Injection Rates - The Effect of Mobility Ratio, Area Swept, and Pattern

1961 ◽  
Vol 1 (02) ◽  
pp. 81-91 ◽  
Author(s):  
John C. Deppe
Keyword(s):  
Radial Flow ◽  
Initial Rate ◽  
Approximate Equation ◽  
Mobility Ratio ◽  
Rate Equations ◽  
Sweep Efficiency ◽  
Linear Flow ◽  
Flow Equations ◽  
Well Pattern ◽  
Injection Rates

Abstract A method is presented for calculating approximate injection rates in secondary recovery operations. The method can he applied to cases of unequal fluid mobilities, irregular well patterns and boundary patterns. The steady-state pressure distributions for the four flood patterns reported by Muskat and for five additional patterns reveal that most of the difference in pressure between the injection and producing wells occurs in regions around the wells which can adequately he described as regions of radial flow. This leads to a method of calculating injectivity by approximating the flood pattern with radial flow elements (or a combination of radial- and linear-flow elements for some patterns such as the direct line drive). Irregular and boundary patterns can also be approximated by radial and linear elements. Each of these elements can be described by radial- and linear-flow equations and the results combined as series flow resistances to give an approximate equation for the initial injection rate. The mobility ratio does not affect the initial rate; therefore, if the well pattern is one of those regular patterns for which theoretical rate equations have been derived for unity mobility ratio, the approximate initial rate equation can be improved by adjusting it to match the theoretical equation. The available theoretical rate equations are listed, including five new cases. As the flood progresses, the injectivity changes because in general the flood front will divide the pattern into areas of different fluid mobilities. Simple shapes can be assumed for the flood front so that both areas can be divided into radial- and linear-flow elements. Radial- and linear-flow equations are applied to these elements to account for the change in flow resistance behind the front. To calculate injection rates after breakthrough, it is necessary to know the sweep efficiency at breakthrough. Areal sweep data are available in the literature for a number of patterns, and sufficiently accurate breakthrough sweep efficiency can be estimated from these data if it is not otherwise available.

A Potentiometric Study of the Effects of Mobility Ratio on Reservoir Flow Patterns

1961 ◽  
Vol 1 (03) ◽  
pp. 125-129 ◽  
Author(s):  
H.B. Bradley ◽  
J.P. Heller ◽  
A.S. Odeh
Keyword(s):  
Mobility Ratio ◽  
Prototype System ◽  
Sweep Efficiency ◽  
X Ray ◽  
Reservoir Flow ◽  
Well Pattern ◽  
Before And After ◽  
Production Wells ◽  
Production History ◽  
Water Cut

Abstract A potentiometric model technique is presented for determining the a real sweep efficiency of a five-spot well pattern, at and beyond breakthrough. A sharp interface between displaced and displacing fluids is assumed. Although the prototype system is referred to as a water flood operation, obvious changes in the notation and computations will adapt the results to other displacement processes. The results of the study include the following.Areal sweep efficiencies for a five-spot well pattern, at and beyond breakthrough, for mobility ratios (displacing to displaced fluid) of 4:1, 2:1, 1:1 and 1:4.Extension of potentiometric analysis to the investigation of the areal sweep-efficiency beyond breakthrough in a five-spot pattern by the application of conformal mapping and conductive-solid models.The use of layers of conductive fabric in representing mobility ratio changes in potentiometric models.The development of a probe mechanism for probing conductive solids. The results obtained by conductive-cloth models agree with earlier areal sweep efficiencies at breakthrough obtained by Aronofsky and Ramey on the potentiometric analyzer using electrolytic-tank models. Results beyond breakthrough differ from those obtained by the X-Ray Shadowgraph technique. Data from this study show that, for mobility ratios greater than one, water cut rises rapidly as fluid is produced after breakthrough. However, for mobility ratios smaller than one, a large increase in area swept resulted with only a small increase in water cut. Introduction In calculating reservoir performance of waterflooding operations and other fluid-injection programs, it is necessary to estimate the areal sweep efficiency before and after injected-fluid breakthrough into production wells. Influence of mobility ratio on oil-production history, before and after breakthrough for a five-spot well pattern, has been studied by X-Ray Shadowgraph techniques and by gelatin models. In none of these investigations was the transition zone controlled experimentally.

