A Mathematical Model Water Movement about Bottom-Water-Drive Reservoirs

1962 ◽  
Vol 2 (01) ◽  
pp. 44-52 ◽  
Author(s):  
Keith H. Coats
Keyword(s):  
Pore Volume ◽  
Bottom Water ◽  
Flow Model ◽  
Radial Flow ◽  
Water Movement ◽  
Gas Storage ◽  
Reservoir Pressure ◽  
Influx Rate ◽  
Aquifer System ◽  
Water Influx

Abstract This paper presents the development and solution of a mathematical model for aquifer water movement about bottom-water-drive reservoirs. Pressure gradients in the vertical direction due to water flow are taken into account. A vertical permeability equal to a fraction of the horizontal permeability is also included in the model. The solution is given in the form of a dimensionless pressure-drop quantity tabulated as a function of dimensionless time. This quantity can be used in given equations to compute reservoir pressure from a known water-influx rate, to predict water- in flux rate (or cumulative amount) from a reservoir- pressure schedule or to predict gas reservoir pressure and pore-volume performance from a given gas-in-place schedule. The model is applied in example problems to gas-storage reservoirs, and the difference between reservoir performances predicted by the thick sand model of this paper and the horizontal, radial-flow model is shown to be appreciable. Introduction The calculation of aquifer water movement into or out of oil and gas reservoirs situated on aquifers is important in pressure maintenance studies, material-balance and well-flooding calculations. In gas storage operations, a knowledge of the water movement is especially important in predicting pressure and pore-volume behavior. Throughout this paper the term "pore volume" denotes volume occupied by the reservoir fluid, while the term "flow model" refers to the idealized or mathematical representation of water flow in the reservoir-aquifer system. The prediction of water movement requires selection of a flow model for the reservoir-aquifer system. A physically reasonable flow model treated in detail to date is the radial-flow model considered by van Everdingen and Hurst. In many cases the reservoir is situated on top of the aquifer with a continuous horizontal interface between reservoir fluid and aquifer water and with a significant depth of aquifer underlying the reservoir. In these cases, bottom-water drive will occur, and a three-dimensional model accounting for the pressure gradient and water flow in the vertical direction should be employed. This paper treats such a model in detail--from the description of the model through formulation of the governing partial differential equation to solution of the equation and preparation of tables giving dimensionless pressure drop as a function of dimensionless time. The model rigorously accounts for the practical case of a vertical permeability equal to some fraction of the horizontal permeability. The pressure-drop values can be used in given equations to predict reservoir pressure from a known water-influx rate or to predict water-influx rate (or cumulative amount) when the reservoir pressure is known. The inclusion of gravity in this analysis is actually trivial since gravity has virtually no effect on the flow of a homogeneous, slightly compressible fluid in a fixed-boundary system subject to the boundary conditions imposed in this study. Thus, if the acceleration of gravity is set equal to zero in the following equations, the final result is unchanged. The pressure distribution is altered by inclusion of gravity in the analysis, but only by the time-constant hydrostatic head. The equations developed are applied in an example case study to predict the pressure and pore-volume behavior of a gas storage reservoir. The prediction of reservoir performance based on the bottom-water-drive model is shown to differ significantly from that based on van Everdingen and Hurst's horizontal-flow model. DESCRIPTION OF FLOW MODEL The edge-water-drive flow model treated by van Everdingen and Hurst is shown in Fig. 1a. The aquifer thickness is small in relation to reservoir radius water invades or recedes from the field at the latter's edges, and only horizontal radial flow is considered as shown in Fig. 1b. The bottom-water-drive reservoir-aquifer system treated herein is sketched in Fig. 2a and 2b. SPEJ P. 44^

A New Approach to Determine Water Influx Rate for Reservoir with Bottom Water

2011 ◽  
Vol 130-134 ◽  
pp. 3843-3846
Author(s):  
Wen Zhong Zhang ◽  
Jian Ping Wang
Keyword(s):  
Sensitivity Analysis ◽  
Bottom Water ◽  
Solution Method ◽  
Material Balance ◽  
The Other ◽  
Reservoir Pressure ◽  
Affecting Factors ◽  
Influx Rate ◽  
New Approach ◽  
Water Influx

The average reservoir pressure and water influx rate are two important parameters for reservoir with active bottom water. The paper, based on the model of KEITH, provides a solution by inversion of Laplace and Hankel, and obtains the relation between the average dimensionless reservoir pressure and water influx. If one is known, then the other can to be confirmed correctly. Compared with solution method of material balance, the affection of anisotropy is considered. In addition, sensitivity analysis of affecting factors on reservoir pressure is presented.

