Roland N. Horne

Oscillatory convection in a porous medium heated from below

The stability of natural convective flow in a porous medium heated both uniformly and non-uniformly from below is studied in order to determine the possibility of oscillatory and other unsteady flows, and to explore the conditions under which they may occur. The results of the numerical work are directly comparable with experiments using a Hele Shaw cell and also, in the uniformly heated case, with the results of Combarnous & Le Fur (1969) and Caltagirone, Cloupeau & Combarnous (1971). It is shown that for the uniformly heated problem there exist, in certain cases, two distinct possible modes of flow, one of which is fluctuating, the other being steady. However in the non-uniformly heated case the boundary conditions force the solution into a unique mode of flow which is regularly oscillatory when there is considerable non-uniformity in the heat input at the lower boundary, provided that the Rayleigh number is sufficiently high.

Reservoir Development and Design Optimization

Optimization of reservoir development requires many evaluations of the possible combinations of the decision variables, such as the reservoir properties, well locations and production scheduling parameters, to obtain the best economical strategies. Running a simulator for such a large number of evaluations may be impractical due to the computation time involved. In this study, a hybrid Genetic Algorithm (GA) was developed. The optimization algorithm integrated economic analysis, simulation and project design. The layout of 33 new wells for a real oil field development project was proposed by the company’s project team. The algorithm was used to determine an optimal relocation of the wells, by evaluating an objective function from a cash flow analysis for the production profile obtained from simulation at each iteration. The wells were allowed to be placed anywhere in the reservoir and could be vertical or horizontal and, if horizontal, any direction in the same layer could be considered. A total of 99 decision variables were used to solve this problem. All the real restrictions were included. Based on the cost of the sea bottom flowlines a second level optimization found the best platform location. The results were compared against the proposed solution, and showed that the algorithm performed very well finding an optimized well distribution. A reduction of the total number of new wells was found as part of the solution. An improvement of about 6% in the project profit was found, representing about US

A Procedure to Integrate Well Test Data, Reservoir Performance History and 4-D Seismic Information into a Reservoir Description

70 million additional income. Introduction The main task of a reservoir engineer is to develop a scheme to produce as much hydrocarbon as possible within economic and physical limits. The solution of this kind of problem encompasses two main entities: the field production system and the geological reservoir. Each of these entities presents a wide set of decision variables and the choice of their values is an optimization problem. In view of the large number of decision variables it is not feasible to try to enumerate all possible combinations. Analysis tools encoded in computer programs can spend hours or days of processing for a single run, depending on their sophistication and features. Also, it can be costly to prepare the input data if many hypotheses are going to be considered. A typical reservoir development involves many variables that affect the operational schedule involved in its management. These variables are usually used as input to a reservoir simulator that generates a forecast of the production profile. Using this forecast, the production engineer has to consider several hypotheses to achieve the best strategy for the field development. Also, each hypothesis can generate others, and so the overall process is one of generating a hypothesis tree. More and more data are generated and analyzed. The solution of these problems requires the effort of several people as well as considerable computer work and physical time. An optimization procedure requires the characterization of the function to be optimized (minimized or maximized), known as the objective function, as well as the choice of an appropriate optimizing method. The complexity of predicting hydrocarbon production profiles requires the use of reservoir simulators. So, the simulator must be part of the evaluation of the objective function. This work concerns the optimization of characteristic petroleum production problems considering economic factors. A hybrid algorithm based on direct methods such as Genetic Algorithm (GA), polytope search and Tabu search was developed. Hybrid techniques were found to improve the overall method. The objective function consisted of a cash flow analysis for production profiles obtained from simulation runs. The optimizing procedure was able to interface with commercial simulators (generating their input data and retrieving the results) that worked as data generators for the objective function evaluation. These hybrid mathematical approaches were found to be successful in obtaining the optimal solution with less time and work than existing techniques. These approaches can speed up the study of a hydrocarbon reservoir development plan and SPE 38895 Reservoir Development and Design Optimization Antonio C. Bittencourt, SPE, Petrobras, and Roland N. Horne, SPE, Stanford University 2 A.C.BITTENCOURT, R.N. HORNE SPE 38895 allow consideration of a wider range of hypotheses. The engineer can also keep track of economics during the study, allowing better project decisions. A real project was optimized using two approaches: the first one had the proposed solution inserted in the initial population and the second one did not. The first approach achieved the better solution, albeit at the cost of a larger number of simulations due to premature convergence earlier in the calculation. The Hybrid Algorithm Approach Optimization concerns the optimal solution determination using an oriented search towards the best possible value. Algorithms used in the optimization procedure are problem dependent, so it is necessary to investigate the strengths, weaknesses and ranges of applicability of each. One of the objectives of this work was to develop a hybrid algorithm to overcome the limitations of individual approaches, and to take advantage of the particular strengths of each. The principal procedure used in the hybrid was the Genetic Algorithm, combined with a polytope search between generations and an initial distribution based on the Fang algorithm. Brief descriptions of the component algorithms will be summarized here. Genetic Algorithm (GA) is a robust search method based on analogies to biology and genetics. Survival of the fittest among a population of individuals, selection criteria, and reproduction strategies are concepts copied from natural life and used as operators in this artificial environment. Applications have been found for GA in business, engineering and science. The GA has four advantageous features: 1. GA begins the search with a population of parameter realizations, rather than a single realization as most of the conventional optimization methods might. In this way, the search domain is covered in a random distribution. 2. The realizations are perturbed by probabilistic rules rather than deterministic ones. 3. The parameter itself is not manipulated directly by the GA operators. GA would alter the chromosome (or individual or string) that is a pattern of zeros and ones representing the whole set of parameters put all together in one binary entity. For binary alphabets, the smaller piece of a chromosome is called a bit. 4. Only function evaluations are used rather than derivatives or other secondary descriptors. The fitness is defined as