scholarly journals A Single Equation to Depict Bottomhole Pressure Behavior for a Uniform Flux Hydraulic Fractured Well

2022 ◽  
Vol 12 (2) ◽  
pp. 817
Author(s):  
Jang Hyun Lee ◽  
Juhairi Aris Bin Muhamad Shuhili
Keyword(s):  
Radial Flow ◽  
Early Stage ◽  
Error Function ◽  
Well Test ◽  
Transition Flow ◽  
Linear Flow ◽  
Pressure Transient ◽  
Flow Equations ◽  
Transition Flow Regime ◽  
Fractured Well

Pressure transient analysis for a vertically hydraulically fractured well is evaluated using two different equations, which cater for linear flow at the early stage and radial flow in the later stage. However, there are three different stages that take place for an analysis of pressure transient, namely linear, transition and pseudo-radial flow. The transition flow regime is usually studied by numerical, inclusive methods or approximated analytically, for which no specific equation has been built, using the linear and radial equations. Neither of the approaches are fully analytical. The numerical, inclusive approach results in separate calculations for the different flow regimes because the equation cannot cater for all of the regimes, while the analytical approach results in a difficult inversion process to compute well test-derived properties such as permeability. There are two types of flow patterns in the fracture, which are uniform and non-uniform, called infinite conductivity in a high conductivity fracture. The study was conducted by utilizing an analogous study of linear flow equations. Instead of using the conventional error function, the exponential integral with an infinite number of wells was used. The results obtained from the developed analytical solution matched the numerical results, which proved that the equation was representative of the case. In conclusion, the generated analytical equation can be directly used as a substitute for current methods of analyzing uniform flow in a hydraulically fractured well.

Unit Mobility Ratio Displacement Calculations for Pattern Floods in Homogeneous Medium

1966 ◽  
Vol 6 (03) ◽  
pp. 217-227 ◽  
Author(s):  
Hubert J. Morel-Seytoux
Keyword(s):  
Oil Recovery ◽  
Numerical Solutions ◽  
Mobility Ratio ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Numerical Techniques ◽  
Sweep Efficiency ◽  
Closed Form Solutions ◽  
Regular Patterns ◽  
Quantities Of Interest

Abstract The influence of pattern geometry on assisted oil recovery for a particular displacement mechanism is the object of investigation in this paper. The displacement is assumed to be of unit mobility ratio and piston-like. Fluids are assumed incompressible and gravity and capillary effects are neglected. With these assumptions it is possible to calculate by analytical methods the quantities of interest to the reservoir engineer for a great variety of patterns. Specifically, this paper presentsvery briefly, the methods and mathematical derivations required to obtain the results of engineering concern, andtypical results in the form of graphs or formulae that can be used readily without prior study of the methods. Results of this work provide checks for solutions obtained from programmed numerical techniques. They also reveal the effect of pattern geometry and, even though the assumptions of piston-like displacement and of unit mobility ratio are restrictive, they can nevertheless be used for rather crude but quick, cheap estimates. These estimates can be refined to account for non-unit mobility ratio and two-phase flow by correlating analytical results in the case M=1 and the numerical results for non-Piston, non-unit mobility ratio displacements. In an earlier paper1 it was also shown that from the knowledge of closed form solutions for unit mobility ratio, quantities called "scale factors" could be readily calculated, increasing considerably the flexibility of the numerical techniques. Many new closed form solutions are given in this paper. INTRODUCTION BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected. BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected.

