An Adjoint Algorithm for Computing Three-Dimensional Capture ...

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An Adjoint Algorithm for Computing Three-Dimensional Capture Functions for Management of Heterogeneous Regional Aquifer Systems Tom Clemo1, Banda S. Ramarao2, Van A Kelly2, Marsh Lavenue2 1

INTERA, Incorporated, 1933 Jadwin Ave., Suite 130 Richland, WA 99354; PH (509) 946-9898; FAX (509) 946-7878 2 INTERA, Incorporated, 1812 Centre Creek Drive, Suite 300, Austin, TX 78754; PH (512) 425-2000; FAX (512) 425-2099 ABSTRACT A recent publication in Ground Water (Sept-Oct., 2010) highlighted the application of capture functions in the management of groundwater, subject to the constraints on the depletion of surface water supplies. Capture maps are used for optimal location of the pumping wells, their rates of withdrawals, and their timing. The computation of capture functions, in the cited paper, is based on perturbation approach with finite differences. An alternative computational strategy for capture functions, based on the adjoint states, is proposed here and is developed for MODFLOW, a groundwater flow simulator. The new methodology is implemented for one of the examples cited in the above paper, namely that of the San Pedro Model, with over 700,000 nodes, developed by the United States Geological Survey, and compared with their results for capture functions based on their perturbation approach. The comparison shows good agreement between the two methods. The proposed adjoint formulation just uses the same computational time, as for one simulation for the heads, and thus saves computational time, relative to the perturbation approach, by a factor equal to the number of nodes in the model, which is of the order of several hundreds of thousands. Because of its immense savings in computational times, this new strategy for the capture functions makes it feasible to embed the groundwater management problem in a stochastic framework (probabilistic approach) to address the uncertainties in the groundwater model. INTRODUCTION Some problems of water management require a study of the effects of groundwater withdrawal on surface-water supplies. The source of water derived by pumping wells is initially from storage in the aquifer, and later, from “capture” from surface water sources, in the form of increased inflow to and decreased outflow from the aquifer. The size of sustainable groundwater development (i.e., pumping rates) should be based on the surface water that can be “captured” with permissible consequences, and not on the recharge rate to the aquifer, as may have been fallaciously believed (Bredehoeft et al., 1982). CAPTURE FUNCTIONS Definition Capture function is so formulated as to quantify the changes in boundary fluxes of a groundwater model in response to a pumping well over a period of time.

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Numerically, the derivative of the boundary flux with respect to the pumping rate in a well at a selected location gives the capture function for that location (of the well). Figure-1 illustrates this definition. The boundary flux (F) in Figure-1 is expressed as a sum of fluxes across selected grid blocks in a boundary, or boundaries. The boundaries may be of different types, such as rivers, drains, streams, etc. The boundary flux (F) may be in steady or transient state.

Figure 1. Capture Function: Definition. CAPTURE MAPS: APPLICATION If the capture functions are evaluated for every grid block (i, j, k) in the model, capture maps can be presented. Capture maps are used in the management of aquifers and aquifer-stream systems, in the optimal location of pumping wells, their rates of withdrawal and their timing. Leake et al. (2010) present an excellent introduction to the importance of capture functions in the management of aquifers. CAPTURE MAPS: PARAMETER PERTURBATION METHODS Preparation of the capture maps, involves the calculation of the capture functions (Ci,j,k ) for a very large number of grid blocks. If the calculation is needed for N grid blocks, it requires (N+1) simulations of the underlying groundwater model (e.g., MODFLOW). For typical large scale models, N is of the order of a million. As an example, the USGS model of San Pedro basin has 704,000 grid blocks. Such large number of repeated runs of the model requires prohibitively enormous computational effort and time. In practice, one limits N to a small number, so as to make this procedure a feasible one. As an example, USGS limited N to about 1600 for their San Pedro model, which had 704,000 grid blocks. At nearly 3 minutes of time for one run, even the severely restricted the computation by USGS requires 80

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hours of time. Of course, one shudders to compute the required time for all the 704,000 nodes (which is 4 Years)!!! One may also have to regenerate the capture maps for a different time frame, or for a revised model. CAPTURE FUNCTIONS: ADJOINT STATE FORMULATION An alternative algorithm, based on Adjoint State formulation, can produce the capture maps (with all the grid blocks in the model included), with just two runs of the model, one for the base case, and another for the solution of the adjoint state. Figure 2 highlights the alternative strategies in the direct and adjoint methodologies.

Figure 2.

Perturbation and Adjoint Methods for Capture Function.

A MODFLOW model, with an Adjoint Sensitivity capability has been prepared by INTERA. This Adjoint Sensitivity code can generate the capture function map in the same time as required for one additional simulation run, after the base case run for groundwater model. The computational advantage is of the order of N, in terms of the computer time. For a model with 704,000 grid blocks (San Pedro Model, USGS), the computational advantage over Finite Difference formulation is a stunning 704,000!