Characterization of Spontaneous Water Imbibition Into Gas-Saturated Rocks

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Abstract This paper deals with the problem of estimating the distributions of permeability and porosity in heterogeneous and multiphase petroleum reservoirs by matching the dynamic behavior. The dynamic data is in the form of field measurements from well testing, production history, interpreted 4-D seismic information, and other data such as correlations between permeability and porosity, geostatistics in the form of a variogram model and the inference of large scale geological structure. The issue was posed as an inverse problem and solved by using nonlinear parameter estimation. The procedure developed here is capable of processing all the information simultaneously and this results in a fast and efficient method. The procedure is also able to determine the uncertainty associated with the estimated permeability and porosity fields. Examples of different parameter types that may be estimated by this approach include: (a) individual block permeabilities and porosities; (b) geological objects such as channels and faults; (c) pilot points that form the basis of a kriged distribution; and (d) seismic attenuation values from 3-D seismic images. An important conclusion of this work is that the value of each piece of information does not reside in its isolated use but in the value it adds to integrated analysis of the complete set of information. Thus data that traditionally was considered to be of low information content for reservoir characterization becomes useful and enhances the value of the data set as a whole. Introduction Devising the optimal strategy for the development of an oil or gas reservoir is an …

Three-dimensional natural convection in a confined porous medium heated from below

Spontaneous water imbibition into gas-saturated rocks is an important physical process during water injection into highly fractured petroleum and geothermal reservoirs. Few methods, however, are available for characterizing the process of spontaneous water imbibition into gas-saturated rocks. To this end, a method has been developed. Water relative permeability and capillary pressure can be calculated separately from water imbibition data using this method. A linear relationship between imbibition rate and the reciprocal of the gas recovery by spontaneous water imbibition was found and confirmed both theoretically and experimentally, even at different initial water saturations. The effect of initial water saturation on imbibition rate, residual gas saturation, and the gas recovery has been investigated. There was almost no effect of initial water saturation on residual gas saturation by spontaneous water imbibition. The higher the initial water saturation, the lower the water imbibition rate and the ultimate gas recovery. It was found that the capillary pressure did not vary with initial water saturation in a certain range. The capillary pressure calculated using the new method was approximately equal to the values measured using an X-ray CT technique in a glass-bead pack. The computed water relative permeability was consistent with published experimental results. The method developed in this paper is also of importance for scaling-up experimental data. Introduction A lot of oil and gas reservoirs are being developed by water injection or associated with active aquifers. The reinjection of produced water into geothermal reservoirs, which are usually highly fractured, is also a practical solution to the problems of reservoir pressure decline and environment pollution. Spontaneous water imbibition phenomena exist in most of these reservoirs except those that are not water-wet; it is an important process driven by capillary forces. The study of spontaneous water imbibition is essential to predict the production performance in these reservoirs developed by water flooding, especially in highly stratified formations and fractured reservoirs where the amount and rate of mass transfer between the matrix and the fracture influence the recovery and the production rate. Another significance is to determine the wettability of the rock-fluid systems from the spontaneous water imbibition measurents. Since the spontaneous imbibition is a capillary pressure dominated process, the imbibition rate is significantly dependent on the properties of the porous media, fluids, and their interactions. These include permeability and relative permeability of the porous media, pore structure, matrix sizes, shapes and boundary conditions 5, , fluid viscosities , initial water saturation, the wettability of the rock-fluid systems, and the interfacial tension between the imbibed phase and the resident phase . Most of the studies in the literature focused mainly on oil-water-rock systems. There are a lot of naturally fractured gas reservoirs positioned over active aquifers. The water coning in these gas reservoirs often results in excessive water production, which may kill a well or severely curtail its economic life due to water handling costs. The understanding of the mechanisms that govern spontaneous water imbibition in gas-water-rock systems is important to the development of naturally fractured gas reservoirs with active aquifers. Few methods, however, are available for characterizing the process of spontaneous water imbibition into gas-saturated rocks. The method used usually is the Handy equation, that is, the weight or the volume of the imbibed water is proportional to the square root of the imbibition time:

Improved Methods for Multivariate Optimization of Field Development Scheduling and Well Placement Design

Previous analyses of natural convection in a porous medium have drawn seemingly contradictory conclusions as to whether the motion is two- or three-dimensional. This investigation uses numerical results to show the relationship between previous contending observations, and demonstrates that there exists more than one mode of convection for any particular physical configuration and Rayleigh number. In some cases, a particular flow situation may be stable even though it does not maximize the energy transfer across the system. The methods used are based on the efficient numerical solution of the governing equations, formulated with the definition of a vector potential. This approach is shown to be superior to formulating the equations in terms of pressure. For a cubic region the flow pattern at a particular value of the Rayleigh number is not unique and is determined by the initial conditions. In some cases there exist four alternatives, two- and three-dimensional, steady and unsteady.

Geothermal Reinjection Experience in Japan

Optimization of reservoir development requires many evaluations of the possible combinations of the decision variables, such as well locations and production scheduling parameters, to obtain the best economic strategies. Running a simulator for such a large number of evaluations may be infeasible due to the computation time involved. This study investigated two multivariate interpolation algorithms, Least Squares and Kriging, to generate new realizations from a limited number of simulations in order to predict the optimal strategies in a field development scheduling project and a waterflood project. The result was a significant reduction in the simulation effort required. The recommended solutions were obtained by searching for the optimal objective function values among the interpolation realizations. Additional simulation runs were performed to refine the search for the final optimal solution in the vicinity of the intermediate optimal region. The net present value was used as the objective function in both projects. The field development scheduling simulation was achieved by an economic model taking account of all the costs and profits during the time period being studied. In the waterflooding application, movement of the waterfront was tracked and the oil and water production at each production well was calculated, after which the optimum well placement strategy was determined. The results obtained using the interpolation methods showed that the algorithms are able to reduce the number of simulation runs, to provide a global sketch of the objective function surface, to avoid possible failure at local optima, and to reach the absolute optimum by refining the grids near the intermediate optima. The approaches were shown to be practical, and will be useful in full-scale optimization problems in which simulation costs would be prohibitive for iterative nonlinear optimization methods.

Origin of oscillatory convection in a porous medium heated from below

The five liquid-dominated geothermal fields under production in Japan (Otake, Onuma, Onikobe, Hatchobaru, and Kakkonda) represent more than half the liquid-dominated fields now in production worldwide. All five have total liquid reinjection programs, and thus collectively represent a major fraction of all full-scale water reinjection operations in the world. The implications of the Japanese reinjection experience are therefore of extreme importance to future liquid-dominated geothermal development. The major impact of the experience in these fields is that, with the exception of Otake, the reinjection wells show a rapid interference with the production wells. In most cases, the reinjected water travels to the production zone within a very short time, and there is consequently a discernible thermal drawdown. 18 refs.

Relative Productivities and Pressure Transient Modeling of Horizontal Wells with Multiple Fractures

In order to investigate the significance of cyclic interaction in the evolution of fluctuating convective flow in a rectangular cell of porous material, the behavior of confined and unconfined flows are compared. In the latter case the constant pressure boundary condition at the top surface does not force the fluid to recirculate around the cell. However, regular oscillatory flow similar to that which occurs in an enclosed cell is still observed. This indicates that the oscillatory flow arises from the instability of the thermal boundary layer on the heated bottom surface. In both cases the frequency of the oscillatory flow is proportional to the Rayleigh number to the power of 3/2.

Reservoir Characterization Constrained to Well Test Data: A Field Example

An analytical model has been developed to describe the inflow performance and transient pressure behavior of a horizontal well with multiple hydraulic fractures. The model has been used to compare the relative productivities of multiple fractures, the objective being to determine the conditions under which multiple fracture provide significant improvement over a single fracture. The approach used was to approximate the series of fractures as fully penetrating, uniform flux, vertical fractures in a box-shaped reservoir of closed boundaries. Interference between the multiple fractures was accounted for by the superposition of influence functions. The effect of wellbore storage and skin was incorporated by numerically converting the solution into the Laplace space.