A Novel Infinite-Acting-Radial-Flow Analysis Procedure for Estimating Permeability Anisotropy From an Observation-Probe Pressure Response at a Vertical, Horizontal, or Inclined Wellbore

2014 ◽  
Vol 17 (02) ◽  
pp. 152-164 ◽  
Author(s):  
M.. Onur ◽  
P.S.. S. Hegeman ◽  
I.M.. M. Gök
Keyword(s):  
Radial Flow ◽  
Flow Analysis ◽  
Single Layer ◽  
Flow Equation ◽  
Observation Point ◽  
Analysis Procedure ◽  
Pressure Transient ◽  
Flow Equations ◽  
Vertical Permeability ◽  
Inclination Angles

Summary This paper presents a new infinite-acting-radial-flow (IARF) analysis procedure for estimating horizontal and vertical permeability solely from pressure-transient data acquired at an observation probe during an interval pressure-transient test (IPTT) conducted with a single-probe, dual-probe, or dual-packer module. The procedure is based on new infinite-acting-radial-flow equations that apply for all inclination angles of the wellbore in a single-layer, 3D anisotropic, homogeneous porous medium. The equations for 2D anisotropic cases are also presented and are derived from the general equations given for the 3D anisotropic case. It is shown that the radial-flow equation presented reduces to Prats' (1970) equation assuming infinite-acting radial flow at an observation point along a vertical wellbore in isotropic or 2D anisotropic formations of finite bed thickness. The applicability of the analysis procedure is demonstrated by considering synthetic and field packer/probe IPTT data. The synthetic IPTT examples include horizontal- and slanted-well cases, but the field IPTT is for a vertical well. The results indicate that the procedure provides reliable estimates of horizontal and vertical permeability solely from observation-probe pressure data during radial flow for vertical, horizontal, and inclined wellbores. Most importantly, the analysis does not require that both spherical and radial flow prevail at the observation probe during the test.

scholarly journals Enhanced Oil Recovery: Chemical Flooding

2021 ◽  
Author(s):  
Ahmed Ragab ◽  
Eman M. Mansour
Keyword(s):  
Interfacial Tension ◽  
Enhanced Oil Recovery ◽  
Oil Recovery ◽  
Recovery Phase ◽  
Mobility Ratio ◽  
Chemical Flooding ◽  
Sweep Efficiency ◽  
Oil Displacement ◽  
Remaining Oil ◽  
Oil Displacement Efficiency

The enhanced oil recovery phase of oil reservoirs production usually comes after the water/gas injection (secondary recovery) phase. The main objective of EOR application is to mobilize the remaining oil through enhancing the oil displacement and volumetric sweep efficiency. The oil displacement efficiency enhances by reducing the oil viscosity and/or by reducing the interfacial tension, while the volumetric sweep efficiency improves by developing a favorable mobility ratio between the displacing fluid and the remaining oil. It is important to identify remaining oil and the production mechanisms that are necessary to improve oil recovery prior to implementing an EOR phase. Chemical enhanced oil recovery is one of the major EOR methods that reduces the residual oil saturation by lowering water-oil interfacial tension (surfactant/alkaline) and increases the volumetric sweep efficiency by reducing the water-oil mobility ratio (polymer). In this chapter, the basic mechanisms of different chemical methods have been discussed including the interactions of different chemicals with the reservoir rocks and fluids. In addition, an up-to-date status of chemical flooding at the laboratory scale, pilot projects and field applications have been reported.

Field Implementation of In-Depth Conformance Gel Treatment Prior to Starting an ASP Flooding Pilot

2021 ◽  
Author(s):  
Mohammed T. Al-Murayri ◽  
Abrahim Hassan ◽  
Naser Alajmi ◽  
Jimmy Nesbit ◽  
Bastien Thery ◽  
...  
Keyword(s):  
Polymer Solution ◽  
Injection Pressure ◽  
High Viscosity ◽  
Production Well ◽  
Sweep Efficiency ◽  
Injection Wells ◽  
Appreciable Change ◽  
Well Pattern ◽  
Gel Treatment ◽  
Viscosity Polymer

