<PAGE>   1
================================================================================

                       SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20549

                                    FORM 10-Q

(Mark One)

[X]  QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(D) OF THE SECURITIES EXCHANGE
     ACT OF 1934

For the quarterly period ended June 30, 1999

                                       OR

[ ]  TRANSITION PERIOD PURSUANT TO SECTION 13 OR 15(D) OF THE SECURITIES
     EXCHANGE ACT OF 1934

For the transition period from _____________________ to ________________________

                          Commission File Number 1-2475

                                SHELL OIL COMPANY
             (Exact Name of Registrant as Specified in its Charter)

                 Delaware                               13-1299890
         (State of Incorporation)           (I.R.S. Employer Identification No.)

     One Shell Plaza, Houston, Texas                       77002
(Address of Principal Executive Offices)                 (Zip Code)

        Registrant's Telephone Number, Including Area Code (713) 241-6161

Indicate by check mark whether the registrant (1) has filed all reports required
to be filed by Section 13 or 15(d) of the Securities Exchange Act of 1934 during
the preceding 12 months (or for such shorter period that the registrant was
required to file such reports), and (2) has been subject to such filing
requirements for the past 90 days. Yes [X] No [ ]

The number of shares of Common Stock, $10.00 par value, outstanding as of June
30, 1999 - 1,000 shares.

                                   ----------

                         OMISSION OF CERTAIN INFORMATION

In accordance with General Instruction H of Form 10-Q, the registrant is
omitting Part II, Items 2, 3, and 4 because:

(1) Royal Dutch Petroleum Company, a Netherlands company, and The "Shell"
Transport and Trading Company, p.l.c., an English company, each of which is a
reporting company under the Securities Exchange Act of 1934 that has filed all
material required to be filed by it pursuant to Section 13, 14, or 15(d)
thereof, own directly or indirectly 60 percent and 40 percent, respectively, of
the shares of the companies of the Royal Dutch/Shell Group of Companies,
including all the equity securities of the registrant; and (2) during the
preceding thirty-six calendar months and any subsequent period of days, there
has not been any material default in the payment of principal, interest, sinking
or purchase fund installment, or any other material default not cured within
thirty days with respect to any indebtedness of the registrant or its
subsidiaries, and there has not been any material default in the payment by the
registrant or its subsidiaries of rentals under material long-term leases.


================================================================================

<PAGE>   2

                          PART I. FINANCIAL INFORMATION

                       SHELL OIL COMPANY AND SUBSIDIARIES
                        CONSOLIDATED STATEMENT OF INCOME
                               Millions of Dollars


<TABLE>
<CAPTION>
                                                        SECOND QUARTER        SIX MONTHS
                                                       -----------------   -----------------
                                                        1999      1998      1999      1998
                                                       -------   -------   -------   -------
<S>                                                    <C>       <C>       <C>       <C>
REVENUES
    Sales and other operating revenue ..............   $ 4,326   $ 5,235   $ 7,813   $10,261
    Less:  Consumer excise and sales taxes .........        29       419        51       796
                                                       -------   -------   -------   -------
                                                         4,297     4,816     7,762     9,465
    Equity in income of affiliates .................        34       151       173       271
    Interest and other income ......................        30        63        35        99
                                                       -------   -------   -------   -------
          TOTAL ....................................     4,361     5,030     7,970     9,835
                                                       -------   -------   -------   -------

COSTS AND EXPENSES
    Purchased raw materials and products ...........     2,528     3,056     4,606     6,006
    Operating expenses .............................       913       687     1,561     1,387
    Selling, general and administrative expenses ...        98       195       184       453
    Exploration, including exploratory dry holes ...        49       110        83       179
    Research expenses ..............................        36        40        68        79
    Depreciation, depletion, amortization and
       retirements .................................       102       342       392       738
    Interest and discount amortization .............        80       101       157       186
                                                       -------   -------   -------   -------
          TOTAL ....................................     3,806     4,531     7,051     9,028
                                                       -------   -------   -------   -------

INCOME BEFORE INCOME TAXES AND
MINORITY INTEREST ..................................   $   555   $   499   $   919   $   807

    Federal and other income taxes .................       102       162       127       274
    Minority interest in income
       of subsidiaries .............................        14        21        26        45
                                                       -------   -------   -------   -------

NET INCOME .........................................   $   439   $   316   $   766   $   488
                                                       =======   =======   =======   =======
</TABLE>

Note: Certain 1998 amounts have been reclassified to conform with current year
      presentation.


                                       2
<PAGE>   3

                       SHELL OIL COMPANY AND SUBSIDIARIES
                           CONSOLIDATED BALANCE SHEET
                               Millions of Dollars


<TABLE>
<CAPTION>
                                                                JUNE 30,    DECEMBER 31,
                                                                  1999         1998
                                                                --------    ------------
<S>                                                             <C>          <C>
ASSETS
CURRENT ASSETS
   Cash and cash equivalents .............................      $   284      $   322
   Receivables and prepayments, less allowance for
      doubtful accounts ..................................        3,037        2,235
   Owing by related parties ..............................          700          618
   Inventories of products ...............................          762          723
   Inventories of materials and supplies .................          149          150
                                                                -------      -------
             TOTAL CURRENT ASSETS ........................        4,932        4,048
INVESTMENTS ..............................................        9,140        9,345
LONG-TERM RECEIVABLES AND DEFERRED CHARGES ...............        1,808        1,706
PROPERTY, PLANT AND EQUIPMENT AT COST, LESS
   ACCUMULATED DEPRECIATION, DEPLETION AND
   AMORTIZATION OF $11,368 AT JUNE 30, 1999
   AND $12,735 AT DECEMBER 31, 1998 ......................       11,026       11,444
                                                                -------      -------
             TOTAL .......................................      $26,906      $26,543
                                                                =======      =======


