History Matching Using a Streamline-Based Approach and Gradual Deformation

Abstract

Reservoir engineers often have to deal with history-matching problems. This is time-consuming because of the many numerical simulations that have to be run and also because of the size of the models. Optimization, coupled with gradient-based methods, enables engineers to find efficiently a reservoir representation that respects all static and dynamic data. Nevertheless, for multiphase flow or for compositional problems, only relatively small models can be handled with a finite-volume flow simulator. On the other hand, streamline-based simulation provides accurate and fast estimates to various flow problems. Simulations can be performed on larger models than with a regular fluid flow simulator. Recently, streamline-based simulation has been used for history-matching problems. First, analytical gradients of production data with respect to a geostatistical parameterization have been computed for the tracer-flow case. The saturations along streamlines were moved using an analytical solution of the transport equation. This approach then has been extended to waterflood problems with varying boundary conditions.In our method, we compute gradients of the production data, of the saturation, and of the pressure. We use a 1D numerical simulator to move the saturation along streamlines. This method is more general than when analytical calculations are used; it is more accurate for complex flow patterns. It also can be extended to take into account other parameters such as gravity. The major drawback is that it is more time-consuming; the cost for the computation of each gradient is slightly smaller than the cost of a simulation. Nevertheless, this cost can be dramatically decreased by computing multiple gradients at the same time on different processors using parallelism.These gradients will be used to perform history matching using a nonlinear optimization package in conjunction with a geostatistical method, the gradual deformation method, enables the combination of multiple realizations of a permeability field and the continuous variation from one to another.We will apply our new method to perform a history match on a large-scale 3D synthetic example of a waterflood with a complex injection history. The Streamline-Based Gradient ApproachIn this section, we will present our methodology to perform history matching. We will first introduce the Gradual Deformation Method. We will then describe the calculation of the streamlinebased gradients.

Three-Dimensional Streamline Tracing Method over Tetrahedral Domains

“…The procedure of tracing a streamline segment in a tetrahedron element and the relative formulations in Sections 2.3 and 2.4 are demonstrated, which are verified by an example shown in Figure 3. A tetrahedron is defined in Actual Element with four vertices, which are P 1 (1, 1, 1), P 2 (3, 2, 2), P 3 (1,4,3), and P 4 (1, 5, 1) whose unit is m, respectively, as well as the corresponding velocity vector at four vertices are also given. When a particle starts from a start point which is located at one face of this tetrahedra element, then it will leave this tetrahedron element at an exit point.…”

“…1), P2(3, 2, 2), P3 (1,4,3), and P4(1, 5, 1) whose unit is m, respectively, as well as the corresponding velocity vector at four vertices are also given. When a particle starts from a start point which is located at one face of this tetrahedra element, then it will leave this tetrahedron element at an exit point.…”

“…The dynamic physical processes of fluid flow and transport in underground porous media play an important role in a lot of applications, i.e., fast history matching [1,2], enhanced oil recovery [3][4][5][6][7], polymer flooding simulation [8], fractured reservoir simulation [9,10], CO 2 storage [11,12], and solving the solute transport and reactive transport in underground porous media [13][14][15][16][17][18][19]. The streamline-based simulation can describe dynamic physical processes of fluid flowing and transporting in underground porous media.…”