Abstract Mature carbonate reservoirs under waterflood in Kuwait suffer from relatively low oil recovery due to poor volumetric sweep efficiency, both areal, vertically, and microscopically. An Alkaline-Surfactant-Polymer (ASP) pilot using a regular five-spot well pattern is in progress targeting the Sabriyah Mauddud (SAMA) reservoir in pursuit of reserves growth and production sustainability. SAMA suffers from reservoir heterogeneities mainly associated with permeability contrast which may be improved with a conformance treatment to de-risk pre-mature breakthrough of water and chemical EOR agents in preparation for subsequent ASP injection and to improve reservoir contact by the injected fluids. Each of the four injection wells in the SAMA ASP pilot was treated with a chemical conformance improvement formulation. A high viscosity polymer solution (HVPS) of 200 cP was injected prior to a gelant formulation consisting of P300 polymer and X1050 crosslinker. After a shut-in period, wells were then returned to water injection. Injection of high viscosity polymer solution (HVPS) at the four injection wells showed no increase in injection pressure and occurred higher than expected injection rates. Early breakthrough of polymer was observed at SA-0561 production well from three of the four injection wells. No appreciable change in oil cut was observed. HVPS did not improve volumetric sweep efficiency based on the injection and production data. Gel treatment to improve the volumetric conformance of the four injection wells resulted in all the injection wells showing increased of injection pressure from approximately 3000 psi to 3600 psi while injecting at a constant rate of approximately 2,000 bb/day/well. Injection profiles from each of the injection well ILTs showed increased injection into lower-capacity zones and decreased injection into high-capacity zones. Inter-well tracer testing showed delayed tracer breakthrough at the center SA-0561 production well from each of the four injection wells after gel placement. SA-0561 produced average daily produced temperature increased from approximately 40°C to over 50°C. SA-0561 oil cuts increased up to almost 12% from negligible oil sheen prior to gel treatments. Gel treatment improved volumetric sweep efficiency in the SAMA SAP pilot area.

New Calculation Method of Sweep Efficiency after Pattern Infilling in Low Permeability Reservoir

2014 ◽  
Vol 1010-1012 ◽  
pp. 1713-1718
Author(s):  
Fang Zhao ◽  
Rui Shen ◽  
Gang Xu
Keyword(s):  
Calculation Method ◽  
Statistical Data ◽  
Effective Means ◽  
Low Permeability ◽  
Effect Evaluation ◽  
Sweep Efficiency ◽  
Well Spacing ◽  
Well Pattern ◽  
Development Effectiveness ◽  
Result Show

Sweep efficiency is a very important parameters for development effect evaluation and dynamic analysis of oilfield. For low permeability oilfield, well pattern thickening is one of the most effective means of improving development effectiveness. In this paper, a corrected calculation method is given and well spacing density is introduced as a parameter for the formula correction. The curve of volumetric sweep efficiency and well spacing density was achieved through the formula and statistical data. After the infill adjustment, increasing multiple of sweep efficiency can be calculated. Using the actual data of Changqing oilfield to calculate, result show that the deviation is 1.1% .

Unsteady-State Spherical Flow With Storage and Skin

1985 ◽  
Vol 25 (06) ◽  
pp. 804-822 ◽  
Author(s):  
Jeffrey A. Joseph ◽  
Leonard F. Koederitz
Keyword(s):  
Boundary Condition ◽  
Radial Flow ◽  
Linear Partial Differential Equation ◽  
Flow Profile ◽  
Unsteady State ◽  
Well Test ◽  
Linear Flow ◽  
Dimensionless Groups ◽  
Short Time ◽  
Suitable Expression