LIABILITIES AND SHAREHOLDER'S EQUITY
CURRENT LIABILITIES
   Accounts payable - trade ..............................      $ 1,214      $ 1,490
   Other payables and accruals ...........................        1,576        1,406
   Income, operating and consumer taxes ..................          642           88
   Owing to related parties ..............................          573          553
   Short-term debt .......................................        5,704        5,301
                                                                -------      -------
           TOTAL CURRENT LIABILITIES .....................        9,709        8,838
LONG-TERM DEBT ...........................................          476          490
DEFERRED INCOME TAXES ....................................        1,586        2,000
LONG-TERM LIABILITIES ....................................        2,302        2,287
MINORITY INTEREST AND PREFERRED STOCK OF SUBSIDIARIES ....        1,529        1,542
SHAREHOLDER'S EQUITY
   Common stock - 1,000 shares of $10 per share
       par value .........................................           --           --
   Capital in excess of par value ........................        2,206        2,206
   Earnings reinvested ...................................        9,098        9,180
                                                                -------      -------
           TOTAL SHAREHOLDER'S EQUITY ....................       11,304       11,386
                                                                -------      -------
           TOTAL .........................................      $26,906      $26,543
                                                                =======      =======
</TABLE>


                                       3
<PAGE>   4

                       SHELL OIL COMPANY AND SUBSIDIARIES
                      CONSOLIDATED STATEMENT OF CASH FLOWS
                               Millions of Dollars


<TABLE>
<CAPTION>
                                                                            SIX MONTHS
                                                                        ------------------
                                                                          1999       1998
                                                                        -------    -------
<S>                                                                     <C>        <C>
CASH FLOWS FROM OPERATING ACTIVITIES
   Net Income .......................................................   $   766    $   488
   Adjustments to reconcile net income to net cash
       provided by operating activities:
           Depreciation, depletion, amortization and retirements ....       392        738
           Dividends less than equity income ........................      (105)       (76)
           (Increases) decreases in working capital:
             Receivables and prepayments ............................      (781)      (202)
             Inventories ............................................       (38)      (161)
             Current payables and accruals ..........................       366       (295)
           Deferred income taxes ....................................      (414)      (250)
           Minority interest in income of subsidiaries ..............        26         45
           Other noncurrent items ...................................       (89)       225
                                                                        -------    -------
             Net Cash Provided by Operating Activities ..............       123        512

CASH FLOWS PROVIDED BY (USED FOR) INVESTING ACTIVITIES
   Capital Expenditures .............................................      (917)    (2,564)
   Proceeds from property sales and salvage .........................       965        196
   Other investments ................................................       289        241
                                                                        -------    -------
             Net Cash Provided by (Used for) Investing Activities ...       337     (2,127)

CASH FLOWS PROVIDED BY (USED FOR) FINANCING ACTIVITIES
   Proceeds from issuance of long-term debt .........................        --        324
   Principal payments on long-term debt .............................       (11)      (370)
   Proceeds from sales of securities of subsidiaries ................        (5)       246
   Dividends to shareholder .........................................      (850)      (825)
   Dividends to minority interest ...................................       (32)       (46)
   Increase in short-term obligations ...............................       400      2,230
                                                                        -------    -------
             Net Cash Provided by (Used for) Financing Activities ...      (498)     1,559

NET CASH FLOWS
   Decrease in cash and cash equivalents ............................   $   (38)   $   (56)
                                                                        =======    =======

CASH AND CASH EQUIVALENTS
   Balance at beginning of period ...................................   $   322    $   342
   Decrease in cash and cash equivalents ............................       (38)       (56)
                                                                        -------    -------
             Balance at end of period ...............................   $   284    $   286
                                                                        =======    =======
</TABLE>


                                       4
<PAGE>   5

                       SHELL OIL COMPANY AND SUBSIDIARIES

                      NOTES TO INTERIM FINANCIAL STATEMENTS


A. INTERIM FINANCIAL STATEMENT MATTERS

         The unaudited financial statements and summarized notes of Shell Oil
Company ("the Company") and its consolidated subsidiaries ("Shell Oil") included
in this report do not include complete financial information and should be read
in conjunction with the Consolidated Financial Statements and the Notes to
Consolidated Financial Statements filed with the Securities and Exchange
Commission in the Registrant's Annual Report on Form 10-K for the year ended
December 31, 1998. The financial information presented in the financial
statements included in this report reflects all adjustments which are, in the
opinion of management, necessary for a fair statement of results for the interim
periods presented. Any such adjustments are of a normal recurring nature, except
as may otherwise be described in Management's Discussion and Analysis of
Financial Condition and Results of Operations.

         The results for the second quarter of 1999 should not be construed as
necessarily indicative of future financial results.