Laboratory Models of Oil Reservoirs Produced By Natural Water Drive

1965 ◽  
Vol 5 (01) ◽  
pp. 25-36 ◽  
Author(s):  
B.H. Caudle ◽  
I.H. Silberberg
Keyword(s):  
Fluid Flow ◽  
Water Flow ◽  
Natural Water ◽  
Bottom Water ◽  
Water Movement ◽  
Flow System ◽  
Reservoir Rock ◽  
Oil Reservoir ◽  
Productive Capacity ◽  
Water Influx

Abstract Reservoir depletion by natural water drive is typified by the movement of water from an aquifer into the adjacent oil-bearing formation. Prior studies of this type of water movement have generally neglected the resistance to flow in the aquifer and its effect on the movement of water into the oil bearing zone. A method for designing and operating scaled models of such reservoir systems is presented, Experimental data on a model of an edge-water-drive reservoir are shown and discussed. Introduction Most petroleum reservoirs derive at least a part of their productive capacity from water influx. This water may be injected from the surface or it may come from an aquifer adjacent to the oil zone. In either case, the reservoir engineer must be able to estimate the advance of the water as a function of either the elapsed time or the fluid produced. This paper describes a type of fluid flow model which includes the effect of viscous and gravitational forces in the reservoir and the surrounding aquifer. Natural water influx can be divided into three general (but widely overlapping) categories according to the direction of flow in the aquifer. These are shown schematically in Fig. 1. Fig. 1 (a) illustrates an edge-water-drive mechanism. In this case, water advances updip along the stratum but the movement of water is essentially horizontal and very little of the oil is actually underlain by water. The bottom-water drive (Fig. l (b)) is characterized by a thick aquifer underlying the oil zone. The water movement is generally vertical in the aquifer. The third category is illustrated in 1 (c). Referred to as the "thin oil column", this type of oil accumulation consists of oil over water in a relatively thin stratum. Fluid movement is horizontal in both the oil and water zones, except close to the producing well. One characteristic which all three of these types have in common is that part of the water flow takes place in the water invaded region of the oil reservoir while the remainder of the water flow occurs in the aquifer. The resistance to the flow of water usually will not be the same in these two regions. This occurs because the porous rock normally contains only water in the aquifer, while the water-invaded region always contains microscopic globules of bypassed oil (residual oil) which interfere with the flow of water through the rock. The production history of any type of a water-drive system is a function of two phenomena where the water goes, and how it displaces the oil in the area invaded by the water. It is usually impractical to study these two phenomena at the same time. In most cases one type of analysis is used to predict the gross movement of water in the reservoir, while a different type of analysis is employed to determine the amount of oil to be recovered from the gross volume contacted by the water. The portion of the oil reservoir which is invaded by water is mainly a function of the resistances to fluid flow in the several parts of the flow system. This gross water influx in the three types of natural water drive typically results in water cusping to the production wells either from the side or from below (water coning).In 1947, Muskat described a greatly simplified mathematical model for predicting the area invaded by water in a bottom-water-drive reservoir. This model used the Laplace equation and suitable boundary conditions to describe the isopotentials and streamlines in the flow system. The use of this model assumes thatthe reservoir rock is homogeneous in nature,the oil and water mobilities are equal,there is no oil flowing in the water invaded region, andexternal forces - such as gravity - do not affect the flow. This method has been used for both natural water drives and water injection projects. In this, as in most models, the method of images is used to reduce the size of the model necessary to describe the reservoir. SPEJ P. 25ˆ

scholarly journals Conceptual and Mathematical Modeling of a Coastal Aquifer in Eastern Delta of R. Nestos (N. Greece)