Abstract This paper presents short-time interpretation methods for radial-spherical (or radial-hemispherical) flow in homogeneous and isotropic reservoirs inclusive of wellbore storage, wellbore phase redistribution, and damage skin effects. New dimensionless groups are introduced to facilitate the classic transformation from radial flow in the sphere to linear flow in the rod. Analytical expressions, type curves (in log-log and semilog format), and tabulated solutions are presented, both in terms of pressure and rate, for all flow problems considered. A new empirical equation to estimate the duration of wellbore and near-wellbore effects under spherical flow is also proposed. Introduction The majority of the reported research on unsteady-state flow theory applicable to well testing usually assumes a cylindrical (typically a radial-cylindrical) flow profile because this condition is valid for many test situations. Certain well tests, however, are better modeled by assuming a spherical flow symmetry (e.g., wireline formation testing, vertical interference testing, and perhaps even some tests conducted in wellbores that do not fully penetrate the productive horizon or are selectively penetrate the productive horizon or are selectively completed). Plugged perforations or blockage of a large part of an openhole interval may also promote spherical flow. Numerous solutions are available in the literature for almost every conceivable cylindrical flow problem; unfortunately, the companion spherical problem has not received as much attention, and comparatively few papers have been published on this topic. papers have been published on this topic. The most common inner boundary condition in well test analysis is that of a constant production rate. But with the advent of downhole tools capable of the simultaneous measurement of pressures and flow rates, this idealized inner boundary condition has been refined and more sophisticated models have been proposed. Therefore, similar methods must be developed for spherical flow analysis, especially for short-time interpretations. This general problem has recently been addressed elsewhere. Theory The fundamental linear partial differential equation (PDE) describing fluid flow in an infinite medium characterized by a radial-spherical symmetry is (1) The assumptions incorporated into this diffusion equation are similar to those imposed on the radial-cylindrical diffusivity equation and are discussed at length in Ref. 9. In solving Eq. 1, the classic approach is illustrated by Carslaw and Jaeger (later used by Chatas, and Brigham et al.). According to Carslaw and Jaeger, mapping b=pr will always reduce the problem of radial flow in the sphere (Eq. 1) to an equivalent problem of linear flow in the rod for which general solutions are usually known. (For example, see Ref. 17 for particular solutions in petroleum applications.) Note that in this study, we assumed that the medium is spherically isotropic; hence k in Eq. 1 is the constant spherical permeability. This assumption, however, does not preclude analysis in systems possessing simple anisotropy (i.e., uniform but unequal horizontal and vertical permeability components). In this case, k as used in this paper should be replaced by k, an equivalent or average (but constant) spherical permeability. Chatas presented a suitable expression (his Eq. 10) obtained presented a suitable expression (his Eq. 10) obtained from a volume integral. It is desirable to transform Eq. 1 to a nondimensional form, thereby rendering its applicability universal. The following new, dimensionless groups accomplish this and have the added feature that solutions are obtained directly in terms of the dimensionless pressure drop, PD, not the usual b (or bD) groups. ......................(2) .......................(3) .........................(4) The quantity rsw is an equivalent or pseudospherical wellbore radius used to represent the actual cylindrical sink (or source) of radius rw. SPEJ p. 804

Derivation of Initial Rate Equations for Low-temperature Water Gas Shift Reaction over Pt–Re/ZrO2Catalysts

2010 ◽  
Vol 39 (2) ◽  
pp. 144-145 ◽  
Author(s):  
Hajime Iida ◽  
Takumi Yamamoto ◽  
Masaru Inagaki ◽  
Akira Igarashi
Keyword(s):  
Low Temperature ◽  
Initial Rate ◽  
Water Gas Shift ◽  
Rate Equations ◽  
Water Gas Shift Reaction ◽  
Shift Reaction ◽  
Water Gas ◽  
Temperature Water

scholarly journals Integrated rate equations for enzyme-catalysed first-order and second-order reactions

1984 ◽  
Vol 223 (1) ◽  
pp. 15-22 ◽  
Author(s):  
E A Boeker
Keyword(s):  
Initial Rate ◽  
Second Order ◽  
Rate Equations ◽  
First Order ◽  
Rate Kinetics

Generalized rate equations covering all mechanisms giving hyperbolic initial-rate kinetics with stoichiometry A in equilibrium P, A in equilibrium P + Q, A + B in equilibrium P and A + B in equilibrium P + Q were integrated. The results are regular and reasonably economical.

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