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With such computational advantage, capture maps can be readily generated for several alternative modeling scenarios, to facilitate optimal decision making by the aquifer managers. If warranted, the entire management problem can be embedded in a probabilistic framework, to address the uncertainty in decision making, by exploiting the ease of obtaining the capture maps. ADJOINT SENSITIVITY ANALYSIS Capture functions are sensitivity coefficients and so the methods of sensitivity analysis of groundwater models are applicable to them. They are: (1) Perturbation Approach (Leake et al, 2010), and (2) Adjoint Approach (presented here). A detailed theory of the Adjoint methodology is given for MODFLOW-2005 in Clemo (2007) and more generally in Ramarao et al. (1995). Adjoint method produces the derivative exactly, while the Finite Difference approximates them. Also, the finite difference approach may require some trial runs to determine the appropriate pumping rates to ensure correct evaluation of the derivatives, particularly, where nonlinearity may be encountered. The adjoint method is free from such trial runs. ADJOINT SENSITIVITY CODES INTERA Inc. has been associated with development of Adjoint Sensitivity capability for Groundwater models, such as MODFLOW-2005, in collaboration with USGS (Clemo, 2007). This work is still in progress. The other groundwater flow codes, for which Adjoint Sensitivity capability was developed, by INTERA include: (1) SWENT: Simulator for Water, Energy and Nuclide Transport, 1984; Office of Nuclear Waste Isolation, Battelle Project Management Division, Columbus, OH (Wilson, Ramarao and McNeish, 1986); (2) SWIFT: Sandia Waste Isolation Flow and Transport, 1988; Sandia National Laboratories, Albuquerque, NM ( Ramarao and Reeves, 1990). Adjoint sensitivity capabilities were also developed for codes simulating Multi-phase, Multi-component flow of Fluid mixtures and heat transport, such as TOUGH2; Transport of Unsaturated Groundwater and Heat: LBL National Laboratory, Berkeley, CA; (Ramarao, 1995; Ramarao and Mishra, 1996) and STOMP (Subsurface Transport over Multiple Phases: Pacific Northwest National Laboratories, Richland, WA). The development for STOMP is still in progress. ADJOINT SENSITIVITY MODULES FOR MODFLOW-2005 Adjoint Sensitivity module developed for MODFLOW-2005 supports the following boundary condition related packages: • • • • •

General Head Boundary Condition (GHB) Specified Head Boundary Condition (CHD) Drain Boundary ( DRN) River Boundary (RIV) Stream Boundary (STR)

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CAPTURE MAP USING ADJOINT METHOD: SAN PEDRO MODEL Leake et al (2010) presented the capture maps for the San Pedro Model covering an area of 80 Km* 110 Km. The model grid includes 5 layers, 440 rows, and 320 columns, comprising of 704,000 nodes. The model was simulated for a 10 Year period. A normal MODFLOW-2005 simulation takes three minutes of computer time. Both the Perturbation and Adjoint Method require a base case run of the underlying groundwater model. The additional runs required for the capture function calculations in each gridblock of the model is one for Adjoint Method and N, the number of gridblocks in the model, for the Perturbation approach. The relative computational performance of the perturbation and adjoint approaches is shown in Table 1. Table 1.

Relative Performance of Perturbation and Adjoint Methods. Metric

Perturbation Method

Adjoint Method

Number of Simulations

704,000

1

Computer Time

4 years

6 minutes

Figure-3 shows the map generated by the adjoint method. It compares well with the one presented by Leake et al (2010, p.696) which is not reproduced here.

Figure 3. Capture Function – San Pedro Model – Layer 4 (Adjoint Method).

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CONCLUSION Adjoint methodology is designed specifically to address the situations encountered in the capture map preparation, where the derivatives of one function (such as the boundary flux, in this case) with respect to a large number of (hundreds of thousands) parameters (such as the pumping rates in each of the nodes in the groundwater model, in this case) are sought. This method is recommended for capture map preparation. With this method, one should experience substantial savings in time and effort, which can make it feasible for the modelers to address uncertainty in the management decisions, via a probabilistic approach. REFERENCES Chavent, G. (1971). Analyse fonctionelle et identification de coefficients répartis des équations aux derivées partielles, thèse de Docteur es Sciences, Univ. de Paris, VI, Paris. Chavent, G., Dupuy, M., and Lemonier, P. (1971). History matching by use of optimal control theory, Soc. Pet. Eng. J., 15(1), p. 74-86. Clemo, T. (2007). MODFLOW-2005 GroundWater Model-User Guide to the adjoint state based Sensitivity Process, Tech. report BSU CGISS 07-01. Leake, S.A., Reeves, H.W., and Dickinson, J.E. (2010). A New Capture Fraction Method to Map How pumpage Affects Surface Water Flow. Ground Water, 48(5), Sept-Oct 2010 , pp 690-700. RamaRao, B.S., and Reeves, M. (1990). Theory and verification for the GRASP II code for adjoint-sensitivity analysis of steady-state and transient groundwater flow, Contract. Rep. SAND89-7143J, Sandia Natl. Lab., Albuquerque, NM. RamaRao, B.S., Lavenue, A.M., de Marsily, G., and Marietta, M.G. (1995). Pilot Point Methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields, 1, Theory and computatonal experiments, Water Resources Research, 31(3), pp. 475-493. Ramarao B.S., and Mishra, S. (1996). Adjoint sensitivity analysis for mathematical models of coupled nonlinear physical processes, ModelCare, 1996, IAHS Publication No. 237. TRW (1995). Post-Tough2 Adjoint Sensitivity Analysis, B00000000-01717-020000126, Civilian Radioactive Waste Management System Management and Operating Contractor, Las Vegas, NV, September 1995. Wilson, J.L., RamaRao, B.S., and McNeish, J.A. (1986). GRASP: A Computer Code to Perform Post-SWENT Adjoint Sensitivity Analysis of Steady-State Ground-Water Flow. A Technical Report prepared by INTERA Technologies, Inc. for the Office of Nuclear Waste Isolation, Battelle Memorial Institute, Columbus, Ohio.

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