Hydrology ◽  
2021 ◽  
Vol 8 (1) ◽  
pp. 23
Author(s):  
Ioannis Gkiougkis ◽  
Christos Pouliaris ◽  
Fotios-Konstantinos Pliakas ◽  
Ioannis Diamantis ◽  
Andreas Kallioras
Keyword(s):  
Mathematical Modeling ◽  
Groundwater Flow ◽  
Coastal Aquifer ◽  
Flow Model ◽  
Meteorological Data ◽  
Field Measurements ◽  
Groundwater Flow Model ◽  
Aquifer System ◽  
Geophysical Surveys ◽  
The Mathematical Model

In this paper, the development of the conceptual and groundwater flow model for the coastal aquifer system of the alluvial plain of River Nestos (N. Greece), that suffers from seawater intrusion due to over-pumping for irrigation, is analyzed. The study area is a typical semi-arid hydrogeologic environment, composed of a multi-layer granular aquifers that covers the eastern coastal delta system of R. Nestos. This study demonstrates the results of a series of field measurements (such as geophysical surveys, hydrochemical and isotopical measurements, hydro-meteorological data, land use, irrigation schemes) that were conducted during the period 2009 to 2014. The synthesis of the above resulted in the development of the conceptual model for this aquifer system, that formed the basis for the application of the mathematical model for simulating groundwater flow. The mathematical modeling was achieved using the finite difference method after the application of the USGS code MODFLOW-2005.

scholarly journals Research on Injection-Production Capability and Seepage Characteristics of Multi-Cycle Operation of Underground Gas Storage in Gas Field—Case Study of the Wen 23 Gas Storage

Energies ◽  
2020 ◽  
Vol 13 (15) ◽  
pp. 3829
Author(s):  
Jie Zhang ◽  
Feifei Fang ◽  
Wei Lin ◽  
Shusheng Gao ◽  
Yalong Li ◽  
...  
Keyword(s):  
Transition Zone ◽  
Pore Volume ◽  
High Speed ◽  
Gas Storage ◽  
Production Flow ◽  
Underground Gas Storage ◽  
Gas Recovery ◽  
Gas Drive ◽  
Gas Fields ◽  
Water Gas

With the increasing energy demands of current modern society, underground gas storage (UGS) in gas fields is the most popular type of UGS used to meet the seasonal variation of gas consumption. However, compared with gas fields, UGS in gas fields has the characteristics of periodic high-speed injection and production of exploitation modes and operation rules, which causes the rules of gas-water seepage and utilization of reserves to be more particular and complicated. In this paper, based on Wen 23 gas storage, the rules of multicycle injection and production flow and the utilization of UGS pore volume were investigated. The experimental results showed that variation in porosity and permeability caused by injection and production pressure changes in Wen 23 gas storage can be neglected. The pore volume of gas storage and the degree of gas recovery increased gradually in the pre-UGS gas zone, which was higher than that of reservoirs. In the initial stage of UGS operation, the pore volume of gas storage and the degree of gas recovery were low in the gas-drive-water gas zone as a result of water invasion during the process of reservoir exploitation. During operation of multicycle high-speed injection and production, the seepage conditions in the gas-drive-water gas zone gradually improved. The higher the reservoir permeability, the greater increases in pore volume and degree of gas recovery. In the gas-water transition zone, gas and water were reciprocated and displaced with the multicycle injection-production of UGS, resulting in the gradual deterioration of pore volume and gas recovery, which remained stable at a low value. The negative effects of reservoir heterogeneity on the effective utilization of UGS occurred in the gas-water transition zone. These findings may contribute to a better understanding of the rules of multicycle injection and production flow and utilization of UGS to optimize the injection-production efficiency of Wen 23 gas storage.