B. EQUILON ENTERPRISES LLC

         Summarized unaudited financial information for Equilon Enterprises LLC
("Equilon"), formed January 1, 1998 and jointly owned 56 percent by Shell Oil
and 44 percent by Texaco Inc. ("Texaco") is presented below on a 100 percent
Equilon basis:

<TABLE>
<CAPTION>
                                 Three Months Ended     Six Months Ended
                                       June 30               June 30
                                 ------------------    ------------------
                                   1999       1998       1999       1998
                                 -------    -------    -------    -------
                                          (millions of dollars)
<S>                              <C>        <C>        <C>        <C>
   Gross Revenues.............   $ 8,008    $ 6,070    $13,787    $12,095
   Income(Loss) Before Tax ...      (123)       198         48        310
</TABLE>

         Shell Oil accounts for its interest in Equilon using the equity method
of accounting. Under this method, Shell Oil's share of Equilon's results of
operations is recorded on a one-line basis to "Equity in Income of Affiliates"
in the Consolidated Statement of Income. Additionally, since Equilon is a
limited liability company, Shell Oil records the provision for income taxes and
related liability applicable to its share of Equilon's income in its
consolidated financial statements.

C. MOTIVA ENTERPRISES LLC

         Summarized unaudited financial information for Motiva Enterprises LLC
("Motiva"), formed July 1, 1998 and jointly owned 35 percent by Shell Oil and
32.5 percent each by Texaco and by Saudi Refining, Inc., a corporate affiliate
of Saudi Aramco, is presented below on a 100 percent Motiva basis:

<TABLE>
<CAPTION>
                                 Three Months Ended     Six Months Ended
                                    June 30, 1999        June 30, 1999
                                 ------------------     ----------------
                                          (millions of dollars)
<S>                              <C>                    <C>
   Gross Revenues............          $3,073                $5,315
   Income(Loss) Before Tax...             (21)                   20
</TABLE>


                                       5
<PAGE>   6

         Shell Oil accounts for its interest in Motiva using the equity method
of accounting. Under this method, Shell Oil's share of Motiva's results of
operations is recorded on a one-line basis to "Equity in Income of Affiliates"
in the Consolidated Statement of Income. Additionally, since Motiva is a limited
liability company, Shell Oil records the provision for income taxes and related
liability applicable to its share of Motiva's income in its consolidated
financial statements.


D. INTERIM SEGMENT INFORMATION

         The following segment information has been prepared in accordance with
Statement of Financial Accounting Standards ("SFAS") No. 131, "Disclosure about
Segments of an Enterprise and Related Information," which was adopted by Shell
Oil in 1998. The adoption of SFAS No. 131 did not result in a change in Shell
Oil's reportable segments; however, the net income for the three month and six
month periods ended June 30, 1998 for each segment has been restated from that
previously reported. The restatement is primarily to reflect the allocation to
the segments of interest income and interest expense previously reported as
"Nonallocated Items."

<TABLE>
<CAPTION>
                                                            Three months ended June 30, 1999
                                     --------------------------------------------------------------------------------
                                     Oil and    Downstream     Oil       Chemical      All       Nonalloc.
                                       Gas         Gas       Products    Products     Other        Items      Total
                                     --------   ----------   --------    --------    --------    ---------   --------
                                                                  (millions of dollars)
<S>                                  <C>         <C>         <C>         <C>         <C>         <C>         <C>
Sales and other operating
  revenues, including intersegment   $    927    $  2,081    $    741    $  1,337    $    173    $     22    $  5,281
Intersegment sales and revenues          (442)        (59)       (117)       (229)       (121)        (16)       (984)
                                     --------    --------    --------    --------    --------    --------    --------
Total external revenues              $    485    $  2,022    $    624    $  1,108    $     52    $      6    $  4,297
                                     ========    ========    ========    ========    ========    ========    ========

Segment net income (loss)            $    310    $     24    $     36    $     78    $      2    $    (11)   $    439
Segment assets                          9,969       5,054       5,886       5,182         285         530      26,906

----------------------------------
Segment assets at
  December 31, 1998                  $ 10,106    $  4,307    $  5,970    $  4,608    $    342    $  1,210    $ 26,543
</TABLE>


<TABLE>
<CAPTION>
                                                      Three months ended June 30, 1998 (restated)
                                     --------------------------------------------------------------------------------
                                     Oil and    Downstream     Oil       Chemical      All       Nonalloc.
                                       Gas         Gas       Products    Products     Other        Items      Total
                                     --------   ----------   --------    --------    --------    ---------   --------
                                                                  (millions of dollars)
<S>                                  <C>         <C>         <C>         <C>         <C>         <C>         <C>
Sales and other operating
  revenues, including intersegment   $    985    $  1,712    $  1,694    $  1,346    $    143    $     27    $  5,907
Intersegment sales and revenues          (409)        (25)       (150)       (373)       (122)        (12)     (1,091)
                                     --------    --------    --------    --------    --------    --------    --------
Total external revenues              $    576    $  1,687    $  1,544    $    973    $     21    $     15    $  4,816
                                     ========    ========    ========    ========    ========    ========    ========

Segment net income (loss)            $    155    $      4    $    102    $     72    $      8    $    (25)   $    316
Segment assets                         12,581       4,812       7,934       5,395         154       1,486      32,362

----------------------------------
Segment assets at
  December 31, 1997                  $ 13,123    $     --    $  9,277    $  5,342    $    211    $  1,648    $ 29,601
</TABLE>


