Radial Aquifer Research Articles

EVALUATION OF AKPET GT9 GAS CONDENSATE RESERVOIR PERFORMANCE

To effectively evaluate a gas condensate reservoir performance, the reservoir engineer must have a reasonable amount of knowledge about the reservoir to adequately analyze the reservoir performance and predict future production under various modes of operation. Due to the multiphase flow that exists in the reservoir, characterization of gas condensate reservoirs is often a difficult task with the variation of its overall composition in both space and time during production which complicates well deliverability analysis and the sizing of surface facilities. This study is primarily concern with the evaluation of a gas condensate reservoir performance of Akpet GT 9 Reservoir in the Niger Delta region of Nigeria with material balance analysis tool “MBal” without having to run numerical simulations. The result obtained with MBal on the analysis of Akpet GT 9 reservoir gave 23.934 Bscf of gas initially in place which compares favorably with the volume obtained from volumetric techniques. Results also shows that the most likely aquifer model is the Hurst–Van Everdingen - Dake radial aquifer and the reservoir is supported by a combined drive of water influx and fluid expansion.
 Okotie, S. | Department of Petroleum Engineering, Federal University of Petroleum Resources (FUPRE), Effurun, Delta State, Nigeria.

Report on the Evaluation of Ugua J2 and J3 Reservoir Performance

Globally, there are different techniques applied in the industry such as analog, volumetric, decline curve, material balance and simulation methods to estimate hydrocarbon reserves which depends upon the volume and quality of available data with some levels of uncertainties. The analogy method is applied by comparing factors for the current fields or wells while the volumetric method implies the determination of the extent of the reservoir, pore volume of the reservoir rock and the content of fluid within the reservoir pore volume when production and pressure data from field become available, decline curve analysis and material balance calculations become the predominant methods of calculating reserves since the hydrocarbon reserve estimates is a continuous process for a field that is producing. This study presents a report on the material analysis of Ugua J2 and J3 reservoirs with MBAL prior to the field’s dynamic reservoir simulation. The available data were validated and the analytical plot of pressure and production data indicated that J2 and J3 reservoirs are communicating. Hence multi-tank material balance analysis approach linked with transmissibility was adopted to model the reservoirs. The Hurst-Van Everdingen radial aquifer model was selected as the most likely case. The oil initially in place value obtained for J2 and J3 are 125.006 and 80.689 MMstb.

Non-Darcy displacement in linear composite and radial aquifer during CO<SUB align="right">2 sequestration

This paper presents Buckley-Leverett type analytical solutions for non-Darcy displacement of two immiscible fluids in linear and radial composite porous media. High velocity or non-Darcy flow commonly occurs in the vicinity of the wellbore because of smaller flowing cross-sectional areas. However, the effect of such non-Darcy flow has been traditionally ignored. To examine the physical behaviour of multiphase immiscible fluids in non-Darcy displacement, an extended Buckley-Leverett type of solution is discussed. There exists a Buckley-Leverett type solution for describing non-Darcy displacement in a linear homogeneous reservoir. This work extends the solution to flow in linear and radial composite flow systems. We present several new Buckley-Leverett type analytical solutions for non-Darcy flow in more complicated flow geometries of linear and radial composite reservoirs, based on non-Darcy flow models of Forchheimer and Barree-Conway. As application examples, we use the analytical solutions to verify numerical simulation results as well as to discuss non-Darcy displacement behaviour. This theory of non-Darcy flow displacement is applied to evaluate the flow behaviour near wellbore areas during CO2 sequestration. The results show how non-Darcy displacement during CO2 injection in linear and radial composite systems is controlled not only by relative permeability, but also by non-Darcy coefficients, characteristic length, injection rates, as well as discontinuities in the saturation profile across the interfaces between adjacent composite flow domains.

A perturbation solution of the nonlinear Boussinesq equation: The case of constant injection into a radial aquifer

A generalized version of the Boussinesq equation is solved, using a singular perturbation technique. Flow takes place in radial directions into an initially-dry homogeneous isotropic soil under the assumption of a constant injection rate at the origin. Explicit formulae are derived for the location of the advancing wetting front and the free surface height (or pressure head). Analytical results are compared graphically with a numerical finite-element solution. Excellent agreement is obtained between the results of the two methods.

