Engineering topics
Srinivasan, Shriram
Publications and source records attributed to Srinivasan, Shriram.
Optimization of District Heating Network Parameters in Steady-State Operation
Here we examine the modeling, simulation, and optimization of district heating systems, which are widely used for thermal transport using steam or hot water as a carrier. We propose a generalizable framework to specify network models and scenario parameters, and develop an optimization method for evaluating system states including pressures, fluid flowrates, and temperatures throughout the network. The network modeling includes pipes, thermal plants, pumps, and passive or controllable loads as system components. We propose basic models for thermodynamic fluid transport and enforce the balance of physical quantities in steady-state flow over co-located outgoing and return networks. We formulate an optimization problem with steam and hot water as the outgoing and return carriers, as in legacy twentieth century systems. The physical laws and engineering limitations are specified for each component type, and the thermal network flow optimization problem is formulated and solved for a realistic test network under several scenarios.
On the Existence of Steady-State Solutions to the Equations Governing Fluid Flow in Networks
The steady-state solution of fluid flow in pipeline infrastructure networks driven by junction/node potentials is a crucial ingredient in various decision-support tools for system design and operation. While the nonlinear system is known to have a unique solution (when one exists), the absence of a definite result on the existence of solutions hobbles the development of computational algorithms, for it is not possible to distinguish between algorithm failure and non-existence of a solution. In this letter, we show that for any fluid whose equation of state is a scaled monomial, a unique solution exists for such nonlinear systems if the term solution is interpreted in terms of potentials and flows rather than pressures and flows. However, for gases following the CNGA equation of state, while the question of existence remains open, we construct an alternative system that always has a unique solution and show that the solution to this system is a good approximant of the true solution. Further, the existence result for flow of natural gas in networks also applies to other fluid flow networks such as water distribution networks or networks that transport carbon dioxide in carbon capture and sequestration. Most importantly, our result enables correct diagnosis of algorithmic failure, problem stiffness, and non-convergence in computational algorithms.
What is the gradient of a scalar function defined on a subspace of square matrices?
We illustrate a technique to calculate the gradient of scalar functions that are defined on any arbitrary matrix subspace. It generalizes our earlier work titled “What is the gradient of a scalar function of a symmetric matrix ?”(Indian Journal of Pure and Applied Mathematics (2022), https://doi.org/10.1007/s13226-022-00313-x), in which we considered the special case of the subspace of symmetric matrices. Extant methods to calculate the gradient in such cases have an inherent flaw which leads to spurious results that populate several publications, as well as respected textbooks and handbooks on matrix calculus. Here, we examine these sources and results in a rigorous and concrete mathematical setting of a finite-dimensional inner-product space and discover the inherent flaw and also a remedy. We demonstrate two ways to calculate the derivative/gradient and second derivative for scalar functions of matrices defined over an arbitrary matrix subspace; the first method is by considering any (differentiable) extension to the space of square matrices and projection of its gradient onto the given subspace. The second method utilizes an ordered basis and computes each component of the gradient through evaluation of the directional derivative. All the ideas presented are illustrated by non-trivial examples, namely, considering the subspace of 3 x 3 circulant and Toeplitz matrices and presenting the results of gradient-descent with both the spurious and correct gradients. Moreover, our bibliography makes it clear that a rigorous approach to matrix calculus is not common in practice, and our presentation of matrix calculus in the language of inner-product spaces will be significant and meaningful for applied mathematicians, engineers and researchers working in inter-disciplinary fields to avoid the conceptual pitfalls that exist.
What is the gradient of a scalar function defined on a subspace of square matrices?
We illustrate a technique to calculate the gradient of scalar functions that are defined on any arbitrary matrix subspace. It generalizes our earlier work titled “What is the gradient of a scalar function of a symmetric matrix ?”(Indian Journal of Pure and Applied Mathematics (2022), doi:10.1007/s13226-022-00313-x), in which we considered the special case of the subspace of symmetric matrices. Extant methods to calculate the gradient in such cases have an inherent flaw which leads to spurious results that populate several publications, as well as respected textbooks and handbooks on matrix calculus. We examine these sources and results in a rigorous and concrete mathematical setting of a finite-dimensional inner-product space and discover the inherent flaw and also a remedy. We demonstrate two ways to calculate the derivative/gradient and second derivative for scalar functions of matrices defined over an arbitrary matrix subspace; the first method is by considering any (differentiable) extension to the space of square matrices and projection of its gradient onto the given subspace. The second method utilizes an ordered basis and computes each component of the gradient through evaluation of the directional derivative. All the ideas presented are illustrated by non-trivial examples, namely, considering the subspace of 3 × 3 circulant and Toeplitz matrices and presenting the results of gradient-descent with both the spurious and correct gradients. Moreover, our bibliography makes it clear that a rigorous approach to matrix calculus is not common in practice, and our presentation of matrix calculus in the language of inner-product spaces will be significant and meaningful for applied mathematicians, engineers and researchers working in inter-disciplinary fields to avoid the conceptual pitfalls that exist.