                                       6
<PAGE>   7

<TABLE>
<CAPTION>
                                                             Six months ended June 30, 1999
                                     --------------------------------------------------------------------------------
                                     Oil and    Downstream     Oil       Chemical      All       Nonalloc.
                                       Gas         Gas       Products    Products     Other        Items      Total
                                     --------   ----------   --------    --------    --------    ---------   --------
                                                                  (millions of dollars)
<S>                                  <C>         <C>         <C>         <C>         <C>         <C>         <C>
Sales and other operating
  revenues, including intersegment   $  1,682    $  3,835    $  1,132    $  2,512    $    307    $     51    $  9,519
Intersegment sales and revenues          (784)       (115)       (180)       (442)       (205)        (31)     (1,757)
                                     --------    --------    --------    --------    --------    --------    --------
Total external revenues              $    898    $  3,720    $    952    $  2,070    $    102    $     20    $  7,762
                                     ========    ========    ========    ========    ========    ========    ========

Segment net income (loss)            $    534    $     17    $     89    $    166    $     (4)   $    (36)   $    766
</TABLE>


<TABLE>
<CAPTION>
                                                        Six months ended June 30, 1998 (restated)
                                     --------------------------------------------------------------------------------
                                     Oil and    Downstream     Oil       Chemical      All       Nonalloc.
                                       Gas         Gas       Products    Products     Other        Items      Total
                                     --------   ----------   --------    --------    --------    ---------   --------
                                                                  (millions of dollars)
<S>                                  <C>         <C>         <C>         <C>         <C>         <C>         <C>
Sales and other operating
  revenues, including intersegment   $  1,937    $  3,386    $  3,152    $  2,801    $    275    $     54    $ 11,605
Intersegment sales and revenues          (786)        (53)       (290)       (750)       (238)        (23)     (2,140)
                                     --------    --------    --------    --------    --------    --------    --------
Total external revenues              $  1,151    $  3,333    $  2,862    $  2,051    $     37    $     31    $  9,465
                                     ========    ========    ========    ========    ========    ========    ========

Segment net income (loss)            $    279    $     (1)   $    101    $    170    $     17    $    (78)   $    488
</TABLE>

E. CONTINGENCIES AND OTHER MATTERS

         Shell Oil is subject to a number of possible loss contingencies. These
include actions based upon environmental laws involving present and past
operating and waste disposal locations and related private claims, contract and
product liability actions and federal, state and private actions challenging the
correctness of oil and gas royalty calculations. In addition, federal, state and
local income, property and excise tax returns are being examined and certain
interpretations by Shell Oil of complex tax statutes, regulations and practices
are being challenged.

         Since 1984, the Company has been named with others as a defendant in
numerous product liability cases, including class actions, involving the failure
of residential plumbing systems constructed with polybutylene plastic pipe. The
Company has also been sued regarding failures in polybutylene pipe connecting
users with utility water lines and polybutylene pipe used in municipal water
distribution systems. The Company fabricated the resin for this pipe. Two other
substantial manufacturers made the resins for the polyacetal insert fittings
used in many of the residential plumbing systems (the fittings co-defendants)
and are also defendants in those cases. The Company and the fittings
co-defendants have agreed on a mechanism to fund the payment of most of the
residential plumbing claims in the United States as the result of two class
action settlements (the "class action settlement"). The class action settlement
provides for the creation of an entity to receive and handle claims and for a
$950 million fund to pay such claims, which claims may be filed until 2009,
depending on various factors. The entity, Consumer Plumbing Recovery Center, was
created in 1995. If the settlement funds are exhausted, additional funds may be
provided by the defendants, or claimants who have not received their full


                                       7
<PAGE>   8

benefits under the class action settlements may seek their remedy in a new court
proceeding at that time. Additionally, claims continue to be filed involving
alleged problems with polybutylene pipe used in municipal water distribution
systems. The Company will continue to defend these matters vigorously but it
cannot currently predict when or how polybutylene related matters will finally
be resolved.

          In an October 1997 decision by the United States District Court in
Delaware, certain income tax credits recorded by Shell Oil in previous years
arising out of production of oil from reserves which it believes classify as
"tar sands" for tax credits were denied because the Court determined that Shell
Oil used the wrong definition of tar sands production to calculate the same.
Shell Oil believes that the District Court decision was incorrect and appealed
the decision to the Third Circuit Court of Appeals in September 1998. In June
1999, the Third Circuit held that the government's definition was a correct
definition of the phrase "oil produced from tar sands." Shell Oil is considering
various options available with respect to the Third Circuit opinion. Shell Oil
also believes that many of its tar sands tax credits are validly claimed under
the alternative definition asserted by the government in the District Court
case.

         The Company's assessment of these matters is continuing. Future
provisions may be required as administrative and judicial proceedings progress
and the scope and nature of remediation programs and related costs estimates are
clarified. However, while periodic results may be significantly affected by
costs in excess of provisions related to one or more of these proceedings, based
upon developments to date, the management of the Company anticipates that it
will be able to meet related obligations without a material adverse effect on
its financial position.

                                   ----------


                                       8
<PAGE>   9

                     MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                  FINANCIAL CONDITION AND RESULTS OF OPERATIONS


RESULTS OF OPERATIONS

         AS DISCUSSED IN NOTE D OF THE NOTES TO INTERIM FINANCIAL STATEMENTS,
THE NET INCOME FOR THE THREE MONTH AND SIX MONTH PERIODS ENDING JUNE 30, 1998
FOR EACH OF SHELL OIL'S REPORTABLE OPERATING SEGMENTS HAS BEEN RESTATED FROM
THAT PREVIOUSLY REPORTED. THE 1998 RESTATEMENT WAS MADE TO BE CONSISTENT WITH
THE 1999 PRESENTATION REFLECTING THE ADOPTION OF STATEMENT OF FINANCIAL
ACCOUNTING STANDARDS NO. 131, "DISCLOSURES ABOUT SEGMENTS OF AN ENTERPRISE AND
RELATED INFORMATION."

