## 152 Flow Equations

The general equations for describing fluid flow in a porous medium are shown in Table 15-1 and associated nomenclature is presented in Table 15-2. The molar conservation equation includes a dispersion term, a convection term, a source sink term representing wells, and the time varying accumulation term. The dispersion term is usually neglected in most workhorse simulators such as black oil and compositional simulators. Neglecting dispersion simplifies program coding and is justified when...

## Derivation of the Flow Equations

Many derivations of the oil, water, and gas fluid flow equations exist in the literature for example, see Crichlow, 1977 Peaceman, 1977 . Consequently, only a brief discussion will be presented here. It closely follows the presentation originally published in Fanchi, et al. 1982 . We begin by considering the flow of fluid into and out of a single reservoir block (Figure 4-1). Let the symbol J denote fluid flux. Flux is defined as the rate Figure 4-1. Reservoir block the coordinate convention...

## 163 Model Types

Models may be classified into three different types full field models, window area models, and conceptual models. Full field models are used to match performance of the entire field. They take into account the interaction between all wells and layers. The results of full field models are already matched to field scale and require no further scaling. The disadvantage of using full field models is that the number of grid blocks may need to be large or the grid size may need to be relatively...

## 212 Material Balance

Volumetrics provides one measure of the quality of a reservoir model, but it is based on information that does not change with time. Another estimate of original oil volume can be obtained from a material balance study if a reasonable amount of production data is available, such as the historical data presented in Chapter 20. At this point we have surmised that the reservoir was initially undersaturated, but it may not have aquifer support. The presence of a few barrels of water during the...

## Subsidence

An issue of special importance to reservoir characterization is subsidence. Subsidence is a compressibility effect that depends on the geomechanics of the produced interval and its overburden. Subsidence, or the change in thickness A h of the reservoir, can be estimated from the compressibility and pressure depletion of the system using the equation cB bulk compressibility psia1 cf formation compressibility psia1 h net thickness of reservoir ft < (> porosity frac AP pressure depletion psia...

## 171 Data Preparation

Data must be acquired and evaluated with a focus on its quality and the identification of relevant drive mechanisms that should be included in the model for example, see Crichlow, 1977 Saleri, et al., 1992 Raza, 1992 . Given that information, it is possible to select the type of model that will be needed for the study conceptual, window area, or full field model. In many cases all three of these models may need to be used, as illustrated in Fanchi, et al. 1996 . Data must be acquired for each...

## Pip

FIP fluid in place at time t OFIP original fluid in place Prod cumulative fluid produced at time t Inj cumulative fluid injected at time t Based on this definition, material balance should equal one in an idealized calculation. Actual simulator material balance may not equal one. Material balance error reported by WINB4D is calculated using the formula Error j- --1 1 * 100 (27.18) Material balance can be a sensitive indicator of error. Material balance error is greatest in WINB4D when a...

## 173 Simulator Selection and Ockhams Razor

Several requirements must be considered when selecting a simulator. These requirements can be classified into two general categories reservoir and non-reservoir. From a reservoir perspective, we are interested in fluid type, reservoir architecture, and the types of recovery processes or drive mechanisms that are anticipated. Reservoir architecture encompasses a variety of parameters that have a major impact on model design. Study objectives and the geologic model must be considered in...

## 153 Well and Facilities Modeling

Well and surface facility models are simplified representations of real equipment Williamson and Chappelear, 1981 . The well model, for example, does not account for flow in the wellbore from the reservoir to the surface. This effect can be taken into account by adding a wellbore model. The wellbore model usually consists of a multivariable table relating surface pressure to such parameters as flow rate and GOR. The tables are often calculated using a separate program that performs a nodal...

## 182 Key History Matching Parameters

A fundamental concept ofhistory matching is the concept of a hierarchy of uncertainty. The hierarchy of uncertainty is a ranking of model input data quality that lets the modeler determine which data is most and least reliable. Changes to model input data are then constrained by the principle that the least reliable data should be changed first. The question is which data are least reliable Data reliability is determined when data are collected and evaluated for completeness and validity Raza,...

