# Simulation and Optimization of Subsurface Flow Processes

Subsurface flow simulations are a key component in the design and optimization of oil and gas production operations. Such simulations will have an analogous role in the planning and management of geological carbon sequestration. The equations appearing in these formulations describe the movement of one or more components (such as methane, octane, water, carbon dioxide, etc.) flowing within one or more phases (aqueous, oleic, vapor) through porous formations such as oil reservoirs or aquifers. The governing partial differential equations are derived by writing mass conservation equations for each component. These equations are then combined with appropriate constitutive relations, in this case Darcy’s law for multiphase flow, as well as thermodynamic models for phase behavior. Darcy’s law is widely used in subsurface flow modeling – it relates phase velocities to pressure gradients in terms of rock properties (absolute permeability), fluid properties (viscosities) and rock-fluid interactions (relative permeability functions). In some cases, additional physics must be incorporated – examples include chemical reactions, thermal effects and geomechanical effects – which lead to additional equations and couplings.

The numerical solution of the subsurface flow equations requires discretization on a computational grid. The most commonly used discretization procedures are based on finite volume approaches, though finite elements are used in some cases. The generation of a suitable grid can be a challenge, as geometrically complex geological features such as faults, fractures and layering can strongly impact flow and may therefore need to be resolved. Wells can also lead to gridding challenges, as they can be quite general geometrically (e.g., multibranched) and can intersect geological features at arbitrary angles.

There are a number of key complications associated with practical subsurface flow modeling. These include the multiphysics and multiscale aspects of many problems, the computational requirements associated with optimization, and the need to handle geological uncertainty. We now consider two examples of subsurface flow problems that illustrate some of these challenges.

**In-situ Upgrading**

Increasing demand for oil is expected to result in a shift away from light, easy-to-produce crude oils toward heavy oils and possibly to other unconventional resources such as oil shales. Shell’s in-situ conversion process, which is still in the development stage, is one of the approaches being studied for oil shale production. In this process, heat is introduced into the reservoir through high-temperature (~700 oF) downhole heaters. This facilitates a sequence of chemical reactions, which converts the in-situ high-carbon-number resource into more desirable products (lighter components). Simulating this process is difficult, however, as it entails chemical reaction modeling coupled to multiphase fluid flow, thermal and geomechanical effects, and porosity/permeability evolution. Flow simulations of this and related processes will be essential, however, if these operations are to be fully understood and optimized.

Our group has recently developed a prototype capability for modeling the in-situ upgrading of oil shales. These simulations include multiple components moving in the oleic and vapor phases, solution of the energy equation to determine the temperature field, and the modeling of temperature-dependent chemical reactions which describe the transformation of kerogen (an oil precursor present in the oil shale) to light oil and gas components. This modeling capability has been implemented into Stanford’s General Purpose Research Simulator (GPRS).

Simulation results for this problem are presented in Figure 1, where we show reservoir temperature and kerogen concentration at different times. The locations of the downhole electrical heaters are evident in the temperature plots; the two production wells are near the center. Kerogen is converted to oil first near the heaters and later throughout the formation. Essentially all of the kerogen in the heated zone is converted to oil and gas and produced by 500 days. The timing for oil and gas production, as well as the relative amounts of oil and gas produced, depend on the heater temperature and spacing. Thus, this type of model can be used to optimize in-situ upgrading processes. Although these simulations are highly demanding computationally, there are still many effects that are simplified (e.g., the chemical reaction system) or neglected entirely (geomechanics, porosity/permeability evolution, reservoir heterogeneity) in this model. The development of more comprehensive formulations will therefore be very useful.

**Optimization of subsurface flow processes**

Simulation models are increasingly being used to optimize subsurface operations. Examples include determining the optimal placement of new wells and finding optimal settings (rates, well pressures) for existing wells in order to maximize an objective function such as net present value or cumulative oil production. Optimizations of this type can be very demanding computationally as they typically require many simulations. Both gradient-based and gradient-free techniques are being investigated and applied for these optimizations. Gradient-based procedures can be very efficient, though they converge to local optima. Some gradient-free procedures, by contrast, explore the global search space, though they may require many more simulations. The incorporation of geological uncertainty into these optimizations, which can be achieved by optimizing over many geological realizations, adds substantially to computational demands.

The large computational requirements associated with optimization are a concern in many application areas, and researchers have devised a number of reduced-order modeling (ROM) procedures to accelerate the function evaluations (simulations) required for optimization. The idea in ROM procedures is to establish a simplified representation for the model that is valid over a certain range. Construction of the reduced-order model requires some number of ‘training’ simulations, which are full-resolution, full-physics simulations. Training simulation results are then used to generate very fast surrogate models that can be applied in optimizations.

Here we present an example of this approach for subsurface flow modeling. The particular ROM in this case was constructed using a ‘trajectory piecewise linearization’ (TPWL) in which the simulation model is linearized and then represented in a reduced form using a special projection matrix. The optimization here is multiobjective – it seeks to determine well pressures that maximize the cumulative oil production while minimizing cumulative water injection. This requires a total of about 14,000 simulation runs.

The result, shown in Figure 2 above, is a Pareto front, which defines the set of optimal solutions. In this figure, the ×’s (TPWL) designate the optimization results using the ROM surrogate, while the solid circles and triangles represent optimized solutions that use the fully-resolved model. It is evident that the ROM surrogate provides a reasonable level of accuracy for this optimization. The speedup using the ROM is quite dramatic here, about a factor of 400 relative to the full-resolution model. This is, however, a fairly simple oil-water simulation containing relatively few (24,000) grid blocks. An outstanding challenge is to extend and generalize ROM procedures to more complicated cases such as compositional or thermal simulation models.

There are of course many other interesting issues in subsurface flow modeling that are not considered here. These include multiscale treatments, history matching (inverse problem) procedures, closed-loop reservoir modeling, uncertainty quantification, and the inclusion of additional physics in simulation models.

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