         Shell Oil reported second quarter 1999 net income of $439 million, an
increase of $123 million, or 39 percent over the second quarter of 1998.
Excluding special items in both quarters, adjusted net income in the 1999
quarter totaled $360 million, an increase of $48 million, or 15 percent.

         The key operational elements contributing to improved earnings in the
second quarter of 1999 as compared to 1998 were lower operating costs coupled
with higher average crude oil prices. Domestic average crude oil prices
increased 22 percent over the same period last year. Also contributing to
improved earnings was lower depreciation expense in the 1999 quarter in part due
to asset sales in 1999, and the effect of suspending depreciation on assets held
for sale. Partially offsetting these improvements were lower refined product
margins, natural gas prices, and production of oil and gas.

         For the first six months of 1999, net income was $766 million, an
increase of $278 million, or 57 percent, over the same period last year.
Excluding special items, adjusted net income for 1999 totaled $553 million, up
$70 million, or 14 percent, over 1998. Lower operating costs and depreciation
expense resulting in part from asset sales in 1999, and the effect of suspending
depreciation on assets held for sale, more than offset the effects from lower
natural gas prices and production, and lower margins for oil and chemical
products.

         Special items in the 1999 periods benefited net income $79 million for
the quarter and $213 million for the first half. Special items in the second
quarter of 1999 were comprised primarily of gains from asset sales and a prior
year tax adjustment, partially offset by adjustments to market value for assets
held for sale. Special items increased 1998 net income by $4 million for the
quarter and $5 million in the first six months.


OIL AND GAS EXPLORATION AND PRODUCTION

<TABLE>
<CAPTION>
Income Highlights                          Second Quarter           Six Months
-----------------                          ---------------       ---------------
(millions of dollars)                      1999       1998       1999       1998
                                           ----       ----       ----       ----
<S>                                        <C>        <C>        <C>        <C>
   Segment Net Income ..............       $310       $155       $534       $279
   Special Items ...................         58         39        153         42
                                           ----       ----       ----       ----
   Adjusted Net Income .............        252        116        381        237
</TABLE>

         Oil and gas exploration and production earnings in the second quarter
of 1999 totaled $310 million, an increase of $155 million over 1998. For the
first half of 1999, earnings were $534 million, up $255 million. Excluding
special items in the comparable periods, adjusted net income increased $136


                                       9
<PAGE>   10

million in the 1999 quarter versus 1998 and $144 million in the first half
comparison.

     Lower operating and exploratory costs were the primary factors
contributing to the improvements in both 1999 periods over 1998. Benefits were
also derived during the second quarter of 1999 compared to 1998 from higher
average crude oil prices, partially offset by lower production volumes. For the
second quarter of 1999, domestic crude oil prices averaged $13.68 per barrel,
increasing $2.51 per barrel, or 22 percent, over the 1998 quarter. For the
six-month periods, average domestic crude oil prices were virtually unchanged,
averaging approximately $11.60 per barrel. Natural gas prices averaged $2.08 per
thousand cubic feet in the second quarter of 1999, down 4 percent from the same
period last year. For the first six months of 1999, natural gas prices averaged
$1.89 per thousand cubic feet, down 12 percent from last year. Also contributing
to the improvements in both 1999 periods was lower depreciation expense,
resulting in part from asset sales in 1999 and the effect of suspending
depreciation on assets held for sale.

     As previously reported in the Company's Quarterly Report on Form 10-Q for
the quarter ended March 31, 1999, in April 1999, Shell Oil announced that it had
entered into a definitive agreement to sell to Apache Corporation a significant
number of oil and gas properties located on the outer continental shelf of the
Gulf of Mexico. The agreed upon sales price for the properties was $715 million
in cash, subject to adjustments, plus one million shares of Apache Corporation
common stock. The sale was closed in May 1999. Included in the properties sold
were a portion of the oil and gas properties which in 1998, were identified as
"assets to be sold" and which at that time were written down to their estimated
sales prices.

         Average domestic crude oil production during the 1999 periods was
427,000 barrels per day for the quarter and 429,000 barrels per day for the six
months, decreasing 33,000 and 26,000 barrels per day, respectively, compared to
1998. These reductions are attributable to property sales in the shelf region of
the Gulf of Mexico and onshore, as well as normal production declines. Deepwater
production of crude oil was up in the quarter-to-quarter comparison. Natural gas
production averaged 2,020 million cubic feet daily during the second quarter of
1999, down 56 million cubic feet daily, or 3 percent. Property sales more than
offset increased deepwater production, resulting in the period-to-period natural
gas decline. For the six month period of 1999, natural gas production averaged
2,032 million cubic feet daily, an increase of 45 million, or 3 percent.

         Domestic crude oil and natural gas production numbers include Shell
Oil's net production plus a pro rata share, based on ownership interest, of
domestic equity companies' production. Equity companies are those companies in
which Shell Oil has significant influence but not control.

