Summary
This paper discusses a method which helps identify the geometry of geological features in an oil reservoir by history matching of production data. Following an initial study on single-phase flow and applied to well tests (Rahon, D., Edoa, P. F., and Masmoudi, M.: "Inversion of Geological Shapes in Reservoir Engineering Using Well Tests and History Matching of Production Data," paper SPE 38656 presented at the 1997 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 5–8 October.), the research presented here was conducted in a multiphase flow context.
This method provides information on the limits of a reservoir being explored, the position and size of faults, and the thickness and dimensions of channels.
The approach consists in matching numerical flow simulation results with production measurements. This is achieved by modifying the geometry of the geological model. The identification of geometric parameters is based on the solution of an inverse problem and boils down to minimizing an objective function integrating the production data.
The minimization algorithm is rendered very efficient by calculating the gradients of the objective function with respect to perturbations of these geometric parameters. This leads to a better characterization of the shape, the dimension, and the position of sedimentary bodies.
Several examples are presented in this paper, in particular, an application of the method in a two-phase water/oil case.
Introduction
A number of semiautomatic history matching techniques have been developed in recent years to assist the reservoir engineer in his reservoir characterization task. These techniques are generally based on the resolution of an inverse problem by the minimization of an objective function and require the use of a numerical simulator.
The matching parameters of the inverse problem comprise two types of properties: petrophysical/porosity and permeability and geometric position, shape, and size of the sedimentary bodies present in the reservoir.
To be efficient, minimization algorithms require the calculation of simulated production gradients with respect to matching parameters. Such gradients are usually calculated by deriving discrete state equations solved in the numerical simulator1–5 or by using a so-called adjoint-state method.6,7 Therefore, most of these gradient-based methods only allow the identification of petrophysical parameters which appear explicitly in the discrete equations of state.
The case of geometric parameters is much more complex, as the gradients of the objective function with respect to these parameters cannot be determined directly from the flow equation. Recent works8–10 have handled this problem by defining geological objects using mathematical functions to describe porosity or permeability fields. But, generalizing these solutions to complex geological models remains difficult.
The method proposed in this paper is well suited to complex geometries and heterogeneous environments. The history matching parameters are the geometric elements that describe the geological objects generated, for example, with a geomodeling tool.
A complete description of the method with the calculation of the sensitivities was presented in Ref. 11, within the particular framework of single-phase flow adapted to well-test interpretations. In this paper we will introduce an extension of the method to multiphase equations in order to match production data. Several examples are presented, illustrating the efficiency of this technique in a two-phase context.
Description of the Method
The objective is to develop an automatic or semiautomatic history matching method which allows identification of geometric parameters that describe geological shapes using a numerical simulator. To be efficient, the optimization process requires the calculation of objective function gradients with respect to the parameters.
With usual fluid flow simulators using a regular grid or corner point geometry, the conventional methods for calculating well response gradients on discrete equations are not readily usable when dealing with geometric parameters. These geometric parameters do not appear explicitly in the model equations. With these kinds of structured models the solution is to determine the expression of the sensitivities of the objective function in the continuous problem using mathematical theory and then to calculate a discrete set of gradients.
Sensitivity Calculation.
Here, we present a sensitivity calculation to the displacement of a geological body in a two-phase water/oil flow context.
State Equations.
Let ? be a two- or three-dimensional spatial field, with a boundary ? and let ]0,T[ be the time interval covering the pressure history. We assume that the capillary pressure is negligible. The pressure p and the water saturation S corresponding to a two-phase flow in the domain ? are governed by the following equations:
∂ ϕ ( p ) S ∂ t − ∇ . ( k k r o ( S ) μ o ∇ ( p + ρ o g z ) ) = q o ρ o , ∂ ϕ ( p ) S ∂ t − ∇ . ( k k r w ( S ) μ w ∇ ( p + ρ w g z ) ) = q w ρ w , ( x , y , z ) ∈ Ω , t ∈ ] 0 , T [ , ( 1 )
with a no-flux boundary condition on ? and an initial equilibrium condition
Flowing Material Balance and Rate-Transient Analysis of Horizontal Wells in Under-Saturated Coal Seam Gas Reservoirs: A Case Study from the Qinshui Basin, China
