Summary Gas wells frequently exhibit changing storage during a transient test because of high fluid compressibility. Further complications may arise due to beat exchange between the wellbore fluid and the formation, especially in high-temperature reservoirs. Thus, fluid temperature changes during a transient test, thereby complicating test interpretation when surface measurements must be used in a hostile downhole environment. In this work we present a transient wellbore/reservoir model for testing gas wells that is particularly useful for high- temperature reservoirs. The model can be used in a forward mode, given reservoir and wellbore parameters, to predict pressure and temperature at any depth during a transient test. The wellbore model formulation involves the use of mass, momentum, and energy balances for a single-phase gas together with the PVT relation to generate the constitutive equations. The reservoir fluid flow is modeled using the standard analytic approach, including superposition effects. Heat transport in the wellbore accounts for conductive and convective heat flow through the annul us fluid and conductive heat transport through the tubulars and cement sheaths into the formation. The finite-wellbore radius solution of the thermal diffusivity equation accounts for heat flow in the formation. Energy balance for the fluid accounts for the Joule-Thompson expansion. A sensitivity study provided some insights into the effect of process variables on wellbore pressure and temperature. As expected, fluid flow rate is shown to have a very significant impact on the wellhead pressure and temperature. Clearly, the temperature rating of surface equipment could limit the maximum production from some wells. Heat transport in the annulus between the tubing and casing also strongly influences wellhead fluid temperature. For example, a fluid with a low-heat transfer coefficient, such as a or oil-based mud, would allow the wellbore gas to retain much of the enthalpy, leading to high fluid temperatures at the wellhead. Unlike previous formulations, this model accounts for the energy absorbed (or released) by the tubulars and the cement sheaths, which is a significant fraction of the energy exchange between the wellbore and the formation at early times. A consequence of accounting for heat capacity of the wellbore system is that rapid temperature rise or fall during a test is dampened, mimicking the actual field response. Introduction A routine well-test interpretation or forward modeling invokes the well-known constant-storage model. When a test is associated with either increasing or decreasing storage, one could use the Fair or the Hegeman et al. model. These wellbore models are popular because they are operable in Laplace space and, therefore, can be linked easily with most analytic reservoir models whose solutions are also available in Laplace space.
The Forbes Formation (Santonian and lower Campanian) and Winters Formation (upper Campanian and lower Maastrichtian) are highly productive turbidite reservoirs in the Sacramento forearc basin of northern California. However, their reservoir geometries and architectures differ markedly, chiefly as a result of differences in sediment supply and orientation of sediment input into the basin. The Forbes Formation has been penetrated by thousands of wells and is productive in about 50 gas fields. It consists, in ascending stratigraphic order, of four major architectural elements that can be distinguished in well logs, seismic-reflection data, and outcrop: (1) a basal massive, concretionary basinwide mudstone, the Dobbins Shale Member, interpreted to be a condensed section deposited during a sea-level highstand; (2) an interval of shale with abundant laterally continuous thin-bedded and fine-grained turbidites interpreted to be a basin-plain succession; (3) a thick interval of mudstone with interbedded sandy and silty turbidites, interpreted to be a mud-rich submarine fan dominated by meandering channel-levee complexes that form the principal reservoir units; and (4) an upper interval of mostly mudstone with laterally discontinuous sandstone bodies incised into south-sloping clinoforms, interpreted to be a muddy slope succession containing abundant sandstone-filled gullies. The deep-marine system was fed by a south-prograding fluvial-dominated delta, the Kione Formation. Both units are overlain by a second basinwide massive mudstone, the Sacramento Shale. The meanderform, ribbon-shaped channel-levee complexes, oriented subparallel to the structural axis of the basin, produce from a variety of traps. The overlying Winters Formation in the central and southern Sacramento basin has also been penetrated by thousands of wells, and produces from about 40 gas fields. It overlies the Sacramento Shale, interpreted to have been deposited as a condensed section during a long sea-level highstand. The Winters Formation consists, in ascending stratigraphic order, of (1) an interval of mostly shale interpreted to be a basin-plain succession, (2) a complex of pod-shaped sandstone bodies interpreted to have been deposited as sand-rich submarine fans dominated by suprafan-type sandstone bodies that form the principal reservoir units, and (3) muddy slope deposits that contain southwest-dipping clinoforms incised by prominent sandstone-filled submarine canyons. The deep-marine system was fed by a complex of west- and southwest-prograding wave-dominated deltas of the Starkey Formation. The small, pod-shaped submarine fans and pod-shaped suprafan sandstone bodies produce from a variety of traps, and provide a model for other sand-rich, delta-fed submarine-fan systems deposited in restricted basins with modest amounts of shelf accommodation space.