DOWNSTREAM GAS

<TABLE>
<CAPTION>
Income Highlights                             Second Quarter       Six Months
-----------------                             --------------     --------------
(millions of dollars)                         1999      1998     1999      1998
                                              ----      ----     ----      ----
<S>                                           <C>       <C>      <C>       <C>
   Segment Net Income (Loss) ............     $ 24         4     $ 17        (1)
   Special Items ........................       53        --       53        --
                                              ----      ----     ----      ----
     Adjusted Net Income (Loss) .........      (29)        4      (36)       (1)
</TABLE>

         Downstream gas earnings for the second quarter of 1999 were $24
million, up $20 million over the same period last year. For the first half of
1999, earnings totaled $17 million, up $18 million over the 1998 period.
Excluding a favorable adjustment to market value for assets held for sale,
adjusted net income decreased $33 million from the 1999 quarter versus 1998, and
$35 million in the first-half comparison.


                                       10
<PAGE>   11

         During the second quarter of 1999, earnings benefited from higher
storage margins and from lower depreciation expense reflecting in part the
effect of suspending depreciation on assets held for sale. However, lower
earnings from gas marketing and trading activities due to limited market
volatility and increased allocated financing costs, more than offset the
improvement. Sales of natural gas in the 1999 quarter increased to 9.2 billion
cubic feet per day, up 16 percent over the 1998 quarter and 27 percent in the
first-half of 1999 as compared to 1998; however, margins on gas sales in Canada
were depressed.

         During the second quarter of 1999, Shell Oil finalized an agreement to
sell its Transok LLC affiliate at a sales price of approximately $700 million.
The sale was closed on July 1, 1999. As reported in the Company's Annual Report
on Form 10-K for the year ended December 31, 1998, during the fourth quarter of
1998, Shell Oil announced its intent to sell Transok LLC, a gatherer, processor
and transporter of natural gas in Oklahoma, and wrote down the carrying value of
the asset to its then estimated sales value. Based on the finalized sales
agreement, a favorable adjustment to the carrying value was made during the
second quarter of 1999.

OIL PRODUCTS

<TABLE>
<CAPTION>
Income Highlights                       Second Quarter          Six Months
-----------------                      ----------------       --------------
(millions of dollars)                  1999        1998       1999      1998
                                       ----        ----       ----      ----
<S>                                    <C>         <C>        <C>       <C>
   Segment Net Income ...........      $ 36        $102       $ 89      $101
   Special Items ................       (21)        (29)        15       (30)
                                       -----       ----       ----      ----
     Adjusted Net Income ........        57         131         74       131
</TABLE>

         Shell Oil's oil products segment consists primarily of its share of
results from Equilon, which began operations in January, 1998 (Shell Oil
interest - 56 percent), Motiva, which began operations in July, 1998 (Shell Oil
interest - 35 percent), and the Deer Park Refining Limited Partnership (Shell
Oil interest - 50 percent). Shell Oil accounts for its investment in each of
these companies using the equity method of accounting.

         Oil products earnings totaled $36 million in the second quarter of
1999, a decrease of $66 million from the 1998 period. For the first half of
1999, earnings were $89 million, a decrease of $12 million. Excluding special
items, adjusted net income declined $74 million in the quarter comparison, and
$57 million in the first half of 1999 compared to 1998. Special items in the
quarter primarily included the effect of asset writedowns to their estimated
sales values as a result of the pending sale by Equilon of its El Dorado and
Wood River refineries, partially offset by a favorable prior year tax
adjustment.

         Contribution margins declined quarter-to-quarter, reflecting severely
depressed refining margins on the Gulf Coast and Eastern regions of the United
States. These margin declines, plus refinery downtime more than offset earnings
benefits from West Coast contribution margin improvements and Alliance
synergies.


                                       11
<PAGE>   12

CHEMICAL PRODUCTS

<TABLE>
<CAPTION>
Income Highlights                      Second Quarter            Six Months
-----------------                     ----------------        ----------------
(millions of dollars)                 1999        1998        1999        1998
                                      ----        ----        ----        ----
<S>                                   <C>         <C>         <C>         <C>
   Segment Net Income ..........      $ 78        $ 72        $166        $170
   Special Items ...............       (11)        (11)         (8)         (9)
                                      ----        ----        ----        ----
     Adjusted Net Income .......        89          83         174         179
</TABLE>

         Chemical products earnings were $78 million in the second quarter of
1999, an increase of $6 million over 1998. For the first six months of 1999,
chemical products earnings totaled $166 million, a decrease of $4 million.
Excluding special items in the comparable periods, adjusted net income for the
1999 quarter increased $6 million over the comparable 1998 period, but declined
$5 million for the first six months.

         In the second quarter of 1999, benefits from lower operating costs and
lower depreciation expense reflecting in part the effect of the suspension of
depreciation on assets held for sale were mostly offset by lower margins due to
declining selling prices and higher feedstock costs. For the six months of 1999,
lower contribution margins in the 1999 period was the key factor in the earnings
decline and more than offset benefits from lower operating costs and lower
depreciation expense.

ALL OTHER

         The other operating segments' earnings in the second quarter of 1999
was $2 million, down $6 million from the same quarter last year. For the first
half of 1999, the other operating segments incurred a loss of $4 million, a
decrease of $21 million from 1998. The decline in the first half comparison was
mainly due to lower margins derived from service operations.