Capabilities and Advantages of the GasModels Package
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Modeling and optimization of steady flow of natural gas and hydrogen mixtures in pipeline networks
Here, we extend the canonical problems of simulation and optimization of steady-state gas flows in pipeline networks with compressors to the transport of mixtures of highly heterogeneous gases injected throughout a network. Our study is motivated by proposed projects to blend hydrogen generated using clean energy into existing natural gas pipeline systems as part of efforts to reduce the reliance of energy systems on fossil fuels. Flow in a pipe is related to endpoint pressures by a basic Weymouth equation model, with an ideal gas equation of state, where the wave speed depends on the hydrogen concentration. At vertices, in addition to mass balance, we also consider mixing of incoming flows of varying hydrogen concentrations. The problems of interest are the heterogeneous gas flow simulation (HGFS), which determines system pressures and flows given fixed boundary conditions and compressor settings, as well as the heterogeneous gas flow optimization (HGFO), which extremizes an objective by determining optimal boundary conditions and compressor settings. We examine conditions for uniqueness of solutions to the HGFS, as well as compare and contrast mixed-integer and continuous nonlinear programming formulations for the HGFO. We develop computational methods to solve both problems, and examine their performance using four test networks of increasing complexity.
Numerical Solution of the Steady-State Network Flow Equations for a Non-Ideal Gas
Herein we formulate a steady-state network flow problem for non-ideal gas that relates injection rates and nodal pressures in the network to flows in pipes. For this problem, we present and prove a theorem on uniqueness of generalized solution for a broad class of non-ideal pressure-density relations that satisfy a monotonicity property. Further, we develop a Newton-Raphson algorithm for numerical solution of the steady-state problem, which is made possible by a systematic non-dimensionalization of the equations. The developed algorithm has been extensively tested on benchmark instances and shown to converge robustly to a generalized solution. Previous results [1]-[4], indicate that the steady-state network flow equations for an ideal gas are difficult to solve by the Newton-Raphson method because of its extreme sensitivity to the initial guess. In contrast, we find that non-dimensionalization of the steady-state problem is key to robust convergence of the Newton-Raphson method. We identify criteria based on the uniqueness of solutions under which the existence of a non-physical generalized solution found by a non-linear solver implies non-existence of a physical solution, i.e., infeasibility of the problem. Finally, we compare pressure and flow solutions based on ideal and non-ideal equations of state to demonstrate the need to apply the latter in practice. The solver developed in this article is open-source and is made available for both the academic and research communities as well as the industry.
A machine-learning inverse model framework for rapid forecasting and history matching in unconventional reservoirs
Model-based optimization for real-time forecasting in unconventional reser-voirs requires novel methods and work?ows since the strategies and work?ows used in conventional reservoirs are either inapplicable, or prohibitively expen-sive and time-consuming. Insu?cient site data and computational expense of high-?delity simulations mean that work?ows with high-?delity simulations are not ideal for usage in comprehensive uncertainty quanti?cation stud-ies that require 1000s of forward model runs. We present an alternative, novel work?ow for unconventional reservoirs, based on the interplay between reduced-order models and machine-learning. Our physics-informed machine-learning (PIML) work?ow addresses the challenges to real-time reservoir management in uncoventionals, namely lack of data (the time-frame for which the wells have been producing), and computational expense of high-?delity modeling. We use the machine-learning paradigm of transfer-learning to bind together fast but less accurate reduced-order models with slow, but accurate high-?delity models and circumvent the di?culties inherent in the current state-of-the-art for unconventionals. Such a PIML work?ow, grounded in physics, is a viable candidate for real-time history matching and production forecasting in a fractured shale gas reservoir. The signi?cance of our approach is that while it is developed for a particu-lar well and site in the Marcelus Shale gas reservoir of the Appalachian basin (MSEEL), it is not wedded to it. We expect the same work?ow can be ap-plied to other shale formations (e.g., Woodford, Barnett, Utica, EagleFord) should site-data become available, using the same set of machine-learning techniques from transfer learning. Some ?ne-tuning (or minimal retraining of the neural networks) will be required to transfer knowledge across shale gas sites/formations but it is a clearly superior alternative to developing a new machine-learning model altogether when considering a di?erent site.
