Optimal Control of Two-Phase Stefan Problems — Numerical Solutions

1987 ◽  
pp. 179-206 ◽  
Author(s):  
Irena Pawlow
Keyword(s):  
Optimal Control ◽  
Numerical Solutions ◽  
Two Phase ◽  
Stefan Problems

History Matching in Two-Phase Petroleum Reservoirs

1980 ◽  
Vol 20 (06) ◽  
pp. 521-532 ◽  
Author(s):  
A.T. Watson ◽  
J.H. Seinfeld ◽  
G.R. Gavalas ◽  
P.T. Woo
Keyword(s):  
Optimal Control ◽  
Parameter Estimation ◽  
Objective Function ◽  
History Matching ◽  
Computing Time ◽  
Absolute Permeability ◽  
Two Phase ◽  
Automatic History Matching ◽  
Control Approach ◽  
First Order

Abstract An automatic history-matching algorithm based onan optimal control approach has been formulated forjoint estimation of spatially varying permeability andporosity and coefficients of relative permeabilityfunctions in two-phase reservoirs. The algorithm usespressure and production rate data simultaneously. The performance of the algorithm for thewaterflooding of one- and two-dimensional hypotheticalreservoirs is examined, and properties associatedwith the parameter estimation problem are discussed. Introduction There has been considerable interest in thedevelopment of automatic history-matchingalgorithms. Most of the published work to date onautomatic history matching has been devoted tosingle-phase reservoirs in which the unknownparameters to be estimated are often the reservoirporosity (or storage) and absolute permeability (ortransmissibility). In the single-phase problem, theobjective function usually consists of the deviationsbetween the predicted and measured reservoirpressures at the wells. Parameter estimation, orhistory matching, in multiphase reservoirs isfundamentally more difficult than in single-phasereservoirs. The multiphase equations are nonlinear, and in addition to the porosity and absolutepermeability, the relative permeabilities of each phasemay be unknown and subject to estimation. Measurements of the relative rates of flow of oil, water, and gas at the wells also may be available forthe objective function. The aspect of the reservoir history-matchingproblem that distinguishes it from other parameterestimation problems in science and engineering is thelarge dimensionality of both the system state and theunknown parameters. As a result of this largedimensionality, computational efficiency becomes aprime consideration in the implementation of anautomatic history-matching method. In all parameterestimation methods, a trade-off exists between theamount of computation performed per iteration andthe speed of convergence of the method. Animportant saving in computing time was realized insingle-phase automatic history matching through theintroduction of optimal control theory as a methodfor calculating the gradient of the objective functionwith respect to the unknown parameters. Thistechnique currently is limited to first-order gradientmethods. First-order gradient methods generallyconverge more slowly than those of higher order.Nevertheless, the amount of computation requiredper iteration is significantly less than that requiredfor higher-order optimization methods; thus, first-order methods are attractive for automatic historymatching. The optimal control algorithm forautomatic history matching has been shown toproduce excellent results when applied to field problems. Therefore, the first approach to thedevelopment of a general automatic history-matchingalgorithm for multiphase reservoirs wouldseem to proceed through the development of anoptimal control approach for calculating the gradientof the objective function with respect to theparameters for use in a first-order method. SPEJ P. 521＾

Optimal Control for Two-Phase Flows

2014 ◽  
pp. 347-363
Author(s):  
Malte Braack ◽  
Markus Klein ◽  
Andreas Prohl ◽  
Benjamin Tews
Keyword(s):  
Optimal Control ◽  
Two Phase Flows ◽  
Two Phase

scholarly journals Mathematical Modeling and Optimal Control of the Hand Foot Mouth Disease Affected by Regional Residency in Thailand

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2863
Author(s):  
Napasool Wongvanich ◽  
I-Ming Tang ◽  
Marc-Antoine Dubois ◽  
Puntani Pongsumpun
Keyword(s):  
Optimal Control ◽  
Numerical Solutions ◽  
Foot And Mouth Disease ◽  
Control Policy ◽  
Mouth Disease ◽  
Sensitivity Analyses ◽  
Urban Region ◽  
Control Parameters ◽  
Control Effort ◽  
Logarithmic Relationship

Hand, foot and mouth disease (HFMD) is a virulent disease most commonly found in East and Southeast Asia. Symptoms include ulcers or sores, inside or around the mouth. In this research, we formulate the dynamic model of HFMD by using the SEIQR model. We separated the infection episodes where there is a higher outbreak and a lower outbreak of the disease associated with regional residency, with the higher level of outbreak occurring in the urban region, and a lower outbreak level occurring in the rural region. We developed two different optimal control programs for the types of outbreaks. Optimal Control Policy 1 (OPC1) is limited to the use of treatment only, whereas Optimal Control Policy 2 (OPC2) includes vaccination along with the treatment. The Pontryagin’s maximum principle is used to establish the necessary and optimal conditions for the two policies. Numerical solutions are presented along with numerical sensitivity analyses of the required control efforts needed as the control parameters are changed. Results show that the time tmax required for the optimal control effort to stay at the maximum amount umax exhibits an intrinsic logarithmic relationship with respect to the control parameters.

Formulation of a General Multiphase, Multicomponent Chemical Flood Model

1981 ◽  
Vol 21 (01) ◽  
pp. 63-76 ◽  
Author(s):  
Paul D. Fleming ◽  
Charles P. Thomas ◽  
William K. Winter
Keyword(s):  
Phase Equilibrium ◽  
Arbitrary Number ◽  
Numerical Solutions ◽  
Field Tests ◽  
Reservoir Rock ◽  
Phase System ◽  
Surfactant Flooding ◽  
Two Phase ◽  
Special Cases ◽  
Flood Model

Abstract A general multiphase, multicomponent chemical flood model has been formulated. The set of mass conservation laws for each component in an isothermal system is closed by assuming local thermodynamic (phase) equilibrium, Darcy's law for multiphase flow through porous media, and Fick's law of diffusion. For the special case of binary, two-phase flow of nonmixing incompressible fluids, the equations reduce to those of Buckley and Leverett. The Buckley-Leverett equations also may be obtained for significant fractions of both components in the phases if the two phases are sufficiently incompressible. To illustrate the usefulness of the approach, a simple chemical flood model for a ternary, two-phase system is obtained which can be applied to surfactant flooding, polymer flooding, caustic flooding, etc. Introduction Field tests of various forms of surfactant flooding currently are under way or planned at a number of locations throughout the country.1 The chemical systems used have become quite complicated, often containing up to six components (water, oil, surfactant, alcohol, salt, and polymer). The interactions of these components with each other and with the reservoir rock and fluids are complex and have been the subject of many laboratory investigations.2–22 To aid in organizing and understanding laboratory work, as well as providing a means of extrapolating laboratory results to field situations, a mathematical description of the process is needed. Although it seems certain that mathematical simulations of such processes are being performed, models aimed specifically at the process have been reported only recently in the literature.23–31 It is likely that many such simulations are being performed on variants of immiscible, miscible, and compositional models that do not account for all the facets of a micellar/polymer process. To help put the many factors of such a process in proper perspective, a generalized model has been formulated incorporating an arbitrary number of components and an arbitrary number of phases. The development assumes isothermal conditions and local phase equilibrium. Darcy's law32,33 is assumed to apply to the flow of separate phases, and Fick's law34 of diffusion is applied to components within a phase. The general development also provides for mass transfer of all components between phases, the adsorption of components by the porous medium, compressibility, gravity segregation effects, and pressure differences between phases. With the proper simplifying assumptions, the general model is shown to degenerate into more familiar special cases. Numerical solutions of special cases of interest are presented elsewhere.35

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