Use of Rational and Modified Rational Method in Texas T.G. Cleveland, Texas Tech University X. Fang, Auburn University W. H. Asquith, US Geological Survey D. B. Thompson, R.O. Anderson, Inc. Project 0‐6070. May 28, 2009
Outline • Introduction • Definitions • What We Have Done/What We Have Learned – Activities/Evidence • Summary • Where We are Headed
Introduction • Rational Equation Qp = C i A Q p = peak discharge (L^3/T) C = runoff coefficient i = rainfall intensity (L/T) A = contributing area (L^2)
Introduction • Rational (and Modified Rational) method abridged sequence. – (A) Choose a risk level. – (B) Determine a characteristic time for drainage area. (need to specify area and time) – (C) Rainfall intensity (Needs (A) and (B) a‐priori) – (D) Choose a runoff generation coefficient – (E) Compute Qp (or a hydrograph if using modified rational method – peak value is Qp)
Introduction • Rational Method http://onlinemanuals.txdot.gov/txdotmanuals/hyd/the_rational_method.htm – Determine drainage area. – Determine time of concentration. – Verify appropriate with stated assumptions. – Determine e,b and d values for desired design frequency. – Compute rainfall intensity. – Select runoff coefficients. – Calculate estimated peak discharge.
Introduction • Modified Rational Method – Similar approach but generates a triangular or trapezoidal hydrograph depending on rainfall duration as related to the watershed characteristic time (Tc). – Largest possible peak discharge is same as rational method if: • Td >= Tc
Introduction • Modified Rational Method – Similar approach but generates a triangular or trapezoidal hydrograph depending on rainfall duration as related to the watershed characteristic time (Tc). – Largest possible peak discharge is same as rational method if: • Td >= Tc
Introduction • Assumptions (current) – The rate of runoff resulting from any constant rainfall intensity is maximum when the duration of rainfall equals the time of concentration. • This assumption is a statement of mass conservation in a system where loss rates after some time become constant. It [the assumption] is probably valid for most hydrologic scales with the qualification that requisite inputs may not ever occur in nature, and constant loss rates may never be realized.
Introduction • Assumptions (current) – The frequency of peak discharge is the same as the frequency of the rainfall intensity for the given time of concentration. • Known non‐linearity in hydrologic systems such that response does not scale at the same rate as input.
Introduction • Assumptions (current) – The rainfall intensity is uniformly distributed over the entire drainage area. • In reality, rainfall intensity varies spatially and temporally during a storm. • For small areas, the assumption of uniform distribution is reasonable. • For larger areas partial contribution may be applicable – hard to quantify for design use.
Introduction • Assumptions (current) – The fraction (C) of rainfall that becomes runoff is independent of rainfall intensity or volume. • This assumption is important to be consistent with the first assumption.
Definitions • Time Concepts – Rainfall duration • How long a constant rainfall rate is applied ‐‐‐ numerically the same as Tc, conceptually different. T2 Single pulse of rainfall over T2 5 pulses of rainfall, T1 units long, in a row. T1
Definitions • Time Concepts – Watershed characteristic times • Concentration • Peak • Lag
Definitions • Runoff Coefficients ∫ Q ( t ) dt – Volumetric C V = ∫ p ( t ) dt C R = Q p – Rational i ( t d ) A
Definitions • Runoff Coefficients • Coefficients are related to basin characteristics.
Definitions • Watershed Characteristics – Area (already in rational equation) – Slope – Channel Length – Basin Development Factor – Percent Impervious – Land Use (Descriptive)
Definitions • Basin Development Factor (BDF) – Reflects improvement in drainage network. – Correlates with development, makes no assumptions about land use surrounding the network per‐se
Definitions • BDF – Channel modification – Channel lining – Storm drain – Curb‐gutter
What We Have Done • Table of C std values from an extensive literature review. – These were presented in TM1 – They will be repeated with further commentary in TM2 (in progress, due soon)
What We Have Done • Table of C std values from an extensive literature review. – These were presented in TM1 – They will be repeated with further commentary in TM2 (in progress, due soon)
What We Have Done • Determined (by literature review and numerical experiments) that the modified rational method is a special case of the unit hydrograph method. – A valuable consequence of this determination is to remove a conceptual decoupling of the methods, instead there is a continuum of methodology. – Care is still required with regards to the scale of applicability.
What We Have Done • Rational Method is a Unit Hydrograph – Watershed is represented as a linear, time‐invariant system – Assumptions 1 and 4 of current method.
What We Have Done • Invested in clarifying the different concepts in the rational method especially the time and the runoff coefficients. – These are to be explained in detail in TM2.
What We Have Done • Forward computation of Cv. – Application of rational method as if ungaged watershed. – Assumed that rational method is a unit hydrograph. – Substitute the rectangular kernel in the convolution integral.
What We Have Done Modified Rational Method Gamma UH
What We Have learned • Forward computation of Cv. – Rational method is a unit hydrograph. – Size limitation is arbitrary (from standpoint of linear systems theory). – Current limitation is meaningful for other reasons (uniform rainfall, storage, etc.)
What We Have Done • Backward computation of Cv for Texas watersheds. – Compute best Cv to explain observations. – Same kind of simulations as just presented. – Again assume rational method is a unit hydrograph.
What We Have Done 10000 Modeled Q p (cfs) 1000 100 1:1 line C from the ratio of runoff and rainfall 10 10 100 1000 10000 Observed Q p (cfs)
What We Have Learned • Backward computation of Cv for Texas watersheds. Using Cv – backcomputed Using Cstd
What We Have Learned • Backward computation of Cv for Texas watersheds. – Evidence over‐prediction potential of peak discharge. – Evidence suggests bias towards smaller areas ‐‐‐ exactly where the method is intended to be applied. – Median C value from literature based approach is nearly two times larger than for back computed.
What We Have Done • Back compute Cr values from Texas + ARS. – Used Tr as the characteristic time value, removes ambiguity for specification of Tc T R hrs = A sq . mi . • Not dimensionally homogeneous. • Consistent with a watershed integrated “velocity” of 1 mile per hour. (~1.46 ft/sec)
Recommend
More recommend
