^{}
Unfortunately, a reservoir is difficult to access. It is buried
hundreds or thousands of meters in the subsurface.
A geoscientist only chance to access the reservoir directly
is in a well.
Petroleum engineers carefully observe a well's
production volumes of oil, water, and gas.
Additionally,
the engineers measure pressure in the well or
even conduct pressure tests.
My discussion of reservoir fluid flow processes follows
Landa 1997 and Aziz 1979.

**Black-oil model.**
Petroleum engineers
use various reservoir fluid flow simulators to relate their
observations to reservoir properties, such as permeability,
porosity, pressure distribution, and saturation.
The *standard* black-oil model for a reservoirs mix of
fluids comprises
three distinct phases: oil, water, and gas.
The oil phase dissolves a certain amount of gas; the water does not.
Intuitively,
one might want to describe the reservoir properties at the scales of pores.
However, the traditional approach homogenizes the pore-scale heterogenities
to macroscopic descriptions such as Darcy's law.

The general **equation of mass conservation**

(1) |

**Darcy's law**
expresses the fluid flux in terms of pressure and gravity

(2) |

The combination of Darcy's law with the mass conservation equation 1 yields the fluid flow equations:

(3) | ||

(4) |

The parameters *B*_{j} relate parameter values at surface conditions
- indicated by index S - and
reservoir condition - indicated by index *R*.
The parameters are defined as volume ratio

(5) |

(6) | ||

The *B*_{j} ratios, and the mass transfer *R*_{s},
depend on the pressure *p*_{j} of the
various phases *j*=*o*,*w*,*g*.
Since water compressibility is small, the relationship
*B*_{w} = *f*(*p*_{w}) can be approximated by

The coefficients in equation 4 express transmissibilities and are defined as

Viscosities of oil and gas strongly depend on temperature and pressure. The pressure dependency is again empirical Aziz and Settari (1979). The viscosity's temperature dependence is especially important in thermal-recovery processes such as steam injections. In general, the temperature dependency is well approximated by where andFinally, porosity is often pressure dependent

whereThe equations

(7) | ||

**Well observations.**
In a field experiment,
petroleum engineers control the pressure in the well.
Petroleum engineers conduct well tests by changing the pressure in the
well and observing the reaction of the surrounding reservoir pressure
(respectively its effect on the controlled well pressure).
These experiments attempt to estimate local reservoir transmissibility
but do have to account for the *skin effect*:
the alterations of the rock next to the well.
Additionally,
petroleum engineers record the well's production volumes.
The production volumes depend on the reservoir pressure *p* and the
well bore pressure *p*_{w}*f*:

I hope to combine such well observations with seismic time-lapse data to yield improved estimates of reservoir property maps.

**Simplified well system**
My preliminary implementation is based on the assumption
that the reservoir only contains a single fluid,
e.g. oil but neither water, nor gas.
This assumption might be locally reasonable for large oil fields,
albeit I expect almost all hydrocarbon mixtures to release gas when
a production lowers the reservoir pressure sufficiently.
The assumption of a single
fluid phase collapses the fluid flow equations 4 and its
constraints 4 to a single equation.

Additionally, I neglect gravitational forces (*g* =0).
I assume that the rocks transmissibility varies smoothly
and that the rocks pore volume is independent of pressure (*c*_{R} = 0).
Finally, I suggest that the reservoir fluid is only slightly
compressible and consequently satisfies

(8) |

I doubt that seismic field data can detect reasonable changes in pressure and density of a single fluid phase around a well. Instead, current time lapse surveys concentrate on detecting saturation changes in multi-phase reservoirs, that often involve gas, steam, or highly pressured water. I chose the single fluid phase case since it is the simplest to implement. Should I continue this work, I will seek the collaboration of an experienced petroleum engineer to improve the reservoir flow simulator. The simplicity of the reservoir model, however, might prove beneficial when formulating the inverse problem.

**Parameterization**
My simple reservoir simulator requires as input
a transmissibility map of the reservoir,
an initial pressure distribution within the reservoir
(usually assumed to be constant),
the pressure history at the boundary of the wells.
Given these parameters, the simulator computes
the reservoir pressure distribution over time.
For this preliminary study,
I restrict the reservoir to a two-dimensional plane.
The model is computationally manageable, and the results
can be visualized effectively.

3/9/1999