NONALLOCATED ITEMS

         Nonallocated costs totaled $11 million in the second quarter and $36
million in the first half of 1999, decreasing $14 million and $42 million from
the respective 1998 periods. The declines in both 1999 periods was primarily due
to a high level of costs in 1998 related to a brand initiative program and lower
financing costs in 1999.

FINANCIAL CONDITION

CAPITAL RESOURCES AND LIQUIDITY

         Cash flow provided by operating activities totaled $123 million for the
first six months of 1999, compared with $512 million in the comparable period
last year, a decrease of $389 million. The period to period decrease was
attributable to higher working capital requirements. Cash generated from
operating activities, coupled with an increase in debt of $389 million and
proceeds from property sales and other investments in the first six months of
1999, was used primarily for capital expenditures of $917 million and dividend
payments of $882 million.


                                       12
<PAGE>   13

OTHER MATTERS

REPORT ON YEAR 2000 ISSUES

         Shell Oil has been engaged in the identification, analysis and
remediation of the Year 2000 ("Y2K") computer date problem since 1997. This
problem is associated with the incorrect interpretation by the computer of dates
which are numerically related to the change of the millennium. Large and small
computing devices and associated software used in running large business
applications as well as some kinds of industrial automation systems are
potentially subject to this problem.

         Under the Shell Oil governance model, each business organization has
independent responsibility for identification, analysis and remediation of Y2K
problems within such business, and in 1997, Year 2000 Teams were formed in each
business organization to deal with the Y2K problem. The Teams communicate
through a central focal point committee which tracks and updates Year 2000
progress reports from each business at the overall Shell Oil level. This Shell
Oil Year 2000 Team also provides a vehicle for exchange of ideas and experiences
to achieve improved efficiencies and results. The Chief Information Officer of
each business represents the organization on the Shell Oil Year 2000 Team.
Additionally, confirmation is sought from those equity companies where Shell Oil
has a significant voting interest of an active and appropriate Y2K program.

         Shell Oil Year 2000 Teams have identified and analyzed the use of
computer technology in the respective businesses to determine where there were
business operation issues associated with the Y2K problem. Priority in such
reviews was placed first on health, safety and environmental issues, then on
areas which could materially affect third parties or revenues. These reviews
necessarily included both internal uses of technology as well as key trading
interfaces, which might be adversely affected by third party customers',
suppliers' or service providers' use of computing technology with Y2K problems.

         The Shell Oil businesses divided their analysis of the Y2K problem into
three areas: 1) business information, computing and telecommunication systems
("IT Systems"); 2) industrial automation systems ("Imbedded Technology
Systems"); and 3) potential problems involving the Y2K problems of third party
suppliers, customers and service providers (each a "Third Party" and together
"Third Parties").

         The analysis and inventory of potential Y2K problems has been completed
in all areas involving critical IT Systems and Imbedded Technology Systems, with
remediation requirements identified and remediation plans developed. At June 30,
1999, remediation of Y2K problems in all such areas in which issues have been
identified had been substantially completed with the remaining remediation work
expected to be completed during the third quarter of 1999. Remediation has been
primarily handled by internal company staff, although some outside companies
have been involved in the remediation associated with particular systems. It is
expected that analysis of interfaces with key Third Parties will continue
through the third quarter of 1999 to insure appropriate coordination and, where
necessary or appropriate, development of contingency plans in the event a Third
Party cannot meet its commitments to Shell Oil. Analyses and remediation of
non-critical IT Systems, Imbedded Technology Systems and Third Party interfaces
will continue through the remainder of the year.

         As to the Imbedded Technology Systems used in industrial automation
within Shell Oil, the various Shell Oil businesses have used several approaches
in the identification and analysis of the potential Y2K problem in such systems,
including field installation inspection, and inspection of design blueprints,
with field-testing to confirm the adequacy of design information. At June 30,
1999, remediation of substantially all material Y2K problems with this
technology had been completed with the remaining remediation expected to be
completed as part of other scheduled facility maintenance occurring during the
third quarter of 1999.


                                       13
<PAGE>   14

         Due to a business-driven demand for improved computing systems
throughout Shell Oil, the primary business computing systems in each Shell Oil
business organization are being replaced in this decade. These new systems are
represented to be fully Y2K compliant by their manufacturers, and Shell Oil
believes that it will have no Y2K issues in its important business computing
systems. While acquisition of these new IT Systems for business computing
avoided the necessity of dealing with Y2K issues in the predecessor systems, the
cost of these systems was incurred primarily for other business reasons. The
majority of the costs of each such system are being capitalized and charged
against income over the estimated useful life of the systems. Funds for the
acquisition of these systems are included in the capital budget of the
respective businesses.

         Costs associated with the identification, analysis and remediation of
Y2K problems, including the repair or replacement of imbedded technology
systems, are currently projected by the businesses at approximately $150
million. These costs are being charged to expense as incurred and do not include
Y2K costs of equity companies. At June 30, 1999, approximately 90 percent of
such expenditure had been committed or incurred. Funding for these efforts has
been a part of the normal budgeting of current operations within each business.

         Beyond the internal programs in the Shell Oil businesses, significant
efforts are continuing in order to understand the position of key Third Parties
with respect to possible business interruptions due to Y2K problems in their
systems. Current indications are that all of Shell Oil's major Third Party
customers, suppliers and service providers are well aware of this problem and
have programs underway to address it. Physical testing of key interfaces and
ongoing discussions are continuing with certain Third Parties to best position
Shell Oil to continue its business operations without interruption. Specific
attention is being given to Third Parties that have critical relationships with
Shell Oil to insure that there is not a disruption in the flow of those products
or services that are essential to the ability of Shell Oil to carry on its
business operations. However, since Shell Oil will remain dependent on Third
Party confirmations in certain areas, appropriate contingency plans are being
developed in each of Shell Oil's businesses.

