M. Susana Bidner

Comparison of optimization techniques for automatic history matching

Abstract Reservoir parameters are estimated by adjusting simulation models to match field or laboratory data. Multivariate optimization techniques with physically realistic constraints on the parameters are used in order to obtain these estimates. Two examples are presented. The first example is the analysis of a drawndown test. Permeability and porosity are determined by minimizing an objective function which is the sum of the squares of the differences between theoretical and measured pressure-time distributions at the well. The minimization is performed by applying four different optimization techniques: Davidon-Fletcher-Powell (DFP), Fletcher-Reeves (FR), Quasi-Newton Approximation for the Least-Squares Problem (QNA) and Levenberg-Marquardt (LM). The second example is the simultaneous determination of capillary pressure and relative permeability curves of oil/water systems. It is based on the analysis of transient output data measured from a linear coreflood experiment. QNA and LM are used to match results from a numerical simulator to laboratory coreflood data. The special methods for the least-squares problem (LM, QNA) behave better than the two others (DFP, FR). LM and QNA arrive to the optimal point more frequently than DFP and FR. LM takes less computing time than QNA but is more affected by rounding errors. Therefore, QNA shows the best behavior when finding the optimum. The automatic algorithms are of particular use whenever the equations which govern the flow are too complex to be solved by the traditional analytical-graphical methods.

Statistical analysis of heterogeneities and their effect on build-up and drawdown tests

The objective of this work is to determine the effect of permeability and porosity spatial variations on well test pressure response. Data from three wells and synthetic data are used. Field data consist of permeability and porosity as functions of depth and pressure transient test measurements from the same wells. To achieve the objective, two different tools are applied: statistical characterization of heterogeneities and a well test interpretation method. For the data: (1) permeabilities are represented by exponential distribution functions; (2) constant porosity estimates that fit measured transient pressures are almost equal to the statistical arithmetic mean, while constant permeability estimates lie between the median and the arithmetic mean; and (3) minor permeability variations cause important changes in pressure response.

On some numerical methods for solving 2D radial flow towards an oil well

In this paper we study a family of finite difference schemes in two dimensions to model the single phase flow of oil through heterogeneous porous media. That family depends on one parameter θ, 0 ≤ θ ≤ 1. Using a suitable order of equations and unknowns, a linear system of equations, with a particular structure, is obtained. The corresponding matrix, excluding the first row and column, has up to five elements in each row, arranged in five diagonals. The system of linear equations is solved by a method based on Taylor series of matrix functions (TSMF). The convergence conditions for this technique are established and the most convenient θ is selected to increase the time step Δt. Besides, TSMF is compared with two iterative methods, ADI and block-SOR, usually applied to solve multidimensional equations. Both methods, ADI and block-SOR, are adapted to this particular problem. We conclude that TSMF is the fastest technique using adequate values of θ and Δt, but the time increment Δt must remain small because of the convergence condition. On the other hand, block-SOR converges using large values of Δt, but it uses a large amount of CPU time. ADI is discarded for not presenting advantages over the other two techniques. Therefore, TSMF is recommended when a short period of time must be simulated, and block-SOR is suitable for long simulations and applying a variable time increment.

Mathematical modelling of flow towards an oil well

We describe the numerical approximations and applications of a mathematical model that governs the flow of oil towards a well. The flow of a single-phase fluid in a porous medium is governed by a parabolic equation obtained by combining the Darcys and the continuity equations. In order to account for the spatial variations of porosity and permeability, and for permeability anisotropy, a two-dimensional model is put forward. A mixed initial–boundary value problem is numerically solved by a finite-difference family of numerical schemes, which depends on a parameter θ. The stability—conditional or unconditional, depending on θ—and the convergence of the schemes have been proved. The linear system originated at each time step is solved by the iterative ADI and block-SOR methods, and by a Taylor series of matrix functions (TSMF). These methods are compared and their relative efficiencies are carefully assessed. TSMF is the fastest technique given that adequate values of θ and time step Δt are used—but Δt must remain small. A combination of TSMF and block-SOR with variable Δt seems to be the best policy. Our numerical simulator is tested by reproducing the existing analytical solutions for limiting cases, and then applied in well test analysis. The contributions of this work are: (1) we introduce the TSMF technique to reservoir simulation and (2) vertical permeability and permeability spatial variations are included in a well test simulator for further developments. Copyright