Problem of reservoir estimation. The optimal exploitation of hydrocarbon reservoirs requires the accurate estimation of the reservoir's characteristics. Placement of wells, injection of gas and water, and production rate of hydrocarbons depend on the correct assessment of the reservoir properties. Furthermore, the field's properties change over time as reservoir pore fluid is removed.
I advocate to assess a reservoir's state by an integrated inversion of combine well test, well log, and seismic data. This paper is result of my exploratory reading and discusses the physics of reservoir flow and seismic wave propagation in porous rocks. The paper includes simple finite difference simulations for reservoir fluid flow and wave propagation.
Overview over various sections.
Inversion. The section on inversion considers when inversion problems should be integrated and when they should be separated.
Reservoir flow simulator. The section on reservoir flow simulation canvasses the fundamental equations based on Darcy's law, mass conservation, and gas-oil-water mixtures (black oil model). Additionally, the section introduces a simple simulator for a single phase fluid that I implemented.
Wave equation for porous rocks. The third section considers wave propagation in a porous rock medium. Central to the section is Gassmann's velocity expressions. Additionally, the section discusses Biot's derivation of frequency dependent wave equation for a saturated porous rock. The section discusses a simple implementation of an acoustic wave simulator.
A first experiment. The final section simulates the fluid flow and seismic time-lapse experiments over a simple reservoir model. For lack of time, the example currently simplifies much of the theory discussed in the paper. The historic success of simple models in seismic analysis encourages such a simplified first attempt.
Darcy-Gassmann-Elastic formulation Several competing physical models of porous rocks exist. The models usually vary in their assumptions and their mathematical sophistication. A scientist ought to choose the most parsimonious model for the experiment. The popular Darcy-Gassmann-Elastic formulation combines the feasible state of the art of all three fields into a single model. The formulation of the reservoir flow and Gassmann's elastic parameters are directly based on macroscopic models of porous rock. However, it is unclear if the combination of the three formulations is computationally feasible or parsimonious.
The formulation and implementation of the seismic and well simulation is my first step towards a future inversion of experimental field data. All simulators are implemented in Jest Schwab and Schroeder (1997), an inversion framework. The simulators are semi-linear operators and contain the adjoint operator of their linear projections.
Literature. Landa Landa (1997) discusses a sequence of synthetic two-dimensional test cases that integrate a petroleum engineer's well observations and estimates of pore saturation. While working out the inversion of well data in great detail, Landa assumes that geophysicists hand him the saturation data presumably derived from seismic time-lapse experiments. In contrast, I advocate the integration of the seismic and well experiments into a single inversion of a consistent reservoir model.
Lumley Lumley (1995) independently migrates several time-lapse surveys and shows the effect of changing pore fluids on migrated sections. He finally speculates about the interpretation of the observed amplitude changes in the seismic difference image. The difference section show clear reservoir changes. However, Lumley's field data is unusually well suited: the reservoir is extremely shallow, the target area is small. He does not combine the migrations into a single inversion, nor does he integrate any data from other physical experiments.
Biondo et al Biondi et al. (1996) simulate and invert synthetic data based on a 3-D reservoir model. However, the inversion is not integrated in a single consistent formulation.
Motivation. I personally wrote this paper to learn about reservoir flow and wave propagation in porous rocks. Additionally, I include in the paper a short discussion of simulation and inversion of physical experiments. I believe that inversion theory is the key to integrate the various observations by geoscientists and petroleum engineers into a single earth model.
Wave propagation aspect. How does the integration of seismic and well experiments relate to a course in linear wave propagation? The formulation of the seismic experiment is based on the fundamentals taught in the course. The wave simulator implements the standard acoustic wave equation. The paper extends the course's discussion of homogenous and random media, to fluid filled porous media. Finally, the article's future inversion aspects complement the course's corresponding simulation techniques. I wanted to do this work, ever since reading Landa's thesis and, I admit, the course on wave propagation gave me the necessary excuse and motivation.