A case study is presented in which a two-stage stochastic modeling approach was employed for modeling heterogeneities in a deep-water Gulf of Mexico field. The hydrocarbonbearing intervals of this field were deposited as turbidites and exhibit high net sand/gross interval (net/gross) ratios.In the first stage, a new facies indicator geostatistical method was used to model uncertainties associated with facies maps derived from seismic amplitudes. The concept of pseudo-wells is introduced for capturing geologic features and constraining the stochastic realizations. In the second modeling stage, synthetic stratigraphic columns based on whole core descriptions were used to capture fine-scale heterogeneities in cross-sectional panels. These cross-sections were subjected to flow simulations, yielding scaled-up values of single and two-phase flow parameters. Results from both stages were combined into a full-field 3-D reservoir simulation model. Lx = correlation range in x-direction, L, ft Ly = correlation range in y-direction, L, ft a = orientation angle of correlation, degrees Kh = horizontal absolute permeability, L 2 , md Kv =vertical absolute permeability,L 2 , md ¢J = porosity
This paper describes the application of the transient wellbordresewoir model developed in Part I of this work. The model ahws forward simulation of 'W-eiiheadPrasdlc and -. . . .temperatureand bottomhole pressureas a function of time, given reaenfoir parameters and well completion details. Another simulator has been developed that directly translates measured wellhead pressure(WHP) and temperature(WHT) to bottomhole pressure (BHP) for subsequent aml ysis.Forwardsimulation of wellbore pressureand temperature k a valuable tool for designing transientwell testa. In partkndar, in a hostile downhole environment, the ability to estimate BHP accurately tbm wellhead measurements is an invaluable asset where downhole measurements may not be coat effective. Even in favorable situations, the simulator can aid both test design and interpretation. Application of the simulator to field examples has helped us gain considerable insight into the mechanicsof transientflow in the wellbore. For example, we observe a denser fluid towarda the wellhead ratherthan at the bottom of the well beeause of the tempemture effect. Consequently, decreasing .----..: -:" ...&._,A~.* the hdtom of the We!!, siorage ucliuv al IS WiW. -. . -. . . . ... References and f~urea at the end of papr.Two field examples, obtained from a Gulf Coast gas field, are used to illustrate the capabilities of the simulators. Good agreement is noted between the computed and measured BHPs during both types of simulations. wmputed and the measured BHFs.We are grateful to the Chevron management for permission to publish this work. Nomenclature Flow cross-sectional area, F (m?. Gas formation volume factor, cu ft/Scf Late-time storage coefficient bbl/psi (m'/kPa). Early time storage coefficient, bbl/psi (m3/kPa).Variable-storage pressure pammeter, dimensionless.Heat capacity of formation, Bt@b "F @J/kg~) Heat capacity of gas, Btu/lbm 'T (kJ/kg "C). Joule-Thompson coefficient, dimensionless. Flow string diameter, ft (m).Internal ene~y, Btu/lbm (kJAg). Friction factor, dimensionless. Acceleration due to gravity, ft/sec2 (m/s2). Conversion tictor, 32.17 lbm-iMbf-sec2, unity in S1 utits, dmensioniess. Geothermal temperaturegradient~F/ft~C/rn). Enthdpy, Btwlbm (kJ/kg). Formation permeability, md (mD) Earth conductivity, Btu/ft 'F (kJ/m"C). Relaxation distance, l/tl (m). Relaxation distance after n tirm?steps,l/ft (m). Mass of gas in a wntrol volume, Ibm (kg) Mass per unit length of tubing/casing/cement system, Ibm (kg) Semilog slope (= 1626 Bs #5.615 kb) Molecular weight of gas, lb-moleflbm (kg-moleAg) Number of gas moles in the wellbore, lb-mole (kg-mole)
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