Operation of Natural Gas Pipeline Networks With Storage Under Transient Flow Conditions
Here, we formulate a nonlinear optimal control problem for intraday operation of a natural gas pipeline network that includes storage reservoirs. The dynamics of compressible gas flow through pipes, compressors, reservoirs, and wells are considered. In particular, a reservoir is modeled as a rigid, hollow container that stores gas under isothermal conditions and uniform density, and a well is modeled as a vertical pipe. For each pipe, flow dynamics are described by a coupled partial differential equation (PDE) system in density and mass flux variables, with momentum dissipation modeled using the Darcy–Wiesbach friction approximation. Compressors are modeled as scaling up the pressure of gas between the inlet and outlet. The governing equations for all network components are spatially discretized and assembled into a nonlinear differential-algebraic equation (DAE) system, which synthesizes above-ground pipeline and subsurface reservoir dynamics into a single reduced-order model. We seek to maximize an objective function that quantifies economic profit and network efficiency subject to the flow equations and inequalities that represent operating limitations. The problem is solved using a primal–dual interior point solver, and the solutions are validated in computational experiments and simulations on several pipeline test networks to demonstrate the effectiveness of the proposed methodology.
Demand Response Analogues for Residential Loads in Natural Gas Networks
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Transient flow modeling in fractured media using graphs
In this work, we describe a method to simulate transient fluid flows in fractured media using an approach based on graph theory. Our approach builds on past work where the graph-based approach was successfully used to simulate steady-state fluid flows in fractured media. We find a mean computational speedup of the order of 1400 from an ensemble of a 100 discrete fracture networks in contrast to the O(10 4 ) speedup that was obtained for steady-state flows earlier. However, the transient flows considered here involve an additional degree of complexity that was not present in the steady-state flows considered previously with a graph-based approach, that of time marching and solution of the flow equations within a time-stepping scheme. We verify our method with an analytical test case and demonstrate its use on a practical problem related to fluid flows in hydraulically fractured reservoirs. By enabling the study of transient flows, we create an opportunity for a wide set of possibilities where a steady-state approximation is not sufficient, such as the example motivated by hydraulic fracturing that we present here. This work validates the concept that graphs are able to reliably capture the topological properties of the fracture network and serve as effective surrogates in an uncertainty-quantification framework.
Towards real-time forecasting of natural gas production by harnessing graph theory for stochastic discrete fracture networks
In this work, we compare hydrocarbon production curves obtained from a graph-based reduced-order model with the high-fidelity Discrete Fracture Network (DFN) predictions for a fracture network created using data from a real shale site. We observe that the bounds for the high fidelity DFN model lie within the bounds for the reduced order model, implying that the reduced-order model provides a conservative estimate. Moreover, we found that except for first-passage times and late arriving mass, the production curves from the reduced-order model predict transport accurately. However, it is to be noted that the results are inspite of trading a three-dimensional geometry for a reduced system in the form of a graph, one that is 500–1000 times faster in terms of computational efficiency (for this particular application). In addition, we also compare the production curves for large drawdown and small drawdown using our graph approach. The reduced-order model is successful in showing that the long term productivity is higher in case of small drawdown although the initial productivity is higher for large drawdown. Thus, this reduced-order model offers great potential in uncertainty quantification for production, as well as in providing operators with information to make real-time decisions for optimal production.
Physics-informed Machine Learning for Real-time Unconventional Reservoir Management
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Matrix Diffusion in Fractured Media: New Insights Into Power Law Scaling of Breakthrough Curves
Here, we develop a theoretical model for power law tailing behavior of transport in fractured rock based on the relative dominance of the decay rate of the advective travel time distribution, modeled using a Pareto distribution (with tail decaying as ~ time -(1+α) ), versus matrix diffusion, modeled using a Lévy distribution. The theory predicts that when the advective travel time distribution decays sufficiently slowly (α<1), the late-time decay rate of the breakthrough curve is -(1+α/2) rather than the classical -3/2. Yet, if α>1, the -3/2 decay rate is recovered. For weak matrix diffusion or short advective first breakthrough times, we identify an early-time regime where the breakthrough curve follows the Pareto distribution, before transitioning to the late-time decay rate. The theoretical predictions are validated against particle tracking simulations in the three-dimensional discrete fracture network simulator dfnWorks, where matrix diffusion is incorporated using a time domain random walk.