         The key objective of Shell Oil's extensive program to address the Y2K
computer technology problem is the significant reduction or complete elimination
of the risk to business operations. Internal audits completed during the first
half of 1999 of Shell Oil's major business units have further validated the
coverage of the Y2K remediation programs in meeting this objective. These audits
will continue during third quarter. As a contingency plan, steps are being
developed to manage this issue around key dates where unanticipated Y2K problems
may arise. Current plans include the positioning of skilled computer staff in
the various areas of business operations to insure quick response to any
unexpected Y2K problems arising due to failure to previously detect the problem
or due to inadequate remediation. Shell Oil is continuing to review and develop
its current business operation contingency plans in order to understand what
additions may be appropriate to those plans to deal with residual risk
associated with Y2K problems. These plans include decisions on operating
schedules and staff deployment during time periods in which Y2K-related problems
are most likely to occur.

         Shell Oil realizes that its failure or the failure of critical Third
Parties to identify and correct a material Y2K problem could result in an
interruption in or failure of certain normal business activities or operations.
While such failures could materially and adversely affect Shell Oil's operations
and relationships with Third Parties, the objective of Shell Oil's Year 2000
Team is the significant reduction or complete elimination of these risks. Shell
Oil expects to achieve this goal. As stated above, steps are being taken to
manage this issue around key dates where the problem may arise.

         The discussion of Shell Oil's Y2K efforts and expectations regarding
the same includes forward-looking statements within the meaning of Section 27A
of the Securities Act of 1933. Shell Oil's ability to


                                       14
<PAGE>   15

achieve Year 2000 readiness and the level of costs associated therewith could be
adversely impacted by, among other things: the availability and cost of
programming and testing resources; vendors' ability to install or modify
hardware, software or other computer technology in Shell Oil systems; and
unanticipated problems identified as the Year 2000 Teams conclude their analysis
and verification of IT and Imbedded Technology Systems. Additionally, no
precedent exists as to the manner in which to fully detect and eliminate Y2K
risks. Thus, it is possible that despite all efforts to identify, analyze and
remediate Y2K related problems, not all potential problems may be detected or
all remediation efforts operate as intended; it is impossible to accurately
predict the impact on Shell Oil in any such case. Finally, the failure of Third
Parties to achieve acceptable Y2K compliance and the absence of available
contingent suppliers and resources could also materially adversely affect Shell
Oil.

STATEMENTS OF FINANCIAL ACCOUNTING STANDARDS NOT YET ADOPTED

         In June 1998 the Financial Accounting Standards Board issued Statement
of Financial Accounting Standards No. 133, "Accounting for Derivative
Instruments and Hedging Activities" ("SFAS 133"). As issued, this new standard
is effective for fiscal years beginning after June 15, 1999. In June 1999, the
Financial Accounting Standard Board issued SFAS 137 which deferred the effective
date of SFAS 133 by one year to be effective for fiscal years beginning after
June 15, 2000 (January 1, 2001 for the Company). SFAS 133 requires that all
derivative instruments be recorded on the balance sheet at their fair value.
Changes in the fair value of derivatives are recorded each period in current
earnings or other comprehensive income, depending on whether a derivative is
designated as part of a hedge transaction and, if it is, the type of hedge
transaction. The Company has not yet completed its evaluation of the impact of
the adoption of this new standard.

         In addition to the economic conditions and other matters discussed
above affecting Shell Oil, the operations, earnings and financial condition of
Shell Oil may be affected by political developments; litigation; and
legislation, regulation and other actions taken by federal, state, local and
foreign governmental entities, including those matters discussed in Note E of
the Notes to Interim Financial Statements.

                                   ----------


                                       15
<PAGE>   16

                           PART II. OTHER INFORMATION


ITEM 1. LEGAL PROCEEDINGS.

         The Company's subsidiary, Shell Chemical Company, received a Notice of
Violation and Proposed Penalty from the Louisiana Department of Environmental
Quality ("LADEQ") alleging violations in 1998 at the Norco, Louisiana chemical
complex of the leak detection and repair regulations under the Louisiana
Environmental Quality Act. Shell Chemical Company and the LADEQ are engaged in
discussions regarding the allegations of the notice.


ITEM 6. EXHIBITS AND REPORTS ON FORM 8-K

         (a)    Exhibits.

                27.  Financial Data Schedule.

         (b)    Reports on Form 8-K.

                None


                                   SIGNATURES

         Pursuant to the requirements of the Securities Exchange Act of 1934,
the Registrant has duly caused this report to be signed on its behalf by the
undersigned thereunto duly authorized.


                                          SHELL OIL COMPANY



                                       By             N. J. CARUSO
                                          -------------------------------------
                                          N. J. Caruso, Chief Financial Officer
                                          and Controller (Principal Accounting
                                          and Duly Authorized Officer)


Date:  August 5, 1999


                                       16
<PAGE>   17

                                INDEX TO EXHIBITS


<TABLE>
<CAPTION>
EXHIBIT
NUMBER        DESCRIPTION
-------       -----------
<S>           <C>
  27          Financial Data Schedule
</TABLE>