## Fluid Displacement

Fluid displacement processes require contact between the displacing fluid and the displaced fluid. The movement of the interface between displacing and displaced fluids and the breakthrough time associated with the production of injected fluids at producing wells are indicators of sweep efficiency. This chapter shows how to calculate such indicators using two analytical techniques Buckley-Leverett theory with Welge's method for immiscible fluid displacement, and solution of the...

## Frontal Stability

The stability of a flood front can influence the efficiency of fluid displacement. A front is stable if it retains the shape of the interface between displaced and displacing fluids as the front moves through the medium. An analysis of frontal stability is presented in this chapter in terms of a specific example - the advance of a water-oil displacement front in the absence of gravity and in the presence of gravity. The stability of the front is then studied using linear stability analysis.

## 234 Example Input Data Sets

Several example input data sets are included with the book. A few are Primary depletion of an under-saturated oil reservoir (high GOR) Primary depletion of an under-saturated oil reservoir (moderate GOR) Primary depletion of an undersaturated oil reservoir (high GOR) Multi-layer waterflood of an undersaturated oil reservoir (high GOR) 5-spot waterflood of an under-saturated oil reservoir (high GOR) Gas injection into undersaturated oil reservoir (high GOR) - Odeh example Depletion of gas...

## 162 Grid Preparation

Reservoir grids may be designed in several different ways. For a review of different types of grids, see Aziz 1993 . Definitions of coordinate system orientation vary from one simulator to another and must be clearly defined for effective use in a simulator. Reservoir grids can often be constructed in one- two-, or three-dimensions, and in Cartesian or cylindrical coordinates. Horizontal 1D models are used to model linear systems that do not include gravity effects. Examples ofhorizontal ID...

## 7 Kip Qo Qw Qg Qt

KIP Code for specifying well operating characteristics. Rate Controlled Well (KIP > 0) QO Oil rate (STB D). QW Water rate (STB D). QG Gas rate (MSCF D). QT Total fluid voidage rate (RB D). NOTE The total fluid rate given by QT is the oil plus water plus gas production for the well or the total reservoir voidage rate at reservoir conditions. For multi-layer systems, QT is a target rate. BHP Controlled Production Well with Optional Rate Constraints (KIP -1) QO...

## Ksol Mitr Omega Tol Toll Dsmax Dpmax Numdis

Maximum number of LSOR iterations per timestep. A typical value is 100. Initial LSOR acceleration parameter. Values of OMEGA should be between 1.0 and 2.0. A typical initial value is 1.5. Maximum acceptable pressure change for convergence of LSOR iterations (psia). A typical value is 0.1. Parameter for determining when to change OMEGA. A typical value is 0.001. If TOLl 0.0, the initial value of OMEGA will be used for the entire run. Maximum saturation change allowed per timestep (fraction). The...

## Basic Reservoir Analysis

The tasks associated with basic reservoir analyses provide information that is needed to prepare input data for a simulation study. These tasks include volumetric analysis, material balance analysis, and decline curve analysis. In addition to providing estimates of fluids in place and forecasts of fieldwide production, they also provide an initial concept of the reservoir which can be used to design a model study. Each of these tasks is outlined below. Fluid volumes in a reservoir are values...

## Compressional and Shear Velocities

Seismic compressional velocity and shear velocity are calculated from the expressions Schcjn, 1996 McQuillin, et al., 1984 pj, effective bulk density (l-( ))pMa + < )p pma density of grains (solid matrix material) pf fluid density p0S0 +pwSw +pgSg Gassman 1951 derived an expression for AT* from the theory of elasticity of porous media Sch n, 1996 McQuillin, et al., 1984 KB - bulk modulus of empty reservoir, that is, dry rock or porous matrix material Kg bulk modulus of grains (solid matrix...

## Multiphase Flow Concepts

This chapter summarizes the basic concepts of multiphase flow including interfacial tension, wettability, and contact angle. These concepts lead naturally to a discussion of capillary pressure, mobility, and fractional flow. Some basic concepts must be introduced as prerequisites for understanding capillary pressure. The concepts are interfacial tension, wettability, and contact angle. They are defined here. On all interfaces between solids and fluids, and between immiscible fluids, there is a...