A Generalized Method for Making Optimal Estimates of Reservoir-Aquifer Parameters

Abstract This paper presents a new, faster method for predicting performance of reservoir aquifer systems predicting performance of reservoir aquifer systems and making optimal estimates of aquifer geometry, reservoir size, aquifer size, and reservoir-aquifer fluid conductivity. The method uses existing analytical solutions giving pressure-production performance of a radial or linear aquifer connected performance of a radial or linear aquifer connected to a reservoir with constant compressibility. The novelty of the method is the use of these solutions to predict behavior of a reservoir with compressibility changes. The method's accuracy was verified by comparing results with those calculated using a limite-difference simulator. The method is built into a computer program that uses an efficient optimization technique to estimate parameters. Results of application to a hypothetical gas reservoir and to an actual oil field are given. In the latter case, a comparison is made with results obtained using the Hurst-van Everdingen water influx calculation method Introduction Aquifers, linear or radial, surround many oil and gas reservoirs. Knowledge of the aquifer's strength is essential to predict future performance of a water-drive reservoir. Strength is influenced by geometric configuration, which may be difficult to determine. Often, geological interpretation suggests that the system is neither radial nor linear. Because it is possible that either aquifer shape might approximate the behavior of an actual system, it is frequently desirable to test both in a given problem. Several methods have been developed for predicting water influx into a reservoir. Of these, predicting water influx into a reservoir. Of these, Hurst's is closest to the method presented here. Hurst used the Laplace transformation to convert standard material balance equations for water-drive reservoirs into a form giving pressure explicitly as a function of other reservoir parameters. To apply his method for predicting reservoir pressure change, it is necessary to approximate all pressure-dependent functions as linear functions of pressure. Hurst gave no indication of the range of applicability of such an approximation and provided no convenient means for updating the approximations. This study develops an improved method for estimating groups of parameters that control behavior of a reservoir-aquifer system. The system is modeled as a single, homogeneous reservoir block with an active (radial or linear) aquifer. The steps taken to develop the method axe given and then its application to two practical field problems is shown. DEVELOPMENT OF METHOD This section presents the steps taken in developing the method. For each step, the main thrust of the work conducted and results obtained are presented; mathematical details appear in appendixes. Step 1 considered the equations describing the behavior of reservoir-aquifer systems. The required solutions gave p vs t in a reservoir with constant compressibility being produced at a constant rate and receiving water influx from either a radial or a linear aquifer. Stop 2 modified the solutions to include changes in production rate and reservoir compressibility during the reservoir's history. A computer program was written to predict reservoir behavior using the method derived in Steps 1 and 2. In Step 3 the method's accuracy was verified by comparing results with those obtained with a more exact tank-type simulator. In Step 4 a computer program was written to estimate values for the reservoir parameters that best matched the given history. STEP 1: MATHEMATICAL MODEL This step is concerned with the equations describing behavior of reservoir-aquifer systems. First, a radial aquifer is discussed, and then linear aquifer theory is presented. SPEJ P. 123

