.

Compared to many northern cities, Houston remains in good shape, and as long as oil is being pumped around the world, the city will continue to benefit from being the international capital of the oil field servicing and supply business.

But the city's recent period of dynamic growth may be over, and some economists believe that, if oil prices decline drastically, Houston's economy might even shrink over the next few years. Increasingly, Houston's leaders are talking about the need to diversify the local economy.

"Our economy is not going to be as healthy and as vibrant as it was in the mid 1970s," city controller Lance Lalor said last week.

Like most cities, Houston, with an unemployment rate of about 8 percent, has been hurt by the national recession. But until recently, local business leaders and economists expected that the recession's effects would be minimal and short lived.

Now, with world and domestic oil prices beginning to fall, and with worst-case projections calling for a decline that could take the price of crude oil down from about $30 to $20 a barrel, the outlook for Houston is, at best, for a sustained period of only modest growth.

Since the recent failure by OPEC nations to agree on prices and production quotas, at least one bank here has launched a study to determine just how dependent the Houston economy is on oil and how badly it may be hurt by a sustained slump in world oil markets.

"While the national economy is starting to recover, the oil industry has other problems," said Paul Casperson, senior economist at First City National Bank. "We want to look at what will happen if the oil industry stays in a depression."

Last month, controller Lalor, with the approval of Mayor Kathy Whitmire, instituted interim controls on city spending because of a projected $30 million shortfall in revenues--the first significant deficit in several decades. Lalor said the controls were imposed in part to protect the city's AAA bond rating until the mayor and city council agree on specific budget cuts.

Over the last decade, employment in Houston grew by an average of about 80,000 annually, with an increase of 130,000 in 1978 and 100,600 in 1981. But in 1982, total employment rose by only 23,000, and local economists now anticipate that the 1983 increase may be no better. If oil prices fall well below $25 a barrel, total employment could show no growth at all.

"Houston is certainly not in a bust time," said Francis J. Magrino of Texas Commerce Bank. "It's still in a growth phase, but it will be more modest."

Two years ago, Chase Econometrics, a national economic forecasting firm, predicted that Houston would lead the nation in the number of new jobs created through 1990. Last spring, in a revision, Houston was just seventh best, and it could fall lower when the estimates are updated this year.

"You have a situation where there could be a contraction of the Houston economy for a couple of years," said Charles Renfro, director of regional forecasting for Chase.

When the domestic oil drilling industry declined sharply last year, Houston bore a disproportionate share of the burden because of its role as America's oil capital. There were widespread layoffs in oil-related firms and a rise in the number of business bankruptcies.

Local banks, heavily invested in oil loans, are watching the situation carefully. Many have outstanding loans in Mexico, whose already weakened economy would be further damaged by a sharp decline in oil prices.

Other once-healthy sectors of the Houston economy also have slowed. The construction boom of recent years has left the city with an oversupply of office space, which some developers are reportedly offering at discounted prices. Major construction projects have been delayed or stretched out.

The vacancy rate for commercial space in the central city recently rose above 5 percent, compared to the normal one-half percent of recent years. On Houston's east side, where the oil field servicing and petrochemical plants are centered, there are empty buildings for sale or lease, which a Chamber of Commerce economist said has not been true in the past.

The recession apparently has slowed migration to Houston. After adding about 80,000 new customers annually in recent years, Southwestern Bell in Houston gained only about 19,000 last year, and in the last six months of 1982 the company actually lost more than 16,000 customers.

Economists say the apartment vacancy rate has increased, and a spokesman for the U-Haul Co. of Texas said there are now as many trailers leaving the city as there are coming in. "They're coming in, but they're going right back out," said Jerry King.

While oil prices were climbing, Houston's energy-dependent economy appeared to be a blessing because it was resistant to the national economic cycles. But in the current recession, Houston has been more affected than have more economically diversified Sun Belt cities.

There is more discussion than ever in Houston about the need to attract other kinds of industries.

"The Chamber of Commerce has tried to foster the notion that Houston is more diversified than in the past," said one bank economist. "But this recession has brought it home that we're as much an energy economy as Detroit is autos. It will take a generation to change that."

Other cities appear more attractive to new industry--especially high technology firms--than Houston, which suffers from traffic congestion and pollution. Houston also may be at a disadvantage because its wage scale is higher than that of many other Sun Belt cities.

Houston leaders, however, remain relatively optimistic about an economic future tied to the oil industry, and some believe a period of slower growth will be good for the city.

"We are in a period of digesting growth that has already been experienced, and at the same time anticipating a continued, more steady level of growth in the city," said Whitmire, who was elected in 1981 by promising to improve city services overwhelmed by rapid growth.

"Houston followed the Southwest boom and on top of that had the oil boom," said economist Paul Casperson. "It created an almost unhealthy rate of growth."