## 2pbo Pbodat Pbgrad

PBO Initial bubble point pressure (psia). PBODAT Depth at which PBO applies (ft). PBGRAD Constant bubble point pressure gradient (psia ft). VSLOPE, BSLOPE, RSLOPE, PMAX, IREPRS VSLOPE Slope of the oil viscosity versus pressure curve for undersaturated oil, i.e. for pressures above PBO. The slope (A i0 A 0) should be in cp psia. Slope of the oil formation volume factor versus pressure curve for undersaturated oil. The slope (AB0 AP0) should be in RB STB psia and should be negative or zero....

## 104 Major Elements of a Reservoir Simulation Study

The essential elements of a simulation study include matching field history making predictions, including a forecast based on the existing operating strategy and evaluating alternative operating scenarios Mattax and Dalton, 1990 Thomas, 1982 . During the history match, the modeler will verify and refine the reservoir description. Starting with an initial reservoir description, the model is used to match and predict reservoir performance. If necessary, the modeler will modify the reservoir...

## 72 Patterns and Spacing

The displacement processes discussed in Chapters 5 and 6 study fluid displacement between one injection well and one production well. The alignment of the injector-producer pair represents a linear displacement process. It is the simplest pattern involving injection and production wells. A variety of other patterns may be defined. Several examples are shown in Figure 7-1. A representative pattern element for the five-spot pattern is shown using lines between wells to denote boundary wells. a...

## 61 Frontal Advance Neglecting Gravity

The displacement of one phase by another may be analytically studied if a linear, homogenous porous medium is assumed. Let us first consider the displacement of oil by water in a horizontal porous medium of length L. We assume piston-like displacement of a front located at . Application of Darcy's law and the continuity equation leads to a pressure distribution described by Poisson's equation. The absence of sources or sinks in the medium reduces Poisson's equation to the Laplace equation for...

## The Impes Formulation

The following section from Fanchi, et al. 1982 shows how the flow equations for a black oil simulator can be recast in a form that is suitable for solution by a numerical technique. The numerical technique is based on the formulation originally presented by Sawyer and Mercer 1978 . The form of the Darcy velocities (Eqs. (4.10) through (4.12)) may be simplified by defining the potential > p of phase p as and we have used the assumption that g gc. In this notation, includingx, y, and z...

## Primary Production

Primary production is ordinarily the first stage of production. It relies entirely on natural energy sources. To remove petroleum from the pore space it occupies, the petroleum must be replaced by another fluid, such as water, natural gas, or air. Oil displacement is caused by the expansion of in situ fluids as pressure declines during primary reservoir depletion. The natural forces involved in the displacement of oil during primary production are called reservoir drives. The most common...

## V

Solution gas in both oil and water is neglected when a gas production rate is specified. This is a reasonable assumption for wells producing primarily free gas. Case 4 Total Production Rate Specified When the total reservoir voidage rate QT is specified, we first compute the phase mobility ratio for all connections Oil Mobility Ratio is the average oil formation volume factor for all connections in which the well is completed. Given Eq. (30.13), we simply proceed as in Eqs. (30.1) through...

## 172 Pressure Correction

When pressures are matched in a model study, the calculated and observed pressures should be compared at a common datum. In addition, pressures from well tests should be corrected for comparison with model block pressures. A widely used pressure correction is the Peaceman 1978, 1983 correction. Figure 17-1 illustrates a pressure buildup curve as a function of radial distance from the center of a wellbore with radius rw. To obtain a well block pressure P0 from a pressure buildup (PBU), Peaceman...

## Oil Well FBHP Estimate

The production well model needs a flowing bottomhole pressure (FBHP). Assuming an oil column in the wellbore, we can prepare a quick estimate of FBHP for a single-phase oil well that is completed at a 9500 ft depth by assuming FBHP oil head. Consequently, oil head is approximated by where yo is the oil pressure gradient and A z is the height of the oil column. An estimate of average oil pressure gradient for the oil column is found by averaging the pressure gradient at surface and reservoir...

## 142 Relative Permeability and Capillary Pressure

Reservoir models calculate saturation as a function of time. Consider the case of water displacing oil. Initially, oil occupies the interior of pore spaces, and connate water is adjacent to the rock surface of a water-wet reservoir. When the flood begins, water displaces oil through the interconnected pore space. The measure of intereonnectedness is permeability. The oil left behind after the waterflood is residual or irreducible oil saturation. Similar behavior is seen for other combinations...