Simulation of Naturally Fractured Reservoirs With Semi-Implicit Source Terms

Abstract Conventional reservoir simulation techniques prove to be inadequate when applied directly to the prove to be inadequate when applied directly to the study of fractured reservoir systems. Such systems are characterized by extremes in porosity, permeability, and saturation. The vast bulk of the permeability, and saturation. The vast bulk of the reservoir volume is occupied by relatively low-permeability, disjoint matrix blocks of various sizes surrounded by a small volume of high-permeability, interconnected fracture space. Our approach to this complex problem bas been to treat the matrix blocks as source and sink terms in an otherwise conventional simulation that models only the fracture system. The source/sink terms are functions of matrix rock and fluid properties with fracture saturation and pressure defining the boundary conditions. These functions are derived either by history-matching simulations or independently by laboratory experiments or single matrix-block simulation. This basic concept of a source/sink treatment is not unique to this work. However, the numerical formulation and implementation of these terms in the fracture simulation offers significant advantages over existing modeling procedures. The fundamental advantage of our approach is that these source terms are handled semi-implicitly in both the pressure and saturation calculations involved in pressure and saturation calculations involved in the fracture simulation. This avoids instability problems that are inherent in a sequential problems that are inherent in a sequential fracture-matrix solution and links more closely the behavior of the matrix and the fracture. Special techniques are developed for modeling the effects of fluid contact movement within a large simulation grid block and for treating receding gas-oil and water-oil contacts. Hysteresis effects are included in matrix blocks that begin to imbibe oil after drainage has begun. Introduction Conventional reservoir simulation techniques are not capable of adequately modeling large, naturally fractured reservoir systems. Extreme discontinuities in porosity, permeability, and saturation exist throughout the reservoir. Most of the fluids are found in very low-permeability, disjoint matrix blocks of various sizes, while most of the fluid mobility is in a small volume of high-permeability, interconnected fracture space. The distribution of fluids within the fracture is usually governed primarily by gravity segregation while the behavior of the individual matrix blocks depends on pressure, fluid environment, and matrix fluid saturation. Fluids can move readily throughout the reservoir in the fracture space, but fluids that reside in matrix rock must enter the fracture to move any great distance. The behavior of individual matrix blocks in response to various drive mechanisms has been studied experimentally by Crawford and Yazdil and has been simulated in two dimensions by Kleppe and Morse and Yamamoto et al. Other investigators have studied the single-phase pressure behavior of fractured reservoirs and its effect on pressure buildup curves. Kazemi investigated single-phase flow in a radial reservoir dominated by horizontal fractures. Closman simultaneously solved equations for flow between matrix and fracture and for flow along the fracture planes for a radial aquifer to develop relationships similar to those developed by van Everdingen and Hursts for water influx. Simulation of an entire reservoir system with multiple phases further complicates the problem and makes additional simulator modifications necessary. Asfari and Witherspoon have developed a modeling approach for reservoirs with a regular pattern of noncommunicating vertical fractures by pattern of noncommunicating vertical fractures by assigning constant pressures along each fracture. Several investigators have applied finiteelement techniques to the fracture-matrix flow problem. problem. Our approach to this complex problem has been to model the flow in the fracture system and to treat fluid transfer to and from the matrix much as injection and production are modeled in conventional simulators. Transfer of fluid into the fracture will be represented by a "source" term and transfer from the fracture to the matrix will be represented by a "sink" (or negative source) term. SPEJ P. 201

Transient Aquifer Flows by Characteristics Methods

A numerical model is presented to analyze nonsteady radial flow in a finite or infinite leaky aquifer system. An equation of motion alteration, which involves the introduction of an inertia multiplier, converts the standard equations of transient ground-water flow to a form suitable for efficient solution by the method of characteristics. The result is a numerical method that should be attractively competitive in computer time usage with other numerical procedures currently in use. The equations are presented and the procedure developed for axisymmetrical flow. Several examples are used to expose the influence of the size of the distance intervals and the size of the inertia multiplier on the accuracy of the numerical process. Either constant or time-variable discharge or drawdown rates may be used as boundary conditions in infinite or bounded radial aquifer systems. Radially varying aquifer properties and leakage may also be treated.

Scheduling Gas Field Production for Maximum Profit

Abstract A mathematical procedure for finding the most profitable gas field production policy to meet a gas profitable gas field production policy to meet a gas sales contract has been developed. Results indicate profits can be increased through combinations of profits can be increased through combinations of earlier liquids production, increased gas recovery, and investment reduction. The optimal plan for operating a gas field is found by applying nonlinear programming to the over-all problem o production programming to the over-all problem o production rate scheduling. Risk is accounted for by specifying that any investment yield some minimum incremental profit-to-investment ratio. Computed results, an profit-to-investment ratio. Computed results, an illustrated by several example problems, include average and peak production rate schedules for each reservoir, well-drilling schedules in each reservoirs, a list of recompletions, and compression purchases for the field. Compression can be assigned purchases for the field. Compression can be assigned to each reservoir or it can be pooled in the field. Dynamic programming is used to find the best investment schedule in each reservoir. Each gas reservoir is assumed to have a uniform pressure distribution throughout. Water influx is described by the van Everdingen-Hurst analytic solution for a radial aquifer. In each reservoir, one set of gas-well deliverability curves for production to four delivery pressures is employed, and in-place facilities are always operated to minimize compression requirements. The Single-Reservoir Problem We shall solve the following optimization problem for a single gas reservoir: Given a desired gas delivery schedule and a specified peak delivery capability, find the optimal schedule for drilling new wells, the optimal plan for installing compressor horsepower, a detailed operating plan, and the total discounted profit for the best policy. We make the following assumptions in the analysis. 1. Pressure is uniform throughout the reservoir. This implies that well location is immaterial. 2. A single set of well deliverability curves is adequate. 3. The unsteady-state water influx equations satisfactorily describe aquifer behavior. 4. The optimum policy can be broken into two stages. In the first period, desired peak delivery capability is maintained through the addition of wells and compressive horsepower In the second period no new wells are drilled and no compression period no new wells are drilled and no compression is added; the reservoir flow rate declines.