Model reduction for fractured porous media: a machine learning approach for identifying main flow pathways
Discrete fracture networks (DFN) are often used to model flow and transport in fractured porous media. The accurate resolution of flow and transport behavior on a large DFN involving thousands of fractures is computationally expensive. This makes uncertainty quantification studies of quantities of interest such as travel time through the network computationally intractable, since hundreds to thousands of runs of the DFN model are required to get bounds on the uncertainty of the predictions. Prior works on the subject demonstrated that the complexity of a DFN could be reduced by considering a sub-network of it (often termed a “backbone” sub-network), one whose flow and transport properties were then shown to be similar to that of the full network. The technique is tantamount to partitioning the complete set of fractures of a network into two disjoint sets, one of which is the backbone sub-network while the other its complement. It is in this context that we present a system-reduction technique for DFNs using supervised machine learning via a Random Forest Classifier that selects a backbone sub-network from the full set of fractures. The in-sample errors (in terms of precision and recall scores) of the trained classifier are found to be very accurate indicators of the out-of-sample errors, thus exhibiting that the classifier generalizes well to test data. Moreover, this system-reduction technique yields sub-networks as small as 12% of the full DFN that still recover transport characteristics of the full network such as the peak dosage and tailing behavior for late times. Most importantly, the sub-networks remain connected, and their size can be controlled by a single dimensionless parameter. Furthermore, measures of KL-divergence and KS-statistic for the breakthrough curves of the sub-networks with respect to the full network show physically realistic trends in that the measures decrease monotonically as the size of the sub-networks increase. In conclusion, the computational efficiency gained by this technique depends on the size of the sub-network, but large reductions in computational time can be expected for small sub-networks, yielding as much as 90% computational savings for sub-networks that are as small as 10-12% of the full network.
Identifying Backbones in Three-Dimensional Discrete Fracture Networks: A Bipartite Graph-Based Approach
In this paper, we present a graph-based method to identify primary flow and transport sub-networks in three-dimensional discrete fracture networks (DFNs). The structure of a DFN lends itself to the use of graphs as a coarse-scale representation that retains the multiscale nature of flow and transport through fracture media. We develop a bipartite graph representation that integrates fracture network topology, fracture geometry, and hydraulic properties. We show that the two most common graph-representations of DFNs, vertices representing intersections and vertices representing fractures, are projections of this bipartite graph thereby providing a generalization of previous DFN-graph frameworks. The primary subnetworks in each DFN are identified by running a heuristic algorithm that determines the edge-disjoint shortest paths through the graph which correspond to the regions where the fastest transport occurs. The method does not have any user-defined parameters and terminates in a finite number of steps. The quality of the method is demonstrated by comparing transport simulations on the identified primary subnetwork and full network, which are in good agreement for early and middle times. In conclusion, these estimates of the first passage times can be achieved with close to an order of magnitude reduction of computational expense using the proposed method.
Robust system size reduction of discrete fracture networks: a multi-fidelity method that preserves transport characteristics
In this paper, we propose a multi-fidelity system reduction technique that uses weighted graphs paired with three-dimensional discrete fracture network (DFN) modelling for efficient simulation of subsurface flow and transport in fractured media. DFN models are used to simulate flow and transport in subsurface fractured rock with low-permeability. One method to alleviate the heavy computational overhead associated with these simulations is to reduce the size of the DFN using a graph representation of it to identify the primary flow sub-network and only simulate flow and transport thereon. The first of these methods used unweighted graphs constructed solely on DFN topology and could be used for accurate predictions of first-passage times. However, these techniques perform poorly when predicting later stages of the mass breakthrough. We utilize a weighted-graph representation of the DFN where edge weights are based on hydrological parameters in the DFN that allows us to exploit the kinematic quantities derivable a posteriori from the flow solution obtained on the graph representation of the DFN to perform system reduction and predict the later stages of the breakthrough curve with high fidelity. We also propose and demonstrate the use of an adaptive pruning algorithm with error control that produces a pruned DFN sub-network whose predicted mass breakthrough agrees with the original DFN within a user-specified tolerance. Lastly, the method allows for the level of accuracy to be a user-controlled parameter.