## JL Iv L kH ki

The Dykstra-Parsons coefficient should be in the range 0 < VDP < 1. For a perfectly homogeneous reservoir, VDP 0 because kA kH. An increase in reservoir heterogeneity increases VDP. Typical values of the Dykstra-Parsons coefficient are in the range 0.4 < VDP < 0.9. Correlations of volumetric sweep efficiency with mobility ratio and permeability variation show that volumetric sweep efficiency declines as reservoir heterogeneity increases or mobility ratio increases, particularly for...

## 181 Illustrative History Matching Strategies

A universally accepted strategy for performing a history match does not exist. History matching is as much art as science because of the complexity of the problem. Nevertheless, there are some general guidelines that can help move a history match toward successful completion. These guidelines have been presented by such authors as Crichlow 1977 , Mattax and Dalton 1990 , Thomas 1982 , and Saleri, et al. 1992 . One set of guidelines is presented in Table 18-1. The first two steps in the table...

## WINB4D Block Index

Reservoir performance with Sgr 0 . influx for a system with an irreducible gas saturation (Sgr) of 0 . The change in gas saturation shows the influx of aquifer water. The change in fluid content changes fluid bulk modulus. As a consequence, the ratio V VS changes significantly in the waterflooded part of the reservoir. If we rerun the example with an irreducible gas saturation of three percent, we obtain the results shown in Figure 12-8. The large change in V IVs is no Figure 12-8....

## Rock and Fluid Models

The interaction between reservoir rock and in situ fluids is modeled with relative permeability and capillary pressure data. This chapter defines the three-phase oil relative permeability model used in WINB4D and its use in transmissibility calculations. It then presents additional details of the fluid property model after reviewing a few commonly used thermodynamic terms. 28.1 Three-Phase Relative Permeability Relative permeability curves are some of the most critical data in the simulator...

## Initialization

It is important when making cross-section or 3D runs that the pressures in the model are correctly initialized. If not, phase potential differences due to gravity terms could cause fluid migration even though no wells are active. Consequently, a simple pressure initialization algorithm is used in WINB4D. It is reviewed below along with an option to correct pressures to a user-specified datum and an option to initialize saturations using gravity segregation. Consider a gridblock that may have a...

## Well Flow Index PID

Productivity index PI is defined as the ratio of rate Q to pressure drop AP, or PI Q AP, where AP Pe- Pw, Pe average reservoir pressure, and Pw wellbore bottomhole pressure BHP. From Darcy's Law for radial oil flow we can write PI as The meaning and units of all terms are given as follows The meaning and units of all terms are given as follows effective permeability md kro Kabs Some of the terms in Eq. 31.1 depend on time-varying pressure and saturation, while other factors change relatively...

## Contents

1 Introduction to Reservoir Management 1 1.2 Management of Simulation Studies 4 1.3 Outline of the Text 6 Exercises 7 Part I - Reservoir Engineering Primer 2 Basic Reservoir Analysis 11 2.3 Decline Curve Analysis 16 Exercises 17 3 Multiphase Flow Concepts 19 3.4 Fractional Flow 26 Exercises 30 4 Derivation of the Flow 4.1 Conservation of Mass 31 4.2 Flow Equations for Three-Phase Flow 33 4.3 Flow Equations in Vector Notation 36 Exercises 37 5 Fluid 5.1 Buckley-Leverett Theory 39 5.3 Miscible...

## 53 Miscible Displacement

Buckley-Leverett theory treats the displacement of one fluid by another under immiscible, piston-like conditions. An immiscible displacement occurs when the displaced and displacing fluids do not mix. The result is a readily discernible interface between the two fluids. In a miscible displacement, the fluids mix and the interfacial tension approaches zero at the interface. A miscible displacement system is described by a convection-dispersion C-D equation. As an illustration, consider the...

## Introduction to Reservoir Management

Reservoir modeling exists within the context of the reservoir management function. Although not universally adopted, reservoir management is often defined as the allocation of resources to optimize hydrocarbon recovery from a reservoir while minimizing capital investments and operating expenses Wiggins and Startzman, 1990 Satter and Thakur, 1994 Al-Hussainy and Humphreys, 1996 Thakur, 1996 . These two outcomes - optimizing recovery and minimizing cost - often conflict with each other....