A Simplified Approach to Water Influx Calculations-Finite Aquifer Systems

This approach to water influx calculations offers a useful and flexible method of forecasting and analyzing the performance of water drive reservoirs. The separation of the water influx problem into a rate equation and a material balance equation, not requiring superposition, makes the concepts and calculations quite simple and easy to apply. Introduction All gas and oil reservoirs are associated to varying extents with formation waters. The inclusion of the effects of expansion or invasion of this water into oil and gas reservoirs has taken many forms, from recognizing the effects of the expansion of the connote water within the gas or oil reservoir itself, to calculating water influx or efflux across a boundary (with the boundary usually being that of an oil or gas reservoir). There are four currently popular methods used for calculating water influx into reservoirs. They are: Schilthuis, steady state Hurst Simplified, unsteady state Resistance or Influence Function, unsteadystate van Everdingen-Hurst Radial, unsteady state The first three methods have proved useful for predicting water drive performance after sufficient predicting water drive performance after sufficient historical data have been obtained to fix the necessary influx constants. With what some consider to be disappointing results, the van Everdingen-Hurst Radial method is often used with geological and core data when little or no performance history is available. It has also been used to predict reservoir performance after enough historical data have been accumulated to develop values of the influx constants, tD and C. In an attempt to include geometries other than radial, derivations for both limited and infinite systems have been made to cover linear spherical, elliptical, thick-sand, and wedge-shaped reservoir-aquifer models. The many rigorous geometrical representations that have been developed cannot readily handle the effect of interference between reservoirs. Electric analyzer studies of the Smackover Limestone aquifer in Arkansas by Bruce, of the Woodbine aquifer in East Texas by Rumble et al., and of the Ellenberger in West Texas by Moore and Truby have shown that reservoirs sharing a common aquifer can severely interfere with each other, and that, for individual reservoirs in a common aquifer, water drive performance calculations that do not consider interference performance calculations that do not consider interference can be greatly in error. Mortada developed a mathematical method with which to handle interference in a basically infinite radial aquifer system. The method has been applied to field cases. Coats concluded from his own study that, "In predicting the pressure-volume behavior of gas reservoirs situated on the common aquifer the effect of interference from other reservoirs on the common aquifer must be accounted for." Another aquifer problem more recently presented in the literature is that of flank water injection for pressure maintenance, either to initiate or to pressure maintenance, either to initiate or to supplement edge-water influx. A case history shows that we need to be able to study the effects of injecting water into the aquifer instead of merely including it in the hydrocarbon material balance equation. JPT P. 814

Evaluating a Slightly Permeable Caprock in Aquifer Gas Storage: 1. Caprock of Infinite Thickness