Investigation of Fluid Conductive Faults and Modeling of Complex Water Influx in the Khafji Oil Field, Arabian Gulf

2000 ◽  
Vol 3 (05) ◽  
pp. 401-407 ◽  
Author(s):  
N. Nishikiori ◽  
Y. Hayashida
Keyword(s):  
Water Movement ◽  
Arabian Gulf ◽  
Oil Field ◽  
Water Production ◽  
Water Contact ◽  
Water Influx ◽  
Model Complex ◽  
Oil Water ◽  
Sandstone Reservoir ◽  
Sandstone Reservoirs

Summary This paper describes the multidisciplinary approach taken to investigate and model complex water influx into a water-driven sandstone reservoir, taking into account vertical water flux from the lower sand as a suspected supplemental source. The Khafji oil field is located offshore in the Arabian Gulf. Two Middle Cretaceous sandstone reservoirs are investigated to understand water movement during production. Both reservoirs are supported by a huge aquifer and had the same original oil-water contact. The reservoirs are separated by a thick and continuous shale so that the upper sand is categorized as edge water drive and the lower sand as bottomwater drive. Water production was observed at the central up structure wells of the upper sand much earlier than expected. This makes the modeling of water influx complicated because it is difficult to explain this phenomenon only by edge water influx. In this study, a technical study was performed to investigate water influx into the upper sand. A comprehensive review of pressure and production history indicated anomalous higher-pressure areas in the upper sand. Moreover, anomalous temperature profiles were observed in some wells in the same area. At the same time, watered zones were trailed through thermal-neutron decay time(TDT) where a thick water column was observed in the central area of the reservoir. In addition, a three-dimensional (3D) seismic survey has been conducted recently, revealing faults passing through the two reservoirs. Therefore, as a result of data review and subsequent investigation, conductive faults from the lower sand were suspected as supplemental fluid conduits. A pressure transient test was then designed and implemented, which suggested possible leakage from the nearby fault. Interference of the two reservoirs and an estimate of supplemental volume of water influx was made by material balance. Finally, an improved full-scale numerical reservoir model was constructed to model complex water movement, which includes suspected supplemental water from the lower sand. Employment of two kinds of water influx—one a conventional edge water and another a supplemental water invasion from the aquifer of the lowers and through conductive faults—achieved a water breakthrough match. Introduction The Khafji oil field is located in the Arabian Gulf about 40 km offshore Al-Khafji as shown by Fig. 1. The length and width of the field are about 20 and 8 km, respectively. The upper sandstone reservoir, the subject of this study, lies at a depth of about 5,000 ft subsea and was discovered in1960. The average thickness of the reservoir is about 190 ft. The reservoir is of Middle Cretaceous geologic age. Underlying the upper sandstone reservoir is another sandstone reservoir at a depth of about 5,400 ft. It has an average gross thickness of about 650 ft and is separated from the upper sand by a thick shale bed of about 200 ft. Both reservoirs had the same original oil-water contact level as shown by the subsurface reservoir profile in Fig. 2. Both sandstone reservoirs are categorized as strong waterdrive that can maintain reservoir pressure well above the bubblepoint. On the other hand, water production cannot be avoided because of an unfavorable water-to-oil mobility ratio of 2 to 4 and high formation permeability in conjunction with a strong waterdrive mechanism. In a typical edge water drive reservoir, water production normally begins from the peripheral wells located near the oil-water contact and water encroaches as oil production proceeds. However, some production wells located in the central up structure area of the upper sand started to produce formation water before the wells located in the flank area near the water level. In 1996, we started an integrated geological and reservoir study to maximize oil recovery, to enhance reservoir management, and to optimize the production scheme for both sandstone reservoirs. This paper describes a part of the integrated study, which focused on the modeling of water movement in the upper sand. The contents of the study described in this paper are outlined as:diagnosis and description of the reservoir by fully utilizing available data, which include comprehensive review of production history, TDT logs, formation temperatures, pressures, and 3D seismic; introduction of fluid conductive faults as a suspected supplemental water source in the central upstructure area; design and implementation of a pressure transient test to investigate communication between the reservoirs and conductivity of faults; running of material balance for the two reservoirs simultaneously to assess their interference; and construction of an improved full-scale reservoir simulation model and precise modeling of complex water movement. Brief Geological Description of the Upper Sand The structure of the upper sand is anticline with the major axis running northeast to southwest. The structure dip is gentle (Fig. 3) at about3° on the northwestern flank and 2° on the southeastern flank. The upper sand is composed mainly of sandstone-dominated sandstone and shale sequences. It is interpreted that the depositional environment is complex, consisting of shoreface and tide-influenced fluvial channels.

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