Abstract Evaluating the permeability of a caprock overlying a potential gas storage reservoir is a very critical problem. Pumping water from the reservoir can be used as an evaluation tool in analyzing this problem. Fluid level changes that occur in the aquifer as well as in the caprock can be measured with appropriately placed wells. If the leakage of water from the caprock into the aquifer is considerable, the effects will be apparent in the aquifer. If the leakage is slight, however, it will not be possible to detect it with certainty from observations in the aquifer alone. Fluid level measurements in the caprock must be relied upon, and improved methods of analyzing such effects have been developed which are based on a theoretical analysis of fluid flow through a caprock of infinite thickness. An example applying these methods to field data is discussed. Introduction One of the most critical problems in evaluating an aquifer gas storage project is determining the tightness of the caprock overlying the formation to be used as the storage reservoir. A formation that has previously held oil or gas obviously has a suitable caprock, but an aquifer that contains only water gives no such assurance. A number of aquifer projects in the United States have been troubled by gas leaking out of the intended storage zone, and the ensuing difficulties have led to the development of new evaluation methods. One of these new methods is pump testing wherein water is removed from the aquifer at some controlled rate prior to injection of gas. This fluid withdrawal causes a pressure drop to move out through the aquifer for considerable distances in a matter of days or weeks. Depending on the properties of the caprock, a pressure transient can also pass upward (as well as downward) through the caprock layers adjacent to the aquifer. Thus, if the operator has placed observation wells at appropriate distances from the pumping well, the rapidity with which the pressure transients reach different points in the system can be used to investigate the fluid transport properties of both the aquifer and its caprock. The usefulness of pump testing has been recognized by groundwater hydrologists for many years as a means of determining the potential yield and properties of aquifers used in water supply. They have introduced the term "leaky aquifer" for a system in which an aquifer is overlain (or underlain) by semipermeable caprock layers. The ease with which water leaks into the aquifer during pumping can, of course, be very beneficial in bringing additional water to the pumped well. Hydrologists have therefore devoted considerable attention to this problem. From the gas storage standpoint, however, the tighter the caprock layers that overlie the intended storage reservoir, the better are the conditions for minimizing or eliminating any vertical migration of gas. Thus, after a suitable geologic structure has been found, the emphasis in aquifer storage projects is in determining that the caprock is tight. Attention has recently been focused on the use of pump testing as one approach to solving this problem. paper presents a further development on evaluating the permeability of a slightly leaky caprock when the caprock is of infinite thickness. From the practical standpoint, this means that the caprock layers are thick enough that pressure transients do not reach the outer boundaries of the system during the pumping test. In a subsequent paper, an analysis of the case where the caprock is of finite thickness will be presented. PREVIOUS WORK ON LEAKY AQUIFERS Jacob developed a partial differential equation describing the flow of water in an aquifer of permeability k that is overlain by a leaky caprock of permeability k'. Fig. 1 shows a schematic cross-section of the system under consideration. One of his principle assumptions was that if k »k', the direction of flow is essentially vertical in the caprock and horizontal in the aquifer. Neuman confirmed the validity of Jacob's assumption using a mathematical model. Another assumption was that a permeable source layer overlies the caprock (Fig. 1) and is able to maintain a constant hydraulic head at the upper boundary of the caprock. By neglecting the effects of compressibility within the caprock, Jacob developed a solution for bounded circular aquifer. Later, Hantush and Jacob used the same assumptions to solve the case of an infinite radial aquifer that is pumped at a constant rate. JPT P. 949ˆ

A Reservoir Study of the Sturgeon Lake South D-3 Pool

Abstract This study illustrates the use of the expanded material balance to determinethe volume of oil originally in place and to define the aquifer system. The expanded material balance, which incorporates an unsteady-state waterinflux term, contains two explicit and several implicit unknowns. The solutionof such a material balance is accomplished by algebraically rearranging theterms to define the equation of a straight line. Such a straight-line conceptis then used to define the various unknowns. Both linear and radial aquiferconfigurations are considered and the acceptability of the results arestatistically tested. Due to the installation and subsequent extension of a gas gathering systemin the field, there have been two major changes in the field separatorpressure. These variations in separator pressure are accounted for in thecalculations. The study concludes that the performance of the Sturgeon Lake South D-3pool, to date, has been consistent with a radial aquifer system which has notyet felt boundary effects. The volume of oil originally in place as determinedin this paper is in excellent agreement with the volumetric estimate. Introduction The Sturgeon Lake South field is situated some 160 miles northwest of theCity of Edmonton. Alberta. The D-3 pool the only productive horizon of anyconsequence in the field, was discovered in early 1953 with the drilling of thewell Amerada HB Union Crown "R" 5–19 (Lsd 5–19–69–22 W5). While sporadicdrilling in the field is continuing, development was substantially completed bythe end of 1957 when some 100 producing oil wells had been completed. There arecurrently 112 wells capable of oil production from this reservoir. Several calculations have been performed in the past to establish avolumetric estimate of the original oil in place; however, due to limited wellcontrol in the southwestern part of the pool (see the attachedstructure-isopach map of Figure1), and a controversy concerning theconnate water saturation, the volumetric estimates for this reservoir haveshown a wide variation in values. These differences in the volumetric estimatesprompted the present detailed investigation.